1932:. Two players may choose to hunt a stag or a rabbit, the stag providing more meat (4 utility units, 2 for each player) than the rabbit (1 utility unit). The caveat is that the stag must be cooperatively hunted, so if one player attempts to hunt the stag, while the other hunts the rabbit, the stag hunter will totally fail, for a payoff of 0, whereas the rabbit hunter will succeed, for a payoff of 1. The game has two equilibria, (stag, stag) and (rabbit, rabbit), because a player's optimal strategy depends on their expectation on what the other player will do. If one hunter trusts that the other will hunt the stag, they should hunt the stag; however if they think the other will hunt the rabbit, they too will hunt the rabbit. This game is used as an analogy for social cooperation, since much of the benefit that people gain in society depends upon people cooperating and implicitly trusting one another to act in a manner corresponding with cooperation.
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6263:{\displaystyle {\begin{aligned}\sigma ^{*}=f(\sigma ^{*})&\Rightarrow \sigma _{i}^{*}=f_{i}(\sigma ^{*})\\&\Rightarrow \sigma _{i}^{*}={\frac {g_{i}(\sigma ^{*})}{\sum _{a\in A_{i}}g_{i}(\sigma ^{*})(a)}}\\&\Rightarrow \sigma _{i}^{*}={\frac {1}{C}}\left(\sigma _{i}^{*}+{\text{Gain}}_{i}(\sigma ^{*},\cdot )\right)\\&\Rightarrow C\sigma _{i}^{*}=\sigma _{i}^{*}+{\text{Gain}}_{i}(\sigma ^{*},\cdot )\\&\Rightarrow \left(C-1\right)\sigma _{i}^{*}={\text{Gain}}_{i}(\sigma ^{*},\cdot )\\&\Rightarrow \sigma _{i}^{*}=\left({\frac {1}{C-1}}\right){\text{Gain}}_{i}(\sigma ^{*},\cdot ).\end{aligned}}}
7534:{\displaystyle {\begin{aligned}0&=u_{i}(\sigma _{i}^{*},\sigma _{-i}^{*})-u_{i}(\sigma _{i}^{*},\sigma _{-i}^{*})\\&=\left(\sum _{a\in A_{i}}\sigma _{i}^{*}(a)u_{i}(a_{i},\sigma _{-i}^{*})\right)-u_{i}(\sigma _{i}^{*},\sigma _{-i}^{*})\\&=\sum _{a\in A_{i}}\sigma _{i}^{*}(a)(u_{i}(a_{i},\sigma _{-i}^{*})-u_{i}(\sigma _{i}^{*},\sigma _{-i}^{*}))\\&=\sum _{a\in A_{i}}\sigma _{i}^{*}(a){\text{Gain}}_{i}(\sigma ^{*},a)&&{\text{ by the previous statements }}\\&=\sum _{a\in A_{i}}\left(C-1\right)\sigma _{i}^{*}(a)^{2}>0\end{aligned}}}
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two pure-strategy Nash equilibria, (yes, yes) and (no, no), and no mixed strategy equilibria, because the strategy "yes" weakly dominates "no". "Yes" is as good as "no" regardless of the other player's action, but if there is any chance the other player chooses "yes" then "yes" is the best reply. Under a small random perturbation of the payoffs, however, the probability that any two payoffs would remain tied, whether at 0 or some other number, is vanishingly small, and the game would have either one or three equilibria instead.
297:. In a strategic interaction, the outcome for each decision-maker depends on the decisions of the others as well as their own. The simple insight underlying Nash's idea is that one cannot predict the choices of multiple decision makers if one analyzes those decisions in isolation. Instead, one must ask what each player would do taking into account what the player expects the others to do. Nash equilibrium requires that one's choices be consistent: no players wish to undo their decision given what the others are deciding.
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least one (mixed-strategy) Nash equilibrium must exist in such a game. The key to Nash's ability to prove existence far more generally than von
Neumann lay in his definition of equilibrium. According to Nash, "an equilibrium point is an n-tuple such that each player's mixed strategy maximizes payoff if the strategies of the others are held fixed. Thus each player's strategy is optimal against those of the others." Putting the problem in this framework allowed Nash to employ the
2712:, Econometrica, 63, 1161-1180 who interpreted each player's mixed strategy as a conjecture about the behaviour of other players and have shown that if the game and the rationality of players is mutually known and these conjectures are commonly known, then the conjectures must be a Nash equilibrium (a common prior assumption is needed for this result in general, but not in the case of two players. In this case, the conjectures need only be mutually known).
2747:, the NE has explanatory power. The payoff in economics is utility (or sometimes money), and in evolutionary biology is gene transmission; both are the fundamental bottom line of survival. Researchers who apply games theory in these fields claim that strategies failing to maximize these for whatever reason will be competed out of the market or environment, which are ascribed the ability to test all strategies. This conclusion is drawn from the "
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2172:. Thus, payoffs for any given strategy depend on the choices of the other players, as is usual. However, the goal, in this case, is to minimize travel time, not maximize it. Equilibrium will occur when the time on all paths is exactly the same. When that happens, no single driver has any incentive to switch routes, since it can only add to their travel time. For the graph on the right, if, for example, 100 cars are travelling from
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to gain from being unkind if player one goes left. If player one goes right the rational player two would de facto be kind to her/him in that subgame. However, The non-credible threat of being unkind at 2(2) is still part of the blue (L, (U,U)) Nash equilibrium. Therefore, if rational behavior can be expected by both parties the subgame perfect Nash equilibrium may be a more meaningful solution concept when such
2008:(where a pure strategy is chosen at random, subject to some fixed probability), then there are three Nash equilibria for the same case: two we have seen from the pure-strategy form, where the probabilities are (0%, 100%) for player one, (0%, 100%) for player two; and (100%, 0%) for player one, (100%, 0%) for player two respectively. We add another where the probabilities for each player are (50%, 50%).
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1's interest to move to the purple square and it is in player 2's interest to move to the blue square. Although it would not fit the definition of a competition game, if the game is modified so that the two players win the named amount if they both choose the same number, and otherwise win nothing, then there are 4 Nash equilibria: (0,0), (1,1), (2,2), and (3,3).
8492:{\displaystyle {\begin{aligned}&\mathbb {E} =(-1)q+(+1)(1-q)=1-2q\\&\mathbb {E} =(+1)q+(-1)(1-q)=2q-1\\&\mathbb {E} =\mathbb {E} \implies 1-2q=2q-1\implies q={\frac {1}{2}}\\&\mathbb {E} =(+1)p+(-1)(1-p)=2p-1\\&\mathbb {E} =(-1)p+(+1)(1-p)=1-2p\\&\mathbb {E} =\mathbb {E} \implies 2p-1=1-2p\implies p={\frac {1}{2}}\\\end{aligned}}}
1558:(where a player chooses probabilities of using various pure strategies) are allowed, then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium, which might be a pure strategy for each player or might be a probability distribution over strategies for each player.
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In the "driving game" example above there are both stable and unstable equilibria. The equilibria involving mixed strategies with 100% probabilities are stable. If either player changes their probabilities slightly, they will be both at a disadvantage, and their opponent will have no reason to change
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A strategy profile is a set of strategies, one for each player. Informally, a strategy profile is a Nash equilibrium if no player can do better by unilaterally changing their strategy. To see what this means, imagine that each player is told the strategies of the others. Suppose then that each player
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choose strategies A and B, (A, B) is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to Bob choosing B, and Bob has no other strategy available that does better than B at maximizing his payoff in response to Alice choosing A. In
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The first condition is not met if the game does not correctly describe the quantities a player wishes to maximize. In this case there is no particular reason for that player to adopt an equilibrium strategy. For instance, the prisoner's dilemma is not a dilemma if either player is happy to be jailed
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Driving on a road against an oncoming car, and having to choose either to swerve on the left or to swerve on the right of the road, is also a coordination game. For example, with payoffs 10 meaning no crash and 0 meaning a crash, the coordination game can be defined with the following payoff matrix:
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has three—two pure and one mixed, and this remains true even if the payoffs change slightly. The free money game is an example of a "special" game with an even number of equilibria. In it, two players have to both vote "yes" rather than "no" to get a reward and the votes are simultaneous. There are
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In 1971, Robert Wilson came up with the
Oddness Theorem, which says that "almost all" finite games have a finite and odd number of Nash equilibria. In 1993, Harsanyi published an alternative proof of the result. "Almost all" here means that any game with an infinite or even number of equilibria is
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The image to the right shows a simple sequential game that illustrates the issue with subgame imperfect Nash equilibria. In this game player one chooses left(L) or right(R), which is followed by player two being called upon to be kind (K) or unkind (U) to player one, However, player two only stands
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If these cases are both met, then a player with the small change in their mixed strategy will return immediately to the Nash equilibrium. The equilibrium is said to be stable. If condition one does not hold then the equilibrium is unstable. If only condition one holds then there are likely to be an
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Using the rule, we can very quickly (much faster than with formal analysis) see that the Nash equilibria cells are (B,A), (A,B), and (C,C). Indeed, for cell (B,A), 40 is the maximum of the first column and 25 is the maximum of the second row. For (A,B), 25 is the maximum of the second column and 40
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There is an easy numerical way to identify Nash equilibria on a payoff matrix. It is especially helpful in two-person games where players have more than two strategies. In this case formal analysis may become too long. This rule does not apply to the case where mixed (stochastic) strategies are of
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This game has a unique pure-strategy Nash equilibrium: both players choosing 0 (highlighted in light red). Any other strategy can be improved by a player switching their number to one less than that of the other player. In the adjacent table, if the game begins at the green square, it is in player
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Each of two players chooses a real number strictly less than 5 and the winner is whoever has the biggest number; no biggest number strictly less than 5 exists (if the number could equal 5, the Nash equilibrium would have both players choosing 5 and tying the game). Here, the set of choices is not
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allows for deviations by every conceivable coalition. Formally, a strong Nash equilibrium is a Nash equilibrium in which no coalition, taking the actions of its complements as given, can cooperatively deviate in a way that benefits all of its members. However, the strong Nash concept is sometimes
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games. They showed that a mixed-strategy Nash equilibrium will exist for any zero-sum game with a finite set of actions. The contribution of Nash in his 1951 article "Non-Cooperative Games" was to define a mixed-strategy Nash equilibrium for any game with a finite set of actions and prove that at
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Intentional or accidental imperfection in execution. For example, a computer capable of flawless logical play facing a second flawless computer will result in equilibrium. Introduction of imperfection will lead to its disruption either through loss to the player who makes the mistake, or through
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Stability is crucial in practical applications of Nash equilibria, since the mixed strategy of each player is not perfectly known, but has to be inferred from statistical distribution of their actions in the game. In this case unstable equilibria are very unlikely to arise in practice, since any
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This said, the actual mechanics of finding equilibrium cells is obvious: find the maximum of a column and check if the second member of the pair is the maximum of the row. If these conditions are met, the cell represents a Nash equilibrium. Check all columns this way to find all NE cells. An N×N
266: – an action plan based on what has happened so far in the game – and no one can increase one's own expected payoff by changing one's strategy while the other players keep theirs unchanged, then the current set of strategy choices constitutes a Nash equilibrium.
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In games with mixed-strategy Nash equilibria, the probability of a player choosing any particular (so pure) strategy can be computed by assigning a variable to each strategy that represents a fixed probability for choosing that strategy. In order for a player to be willing to randomize, their
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This can be illustrated by a two-player game in which both players simultaneously choose an integer from 0 to 3 and they both win the smaller of the two numbers in points. In addition, if one player chooses a larger number than the other, then they have to give up two points to the other.
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What is assumed is that there is a population of participants for each position in the game, which will be played throughout time by participants drawn at random from the different populations. If there is a stable average frequency with which each pure strategy is employed by the
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The criterion of common knowledge may not be met even if all players do, in fact, meet all the other criteria. Players wrongly distrusting each other's rationality may adopt counter-strategies to expected irrational play on their opponents’ behalf. This is a major consideration in
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to the right. There are two pure-strategy equilibria, (A,A) with payoff 4 for each player and (B,B) with payoff 2 for each. The combination (B,B) is a Nash equilibrium because if either player unilaterally changes their strategy from B to A, their payoff will fall from 2 to 1.
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For instance if a player prefers "Yes", then that set of strategies is not a Nash equilibrium. But if every player prefers not to switch (or is indifferent between switching and not) then the strategy profile is a Nash equilibrium. Thus, each strategy in a Nash equilibrium is a
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that all players meet these conditions, including this one. So, not only must each player know the other players meet the conditions, but also they must know that they all know that they meet them, and know that they know that they know that they meet them, and so
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expected payoff for each (pure) strategy should be the same. In addition, the sum of the probabilities for each strategy of a particular player should be 1. This creates a system of equations from which the probabilities of choosing each strategy can be derived.
395:, where players choose a probability distribution over possible pure strategies (which might put 100% of the probability on one pure strategy; such pure strategies are a subset of mixed strategies). The concept of a mixed-strategy equilibrium was introduced by
9533:– it can be represented as a strategy complying with his original conditions for a game with a NE. Such games may not have unique NE, but at least one of the many equilibrium strategies would be played by hypothetical players having perfect knowledge of all
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If instead, for some player, there is exact equality between the strategy in Nash equilibrium and some other strategy that gives exactly the same payout (i.e. the player is indifferent between switching and not), then the equilibrium is classified as a
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The Nash equilibrium is a superset of the subgame perfect Nash equilibrium. The subgame perfect equilibrium in addition to the Nash equilibrium requires that the strategy also is a Nash equilibrium in every subgame of that game. This eliminates all
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1522:. As a result of these requirements, strong Nash is too rare to be useful in many branches of game theory. However, in games such as elections with many more players than possible outcomes, it can be more common than a stable equilibrium.
451:. However, subsequent refinements and extensions of Nash equilibrium share the main insight on which Nash's concept rests: the equilibrium is a set of strategies such that each player's strategy is optimal given the choices of the others.
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Suppose that in the Nash equilibrium, each player asks themselves: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, would I suffer a loss by changing my strategy?"
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One interpretation is rationalistic: if we assume that players are rational, know the full structure of the game, the game is played just once, and there is just one Nash equilibrium, then players will play according to that
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In the matching pennies game, player A loses a point to B if A and B play the same strategy and wins a point from B if they play different strategies. To compute the mixed-strategy Nash equilibrium, assign A the probability
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The rule goes as follows: if the first payoff number, in the payoff pair of the cell, is the maximum of the column of the cell and if the second number is the maximum of the row of the cell - then the cell represents a Nash
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Due to the limited conditions in which NE can actually be observed, they are rarely treated as a guide to day-to-day behaviour, or observed in practice in human negotiations. However, as a theoretical concept in
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Extensive and Normal form illustrations that show the difference between SPNE and other NE. The blue equilibrium is not subgame perfect because player two makes a non-credible threat at 2(2) to be unkind
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247:. A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed). The idea of Nash equilibrium dates back to the time of
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A Nash equilibrium for a mixed-strategy game is stable if a small change (specifically, an infinitesimal change) in probabilities for one player leads to a situation where two conditions hold:
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In his Ph.D. dissertation, John Nash proposed two interpretations of his equilibrium concept, with the objective of showing how equilibrium points can be connected with observable phenomenon.
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Nash equilibrium and is played among players under certain conditions, then the NE strategy set will be adopted. Sufficient conditions to guarantee that the Nash equilibrium is played are:
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427:('refinements' of Nash equilibria) designed to rule out implausible Nash equilibria. One particularly important issue is that some Nash equilibria may be based on threats that are not '
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of the player who did the change, if the other player's mixed strategy is still (50%,50%)), then the other player immediately has a better strategy at either (0%, 100%) or (100%, 0%).
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4789:{\displaystyle {\begin{cases}f=(f_{1},\cdots ,f_{N}):\Delta \to \Delta \\f_{i}(\sigma )(a)={\frac {g_{i}(\sigma )(a)}{\sum _{b\in A_{i}}g_{i}(\sigma )(b)}}&a\in A_{i}\end{cases}}}
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t is unnecessary to assume that the participants have full knowledge of the total structure of the game, or the ability and inclination to go through any complex reasoning processes.
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is the maximum of the first row; the same applies for cell (C,C). For other cells, either one or both of the duplet members are not the maximum of the corresponding rows and columns.
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6611:{\displaystyle \forall a\in A_{i}:\quad \sigma _{i}^{*}(a)(u_{i}(a_{i},\sigma _{-i}^{*})-u_{i}(\sigma _{i}^{*},\sigma _{-i}^{*}))=\sigma _{i}^{*}(a){\text{Gain}}_{i}(\sigma ^{*},a)}
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A game where two players simultaneously name a number and the player naming the larger number wins does not have a NE, as the set of choices is not compact because it is unbounded.
274:
a game in which Carol and Dan are also players, (A, B, C, D) is a Nash equilibrium if A is Alice's best response to (B, C, D), B is Bob's best response to (A, C, D), and so forth.
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376:. In Cournot's theory, each of several firms choose how much output to produce to maximize its profit. The best output for one firm depends on the outputs of the others. A
1529:(CPNE) occurs when players cannot do better even if they are allowed to communicate and make "self-enforcing" agreement to deviate. Every correlated strategy supported by
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asks themselves: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?"
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in all Nash equilibria. If both A and B have strictly dominant strategies, there exists a unique Nash equilibrium in which each plays their strictly dominant strategy.
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4558:{\displaystyle \sum _{a\in A_{i}}g_{i}(\sigma )(a)=\sum _{a\in A_{i}}\sigma _{i}(a)+{\text{Gain}}_{i}(\sigma ,a)=1+\sum _{a\in A_{i}}{\text{Gain}}_{i}(\sigma ,a)>0.}
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is a CPNE. Further, it is possible for a game to have a Nash equilibrium that is resilient against coalitions less than a specified size, k. CPNE is related to the
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can be different for different players, and its elements can be a variety of mathematical objects. Most simply, a player might choose between two strategies, e.g.
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Game theorists have discovered that in some circumstances Nash equilibrium makes invalid predictions or fails to make a unique prediction. They have proposed many
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This term is dispreferred, as it can also mean the opposite of a "strong" Nash equilibrium (i.e. a Nash equilibrium that is vulnerable to manipulation by groups).
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In many cases, the third condition is not met because, even though the equilibrium must exist, it is unknown due to the complexity of the game, for instance in
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This simply states that each player gains no benefit by unilaterally changing their strategy, which is exactly the necessary condition for a Nash equilibrium.
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very special in the sense that if its payoffs were even slightly randomly perturbed, with probability one it would have an odd number of equilibria instead.
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An application of Nash equilibria is in determining the expected flow of traffic in a network. Consider the graph on the right. If we assume that there are
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Nash's original proof (in his thesis) used
Brouwer's fixed-point theorem (e.g., see below for a variant). This section presents a simpler proof via the
1308:
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their strategy in turn. The (50%,50%) equilibrium is unstable. If either player changes their probabilities (which would neither benefit or damage the
1626:. This means that the actions of players may potentially be constrained based on actions of other players. A common special case of the model is when
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The Nash equilibrium defines stability only in terms of individual player deviations. In cooperative games such a concept is not convincing enough.
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dynamics in his analysis of the stability of equilibrium. Cournot did not use the idea in any other applications, however, or define it generally.
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340:), the outcome of efforts exerted by multiple parties in the education process, regulatory legislation such as environmental regulations (see
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is a non-negative real number. Nash's existing proofs assume a finite strategy set, but the concept of Nash equilibrium does not require it.
2211:, then travel time for any single car would actually be 3.5, which is less than 3.75. This is also the Nash equilibrium if the path between
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The Nash equilibrium may sometimes appear non-rational in a third-person perspective. This is because a Nash equilibrium is not necessarily
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9097:
Muchen Sun; Francesca
Baldini; Katie Hughes; Peter Trautman; Todd Murphey (2024). "Mixed-Strategy Nash Equilibrium for Crowd Navigation".
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perceived as too "strong" in that the environment allows for unlimited private communication. In fact, strong Nash equilibrium has to be
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2751:" theory above. In these situations the assumption that the strategy observed is actually a NE has often been borne out by research.
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In this case there are two pure-strategy Nash equilibria, when both choose to either drive on the left or on the right. If we admit
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criterion leading to possible victory for the player. (An example would be a player suddenly putting the car into reverse in the
2976:{\displaystyle r_{i}(\sigma _{-i})=\mathop {\underset {\sigma _{i}}{\operatorname {arg\,max} }} u_{i}(\sigma _{i},\sigma _{-i})}
1208:{\displaystyle u_{i}(s_{i}^{*},s_{-i}^{*})>u_{i}(s_{i},s_{-i}^{*})\;\;{\rm {for\;all}}\;\;s_{i}\in S_{i},s_{i}\neq s_{i}^{*}}
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minute change in the proportions of each strategy seen will lead to a change in strategy and the breakdown of the equilibrium.
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Rosen extended Nash's existence theorem in several ways. He considers an n-player game, in which the strategy of each player
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is removed, which means that adding another possible route can decrease the efficiency of the system, a phenomenon known as
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De Fraja, G.; Oliveira, T.; Zanchi, L. (2010). "Must Try Harder: Evaluating the Role of Effort in
Educational Attainment".
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Kakutani's fixed point theorem guarantees the existence of a fixed point if the following four conditions are satisfied.
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1706:. Rosen also proves that, under certain technical conditions which include strict concavity, the equilibrium is unique.
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1000:: a player might be indifferent among several strategies given the other players' choices. It is unique and called a
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2105:). The "payoff" of each strategy is the travel time of each route. In the graph on the right, a car travelling via
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5496:{\displaystyle \sum _{a\in A_{i}}g_{i}(\sigma ^{*},a)=1+\sum _{a\in A_{i}}{\text{Gain}}_{i}(\sigma ^{*},a)>1.}
2773:, that is, strategies that contain non-rational moves in order to make the counter-player change their strategy.
17:
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of the appropriate population, then this stable average frequency constitutes a mixed strategy Nash equilibrium.
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A second interpretation, that Nash referred to by the mass action interpretation, is less demanding on players:
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The gain function represents the benefit a player gets by unilaterally changing their strategy. We now define
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For a formal result along these lines, see Kuhn, H. and et al., 1996, "The Work of John Nash in Game Theory",
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Aviad
Rubinstein: "Hardness of Approximation Between P and NP", ACM, ISBN 978-1-947487-23-9 (May 2019), DOI:
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4138:{\displaystyle {\text{Gain}}_{i}(\sigma ,a)=\max\{0,u_{i}(a,\sigma _{-i})-u_{i}(\sigma _{i},\sigma _{-i})\}.}
3631:. i.e. if two strategies maximize payoffs, then a mix between the two strategies will yield the same payoff.
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because players may "threaten" each other with threats they would not actually carry out. For such games the
978:{\displaystyle u_{i}(s_{i}^{*},s_{-i}^{*})\geq u_{i}(s_{i},s_{-i}^{*})\;\;{\rm {for\;all}}\;\;s_{i}\in S_{i}}
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Notice that this distribution is not, actually, socially optimal. If the 100 cars agreed that 50 travel via
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Djehiche, Boualem; Tcheukam, Alain; Tembine, Hamidou (2017-09-27). "Mean-Field-Type Games in
Engineering".
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Sample network graph. Values on edges are the travel time experienced by a "car" traveling down that edge.
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Djehiche, B.; Tcheukam, A.; Tembine, H. (2017). "A Mean-Field Game of
Evacuation in Multilevel Building".
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is a non-zero vector. But this is a clear contradiction, so all the gains must indeed be zero. Therefore,
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The players believe that a deviation in their own strategy will not cause deviations by any other players.
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5203:{\displaystyle \forall i\in \{1,\cdots ,N\},\forall a\in A_{i}:\quad {\text{Gain}}_{i}(\sigma ^{*},a)=0.}
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9055:"Testing Mixed-Strategy Equilibria when Players Are Heterogeneous: The Case of Penalty Kicks in Soccer"
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Thus, a mixed-strategy Nash equilibrium in this game is for each player to randomly choose H or T with
1305:
Or, the strategy set might be a finite set of conditional strategies responding to other players, e.g.
8657: – Methods and processes involved in facilitating the peaceful ending of conflict and retribution
8538:
8505:
1473:
Nash equilibrium. In the latter, not every player always plays the same strategy. Instead, there is a
308:). It has also been used to study to what extent people with different preferences can cooperate (see
11057:
10980:
10716:
10272:
10197:
10054:
9946:
8639:
7547:
6302:
2574:
1720:(representing all possible mixtures of pure strategies), and the payoff functions of all players are
380:
occurs when each firm's output maximizes its profits given the output of the other firms, which is a
9074:
4583:
3661:
in 1949, von
Neumann famously dismissed it with the words, "That's trivial, you know. That's just a
3437:
1433:
356:
in crowds, energy systems, transportation systems, evacuation problems and wireless communications.
11072:
10805:
10691:
10488:
10282:
10100:
9801:
9299:
2656:
2629:
1474:
526:
369:
248:
10875:
6841:{\displaystyle \sigma _{i}^{*}(a)=\left({\frac {1}{C-1}}\right){\text{Gain}}_{i}(\sigma ^{*},a)=0}
11077:
10676:
10646:
10302:
10090:
9978:
9823:
9697:
8962:
Ward, H. (1996). "Game Theory and the
Politics of Global Warming: The State of Play and Beyond".
8740:
8677:
7579:
5660:
5050:
5023:
4829:
1530:
1514:
197:
46:
5269:
3967:
3330:
3296:
3265:
2613:
The players all will do their utmost to maximize their expected payoff as described by the game.
636:
11102:
11082:
11062:
11011:
10681:
10586:
10445:
10390:
10322:
10292:
10212:
10140:
9069:
8746:
8645:
3676:
1852:
1561:
Nash equilibria need not exist if the set of choices is infinite and non-compact. For example:
996:
A game can have more than one Nash equilibrium. Even if the equilibrium is unique, it might be
341:
333:
263:
167:
2543:, useful in the analysis of many kinds of equilibria, can also be applied to Nash equilibria.
93:
10561:
10546:
10120:
9270:
8666:
8621:
5003:
4963:
4923:
4883:
3898:
3383:
3363:
3243:
2778:
2192:. Every driver now has a total travel time of 3.75 (to see this, a total of 75 cars take the
2085:", where every traveler has a choice of 3 strategies and where each strategy is a route from
301:
6276:
443:. Other extensions of the Nash equilibrium concept have addressed what happens if a game is
10895:
10880:
10767:
10762:
10666:
10651:
10616:
10581:
10180:
10125:
10047:
9536:
9522:
9355:
B. D. Bernheim; B. Peleg; M. D. Whinston (1987), "Coalition-Proof Equilibria I. Concepts",
8848:
7924:
7872:
7772:
7745:
7718:
7691:
7664:
7637:
5633:
4856:
4802:
3871:
3792:
3410:
3064:
2770:
2744:
2606:
1659:
are constrained independently of other players' actions. If the following conditions hold:
1229:
781:
479:
440:
365:
345:
244:
223:
212:
4920:
is a continuous function. As the cross product of a finite number of compact convex sets,
3380:
is a simplex and thus compact. Convexity follows from players' ability to mix strategies.
8:
11052:
10671:
10621:
10458:
10385:
10365:
10222:
10105:
9983:
9529:
8654:
3662:
3089:
2220:
2017:
671:
377:
300:
The concept has been used to analyze hostile situations such as wars and arms races (see
252:
163:
10711:
9526:, copyright 2001, Texas A&M University, London School of Economics, pages 141-144.
8852:
11031:
10890:
10721:
10701:
10551:
10430:
10335:
10262:
10207:
9665:
9503:
9188:
9170:
9143:
9098:
8979:
8975:
8944:
8871:
8836:
8683:
7904:
7852:
7688:. In the case of two players A and B, there exists a Nash equilibrium in which A plays
7606:
6854:
5077:
4983:
4943:
4903:
3947:
3720:
3637:
3217:
2586:
2151:
2053:
2029:
697:
506:
317:
11016:
10985:
10940:
10835:
10706:
10661:
10636:
10566:
10440:
10370:
10360:
10252:
10202:
10150:
9903:
9878:
9853:
9827:
9805:
9778:
9765:
9731:
9701:
9669:
9657:
9618:
9577:
9495:
9387:
9368:
9266:
9135:
9096:
8983:
8900:
8876:
8799:
7632:
2594:
2582:
1761:
1733:
1721:
1618:; so a strategy-tuple is a vector in R. Part of the definition of a game is a subset
400:
329:
159:
9551:
9231:
9192:
9147:
8948:
2759:
416:
in his 1950 paper to prove existence of equilibria. His 1951 paper used the simpler
344:), natural resource management, analysing strategies in marketing, penalty kicks in
11097:
11092:
11026:
10990:
10970:
10930:
10900:
10855:
10810:
10795:
10606:
10184:
10170:
10135:
9941:
9929:
9866:
9782:
9775:. Lucid and detailed introduction to game theory in an explicitly economic context.
9649:
9610:
9485:
9477:
9450:
MIT OpenCourseWare. 6.254: Game Theory with Engineering Applications, Spring 2010.
9431:
9423:
9364:
9256:
9248:
9205:
Cournot A. (1838) Researches on the Mathematical Principles of the Theory of Wealth
9180:
9127:
9079:
9011:
8971:
8934:
8926:
8866:
8856:
8791:
8734:
8700:
3658:
2578:
2540:
1678:
1538:
1519:
424:
396:
353:
349:
278:
240:
140:
9000:"Risks and benefits of catching pretty good yield in multispecies mixed fisheries"
8709: – Gives conditions that guarantee the max–min inequality is also an equality
1382:{\displaystyle S_{i}=\{{\text{Yes}}|p={\text{Low}},{\text{No}}|p={\text{High}}\}.}
10995:
10955:
10910:
10825:
10820:
10541:
10493:
10380:
10145:
10115:
10085:
10034:
9598:
9562:, copyright 1997, Texas A&M University, University of Arizona, pages 141-144
9558:
8706:
3404:
2679:
2660:
1740:
1534:
432:
325:
294:
10860:
9184:
10935:
10925:
10915:
10850:
10840:
10830:
10815:
10611:
10591:
10576:
10571:
10531:
10498:
10483:
10478:
10468:
10277:
8818:
3624:{\displaystyle \lambda \sigma _{i}+(1-\lambda )\sigma '_{i}\in r(\sigma _{-i})}
3471:
3323:
2563:
2005:
1470:
633:
be a strategy profile, a set consisting of one strategy for each player, where
392:
337:
9914:
9382:
Aumann, R. (1959). "Acceptable points in general cooperative n-person games".
9252:
9083:
2622:
The players know the planned equilibrium strategy of all of the other players.
11117:
10975:
10965:
10920:
10905:
10885:
10656:
10631:
10503:
10473:
10463:
10450:
10355:
10297:
10232:
10165:
9913:. A comprehensive reference from a computational perspective; see Chapter 3.
9793:
9661:
9622:
9499:
9139:
9131:
9016:
8999:
8689:
4299:{\displaystyle g_{i}(\sigma )(a)=\sigma _{i}(a)+{\text{Gain}}_{i}(\sigma ,a)}
2674:
with a small child who desperately wants to win (meeting the other criteria).
2667:
2528:
1856:
1574:
1466:
470:
444:
385:
381:
270:
9718:. W.W. Norton & Company. (Third edition in 2009.) An undergraduate text.
9490:
2550:
the player who did not change has no better strategy in the new circumstance
10950:
10945:
10800:
10375:
9973:
9466:"Existence and Uniqueness of Equilibrium Points for Concave N-Person Games"
9427:
8880:
1767:
A coordination game showing payoffs for player 1 (row) \ player 2 (column)
1577:
with each player's payoff continuous in the strategies of all the players.
1489:
If every player's answer is "Yes", then the equilibrium is classified as a
778:
be player i's payoff as a function of the strategies. The strategy profile
3868:
denote the set of mixed strategies for the players. The finiteness of the
1004:
if the inequality is strict so one strategy is the unique best response:
312:), and whether they will take risks to achieve a cooperative outcome (see
11067:
10870:
10865:
10845:
10641:
10626:
10435:
10405:
10340:
10330:
10160:
10095:
10071:
10027:
9745:
8930:
8861:
8660:
2851:
be the best response of player i to the strategies of all other players.
2671:
2643:
2590:
2082:
2022:
448:
428:
305:
232:
144:
10039:
9895:
Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations
4940:
is also compact and convex. Applying the Brouwer fixed point theorem to
3668:
2553:
the player who did change is now playing with a strictly worse strategy.
10696:
10350:
9653:
9507:
9465:
9261:
2799:
1664:
304:), and also how conflict may be mitigated by repeated interaction (see
9436:
9331:
8939:
6851:
and so the left term is zero, giving us that the entire expression is
6678:
then this is true by definition of the gain function. Now assume that
391:
The modern concept of Nash equilibrium is instead defined in terms of
10601:
10521:
10345:
9874:
9727:
9637:
8680: – List of definitions of terms and concepts used in game theory
3061:, is a mixed-strategy profile in the set of all mixed strategies and
2740:
2683:
1929:
373:
313:
256:
9993:. # Explains the Nash Equilibrium is a hard problem in computation.
9614:
9481:
9354:
2670:. Or, if known, it may not be known to all players, as when playing
35:
11036:
10536:
9175:
9103:
3861:{\displaystyle \Delta =\Delta _{1}\times \cdots \times \Delta _{N}}
3477:
Condition 4. is satisfied as a result of mixed strategies. Suppose
408:
321:
290:
9990:
9232:"On the Existence of Pure Strategy Nash Equilibria in Large Games"
2708:
This idea was formalized by R. Aumann and A. Brandenburger, 1995,
2558:
infinite number of optimal strategies for the player who changed.
289:
Game theorists use Nash equilibrium to analyze the outcome of the
10757:
10747:
10425:
2573:
Finally in the eighties, building with great depth on such ideas
1717:
9893:
2619:
The players have sufficient intelligence to deduce the solution.
1389:
Or, it might be an infinite set, a continuum or unbounded, e.g.
368:. The same idea was used in a particular application in 1838 by
3207:{\displaystyle r=r_{i}(\sigma _{-i})\times r_{-i}(\sigma _{i})}
2789:
2078:, what is the expected distribution of traffic in the network?
2802:
with the observation that such a simplification is possible).
10526:
9528:
Nash proved that a perfect NE exists for this type of finite
9386:. Vol. IV. Princeton, N.J.: Princeton University Press.
8726: – Bridge scoring terms in the card game contract bridge
8998:
Thorpe, Robert B.; Jennings, Simon; Dolder, Paul J. (2017).
8737: – Formal rule for predicting how a game will be played
1739:
Nash equilibrium may also have non-rational consequences in
1573:
However, a Nash equilibrium exists if the set of choices is
4782:
3918:
We can now define the gain functions. For a mixed strategy
3538:{\displaystyle \sigma _{i},\sigma '_{i}\in r(\sigma _{-i})}
1655:. This represents the case that the actions of each player
439:
as a refinement that eliminates equilibria which depend on
407:, but their analysis was restricted to the special case of
9638:"Oddness of the Number of Equilibrium Points: A New Proof"
8719:
Extended Mathematical Programming for Equilibrium Problems
3789:
is the action set for the players. All of the action sets
277:
Nash showed that there is a Nash equilibrium, possibly in
9052:
5583:{\displaystyle C=\sum _{a\in A_{i}}g_{i}(\sigma ^{*},a).}
2180:, then equilibrium will occur when 25 drivers travel via
384:
Nash equilibrium. Cournot also introduced the concept of
8821:(1987) "Nash Equilibrium." In: Palgrave Macmillan (eds)
9160:
9117:
8751:
Pages displaying short descriptions of redirect targets
5216:
Now assume that the gains are not all zero. Therefore,
3214:. The existence of a Nash equilibrium is equivalent to
1709:
Nash's result refers to the special case in which each
364:
Nash equilibrium is named after American mathematician
9846:
Game Theory for Business: A Primer in Strategic Gaming
9053:
Chiappori, P. -A.; Levitt, S.; Groseclose, T. (2002).
8916:
7661:
then there exists a Nash equilibrium in which A plays
3123:{\displaystyle r\colon \Sigma \rightarrow 2^{\Sigma }}
3054:{\displaystyle \Sigma =\Sigma _{i}\times \Sigma _{-i}}
473:
to the other players' strategies in that equilibrium.
8541:
8508:
7962:
7927:
7907:
7875:
7855:
7775:
7748:
7721:
7694:
7667:
7640:
7609:
7582:
7550:
6883:
6857:
6744:
6684:
6671:{\displaystyle {\text{Gain}}_{i}(\sigma ^{*},a)>0}
6627:
6391:
6375:{\displaystyle {\text{Gain}}_{i}(\sigma ^{*},\cdot )}
6337:
6305:
6279:
5693:
5663:
5636:
5599:
5515:
5365:
5349:{\displaystyle {\text{Gain}}_{i}(\sigma ^{*},a)>0}
5305:
5272:
5222:
5103:
5080:
5053:
5026:
5006:
4986:
4966:
4946:
4926:
4906:
4886:
4859:
4832:
4805:
4577:
4363:
4315:
4215:
4154:
4006:
3970:
3950:
3924:
3901:
3874:
3822:
3795:
3743:
3723:
3679:
3669:
Alternate proof using the Brouwer fixed-point theorem
3640:
3551:
3483:
3440:
3413:
3386:
3366:
3333:
3299:
3268:
3246:
3220:
3136:
3097:
3067:
3018:
2992:
2860:
2811:
2432:
2154:
2115:
2056:
2032:
1436:
1395:
1311:
1259:
1232:
1021:
822:
784:
720:
700:
674:
639:
567:
529:
509:
482:
8997:
8650:
Pages displaying wikidata descriptions as a fallback
3403:
Condition 2. and 3. are satisfied by way of Berge's
1677:
is continuous in the strategies of all players, and
9741:. Suitable for undergraduate and business students.
9406:
8749: – Major theorist of traffic flow equilibrium
2637:
1702:Then a Nash equilibrium exists. The proof uses the
1298:{\displaystyle S_{i}=\{{\text{Yes}},{\text{No}}\}.}
60:. Unsourced material may be challenged and removed.
9891:
9220:, copyright 1944, 1953, Princeton University Press
8899:, copyright 1960, 1980, Harvard University Press,
8560:
8527:
8491:
7945:
7913:
7893:
7861:
7788:
7761:
7734:
7707:
7680:
7653:
7615:
7595:
7568:
7533:
6863:
6840:
6728:{\displaystyle {\text{Gain}}_{i}(\sigma ^{*},a)=0}
6727:
6670:
6610:
6374:
6323:
6291:
6262:
5676:
5649:
5622:
5582:
5495:
5348:
5291:
5258:
5202:
5086:
5066:
5039:
5012:
4992:
4972:
4952:
4932:
4912:
4892:
4872:
4845:
4818:
4788:
4557:
4346:
4298:
4198:
4137:
3989:
3956:
3936:
3907:
3887:
3860:
3808:
3781:
3729:
3709:
3646:
3623:
3537:
3462:
3426:
3392:
3372:
3348:
3314:
3283:
3252:
3226:
3206:
3122:
3080:
3053:
3004:
2975:
2843:
2805:To prove the existence of a Nash equilibrium, let
2160:
2140:
2062:
2038:
1444:
1422:
1381:
1297:
1245:
1207:
977:
797:
770:
706:
686:
660:
625:
553:
515:
495:
9964:The Game's Afoot! Game Theory in Myth and Paradox
9865:
9350:
9348:
8789:
8648: – type of mathematical optimization problem
3782:{\displaystyle A=A_{1}\times \cdots \times A_{N}}
503:be the set of all possible strategies for player
11115:
9282:
9280:
9033:"Marketing Lessons from Dr. Nash - Andrew Frank"
8571:
4037:
3400:is nonempty as long as players have strategies.
2646:problems in which these conditions are not met:
2597:now usually refer to Mertens stable equilibria.
1480:
332:). Other applications include traffic flow (see
9935:Proceedings of the National Academy of Sciences
9932:(1950) "Equilibrium points in n-person games"
9714:Dixit, Avinash, Susan Skeath and David Reiley.
9571:
9229:
2690:
1928:A famous example of a coordination game is the
10023:Complete Proof of Existence of Nash Equilibria
9888:. A modern introduction at the graduate level.
9345:
8624:, for example, has one equilibrium, while the
3088:is the payoff function for player i. Define a
2754:
2046:is the number of cars traveling via that edge.
1747:may be more meaningful as a tool of analysis.
10055:
9452:Lecture 6: Continuous and Discontinuous Games
9277:
9230:Carmona, Guilherme; Podczeck, Konrad (2009).
5259:{\displaystyle \exists i\in \{1,\cdots ,N\},}
5094:. For this purpose, it suffices to show that
4347:{\displaystyle \sigma \in \Delta ,a\in A_{i}}
3360:Condition 1. is satisfied from the fact that
1622:of R such that the strategy-tuple must be in
447:, or what happens if a game is played in the
316:). It has been used to study the adoption of
7626:
5250:
5232:
5131:
5113:
4129:
4040:
2790:Proof using the Kakutani fixed-point theorem
1580:
1508:
1417:
1409:
1373:
1325:
1289:
1273:
626:{\displaystyle s^{*}=(s_{i}^{*},s_{-i}^{*})}
9552:Learning to Play Cournot Duoploy Strategies
9163:AIMS Electronics and Electrical Engineering
7901:of playing T, and assign B the probability
1549:
10062:
10048:
9892:Shoham, Yoav; Leyton-Brown, Kevin (2009),
9840:
8642: – Method of fairly dividing property
8468:
8464:
8436:
8432:
8207:
8203:
8175:
8171:
6735:. By our previous statements we have that
2798:, following Nash's 1950 paper (he credits
2446:A payoff matrix – Nash equilibria in bold
1150:
1149:
1137:
1125:
1124:
951:
950:
938:
926:
925:
27:Solution concept of a non-cooperative game
10069:
9787:The Theory of Games and Economic Behavior
9724:Strategies and games: theory and practice
9489:
9435:
9260:
9174:
9102:
9073:
9015:
8938:
8870:
8860:
8663: – Groups working or acting together
8417:
8398:
8314:
8230:
8156:
8137:
8053:
7969:
7769:is a strictly dominant strategy, A plays
5630:as the gain vector indexed by actions in
3634:Therefore, there exists a fixed point in
2905:
2710:Epistemic Conditions for Nash Equilibrium
2581:. Mertens stable equilibria satisfy both
1460:
405:The Theory of Games and Economic Behavior
120:Learn how and when to remove this message
9684:
9635:
9599:"Computing Equilibria of N-Person Games"
8823:The New Palgrave Dictionary of Economics
7544:where the last inequality follows since
2758:
2519:We can apply this rule to a 3×3 matrix:
2168:is the number of cars traveling on edge
2021:
1423:{\displaystyle S_{i}=\{{\text{Price}}\}}
9922:
9817:
9792:
9755:
9694:Playing for Real: A Text on Game Theory
9691:
8674: – Solution concept in game theory
6331:is some positive scaling of the vector
5623:{\displaystyle {\text{Gain}}(i,\cdot )}
4199:{\displaystyle g=(g_{1},\dotsc ,g_{N})}
771:{\displaystyle u_{i}(s_{i},s_{-i}^{*})}
14:
11116:
9596:
9572:Fudenburg, Drew; Tirole, Jean (1991).
9381:
9120:IEEE Transactions on Automatic Control
8837:"Equilibrium points in n-person games"
8783:
7435: by the previous statements
2663:, ensuring a no-loss no-win scenario).
2616:The players are flawless in execution.
1630:is a Cartesian product of convex sets
10043:
9721:
9463:
9217:Theory of Games and Economic Behavior
8715: – Doctrine of military strategy
4853:. It is also easy to check that each
2784:
2453:
2450:
2239:
2236:
2196:edge, and likewise, 75 cars take the
1947:
1944:
1871:
1868:
1774:
1771:
694:strategies of all the players except
9642:International Journal of Game Theory
9400:
9384:Contributions to the Theory of Games
8961:
8834:
2141:{\displaystyle 1+{\frac {x}{100}}+2}
2081:This situation can be modeled as a "
1755:
1525:A refined Nash equilibrium known as
58:adding citations to reliable sources
29:
10028:Simplified Form and Related Results
9955:
9603:SIAM Journal on Applied Mathematics
9375:
9286:
2844:{\displaystyle r_{i}(\sigma _{-i})}
2226:
1644:, such that the strategy of player
459:
24:
10111:First-player and second-player win
9758:Game Theory for Applied Economists
8976:10.1111/j.1467-9248.1996.tb00338.x
8919:Review of Economics and Statistics
8798:. Cambridge, MA: MIT. p. 14.
6392:
5223:
5137:
5104:
5007:
4967:
4927:
4834:
4636:
4630:
4322:
3937:{\displaystyle \sigma \in \Delta }
3931:
3902:
3849:
3830:
3823:
3665:." (See Nasar, 1998, p. 94.)
3387:
3367:
3247:
3115:
3104:
3039:
3026:
3019:
3005:{\displaystyle \sigma \in \Sigma }
2999:
2912:
2909:
2906:
2902:
2899:
2896:
2527:matrix may have between 0 and N×N
2433:Nash equilibria in a payoff matrix
2011:
1144:
1141:
1138:
1134:
1131:
1128:
945:
942:
939:
935:
932:
929:
190:Trembling hand perfect equilibrium
25:
11145:
9998:
9798:Game Theory: Analysis of Conflict
8724:Optimum contract and par contract
3260:is compact, convex, and nonempty.
1569:compact because it is not closed.
336:), how to organize auctions (see
11124:Game theory equilibrium concepts
10218:Coalition-proof Nash equilibrium
9214:J. Von Neumann, O. Morgenstern,
8686: – Observation in economics
8561:{\displaystyle q={\frac {1}{2}}}
8528:{\displaystyle p={\frac {1}{2}}}
2638:Where the conditions are not met
1745:subgame perfect Nash equilibrium
1527:coalition-proof Nash equilibrium
34:
10006:"Nash theorem (in game theory)"
9991:https://doi.org/10.1145/3241304
9944:(1951) "Non-Cooperative Games"
9679:
9629:
9590:
9565:
9543:
9514:
9457:
9444:
9316:
9223:
9208:
9199:
9154:
9111:
9090:
8764:
7715:and B plays a best response to
7569:{\displaystyle \sigma _{i}^{*}}
6414:
6324:{\displaystyle \sigma _{i}^{*}}
5159:
449:absence of complete information
284:
45:needs additional citations for
10228:Evolutionarily stable strategy
9820:An Introduction to Game Theory
9636:Harsanyi, J. C. (1973-12-01).
9520:T. L. Turocy, B. Von Stengel,
9407:D. Moreno; J. Wooders (1996),
9289:"Preliminaries of Game Theory"
9046:
9025:
9004:ICES Journal of Marine Science
8991:
8955:
8910:
8887:
8828:
8812:
8672:Evolutionarily stable strategy
8465:
8433:
8429:
8421:
8410:
8402:
8374:
8362:
8359:
8350:
8341:
8332:
8326:
8318:
8290:
8278:
8275:
8266:
8257:
8248:
8242:
8234:
8204:
8172:
8168:
8160:
8149:
8141:
8113:
8101:
8098:
8089:
8080:
8071:
8065:
8057:
8029:
8017:
8014:
8005:
7996:
7987:
7981:
7973:
7940:
7928:
7888:
7876:
7512:
7505:
7427:
7408:
7393:
7387:
7336:
7333:
7294:
7278:
7244:
7231:
7228:
7222:
7171:
7132:
7111:
7077:
7064:
7058:
7002:
6963:
6947:
6908:
6829:
6810:
6766:
6760:
6716:
6697:
6659:
6640:
6605:
6586:
6571:
6565:
6544:
6541:
6502:
6486:
6452:
6439:
6436:
6430:
6369:
6350:
6250:
6231:
6172:
6162:
6143:
6091:
6081:
6062:
6008:
5993:
5974:
5908:
5895:
5889:
5886:
5873:
5835:
5822:
5788:
5778:
5765:
5734:
5727:
5714:
5617:
5605:
5574:
5555:
5484:
5465:
5418:
5399:
5337:
5318:
5191:
5172:
4755:
4749:
4746:
4740:
4702:
4696:
4693:
4687:
4668:
4662:
4659:
4653:
4633:
4624:
4592:
4546:
4534:
4487:
4475:
4457:
4451:
4412:
4406:
4403:
4397:
4293:
4281:
4263:
4257:
4241:
4235:
4232:
4226:
4193:
4161:
4126:
4097:
4081:
4059:
4031:
4019:
3704:
3686:
3618:
3602:
3580:
3568:
3532:
3516:
3463:{\displaystyle r(\sigma _{i})}
3457:
3444:
3343:
3337:
3309:
3303:
3278:
3272:
3201:
3188:
3169:
3153:
3107:
2970:
2941:
2887:
2871:
2838:
2822:
2748:
1855:game, as shown in the example
1727:
1445:{\displaystyle {\text{Price}}}
1358:
1334:
1121:
1087:
1071:
1032:
922:
888:
872:
833:
765:
731:
620:
581:
454:
193:
178:Evolutionarily stable strategy
13:
1:
10156:Simultaneous action selection
9968:American Mathematical Society
9869:; Osborne, Martin J. (1994),
9597:Wilson, Robert (1971-07-01).
9409:"Coalition-Proof Equilibrium"
8825:. Palgrave Macmillan, London.
8777:
8703: – Type of confrontation
8572:Oddness of equilibrium points
7803:
4826:is a valid mixed strategy in
3944:, we let the gain for player
3895:s ensures the compactness of
3737:is the number of players and
3657:When Nash made this point to
2600:
1851:is a classic two-player, two-
1481:Strict/Non-strict equilibrium
554:{\displaystyle i=1,\ldots ,N}
320:, and also the occurrence of
11088:List of games in game theory
10268:Quantal response equilibrium
10258:Perfect Bayesian equilibrium
10193:Bayes correlated equilibrium
9369:10.1016/0022-0531(87)90099-8
8696:Manipulated Nash equilibrium
8692: – Mathematical concept
8595:
5684:is the fixed point we have:
4880:is a continuous function of
4799:It is easy to see that each
2796:Kakutani fixed-point theorem
2691:Where the conditions are met
2534:
1704:Kakutani fixed-point theorem
1544:
414:Kakutani fixed-point theorem
262:If each player has chosen a
251:, who in 1838 applied it to
186:Perfect Bayesian equilibrium
7:
10557:Optional prisoner's dilemma
10288:Self-confirming equilibrium
10011:Encyclopedia of Mathematics
9789:Princeton University Press.
9416:Games and Economic Behavior
9185:10.3934/ElectrEng.2017.1.18
8730:Self-confirming equilibrium
8632:
7596:{\displaystyle \sigma ^{*}}
5677:{\displaystyle \sigma ^{*}}
5067:{\displaystyle \sigma ^{*}}
5040:{\displaystyle \sigma ^{*}}
4846:{\displaystyle \Delta _{i}}
3434:is continuous and compact,
2755:NE and non-credible threats
2109:experiences travel time of
1817:Player 1 adopts strategy B
1788:Player 1 adopts strategy A
1783:Player 2 adopts strategy B
1780:Player 2 adopts strategy A
1750:
1503:non-strict Nash equilibrium
1477:over different strategies.
1455:
437:subgame perfect equilibrium
418:Brouwer fixed-point theorem
182:Subgame perfect equilibrium
10:
11150:
11129:Fixed points (mathematics)
11022:Principal variation search
10738:Aumann's agreement theorem
10401:Strategy-stealing argument
10313:Trembling hand equilibrium
10243:Markov perfect equilibrium
10238:Mertens-stable equilibrium
9900:Cambridge University Press
9762:Princeton University Press
9357:Journal of Economic Theory
9240:Journal of Economic Theory
8713:Mutual assured destruction
7603:is a Nash equilibrium for
5292:{\displaystyle a\in A_{i}}
3990:{\displaystyle a\in A_{i}}
3349:{\displaystyle r(\sigma )}
3315:{\displaystyle r(\sigma )}
3284:{\displaystyle r(\sigma )}
2733:Journal of Economic Theory
2015:
1759:
805:is a Nash equilibrium if
661:{\displaystyle s_{-i}^{*}}
359:
239:is the most commonly-used
11058:Combinatorial game theory
11045:
11004:
10786:
10730:
10717:Princess and monster game
10512:
10414:
10321:
10273:Quasi-perfect equilibrium
10198:Bayesian Nash equilibrium
10179:
10078:
9947:The Annals of Mathematics
9722:Dutta, Prajit K. (1999),
9253:10.1016/j.jet.2008.11.009
9084:10.1257/00028280260344678
8640:Adjusted winner procedure
8603:
8592:
8587:
8584:
8581:
7835:
7824:
7819:
7816:
7813:
7627:Computing Nash equilibria
5074:is a Nash equilibrium in
3710:{\displaystyle G=(N,A,u)}
2575:Mertens-stable equilibria
2501:
2485:
2469:
2464:
2461:
2458:
2381:
2340:
2299:
2258:
2253:
2250:
2247:
2244:
2207:and the other 50 through
1981:
1960:
1955:
1952:
1905:
1884:
1879:
1876:
1816:
1787:
1782:
1779:
1688:for every fixed value of
1581:Rosen's existence theorem
1531:iterated strict dominance
1509:Equilibria for coalitions
281:, for every finite game.
218:
208:
203:
173:
155:
150:
139:
134:
11134:1951 in economic history
11073:Evolutionary game theory
10806:Antoine Augustin Cournot
10692:Guess 2/3 of the average
10489:Strictly determined game
10283:Satisfaction equilibrium
10101:Escalation of commitment
9915:Downloadable free online
9850:Probabilistic Publishing
9818:Osborne, Martin (2004),
9802:Harvard University Press
9756:Gibbons, Robert (1992),
9132:10.1109/TAC.2017.2679487
9062:American Economic Review
8896:The Strategy of Conflict
8757:
6874:So we finally have that
3654:and a Nash equilibrium.
1596:in the Euclidean space R
1550:Nash's existence theorem
1475:probability distribution
370:Antoine Augustin Cournot
253:his model of competition
11078:Glossary of game theory
10677:Stackelberg competition
10303:Strong Nash equilibrium
9871:A Course in Game Theory
9824:Oxford University Press
9698:Oxford University Press
8796:A Course in Game Theory
8741:Stackelberg competition
8678:Glossary of game theory
5013:{\displaystyle \Delta }
4973:{\displaystyle \Delta }
4933:{\displaystyle \Delta }
4893:{\displaystyle \sigma }
3908:{\displaystyle \Delta }
3393:{\displaystyle \Sigma }
3373:{\displaystyle \Sigma }
3253:{\displaystyle \Sigma }
2779:dynamic inconsistencies
1515:Strong Nash equilibrium
1491:strict Nash equilibrium
1002:strict Nash equilibrium
198:Strong Nash equilibrium
194:Stable Nash equilibrium
11103:Tragedy of the commons
11083:List of game theorists
11063:Confrontation analysis
10773:Sprague–Grundy theorem
10293:Sequential equilibrium
10213:Correlated equilibrium
9549:J. C. Cox, M. Walker,
9428:10.1006/game.1996.0095
9017:10.1093/icesjms/fsx062
8835:Nash, John F. (1950).
8743: – Economic model
8646:Complementarity theory
8562:
8529:
8493:
8426:payoff for B playing T
8407:payoff for B playing H
8323:payoff for B playing T
8239:payoff for B playing H
8165:payoff for A playing T
8146:payoff for A playing H
8062:payoff for A playing T
7978:payoff for A playing H
7947:
7915:
7895:
7863:
7790:
7763:
7736:
7709:
7682:
7655:
7617:
7597:
7570:
7535:
6865:
6842:
6729:
6672:
6621:To see this, first if
6612:
6376:
6325:
6293:
6292:{\displaystyle C>1}
6264:
5678:
5651:
5624:
5584:
5497:
5350:
5293:
5260:
5204:
5088:
5068:
5041:
5014:
4994:
4974:
4954:
4934:
4914:
4894:
4874:
4847:
4820:
4790:
4559:
4348:
4300:
4200:
4139:
3991:
3958:
3938:
3909:
3889:
3862:
3810:
3783:
3731:
3711:
3648:
3625:
3539:
3464:
3428:
3394:
3374:
3350:
3316:
3285:
3254:
3234:having a fixed point.
3228:
3208:
3124:
3082:
3055:
3006:
2977:
2845:
2765:
2729:
2706:
2162:
2142:
2070:"cars" traveling from
2064:
2047:
2040:
1461:Pure/mixed equilibrium
1446:
1424:
1383:
1299:
1247:
1209:
979:
799:
772:
708:
688:
662:
627:
555:
517:
497:
420:for the same purpose.
342:tragedy of the commons
168:Correlated equilibrium
10876:Jean-François Mertens
9692:Binmore, Ken (2007),
9685:Game theory textbooks
9464:Rosen, J. B. (1965).
8667:Equilibrium selection
8563:
8530:
8494:
7948:
7946:{\displaystyle (1-q)}
7916:
7896:
7894:{\displaystyle (1-p)}
7864:
7791:
7789:{\displaystyle s_{A}}
7764:
7762:{\displaystyle s_{A}}
7737:
7735:{\displaystyle s_{A}}
7710:
7708:{\displaystyle s_{A}}
7683:
7681:{\displaystyle s_{A}}
7656:
7654:{\displaystyle s_{A}}
7618:
7598:
7571:
7536:
6866:
6843:
6730:
6673:
6613:
6377:
6326:
6294:
6265:
5679:
5652:
5650:{\displaystyle A_{i}}
5625:
5593:Also we shall denote
5585:
5498:
5351:
5294:
5261:
5205:
5089:
5069:
5042:
5015:
5000:has a fixed point in
4995:
4975:
4955:
4935:
4915:
4895:
4875:
4873:{\displaystyle f_{i}}
4848:
4821:
4819:{\displaystyle f_{i}}
4791:
4560:
4349:
4301:
4201:
4140:
3992:
3959:
3939:
3910:
3890:
3888:{\displaystyle A_{i}}
3863:
3811:
3809:{\displaystyle A_{i}}
3784:
3732:
3712:
3649:
3626:
3540:
3465:
3429:
3427:{\displaystyle u_{i}}
3395:
3375:
3351:
3317:
3286:
3255:
3229:
3209:
3125:
3083:
3081:{\displaystyle u_{i}}
3056:
3007:
2978:
2846:
2762:
2717:
2697:
2577:were introduced as a
2163:
2143:
2065:
2041:
2025:
1670:Each payoff function
1667:, closed and bounded;
1447:
1425:
1384:
1300:
1248:
1246:{\displaystyle S_{i}}
1210:
980:
800:
798:{\displaystyle s^{*}}
773:
709:
689:
663:
628:
556:
518:
498:
496:{\displaystyle S_{i}}
291:strategic interaction
245:non-cooperative games
224:non-cooperative games
11005:Search optimizations
10881:Jennifer Tour Chayes
10768:Revelation principle
10763:Purification theorem
10702:Nash bargaining game
10667:Bertrand competition
10652:El Farol Bar problem
10617:Electronic mail game
10582:Lewis signaling game
10126:Hierarchy of beliefs
9984:Simon & Schuster
9962:Mehlmann, A. (2000)
9923:Original Nash papers
9744:Fudenberg, Drew and
8931:10.1162/REST_a_00013
8862:10.1073/pnas.36.1.48
8790:Osborne, Martin J.;
8539:
8506:
7960:
7925:
7905:
7873:
7853:
7773:
7746:
7719:
7692:
7665:
7638:
7631:If a player A has a
7607:
7580:
7548:
6881:
6855:
6742:
6682:
6625:
6389:
6382:. Now we claim that
6335:
6303:
6277:
5691:
5661:
5634:
5597:
5513:
5363:
5303:
5270:
5220:
5101:
5078:
5051:
5024:
5004:
4984:
4964:
4944:
4924:
4904:
4884:
4857:
4830:
4803:
4575:
4361:
4313:
4213:
4152:
4004:
3968:
3948:
3922:
3899:
3872:
3820:
3793:
3741:
3721:
3677:
3638:
3549:
3481:
3472:upper hemicontinuous
3438:
3411:
3384:
3364:
3331:
3324:upper hemicontinuous
3297:
3266:
3244:
3218:
3134:
3095:
3065:
3016:
2990:
2858:
2809:
2771:non-credible threats
2745:evolutionary biology
2152:
2113:
2054:
2030:
1554:Nash proved that if
1434:
1393:
1309:
1257:
1230:
1019:
820:
782:
718:
698:
672:
637:
565:
527:
507:
480:
441:non-credible threats
213:John Forbes Nash Jr.
54:improve this article
11053:Bounded rationality
10672:Cournot competition
10622:Rock paper scissors
10597:Battle of the sexes
10587:Volunteer's dilemma
10459:Perfect information
10386:Dominant strategies
10223:Epsilon-equilibrium
10106:Extensive-form game
9530:extensive form game
8893:Schelling, Thomas,
8853:1950PNAS...36...48N
8747:Wardrop's principle
8655:Conflict resolution
8626:battle of the sexes
8593:Player A votes Yes
8585:Player B votes Yes
8578:
7810:
7565:
7504:
7386:
7332:
7311:
7277:
7221:
7170:
7149:
7110:
7057:
7001:
6980:
6946:
6925:
6759:
6564:
6540:
6519:
6485:
6429:
6320:
6189:
6127:
6046:
6028:
5958:
5925:
5805:
5751:
3663:fixed-point theorem
3595:
3509:
3090:set-valued function
2447:
2233:
2232:A competition game
1982:Drive on the right
1956:Drive on the right
1941:
1865:
1768:
1724:of the strategies.
1204:
1120:
1070:
1049:
921:
871:
850:
764:
687:{\displaystyle N-1}
657:
619:
598:
403:in their 1944 book
378:Cournot equilibrium
366:John Forbes Nash Jr
334:Wardrop's principle
318:technical standards
310:battle of the sexes
164:Epsilon-equilibrium
11032:Paranoid algorithm
11012:Alpha–beta pruning
10891:John Maynard Smith
10722:Rendezvous problem
10562:Traveler's dilemma
10552:Gift-exchange game
10547:Prisoner's dilemma
10464:Large Poisson game
10431:Bargaining problem
10336:Backward induction
10308:Subgame perfection
10263:Proper equilibrium
10033:2021-07-31 at the
9779:Morgenstern, Oskar
9654:10.1007/BF01737572
9557:2013-12-11 at the
9328:hoylab.cornell.edu
9296:Science of the Web
8622:prisoner's dilemma
8604:Player A votes No
8588:Player B votes No
8576:
8558:
8525:
8489:
8487:
7943:
7911:
7891:
7859:
7808:
7786:
7759:
7732:
7705:
7678:
7651:
7613:
7593:
7566:
7551:
7531:
7529:
7490:
7470:
7372:
7371:
7315:
7297:
7260:
7207:
7206:
7153:
7135:
7093:
7043:
7042:
6984:
6966:
6929:
6911:
6861:
6838:
6745:
6725:
6668:
6608:
6550:
6523:
6505:
6468:
6415:
6372:
6321:
6306:
6289:
6260:
6258:
6175:
6113:
6032:
6014:
5944:
5911:
5862:
5791:
5737:
5674:
5647:
5620:
5580:
5544:
5493:
5452:
5388:
5346:
5289:
5256:
5200:
5084:
5064:
5037:
5010:
4990:
4970:
4950:
4930:
4910:
4890:
4870:
4843:
4816:
4786:
4781:
4729:
4555:
4521:
4440:
4386:
4344:
4296:
4196:
4135:
3987:
3954:
3934:
3905:
3885:
3858:
3806:
3779:
3727:
3707:
3644:
3621:
3583:
3535:
3497:
3460:
3424:
3390:
3370:
3346:
3312:
3281:
3250:
3224:
3204:
3120:
3078:
3051:
3002:
2973:
2926:
2841:
2785:Proof of existence
2766:
2587:backward induction
2454:Player 2 strategy
2451:Player 1 strategy
2445:
2240:Player 2 strategy
2237:Player 1 strategy
2231:
2158:
2138:
2060:
2048:
2036:
1961:Drive on the left
1953:Drive on the left
1948:Player 2 strategy
1945:Player 1 strategy
1939:
1872:Player 2 strategy
1869:Player 1 strategy
1863:
1775:Player 2 strategy
1772:Player 1 strategy
1766:
1722:bilinear functions
1539:theory of the core
1465:A game can have a
1442:
1420:
1379:
1295:
1243:
1205:
1190:
1103:
1053:
1035:
975:
904:
854:
836:
795:
768:
747:
704:
684:
658:
640:
623:
602:
584:
551:
513:
493:
302:prisoner's dilemma
69:"Nash equilibrium"
11111:
11110:
11017:Aspiration window
10986:Suzanne Scotchmer
10941:Oskar Morgenstern
10836:Donald B. Gillies
10778:Zermelo's theorem
10707:Induction puzzles
10662:Fair cake-cutting
10637:Public goods game
10567:Coordination game
10441:Intransitive game
10371:Forward induction
10253:Pareto efficiency
10233:Gibbs equilibrium
10203:Berge equilibrium
10151:Simultaneous game
9909:978-0-521-89943-7
9884:978-0-262-65040-3
9867:Rubinstein, Ariel
9833:978-0-19-512895-6
9811:978-0-674-34116-6
9794:Myerson, Roger B.
9771:978-0-691-00395-5
9764:(July 13, 1992),
9737:978-0-262-04169-0
9716:Games of Strategy
9583:978-0-262-06141-4
9393:978-1-4008-8216-8
9324:"Nash Equilibria"
9126:(10): 5154–5169.
8964:Political Studies
8792:Rubinstein, Ariel
8614:
8613:
8556:
8523:
8483:
8427:
8408:
8324:
8240:
8222:
8166:
8147:
8063:
7979:
7921:of playing H and
7914:{\displaystyle q}
7869:of playing H and
7862:{\displaystyle p}
7846:
7845:
7836:Player A plays T
7825:Player A plays H
7820:Player B plays T
7817:Player B plays H
7809:Matching pennies
7633:dominant strategy
7616:{\displaystyle G}
7448:
7436:
7400:
7349:
7184:
7020:
6864:{\displaystyle 0}
6802:
6792:
6689:
6632:
6578:
6342:
6223:
6213:
6135:
6054:
5966:
5937:
5899:
5840:
5603:
5522:
5457:
5430:
5366:
5310:
5164:
5087:{\displaystyle G}
4993:{\displaystyle f}
4980:we conclude that
4953:{\displaystyle f}
4913:{\displaystyle f}
4759:
4707:
4526:
4499:
4467:
4418:
4364:
4273:
4011:
3957:{\displaystyle i}
3730:{\displaystyle N}
3647:{\displaystyle r}
3470:is non-empty and
3227:{\displaystyle r}
2894:
2595:stable equilibria
2583:forward induction
2531:Nash equilibria.
2517:
2516:
2422:
2421:
2161:{\displaystyle x}
2130:
2063:{\displaystyle x}
2039:{\displaystyle x}
2002:
2001:
1940:The driving game
1926:
1925:
1849:coordination game
1845:
1844:
1762:Coordination game
1756:Coordination game
1440:
1415:
1371:
1355:
1347:
1331:
1287:
1279:
1226:The strategy set
707:{\displaystyle i}
516:{\displaystyle i}
425:solution concepts
401:Oskar Morgenstern
372:in his theory of
330:coordination game
229:
228:
160:Rationalizability
130:
129:
122:
104:
16:(Redirected from
11141:
11098:Topological game
11093:No-win situation
10991:Thomas Schelling
10971:Robert B. Wilson
10931:Merrill M. Flood
10901:John von Neumann
10811:Ariel Rubinstein
10796:Albert W. Tucker
10647:War of attrition
10607:Matching pennies
10248:Nash equilibrium
10171:Mechanism design
10136:Normal-form game
10091:Cooperative game
10064:
10057:
10050:
10041:
10040:
10019:
9979:A Beautiful Mind
9956:Other references
9912:
9887:
9862:
9842:Papayoanou, Paul
9836:
9814:
9783:John von Neumann
9774:
9740:
9710:
9674:
9673:
9633:
9627:
9626:
9594:
9588:
9587:
9569:
9563:
9547:
9541:
9539:
9532:
9518:
9512:
9511:
9493:
9491:2060/19650010164
9461:
9455:
9448:
9442:
9441:
9439:
9413:
9404:
9398:
9397:
9379:
9373:
9372:
9352:
9343:
9342:
9340:
9339:
9330:. Archived from
9320:
9314:
9313:
9311:
9310:
9304:
9298:. Archived from
9293:
9284:
9275:
9274:
9264:
9247:(3): 1300–1319.
9236:
9227:
9221:
9212:
9206:
9203:
9197:
9196:
9178:
9158:
9152:
9151:
9115:
9109:
9108:
9106:
9094:
9088:
9087:
9077:
9059:
9050:
9044:
9043:
9041:
9040:
9029:
9023:
9021:
9019:
9010:(8): 2097–2106.
8995:
8989:
8987:
8959:
8953:
8952:
8942:
8914:
8908:
8891:
8885:
8884:
8874:
8864:
8832:
8826:
8816:
8810:
8809:
8787:
8771:
8768:
8752:
8735:Solution concept
8701:Mexican standoff
8651:
8579:
8577:Free Money Game
8575:
8567:
8565:
8564:
8559:
8557:
8549:
8534:
8532:
8531:
8526:
8524:
8516:
8498:
8496:
8495:
8490:
8488:
8484:
8476:
8428:
8425:
8420:
8409:
8406:
8401:
8395:
8325:
8322:
8317:
8311:
8241:
8238:
8233:
8227:
8223:
8215:
8167:
8164:
8159:
8148:
8145:
8140:
8134:
8064:
8061:
8056:
8050:
7980:
7977:
7972:
7966:
7952:
7950:
7949:
7944:
7920:
7918:
7917:
7912:
7900:
7898:
7897:
7892:
7868:
7866:
7865:
7860:
7811:
7807:
7795:
7793:
7792:
7787:
7785:
7784:
7768:
7766:
7765:
7760:
7758:
7757:
7741:
7739:
7738:
7733:
7731:
7730:
7714:
7712:
7711:
7706:
7704:
7703:
7687:
7685:
7684:
7679:
7677:
7676:
7660:
7658:
7657:
7652:
7650:
7649:
7622:
7620:
7619:
7614:
7602:
7600:
7599:
7594:
7592:
7591:
7575:
7573:
7572:
7567:
7564:
7559:
7540:
7538:
7537:
7532:
7530:
7520:
7519:
7503:
7498:
7489:
7485:
7469:
7468:
7467:
7441:
7437:
7434:
7431:
7420:
7419:
7407:
7406:
7401:
7398:
7385:
7380:
7370:
7369:
7368:
7342:
7331:
7326:
7310:
7305:
7293:
7292:
7276:
7271:
7256:
7255:
7243:
7242:
7220:
7215:
7205:
7204:
7203:
7177:
7169:
7164:
7148:
7143:
7131:
7130:
7118:
7114:
7109:
7104:
7089:
7088:
7076:
7075:
7056:
7051:
7041:
7040:
7039:
7008:
7000:
6995:
6979:
6974:
6962:
6961:
6945:
6940:
6924:
6919:
6907:
6906:
6870:
6868:
6867:
6862:
6847:
6845:
6844:
6839:
6822:
6821:
6809:
6808:
6803:
6800:
6797:
6793:
6791:
6777:
6758:
6753:
6734:
6732:
6731:
6726:
6709:
6708:
6696:
6695:
6690:
6687:
6677:
6675:
6674:
6669:
6652:
6651:
6639:
6638:
6633:
6630:
6617:
6615:
6614:
6609:
6598:
6597:
6585:
6584:
6579:
6576:
6563:
6558:
6539:
6534:
6518:
6513:
6501:
6500:
6484:
6479:
6464:
6463:
6451:
6450:
6428:
6423:
6410:
6409:
6381:
6379:
6378:
6373:
6362:
6361:
6349:
6348:
6343:
6340:
6330:
6328:
6327:
6322:
6319:
6314:
6298:
6296:
6295:
6290:
6269:
6267:
6266:
6261:
6259:
6243:
6242:
6230:
6229:
6224:
6221:
6218:
6214:
6212:
6198:
6188:
6183:
6168:
6155:
6154:
6142:
6141:
6136:
6133:
6126:
6121:
6112:
6108:
6087:
6074:
6073:
6061:
6060:
6055:
6052:
6045:
6040:
6027:
6022:
6004:
6000:
5996:
5986:
5985:
5973:
5972:
5967:
5964:
5957:
5952:
5938:
5930:
5924:
5919:
5904:
5900:
5898:
5885:
5884:
5872:
5871:
5861:
5860:
5859:
5838:
5834:
5833:
5821:
5820:
5810:
5804:
5799:
5784:
5777:
5776:
5764:
5763:
5750:
5745:
5726:
5725:
5707:
5706:
5683:
5681:
5680:
5675:
5673:
5672:
5656:
5654:
5653:
5648:
5646:
5645:
5629:
5627:
5626:
5621:
5604:
5601:
5589:
5587:
5586:
5581:
5567:
5566:
5554:
5553:
5543:
5542:
5541:
5502:
5500:
5499:
5494:
5477:
5476:
5464:
5463:
5458:
5455:
5451:
5450:
5449:
5411:
5410:
5398:
5397:
5387:
5386:
5385:
5355:
5353:
5352:
5347:
5330:
5329:
5317:
5316:
5311:
5308:
5298:
5296:
5295:
5290:
5288:
5287:
5265:
5263:
5262:
5257:
5209:
5207:
5206:
5201:
5184:
5183:
5171:
5170:
5165:
5162:
5155:
5154:
5093:
5091:
5090:
5085:
5073:
5071:
5070:
5065:
5063:
5062:
5047:. We claim that
5046:
5044:
5043:
5038:
5036:
5035:
5019:
5017:
5016:
5011:
4999:
4997:
4996:
4991:
4979:
4977:
4976:
4971:
4959:
4957:
4956:
4951:
4939:
4937:
4936:
4931:
4919:
4917:
4916:
4911:
4899:
4897:
4896:
4891:
4879:
4877:
4876:
4871:
4869:
4868:
4852:
4850:
4849:
4844:
4842:
4841:
4825:
4823:
4822:
4817:
4815:
4814:
4795:
4793:
4792:
4787:
4785:
4784:
4778:
4777:
4760:
4758:
4739:
4738:
4728:
4727:
4726:
4705:
4686:
4685:
4675:
4652:
4651:
4623:
4622:
4604:
4603:
4568:Next we define:
4564:
4562:
4561:
4556:
4533:
4532:
4527:
4524:
4520:
4519:
4518:
4474:
4473:
4468:
4465:
4450:
4449:
4439:
4438:
4437:
4396:
4395:
4385:
4384:
4383:
4353:
4351:
4350:
4345:
4343:
4342:
4305:
4303:
4302:
4297:
4280:
4279:
4274:
4271:
4256:
4255:
4225:
4224:
4205:
4203:
4202:
4197:
4192:
4191:
4173:
4172:
4144:
4142:
4141:
4136:
4125:
4124:
4109:
4108:
4096:
4095:
4080:
4079:
4058:
4057:
4018:
4017:
4012:
4009:
3996:
3994:
3993:
3988:
3986:
3985:
3963:
3961:
3960:
3955:
3943:
3941:
3940:
3935:
3914:
3912:
3911:
3906:
3894:
3892:
3891:
3886:
3884:
3883:
3867:
3865:
3864:
3859:
3857:
3856:
3838:
3837:
3816:are finite. Let
3815:
3813:
3812:
3807:
3805:
3804:
3788:
3786:
3785:
3780:
3778:
3777:
3759:
3758:
3736:
3734:
3733:
3728:
3716:
3714:
3713:
3708:
3659:John von Neumann
3653:
3651:
3650:
3645:
3630:
3628:
3627:
3622:
3617:
3616:
3591:
3564:
3563:
3544:
3542:
3541:
3536:
3531:
3530:
3505:
3493:
3492:
3469:
3467:
3466:
3461:
3456:
3455:
3433:
3431:
3430:
3425:
3423:
3422:
3399:
3397:
3396:
3391:
3379:
3377:
3376:
3371:
3355:
3353:
3352:
3347:
3321:
3319:
3318:
3313:
3290:
3288:
3287:
3282:
3259:
3257:
3256:
3251:
3233:
3231:
3230:
3225:
3213:
3211:
3210:
3205:
3200:
3199:
3187:
3186:
3168:
3167:
3152:
3151:
3129:
3127:
3126:
3121:
3119:
3118:
3087:
3085:
3084:
3079:
3077:
3076:
3060:
3058:
3057:
3052:
3050:
3049:
3034:
3033:
3011:
3009:
3008:
3003:
2982:
2980:
2979:
2974:
2969:
2968:
2953:
2952:
2940:
2939:
2927:
2925:
2924:
2915:
2886:
2885:
2870:
2869:
2850:
2848:
2847:
2842:
2837:
2836:
2821:
2820:
2657:common knowledge
2655:negation of the
2630:common knowledge
2605:If a game has a
2579:solution concept
2448:
2444:
2234:
2230:
2227:Competition game
2221:Braess's paradox
2218:
2214:
2210:
2206:
2199:
2195:
2191:
2187:
2183:
2179:
2175:
2171:
2167:
2165:
2164:
2159:
2147:
2145:
2144:
2139:
2131:
2123:
2108:
2104:
2100:
2096:
2092:
2088:
2077:
2073:
2069:
2067:
2066:
2061:
2045:
2043:
2042:
2037:
2018:Braess's paradox
2006:mixed strategies
1942:
1938:
1866:
1862:
1769:
1765:
1741:sequential games
1556:mixed strategies
1520:Pareto efficient
1451:
1449:
1448:
1443:
1441:
1438:
1429:
1427:
1426:
1421:
1416:
1413:
1405:
1404:
1388:
1386:
1385:
1380:
1372:
1369:
1361:
1356:
1353:
1348:
1345:
1337:
1332:
1329:
1321:
1320:
1304:
1302:
1301:
1296:
1288:
1285:
1280:
1277:
1269:
1268:
1252:
1250:
1249:
1244:
1242:
1241:
1214:
1212:
1211:
1206:
1203:
1198:
1186:
1185:
1173:
1172:
1160:
1159:
1148:
1147:
1119:
1114:
1099:
1098:
1086:
1085:
1069:
1064:
1048:
1043:
1031:
1030:
984:
982:
981:
976:
974:
973:
961:
960:
949:
948:
920:
915:
900:
899:
887:
886:
870:
865:
849:
844:
832:
831:
804:
802:
801:
796:
794:
793:
777:
775:
774:
769:
763:
758:
743:
742:
730:
729:
713:
711:
710:
705:
693:
691:
690:
685:
667:
665:
664:
659:
656:
651:
632:
630:
629:
624:
618:
613:
597:
592:
577:
576:
560:
558:
557:
552:
522:
520:
519:
514:
502:
500:
499:
494:
492:
491:
460:Nash equilibrium
397:John von Neumann
393:mixed strategies
354:robot navigation
350:matching pennies
279:mixed strategies
241:solution concept
237:Nash equilibrium
141:Solution concept
135:Nash equilibrium
132:
131:
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
11149:
11148:
11144:
11143:
11142:
11140:
11139:
11138:
11114:
11113:
11112:
11107:
11041:
11027:max^n algorithm
11000:
10996:William Vickrey
10956:Reinhard Selten
10911:Kenneth Binmore
10826:David K. Levine
10821:Daniel Kahneman
10788:
10782:
10758:Negamax theorem
10748:Minimax theorem
10726:
10687:Diner's dilemma
10542:All-pay auction
10508:
10494:Stochastic game
10446:Mean-field game
10417:
10410:
10381:Markov strategy
10317:
10183:
10175:
10146:Sequential game
10131:Information set
10116:Game complexity
10086:Congestion game
10074:
10068:
10035:Wayback Machine
10004:
10001:
9996:
9958:
9953:
9925:
9920:
9910:
9885:
9860:
9834:
9812:
9772:
9738:
9708:
9687:
9682:
9677:
9634:
9630:
9615:10.1137/0121011
9595:
9591:
9584:
9570:
9566:
9559:Wayback Machine
9548:
9544:
9534:
9527:
9519:
9515:
9482:10.2307/1911749
9462:
9458:
9449:
9445:
9411:
9405:
9401:
9394:
9380:
9376:
9353:
9346:
9337:
9335:
9334:on Jun 16, 2019
9322:
9321:
9317:
9308:
9306:
9302:
9291:
9287:von Ahn, Luis.
9285:
9278:
9234:
9228:
9224:
9213:
9209:
9204:
9200:
9159:
9155:
9116:
9112:
9095:
9091:
9075:10.1.1.178.1646
9057:
9051:
9047:
9038:
9036:
9031:
9030:
9026:
8996:
8992:
8960:
8956:
8915:
8911:
8892:
8888:
8833:
8829:
8817:
8813:
8806:
8794:(12 Jul 1994).
8788:
8784:
8780:
8775:
8774:
8769:
8765:
8760:
8755:
8750:
8707:Minimax theorem
8684:Hotelling's law
8649:
8635:
8574:
8548:
8540:
8537:
8536:
8515:
8507:
8504:
8503:
8486:
8485:
8475:
8424:
8416:
8405:
8397:
8393:
8392:
8321:
8313:
8309:
8308:
8237:
8229:
8225:
8224:
8214:
8163:
8155:
8144:
8136:
8132:
8131:
8060:
8052:
8048:
8047:
7976:
7968:
7963:
7961:
7958:
7957:
7926:
7923:
7922:
7906:
7903:
7902:
7874:
7871:
7870:
7854:
7851:
7850:
7806:
7780:
7776:
7774:
7771:
7770:
7753:
7749:
7747:
7744:
7743:
7726:
7722:
7720:
7717:
7716:
7699:
7695:
7693:
7690:
7689:
7672:
7668:
7666:
7663:
7662:
7645:
7641:
7639:
7636:
7635:
7629:
7608:
7605:
7604:
7587:
7583:
7581:
7578:
7577:
7560:
7555:
7549:
7546:
7545:
7528:
7527:
7515:
7511:
7499:
7494:
7475:
7471:
7463:
7459:
7452:
7439:
7438:
7433:
7430:
7415:
7411:
7402:
7397:
7396:
7381:
7376:
7364:
7360:
7353:
7340:
7339:
7327:
7319:
7306:
7301:
7288:
7284:
7272:
7264:
7251:
7247:
7238:
7234:
7216:
7211:
7199:
7195:
7188:
7175:
7174:
7165:
7157:
7144:
7139:
7126:
7122:
7105:
7097:
7084:
7080:
7071:
7067:
7052:
7047:
7035:
7031:
7024:
7019:
7015:
7006:
7005:
6996:
6988:
6975:
6970:
6957:
6953:
6941:
6933:
6920:
6915:
6902:
6898:
6891:
6884:
6882:
6879:
6878:
6856:
6853:
6852:
6817:
6813:
6804:
6799:
6798:
6781:
6776:
6772:
6754:
6749:
6743:
6740:
6739:
6704:
6700:
6691:
6686:
6685:
6683:
6680:
6679:
6647:
6643:
6634:
6629:
6628:
6626:
6623:
6622:
6593:
6589:
6580:
6575:
6574:
6559:
6554:
6535:
6527:
6514:
6509:
6496:
6492:
6480:
6472:
6459:
6455:
6446:
6442:
6424:
6419:
6405:
6401:
6390:
6387:
6386:
6357:
6353:
6344:
6339:
6338:
6336:
6333:
6332:
6315:
6310:
6304:
6301:
6300:
6278:
6275:
6274:
6257:
6256:
6238:
6234:
6225:
6220:
6219:
6202:
6197:
6193:
6184:
6179:
6166:
6165:
6150:
6146:
6137:
6132:
6131:
6122:
6117:
6098:
6094:
6085:
6084:
6069:
6065:
6056:
6051:
6050:
6041:
6036:
6023:
6018:
6002:
6001:
5981:
5977:
5968:
5963:
5962:
5953:
5948:
5943:
5939:
5929:
5920:
5915:
5902:
5901:
5880:
5876:
5867:
5863:
5855:
5851:
5844:
5839:
5829:
5825:
5816:
5812:
5811:
5809:
5800:
5795:
5782:
5781:
5772:
5768:
5759:
5755:
5746:
5741:
5730:
5721:
5717:
5702:
5698:
5694:
5692:
5689:
5688:
5668:
5664:
5662:
5659:
5658:
5641:
5637:
5635:
5632:
5631:
5600:
5598:
5595:
5594:
5562:
5558:
5549:
5545:
5537:
5533:
5526:
5514:
5511:
5510:
5472:
5468:
5459:
5454:
5453:
5445:
5441:
5434:
5406:
5402:
5393:
5389:
5381:
5377:
5370:
5364:
5361:
5360:
5325:
5321:
5312:
5307:
5306:
5304:
5301:
5300:
5283:
5279:
5271:
5268:
5267:
5221:
5218:
5217:
5179:
5175:
5166:
5161:
5160:
5150:
5146:
5102:
5099:
5098:
5079:
5076:
5075:
5058:
5054:
5052:
5049:
5048:
5031:
5027:
5025:
5022:
5021:
5005:
5002:
5001:
4985:
4982:
4981:
4965:
4962:
4961:
4945:
4942:
4941:
4925:
4922:
4921:
4905:
4902:
4901:
4885:
4882:
4881:
4864:
4860:
4858:
4855:
4854:
4837:
4833:
4831:
4828:
4827:
4810:
4806:
4804:
4801:
4800:
4780:
4779:
4773:
4769:
4761:
4734:
4730:
4722:
4718:
4711:
4706:
4681:
4677:
4676:
4674:
4647:
4643:
4640:
4639:
4618:
4614:
4599:
4595:
4579:
4578:
4576:
4573:
4572:
4528:
4523:
4522:
4514:
4510:
4503:
4469:
4464:
4463:
4445:
4441:
4433:
4429:
4422:
4391:
4387:
4379:
4375:
4368:
4362:
4359:
4358:
4338:
4334:
4314:
4311:
4310:
4275:
4270:
4269:
4251:
4247:
4220:
4216:
4214:
4211:
4210:
4187:
4183:
4168:
4164:
4153:
4150:
4149:
4117:
4113:
4104:
4100:
4091:
4087:
4072:
4068:
4053:
4049:
4013:
4008:
4007:
4005:
4002:
4001:
3981:
3977:
3969:
3966:
3965:
3949:
3946:
3945:
3923:
3920:
3919:
3900:
3897:
3896:
3879:
3875:
3873:
3870:
3869:
3852:
3848:
3833:
3829:
3821:
3818:
3817:
3800:
3796:
3794:
3791:
3790:
3773:
3769:
3754:
3750:
3742:
3739:
3738:
3722:
3719:
3718:
3678:
3675:
3674:
3673:We have a game
3671:
3639:
3636:
3635:
3609:
3605:
3587:
3559:
3555:
3550:
3547:
3546:
3523:
3519:
3501:
3488:
3484:
3482:
3479:
3478:
3451:
3447:
3439:
3436:
3435:
3418:
3414:
3412:
3409:
3408:
3405:maximum theorem
3385:
3382:
3381:
3365:
3362:
3361:
3332:
3329:
3328:
3298:
3295:
3294:
3267:
3264:
3263:
3245:
3242:
3241:
3219:
3216:
3215:
3195:
3191:
3179:
3175:
3160:
3156:
3147:
3143:
3135:
3132:
3131:
3114:
3110:
3096:
3093:
3092:
3072:
3068:
3066:
3063:
3062:
3042:
3038:
3029:
3025:
3017:
3014:
3013:
2991:
2988:
2987:
2961:
2957:
2948:
2944:
2935:
2931:
2920:
2916:
2895:
2893:
2878:
2874:
2865:
2861:
2859:
2856:
2855:
2829:
2825:
2816:
2812:
2810:
2807:
2806:
2792:
2787:
2757:
2735:, 69, 153–185.
2693:
2661:game of chicken
2640:
2603:
2539:The concept of
2537:
2435:
2229:
2216:
2212:
2208:
2204:
2197:
2193:
2189:
2185:
2181:
2177:
2173:
2169:
2153:
2150:
2149:
2122:
2114:
2111:
2110:
2106:
2102:
2098:
2094:
2090:
2086:
2075:
2071:
2055:
2052:
2051:
2031:
2028:
2027:
2020:
2014:
2012:Network traffic
1998:
1995:
1990:
1987:
1977:
1974:
1969:
1966:
1922:
1919:
1914:
1911:
1901:
1898:
1893:
1890:
1841:
1836:
1829:
1824:
1812:
1807:
1800:
1795:
1764:
1758:
1753:
1730:
1714:
1697:
1686:
1675:
1653:
1642:
1636:
1616:
1610:
1599:
1594:
1583:
1552:
1547:
1535:Pareto frontier
1511:
1483:
1463:
1458:
1437:
1435:
1432:
1431:
1412:
1400:
1396:
1394:
1391:
1390:
1368:
1357:
1352:
1344:
1333:
1328:
1316:
1312:
1310:
1307:
1306:
1284:
1276:
1264:
1260:
1258:
1255:
1254:
1237:
1233:
1231:
1228:
1227:
1199:
1194:
1181:
1177:
1168:
1164:
1155:
1151:
1127:
1126:
1115:
1107:
1094:
1090:
1081:
1077:
1065:
1057:
1044:
1039:
1026:
1022:
1020:
1017:
1016:
969:
965:
956:
952:
928:
927:
916:
908:
895:
891:
882:
878:
866:
858:
845:
840:
827:
823:
821:
818:
817:
789:
785:
783:
780:
779:
759:
751:
738:
734:
725:
721:
719:
716:
715:
699:
696:
695:
673:
670:
669:
652:
644:
638:
635:
634:
614:
606:
593:
588:
572:
568:
566:
563:
562:
528:
525:
524:
508:
505:
504:
487:
483:
481:
478:
477:
462:
457:
433:Reinhard Selten
362:
326:currency crises
295:decision makers
287:
269:If two players
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
18:Nash equilibria
15:
12:
11:
5:
11147:
11137:
11136:
11131:
11126:
11109:
11108:
11106:
11105:
11100:
11095:
11090:
11085:
11080:
11075:
11070:
11065:
11060:
11055:
11049:
11047:
11043:
11042:
11040:
11039:
11034:
11029:
11024:
11019:
11014:
11008:
11006:
11002:
11001:
10999:
10998:
10993:
10988:
10983:
10978:
10973:
10968:
10963:
10961:Robert Axelrod
10958:
10953:
10948:
10943:
10938:
10936:Olga Bondareva
10933:
10928:
10926:Melvin Dresher
10923:
10918:
10916:Leonid Hurwicz
10913:
10908:
10903:
10898:
10893:
10888:
10883:
10878:
10873:
10868:
10863:
10858:
10853:
10851:Harold W. Kuhn
10848:
10843:
10841:Drew Fudenberg
10838:
10833:
10831:David M. Kreps
10828:
10823:
10818:
10816:Claude Shannon
10813:
10808:
10803:
10798:
10792:
10790:
10784:
10783:
10781:
10780:
10775:
10770:
10765:
10760:
10755:
10753:Nash's theorem
10750:
10745:
10740:
10734:
10732:
10728:
10727:
10725:
10724:
10719:
10714:
10709:
10704:
10699:
10694:
10689:
10684:
10679:
10674:
10669:
10664:
10659:
10654:
10649:
10644:
10639:
10634:
10629:
10624:
10619:
10614:
10612:Ultimatum game
10609:
10604:
10599:
10594:
10592:Dollar auction
10589:
10584:
10579:
10577:Centipede game
10574:
10569:
10564:
10559:
10554:
10549:
10544:
10539:
10534:
10532:Infinite chess
10529:
10524:
10518:
10516:
10510:
10509:
10507:
10506:
10501:
10499:Symmetric game
10496:
10491:
10486:
10484:Signaling game
10481:
10479:Screening game
10476:
10471:
10469:Potential game
10466:
10461:
10456:
10448:
10443:
10438:
10433:
10428:
10422:
10420:
10412:
10411:
10409:
10408:
10403:
10398:
10396:Mixed strategy
10393:
10388:
10383:
10378:
10373:
10368:
10363:
10358:
10353:
10348:
10343:
10338:
10333:
10327:
10325:
10319:
10318:
10316:
10315:
10310:
10305:
10300:
10295:
10290:
10285:
10280:
10278:Risk dominance
10275:
10270:
10265:
10260:
10255:
10250:
10245:
10240:
10235:
10230:
10225:
10220:
10215:
10210:
10205:
10200:
10195:
10189:
10187:
10177:
10176:
10174:
10173:
10168:
10163:
10158:
10153:
10148:
10143:
10138:
10133:
10128:
10123:
10121:Graphical game
10118:
10113:
10108:
10103:
10098:
10093:
10088:
10082:
10080:
10076:
10075:
10067:
10066:
10059:
10052:
10044:
10038:
10037:
10025:
10020:
10000:
9999:External links
9997:
9995:
9994:
9987:
9971:
9959:
9957:
9954:
9952:
9951:
9950:54(2):286-295.
9939:
9926:
9924:
9921:
9919:
9918:
9908:
9889:
9883:
9863:
9859:978-0964793873
9858:
9838:
9832:
9815:
9810:
9790:
9776:
9770:
9753:
9742:
9736:
9719:
9712:
9707:978-0195300574
9706:
9688:
9686:
9683:
9681:
9678:
9676:
9675:
9648:(1): 235–250.
9628:
9589:
9582:
9564:
9542:
9513:
9476:(3): 520–534.
9456:
9443:
9399:
9392:
9374:
9344:
9315:
9276:
9222:
9207:
9198:
9153:
9110:
9089:
9045:
9024:
8990:
8970:(5): 850–871.
8954:
8909:
8886:
8827:
8811:
8804:
8781:
8779:
8776:
8773:
8772:
8762:
8761:
8759:
8756:
8754:
8753:
8744:
8738:
8732:
8727:
8721:
8716:
8710:
8704:
8698:
8693:
8687:
8681:
8675:
8669:
8664:
8658:
8652:
8643:
8636:
8634:
8631:
8612:
8611:
8608:
8605:
8601:
8600:
8597:
8594:
8590:
8589:
8586:
8583:
8573:
8570:
8555:
8552:
8547:
8544:
8522:
8519:
8514:
8511:
8500:
8499:
8482:
8479:
8474:
8471:
8467:
8463:
8460:
8457:
8454:
8451:
8448:
8445:
8442:
8439:
8435:
8431:
8423:
8419:
8415:
8412:
8404:
8400:
8396:
8394:
8391:
8388:
8385:
8382:
8379:
8376:
8373:
8370:
8367:
8364:
8361:
8358:
8355:
8352:
8349:
8346:
8343:
8340:
8337:
8334:
8331:
8328:
8320:
8316:
8312:
8310:
8307:
8304:
8301:
8298:
8295:
8292:
8289:
8286:
8283:
8280:
8277:
8274:
8271:
8268:
8265:
8262:
8259:
8256:
8253:
8250:
8247:
8244:
8236:
8232:
8228:
8226:
8221:
8218:
8213:
8210:
8206:
8202:
8199:
8196:
8193:
8190:
8187:
8184:
8181:
8178:
8174:
8170:
8162:
8158:
8154:
8151:
8143:
8139:
8135:
8133:
8130:
8127:
8124:
8121:
8118:
8115:
8112:
8109:
8106:
8103:
8100:
8097:
8094:
8091:
8088:
8085:
8082:
8079:
8076:
8073:
8070:
8067:
8059:
8055:
8051:
8049:
8046:
8043:
8040:
8037:
8034:
8031:
8028:
8025:
8022:
8019:
8016:
8013:
8010:
8007:
8004:
8001:
7998:
7995:
7992:
7989:
7986:
7983:
7975:
7971:
7967:
7965:
7953:of playing T.
7942:
7939:
7936:
7933:
7930:
7910:
7890:
7887:
7884:
7881:
7878:
7858:
7844:
7843:
7840:
7837:
7833:
7832:
7829:
7826:
7822:
7821:
7818:
7815:
7805:
7802:
7783:
7779:
7756:
7752:
7729:
7725:
7702:
7698:
7675:
7671:
7648:
7644:
7628:
7625:
7612:
7590:
7586:
7563:
7558:
7554:
7542:
7541:
7526:
7523:
7518:
7514:
7510:
7507:
7502:
7497:
7493:
7488:
7484:
7481:
7478:
7474:
7466:
7462:
7458:
7455:
7451:
7447:
7444:
7442:
7440:
7432:
7429:
7426:
7423:
7418:
7414:
7410:
7405:
7395:
7392:
7389:
7384:
7379:
7375:
7367:
7363:
7359:
7356:
7352:
7348:
7345:
7343:
7341:
7338:
7335:
7330:
7325:
7322:
7318:
7314:
7309:
7304:
7300:
7296:
7291:
7287:
7283:
7280:
7275:
7270:
7267:
7263:
7259:
7254:
7250:
7246:
7241:
7237:
7233:
7230:
7227:
7224:
7219:
7214:
7210:
7202:
7198:
7194:
7191:
7187:
7183:
7180:
7178:
7176:
7173:
7168:
7163:
7160:
7156:
7152:
7147:
7142:
7138:
7134:
7129:
7125:
7121:
7117:
7113:
7108:
7103:
7100:
7096:
7092:
7087:
7083:
7079:
7074:
7070:
7066:
7063:
7060:
7055:
7050:
7046:
7038:
7034:
7030:
7027:
7023:
7018:
7014:
7011:
7009:
7007:
7004:
6999:
6994:
6991:
6987:
6983:
6978:
6973:
6969:
6965:
6960:
6956:
6952:
6949:
6944:
6939:
6936:
6932:
6928:
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6918:
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6897:
6894:
6892:
6890:
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6886:
6860:
6849:
6848:
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6831:
6828:
6825:
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6816:
6812:
6807:
6796:
6790:
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6784:
6780:
6775:
6771:
6768:
6765:
6762:
6757:
6752:
6748:
6724:
6721:
6718:
6715:
6712:
6707:
6703:
6699:
6694:
6667:
6664:
6661:
6658:
6655:
6650:
6646:
6642:
6637:
6619:
6618:
6607:
6604:
6601:
6596:
6592:
6588:
6583:
6573:
6570:
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6562:
6557:
6553:
6549:
6546:
6543:
6538:
6533:
6530:
6526:
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6517:
6512:
6508:
6504:
6499:
6495:
6491:
6488:
6483:
6478:
6475:
6471:
6467:
6462:
6458:
6454:
6449:
6445:
6441:
6438:
6435:
6432:
6427:
6422:
6418:
6413:
6408:
6404:
6400:
6397:
6394:
6371:
6368:
6365:
6360:
6356:
6352:
6347:
6318:
6313:
6309:
6288:
6285:
6282:
6271:
6270:
6255:
6252:
6249:
6246:
6241:
6237:
6233:
6228:
6217:
6211:
6208:
6205:
6201:
6196:
6192:
6187:
6182:
6178:
6174:
6171:
6169:
6167:
6164:
6161:
6158:
6153:
6149:
6145:
6140:
6130:
6125:
6120:
6116:
6111:
6107:
6104:
6101:
6097:
6093:
6090:
6088:
6086:
6083:
6080:
6077:
6072:
6068:
6064:
6059:
6049:
6044:
6039:
6035:
6031:
6026:
6021:
6017:
6013:
6010:
6007:
6005:
6003:
5999:
5995:
5992:
5989:
5984:
5980:
5976:
5971:
5961:
5956:
5951:
5947:
5942:
5936:
5933:
5928:
5923:
5918:
5914:
5910:
5907:
5905:
5903:
5897:
5894:
5891:
5888:
5883:
5879:
5875:
5870:
5866:
5858:
5854:
5850:
5847:
5843:
5837:
5832:
5828:
5824:
5819:
5815:
5808:
5803:
5798:
5794:
5790:
5787:
5785:
5783:
5780:
5775:
5771:
5767:
5762:
5758:
5754:
5749:
5744:
5740:
5736:
5733:
5731:
5729:
5724:
5720:
5716:
5713:
5710:
5705:
5701:
5697:
5696:
5671:
5667:
5644:
5640:
5619:
5616:
5613:
5610:
5607:
5591:
5590:
5579:
5576:
5573:
5570:
5565:
5561:
5557:
5552:
5548:
5540:
5536:
5532:
5529:
5525:
5521:
5518:
5504:
5503:
5492:
5489:
5486:
5483:
5480:
5475:
5471:
5467:
5462:
5448:
5444:
5440:
5437:
5433:
5429:
5426:
5423:
5420:
5417:
5414:
5409:
5405:
5401:
5396:
5392:
5384:
5380:
5376:
5373:
5369:
5345:
5342:
5339:
5336:
5333:
5328:
5324:
5320:
5315:
5286:
5282:
5278:
5275:
5255:
5252:
5249:
5246:
5243:
5240:
5237:
5234:
5231:
5228:
5225:
5211:
5210:
5199:
5196:
5193:
5190:
5187:
5182:
5178:
5174:
5169:
5158:
5153:
5149:
5145:
5142:
5139:
5136:
5133:
5130:
5127:
5124:
5121:
5118:
5115:
5112:
5109:
5106:
5083:
5061:
5057:
5034:
5030:
5009:
4989:
4969:
4949:
4929:
4909:
4889:
4867:
4863:
4840:
4836:
4813:
4809:
4797:
4796:
4783:
4776:
4772:
4768:
4765:
4762:
4757:
4754:
4751:
4748:
4745:
4742:
4737:
4733:
4725:
4721:
4717:
4714:
4710:
4704:
4701:
4698:
4695:
4692:
4689:
4684:
4680:
4673:
4670:
4667:
4664:
4661:
4658:
4655:
4650:
4646:
4642:
4641:
4638:
4635:
4632:
4629:
4626:
4621:
4617:
4613:
4610:
4607:
4602:
4598:
4594:
4591:
4588:
4585:
4584:
4582:
4566:
4565:
4554:
4551:
4548:
4545:
4542:
4539:
4536:
4531:
4517:
4513:
4509:
4506:
4502:
4498:
4495:
4492:
4489:
4486:
4483:
4480:
4477:
4472:
4462:
4459:
4456:
4453:
4448:
4444:
4436:
4432:
4428:
4425:
4421:
4417:
4414:
4411:
4408:
4405:
4402:
4399:
4394:
4390:
4382:
4378:
4374:
4371:
4367:
4354:. We see that
4341:
4337:
4333:
4330:
4327:
4324:
4321:
4318:
4307:
4306:
4295:
4292:
4289:
4286:
4283:
4278:
4268:
4265:
4262:
4259:
4254:
4250:
4246:
4243:
4240:
4237:
4234:
4231:
4228:
4223:
4219:
4195:
4190:
4186:
4182:
4179:
4176:
4171:
4167:
4163:
4160:
4157:
4146:
4145:
4134:
4131:
4128:
4123:
4120:
4116:
4112:
4107:
4103:
4099:
4094:
4090:
4086:
4083:
4078:
4075:
4071:
4067:
4064:
4061:
4056:
4052:
4048:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
4016:
3984:
3980:
3976:
3973:
3953:
3933:
3930:
3927:
3904:
3882:
3878:
3855:
3851:
3847:
3844:
3841:
3836:
3832:
3828:
3825:
3803:
3799:
3776:
3772:
3768:
3765:
3762:
3757:
3753:
3749:
3746:
3726:
3706:
3703:
3700:
3697:
3694:
3691:
3688:
3685:
3682:
3670:
3667:
3643:
3620:
3615:
3612:
3608:
3604:
3601:
3598:
3594:
3590:
3586:
3582:
3579:
3576:
3573:
3570:
3567:
3562:
3558:
3554:
3534:
3529:
3526:
3522:
3518:
3515:
3512:
3508:
3504:
3500:
3496:
3491:
3487:
3459:
3454:
3450:
3446:
3443:
3421:
3417:
3389:
3369:
3358:
3357:
3345:
3342:
3339:
3336:
3326:
3311:
3308:
3305:
3302:
3292:
3280:
3277:
3274:
3271:
3261:
3249:
3223:
3203:
3198:
3194:
3190:
3185:
3182:
3178:
3174:
3171:
3166:
3163:
3159:
3155:
3150:
3146:
3142:
3139:
3117:
3113:
3109:
3106:
3103:
3100:
3075:
3071:
3048:
3045:
3041:
3037:
3032:
3028:
3024:
3021:
3001:
2998:
2995:
2984:
2983:
2972:
2967:
2964:
2960:
2956:
2951:
2947:
2943:
2938:
2934:
2930:
2923:
2919:
2914:
2911:
2908:
2904:
2901:
2898:
2892:
2889:
2884:
2881:
2877:
2873:
2868:
2864:
2840:
2835:
2832:
2828:
2824:
2819:
2815:
2791:
2788:
2786:
2783:
2756:
2753:
2724:average member
2692:
2689:
2688:
2687:
2686:, for example.
2675:
2664:
2652:
2639:
2636:
2635:
2634:
2626:
2623:
2620:
2617:
2614:
2602:
2599:
2555:
2554:
2551:
2536:
2533:
2515:
2514:
2509:
2506:
2503:
2499:
2498:
2495:
2492:
2487:
2483:
2482:
2479:
2474:
2471:
2467:
2466:
2463:
2460:
2456:
2455:
2452:
2434:
2431:
2420:
2419:
2410:
2401:
2392:
2383:
2379:
2378:
2369:
2360:
2351:
2342:
2338:
2337:
2328:
2319:
2310:
2301:
2297:
2296:
2287:
2278:
2269:
2260:
2256:
2255:
2252:
2249:
2246:
2242:
2241:
2238:
2228:
2225:
2157:
2137:
2134:
2129:
2126:
2121:
2118:
2059:
2035:
2013:
2010:
2000:
1999:
1996:
1993:
1991:
1988:
1985:
1983:
1979:
1978:
1975:
1972:
1970:
1967:
1964:
1962:
1958:
1957:
1954:
1950:
1949:
1946:
1924:
1923:
1920:
1917:
1915:
1912:
1909:
1907:
1903:
1902:
1899:
1896:
1894:
1891:
1888:
1886:
1882:
1881:
1878:
1874:
1873:
1870:
1864:The Stag Hunt
1843:
1842:
1837:
1832:
1830:
1825:
1820:
1818:
1814:
1813:
1808:
1803:
1801:
1796:
1791:
1789:
1785:
1784:
1781:
1777:
1776:
1773:
1760:Main article:
1757:
1754:
1752:
1749:
1734:Pareto optimal
1729:
1726:
1712:
1700:
1699:
1692:
1684:
1673:
1668:
1651:
1640:
1634:
1614:
1608:
1597:
1592:
1582:
1579:
1571:
1570:
1566:
1551:
1548:
1546:
1543:
1510:
1507:
1482:
1479:
1471:mixed-strategy
1462:
1459:
1457:
1454:
1419:
1411:
1408:
1403:
1399:
1378:
1375:
1367:
1364:
1360:
1351:
1343:
1340:
1336:
1327:
1324:
1319:
1315:
1294:
1291:
1283:
1275:
1272:
1267:
1263:
1240:
1236:
1224:
1223:
1222:
1221:
1220:
1219:
1218:
1217:
1216:
1215:
1202:
1197:
1193:
1189:
1184:
1180:
1176:
1171:
1167:
1163:
1158:
1154:
1146:
1143:
1140:
1136:
1133:
1130:
1123:
1118:
1113:
1110:
1106:
1102:
1097:
1093:
1089:
1084:
1080:
1076:
1073:
1068:
1063:
1060:
1056:
1052:
1047:
1042:
1038:
1034:
1029:
1025:
994:
993:
992:
991:
990:
989:
988:
987:
986:
985:
972:
968:
964:
959:
955:
947:
944:
941:
937:
934:
931:
924:
919:
914:
911:
907:
903:
898:
894:
890:
885:
881:
877:
874:
869:
864:
861:
857:
853:
848:
843:
839:
835:
830:
826:
792:
788:
767:
762:
757:
754:
750:
746:
741:
737:
733:
728:
724:
703:
683:
680:
677:
655:
650:
647:
643:
622:
617:
612:
609:
605:
601:
596:
591:
587:
583:
580:
575:
571:
550:
547:
544:
541:
538:
535:
532:
512:
490:
486:
476:Formally, let
461:
458:
456:
453:
361:
358:
338:auction theory
286:
283:
227:
226:
220:
216:
215:
210:
206:
205:
201:
200:
175:
171:
170:
157:
153:
152:
148:
147:
137:
136:
128:
127:
42:
40:
33:
26:
9:
6:
4:
3:
2:
11146:
11135:
11132:
11130:
11127:
11125:
11122:
11121:
11119:
11104:
11101:
11099:
11096:
11094:
11091:
11089:
11086:
11084:
11081:
11079:
11076:
11074:
11071:
11069:
11066:
11064:
11061:
11059:
11056:
11054:
11051:
11050:
11048:
11046:Miscellaneous
11044:
11038:
11035:
11033:
11030:
11028:
11025:
11023:
11020:
11018:
11015:
11013:
11010:
11009:
11007:
11003:
10997:
10994:
10992:
10989:
10987:
10984:
10982:
10981:Samuel Bowles
10979:
10977:
10976:Roger Myerson
10974:
10972:
10969:
10967:
10966:Robert Aumann
10964:
10962:
10959:
10957:
10954:
10952:
10949:
10947:
10944:
10942:
10939:
10937:
10934:
10932:
10929:
10927:
10924:
10922:
10921:Lloyd Shapley
10919:
10917:
10914:
10912:
10909:
10907:
10906:Kenneth Arrow
10904:
10902:
10899:
10897:
10894:
10892:
10889:
10887:
10886:John Harsanyi
10884:
10882:
10879:
10877:
10874:
10872:
10869:
10867:
10864:
10862:
10859:
10857:
10856:Herbert Simon
10854:
10852:
10849:
10847:
10844:
10842:
10839:
10837:
10834:
10832:
10829:
10827:
10824:
10822:
10819:
10817:
10814:
10812:
10809:
10807:
10804:
10802:
10799:
10797:
10794:
10793:
10791:
10785:
10779:
10776:
10774:
10771:
10769:
10766:
10764:
10761:
10759:
10756:
10754:
10751:
10749:
10746:
10744:
10741:
10739:
10736:
10735:
10733:
10729:
10723:
10720:
10718:
10715:
10713:
10710:
10708:
10705:
10703:
10700:
10698:
10695:
10693:
10690:
10688:
10685:
10683:
10680:
10678:
10675:
10673:
10670:
10668:
10665:
10663:
10660:
10658:
10657:Fair division
10655:
10653:
10650:
10648:
10645:
10643:
10640:
10638:
10635:
10633:
10632:Dictator game
10630:
10628:
10625:
10623:
10620:
10618:
10615:
10613:
10610:
10608:
10605:
10603:
10600:
10598:
10595:
10593:
10590:
10588:
10585:
10583:
10580:
10578:
10575:
10573:
10570:
10568:
10565:
10563:
10560:
10558:
10555:
10553:
10550:
10548:
10545:
10543:
10540:
10538:
10535:
10533:
10530:
10528:
10525:
10523:
10520:
10519:
10517:
10515:
10511:
10505:
10504:Zero-sum game
10502:
10500:
10497:
10495:
10492:
10490:
10487:
10485:
10482:
10480:
10477:
10475:
10474:Repeated game
10472:
10470:
10467:
10465:
10462:
10460:
10457:
10455:
10453:
10449:
10447:
10444:
10442:
10439:
10437:
10434:
10432:
10429:
10427:
10424:
10423:
10421:
10419:
10413:
10407:
10404:
10402:
10399:
10397:
10394:
10392:
10391:Pure strategy
10389:
10387:
10384:
10382:
10379:
10377:
10374:
10372:
10369:
10367:
10364:
10362:
10359:
10357:
10356:De-escalation
10354:
10352:
10349:
10347:
10344:
10342:
10339:
10337:
10334:
10332:
10329:
10328:
10326:
10324:
10320:
10314:
10311:
10309:
10306:
10304:
10301:
10299:
10298:Shapley value
10296:
10294:
10291:
10289:
10286:
10284:
10281:
10279:
10276:
10274:
10271:
10269:
10266:
10264:
10261:
10259:
10256:
10254:
10251:
10249:
10246:
10244:
10241:
10239:
10236:
10234:
10231:
10229:
10226:
10224:
10221:
10219:
10216:
10214:
10211:
10209:
10206:
10204:
10201:
10199:
10196:
10194:
10191:
10190:
10188:
10186:
10182:
10178:
10172:
10169:
10167:
10166:Succinct game
10164:
10162:
10159:
10157:
10154:
10152:
10149:
10147:
10144:
10142:
10139:
10137:
10134:
10132:
10129:
10127:
10124:
10122:
10119:
10117:
10114:
10112:
10109:
10107:
10104:
10102:
10099:
10097:
10094:
10092:
10089:
10087:
10084:
10083:
10081:
10077:
10073:
10065:
10060:
10058:
10053:
10051:
10046:
10045:
10042:
10036:
10032:
10029:
10026:
10024:
10021:
10017:
10013:
10012:
10007:
10003:
10002:
9992:
9988:
9985:
9981:
9980:
9975:
9974:Nasar, Sylvia
9972:
9969:
9965:
9961:
9960:
9949:
9948:
9943:
9940:
9937:
9936:
9931:
9928:
9927:
9916:
9911:
9905:
9901:
9897:
9896:
9890:
9886:
9880:
9876:
9872:
9868:
9864:
9861:
9855:
9851:
9847:
9843:
9839:
9835:
9829:
9825:
9821:
9816:
9813:
9807:
9803:
9799:
9795:
9791:
9788:
9784:
9780:
9777:
9773:
9767:
9763:
9759:
9754:
9751:
9747:
9743:
9739:
9733:
9729:
9725:
9720:
9717:
9713:
9709:
9703:
9699:
9695:
9690:
9689:
9671:
9667:
9663:
9659:
9655:
9651:
9647:
9643:
9639:
9632:
9624:
9620:
9616:
9612:
9608:
9604:
9600:
9593:
9585:
9579:
9576:. MIT Press.
9575:
9568:
9561:
9560:
9556:
9553:
9546:
9538:
9531:
9525:
9524:
9517:
9509:
9505:
9501:
9497:
9492:
9487:
9483:
9479:
9475:
9471:
9467:
9460:
9453:
9447:
9438:
9433:
9429:
9425:
9422:(1): 80–112,
9421:
9417:
9410:
9403:
9395:
9389:
9385:
9378:
9370:
9366:
9362:
9358:
9351:
9349:
9333:
9329:
9325:
9319:
9305:on 2011-10-18
9301:
9297:
9290:
9283:
9281:
9272:
9268:
9263:
9258:
9254:
9250:
9246:
9242:
9241:
9233:
9226:
9219:
9218:
9211:
9202:
9194:
9190:
9186:
9182:
9177:
9172:
9168:
9164:
9157:
9149:
9145:
9141:
9137:
9133:
9129:
9125:
9121:
9114:
9105:
9100:
9093:
9085:
9081:
9076:
9071:
9067:
9063:
9056:
9049:
9034:
9028:
9018:
9013:
9009:
9005:
9001:
8994:
8985:
8981:
8977:
8973:
8969:
8965:
8958:
8950:
8946:
8941:
8936:
8932:
8928:
8924:
8920:
8913:
8906:
8905:0-674-84031-3
8902:
8898:
8897:
8890:
8882:
8878:
8873:
8868:
8863:
8858:
8854:
8850:
8846:
8842:
8838:
8831:
8824:
8820:
8815:
8807:
8805:9780262150415
8801:
8797:
8793:
8786:
8782:
8767:
8763:
8748:
8745:
8742:
8739:
8736:
8733:
8731:
8728:
8725:
8722:
8720:
8717:
8714:
8711:
8708:
8705:
8702:
8699:
8697:
8694:
8691:
8690:M equilibrium
8688:
8685:
8682:
8679:
8676:
8673:
8670:
8668:
8665:
8662:
8659:
8656:
8653:
8647:
8644:
8641:
8638:
8637:
8630:
8627:
8623:
8618:
8609:
8606:
8602:
8598:
8591:
8580:
8569:
8553:
8550:
8545:
8542:
8520:
8517:
8512:
8509:
8480:
8477:
8472:
8469:
8461:
8458:
8455:
8452:
8449:
8446:
8443:
8440:
8437:
8413:
8389:
8386:
8383:
8380:
8377:
8371:
8368:
8365:
8356:
8353:
8347:
8344:
8338:
8335:
8329:
8305:
8302:
8299:
8296:
8293:
8287:
8284:
8281:
8272:
8269:
8263:
8260:
8254:
8251:
8245:
8219:
8216:
8211:
8208:
8200:
8197:
8194:
8191:
8188:
8185:
8182:
8179:
8176:
8152:
8128:
8125:
8122:
8119:
8116:
8110:
8107:
8104:
8095:
8092:
8086:
8083:
8077:
8074:
8068:
8044:
8041:
8038:
8035:
8032:
8026:
8023:
8020:
8011:
8008:
8002:
7999:
7993:
7990:
7984:
7956:
7955:
7954:
7937:
7934:
7931:
7908:
7885:
7882:
7879:
7856:
7841:
7838:
7834:
7830:
7827:
7823:
7812:
7801:
7797:
7781:
7777:
7754:
7750:
7727:
7723:
7700:
7696:
7673:
7669:
7646:
7642:
7634:
7624:
7610:
7588:
7584:
7561:
7556:
7552:
7524:
7521:
7516:
7508:
7500:
7495:
7491:
7486:
7482:
7479:
7476:
7472:
7464:
7460:
7456:
7453:
7449:
7445:
7443:
7424:
7421:
7416:
7412:
7403:
7390:
7382:
7377:
7373:
7365:
7361:
7357:
7354:
7350:
7346:
7344:
7328:
7323:
7320:
7316:
7312:
7307:
7302:
7298:
7289:
7285:
7281:
7273:
7268:
7265:
7261:
7257:
7252:
7248:
7239:
7235:
7225:
7217:
7212:
7208:
7200:
7196:
7192:
7189:
7185:
7181:
7179:
7166:
7161:
7158:
7154:
7150:
7145:
7140:
7136:
7127:
7123:
7119:
7115:
7106:
7101:
7098:
7094:
7090:
7085:
7081:
7072:
7068:
7061:
7053:
7048:
7044:
7036:
7032:
7028:
7025:
7021:
7016:
7012:
7010:
6997:
6992:
6989:
6985:
6981:
6976:
6971:
6967:
6958:
6954:
6950:
6942:
6937:
6934:
6930:
6926:
6921:
6916:
6912:
6903:
6899:
6895:
6893:
6888:
6877:
6876:
6875:
6872:
6858:
6835:
6832:
6826:
6823:
6818:
6814:
6805:
6794:
6788:
6785:
6782:
6778:
6773:
6769:
6763:
6755:
6750:
6746:
6738:
6737:
6736:
6722:
6719:
6713:
6710:
6705:
6701:
6692:
6665:
6662:
6656:
6653:
6648:
6644:
6635:
6602:
6599:
6594:
6590:
6581:
6568:
6560:
6555:
6551:
6547:
6536:
6531:
6528:
6524:
6520:
6515:
6510:
6506:
6497:
6493:
6489:
6481:
6476:
6473:
6469:
6465:
6460:
6456:
6447:
6443:
6433:
6425:
6420:
6416:
6411:
6406:
6402:
6398:
6395:
6385:
6384:
6383:
6366:
6363:
6358:
6354:
6345:
6316:
6311:
6307:
6299:we have that
6286:
6283:
6280:
6253:
6247:
6244:
6239:
6235:
6226:
6215:
6209:
6206:
6203:
6199:
6194:
6190:
6185:
6180:
6176:
6170:
6159:
6156:
6151:
6147:
6138:
6128:
6123:
6118:
6114:
6109:
6105:
6102:
6099:
6095:
6089:
6078:
6075:
6070:
6066:
6057:
6047:
6042:
6037:
6033:
6029:
6024:
6019:
6015:
6011:
6006:
5997:
5990:
5987:
5982:
5978:
5969:
5959:
5954:
5949:
5945:
5940:
5934:
5931:
5926:
5921:
5916:
5912:
5906:
5892:
5881:
5877:
5868:
5864:
5856:
5852:
5848:
5845:
5841:
5830:
5826:
5817:
5813:
5806:
5801:
5796:
5792:
5786:
5773:
5769:
5760:
5756:
5752:
5747:
5742:
5738:
5732:
5722:
5718:
5711:
5708:
5703:
5699:
5687:
5686:
5685:
5669:
5665:
5642:
5638:
5614:
5611:
5608:
5577:
5571:
5568:
5563:
5559:
5550:
5546:
5538:
5534:
5530:
5527:
5523:
5519:
5516:
5509:
5508:
5507:
5490:
5487:
5481:
5478:
5473:
5469:
5460:
5446:
5442:
5438:
5435:
5431:
5427:
5424:
5421:
5415:
5412:
5407:
5403:
5394:
5390:
5382:
5378:
5374:
5371:
5367:
5359:
5358:
5357:
5343:
5340:
5334:
5331:
5326:
5322:
5313:
5284:
5280:
5276:
5273:
5253:
5247:
5244:
5241:
5238:
5235:
5229:
5226:
5214:
5197:
5194:
5188:
5185:
5180:
5176:
5167:
5156:
5151:
5147:
5143:
5140:
5134:
5128:
5125:
5122:
5119:
5116:
5110:
5107:
5097:
5096:
5095:
5081:
5059:
5055:
5032:
5028:
4987:
4947:
4907:
4887:
4865:
4861:
4838:
4811:
4807:
4774:
4770:
4766:
4763:
4752:
4743:
4735:
4731:
4723:
4719:
4715:
4712:
4708:
4699:
4690:
4682:
4678:
4671:
4665:
4656:
4648:
4644:
4627:
4619:
4615:
4611:
4608:
4605:
4600:
4596:
4589:
4586:
4580:
4571:
4570:
4569:
4552:
4549:
4543:
4540:
4537:
4529:
4515:
4511:
4507:
4504:
4500:
4496:
4493:
4490:
4484:
4481:
4478:
4470:
4460:
4454:
4446:
4442:
4434:
4430:
4426:
4423:
4419:
4415:
4409:
4400:
4392:
4388:
4380:
4376:
4372:
4369:
4365:
4357:
4356:
4355:
4339:
4335:
4331:
4328:
4325:
4319:
4316:
4290:
4287:
4284:
4276:
4266:
4260:
4252:
4248:
4244:
4238:
4229:
4221:
4217:
4209:
4208:
4207:
4188:
4184:
4180:
4177:
4174:
4169:
4165:
4158:
4155:
4132:
4121:
4118:
4114:
4110:
4105:
4101:
4092:
4088:
4084:
4076:
4073:
4069:
4065:
4062:
4054:
4050:
4046:
4043:
4034:
4028:
4025:
4022:
4014:
4000:
3999:
3998:
3982:
3978:
3974:
3971:
3951:
3928:
3925:
3916:
3880:
3876:
3853:
3845:
3842:
3839:
3834:
3826:
3801:
3797:
3774:
3770:
3766:
3763:
3760:
3755:
3751:
3747:
3744:
3724:
3701:
3698:
3695:
3692:
3689:
3683:
3680:
3666:
3664:
3660:
3655:
3641:
3632:
3613:
3610:
3606:
3599:
3596:
3592:
3588:
3584:
3577:
3574:
3571:
3565:
3560:
3556:
3552:
3527:
3524:
3520:
3513:
3510:
3506:
3502:
3498:
3494:
3489:
3485:
3475:
3473:
3452:
3448:
3441:
3419:
3415:
3406:
3401:
3340:
3334:
3327:
3325:
3306:
3300:
3293:
3275:
3269:
3262:
3240:
3239:
3238:
3235:
3221:
3196:
3192:
3183:
3180:
3176:
3172:
3164:
3161:
3157:
3148:
3144:
3140:
3137:
3111:
3101:
3098:
3091:
3073:
3069:
3046:
3043:
3035:
3030:
3022:
2996:
2993:
2965:
2962:
2958:
2954:
2949:
2945:
2936:
2932:
2928:
2921:
2917:
2890:
2882:
2879:
2875:
2866:
2862:
2854:
2853:
2852:
2833:
2830:
2826:
2817:
2813:
2803:
2801:
2797:
2782:
2780:
2774:
2772:
2761:
2752:
2750:
2746:
2742:
2736:
2734:
2728:
2727:
2723:
2716:
2713:
2711:
2705:
2703:
2696:
2685:
2681:
2676:
2673:
2669:
2668:Chinese chess
2665:
2662:
2658:
2653:
2651:indefinitely.
2649:
2648:
2647:
2645:
2631:
2627:
2624:
2621:
2618:
2615:
2612:
2611:
2610:
2608:
2598:
2596:
2592:
2588:
2584:
2580:
2576:
2571:
2567:
2565:
2559:
2552:
2549:
2548:
2547:
2544:
2542:
2532:
2530:
2529:pure-strategy
2524:
2520:
2513:
2510:
2507:
2504:
2500:
2496:
2493:
2491:
2488:
2484:
2480:
2478:
2475:
2472:
2468:
2457:
2449:
2443:
2442:
2430:
2426:
2418:
2414:
2411:
2409:
2405:
2402:
2400:
2396:
2393:
2391:
2387:
2384:
2380:
2377:
2373:
2370:
2368:
2364:
2361:
2359:
2355:
2352:
2350:
2346:
2343:
2339:
2336:
2332:
2329:
2327:
2323:
2320:
2318:
2314:
2311:
2309:
2305:
2302:
2298:
2295:
2291:
2288:
2286:
2282:
2279:
2277:
2273:
2270:
2268:
2264:
2261:
2257:
2243:
2235:
2224:
2222:
2201:
2188:, and 25 via
2155:
2135:
2132:
2127:
2124:
2119:
2116:
2084:
2079:
2057:
2033:
2024:
2019:
2009:
2007:
1992:
1984:
1980:
1971:
1963:
1959:
1951:
1943:
1937:
1933:
1931:
1916:
1908:
1904:
1895:
1887:
1883:
1875:
1867:
1861:
1858:
1857:payoff matrix
1854:
1850:
1840:
1835:
1831:
1828:
1823:
1819:
1815:
1811:
1806:
1802:
1799:
1794:
1790:
1786:
1778:
1770:
1763:
1748:
1746:
1742:
1737:
1735:
1725:
1723:
1719:
1715:
1707:
1705:
1696:
1691:
1687:
1680:
1676:
1669:
1666:
1662:
1661:
1660:
1658:
1654:
1647:
1643:
1633:
1629:
1625:
1621:
1617:
1607:
1603:
1595:
1588:
1578:
1576:
1567:
1564:
1563:
1562:
1559:
1557:
1542:
1540:
1536:
1532:
1528:
1523:
1521:
1516:
1506:
1504:
1500:
1494:
1492:
1487:
1478:
1476:
1472:
1468:
1467:pure-strategy
1453:
1406:
1401:
1397:
1376:
1365:
1362:
1349:
1341:
1338:
1322:
1317:
1313:
1292:
1281:
1270:
1265:
1261:
1238:
1234:
1200:
1195:
1191:
1187:
1182:
1178:
1174:
1169:
1165:
1161:
1156:
1152:
1116:
1111:
1108:
1104:
1100:
1095:
1091:
1082:
1078:
1074:
1066:
1061:
1058:
1054:
1050:
1045:
1040:
1036:
1027:
1023:
1015:
1014:
1013:
1012:
1011:
1010:
1009:
1008:
1007:
1006:
1005:
1003:
999:
970:
966:
962:
957:
953:
917:
912:
909:
905:
901:
896:
892:
883:
879:
875:
867:
862:
859:
855:
851:
846:
841:
837:
828:
824:
816:
815:
814:
813:
812:
811:
810:
809:
808:
807:
806:
790:
786:
760:
755:
752:
748:
744:
739:
735:
726:
722:
701:
681:
678:
675:
653:
648:
645:
641:
615:
610:
607:
603:
599:
594:
589:
585:
578:
573:
569:
548:
545:
542:
539:
536:
533:
530:
510:
488:
484:
474:
472:
471:best response
466:
452:
450:
446:
442:
438:
434:
430:
426:
421:
419:
415:
410:
406:
402:
398:
394:
389:
387:
386:best response
383:
382:pure-strategy
379:
375:
371:
367:
357:
355:
351:
347:
343:
339:
335:
331:
327:
323:
319:
315:
311:
307:
303:
298:
296:
292:
282:
280:
275:
272:
271:Alice and Bob
267:
265:
260:
258:
254:
250:
246:
242:
238:
234:
225:
221:
217:
214:
211:
207:
202:
199:
195:
191:
187:
183:
179:
176:
172:
169:
165:
161:
158:
154:
149:
146:
142:
138:
133:
124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
10951:Peyton Young
10946:Paul Milgrom
10861:Hervé Moulin
10801:Amos Tversky
10752:
10743:Folk theorem
10454:-player game
10451:
10376:Grim trigger
10247:
10009:
9977:
9963:
9945:
9938:36(1):48-49.
9933:
9898:, New York:
9894:
9870:
9845:
9819:
9797:
9786:
9757:
9749:
9723:
9715:
9693:
9680:Bibliography
9645:
9641:
9631:
9609:(1): 80–87.
9606:
9602:
9592:
9573:
9567:
9550:
9545:
9521:
9516:
9473:
9470:Econometrica
9469:
9459:
9446:
9419:
9415:
9402:
9383:
9377:
9360:
9356:
9336:. Retrieved
9332:the original
9327:
9318:
9307:. Retrieved
9300:the original
9295:
9244:
9238:
9225:
9215:
9210:
9201:
9166:
9162:
9156:
9123:
9119:
9113:
9092:
9065:
9061:
9048:
9037:. Retrieved
9035:. 2015-05-25
9027:
9007:
9003:
8993:
8967:
8963:
8957:
8922:
8918:
8912:
8894:
8889:
8847:(1): 48–49.
8844:
8840:
8830:
8822:
8814:
8795:
8785:
8766:
8619:
8615:
8501:
7847:
7798:
7630:
7543:
6873:
6850:
6620:
6272:
5592:
5505:
5215:
5212:
4900:, and hence
4798:
4567:
4308:
4147:
3917:
3672:
3656:
3633:
3476:
3402:
3359:
3291:is nonempty.
3236:
2985:
2804:
2793:
2775:
2767:
2737:
2732:
2730:
2725:
2720:
2718:
2714:
2709:
2707:
2700:
2698:
2694:
2642:Examples of
2641:
2604:
2572:
2568:
2560:
2556:
2545:
2538:
2525:
2521:
2518:
2511:
2489:
2476:
2441:equilibrium.
2439:
2436:
2427:
2423:
2416:
2412:
2407:
2403:
2398:
2394:
2389:
2385:
2375:
2371:
2366:
2362:
2357:
2353:
2348:
2344:
2334:
2330:
2325:
2321:
2316:
2312:
2307:
2303:
2293:
2289:
2284:
2280:
2275:
2271:
2266:
2262:
2202:
2080:
2049:
2003:
1934:
1927:
1906:Hunt rabbit
1880:Hunt rabbit
1848:
1846:
1838:
1833:
1826:
1821:
1809:
1804:
1797:
1792:
1738:
1731:
1710:
1708:
1701:
1694:
1689:
1682:
1671:
1656:
1649:
1645:
1638:
1631:
1627:
1623:
1619:
1612:
1605:
1601:
1590:
1589:is a vector
1586:
1584:
1572:
1560:
1553:
1524:
1512:
1502:
1498:
1495:
1490:
1488:
1484:
1464:
1225:
1001:
997:
995:
668:denotes the
475:
467:
463:
422:
404:
390:
363:
299:
288:
285:Applications
276:
268:
261:
236:
230:
204:Significance
151:Relationship
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
11068:Coopetition
10871:Jean Tirole
10866:John Conway
10846:Eric Maskin
10642:Blotto game
10627:Pirate game
10436:Global game
10406:Tit for tat
10341:Bid shading
10331:Appeasement
10181:Equilibrium
10161:Solved game
10096:Determinacy
10079:Definitions
10072:game theory
9750:Game Theory
9746:Jean Tirole
9574:Game Theory
9523:Game Theory
9363:(1): 1–12,
9262:10362/11577
9068:(4): 1138.
8661:Cooperation
7623:as needed.
6871:as needed.
2702:equilibrium
2672:tic-tac-toe
2644:game theory
2591:game theory
2564:expectation
2382:Choose "3"
2341:Choose "2"
2300:Choose "1"
2259:Choose "0"
2254:Choose "3"
2251:Choose "2"
2248:Choose "1"
2245:Choose "0"
1728:Rationality
1648:must be in
1533:and on the
455:Definitions
431:'. In 1965
306:tit-for-tat
293:of several
233:game theory
209:Proposed by
174:Superset of
145:game theory
11118:Categories
10712:Trust game
10697:Kuhn poker
10366:Escalation
10361:Deterrence
10351:Cheap talk
10323:Strategies
10141:Preference
10070:Topics of
9942:Nash, John
9930:Nash, John
9752:MIT Press.
9537:game trees
9437:10016/4408
9338:2019-12-08
9309:2008-11-07
9176:1605.03281
9104:2403.01537
9039:2015-08-30
8940:2108/55644
8925:(3): 577.
8819:Kreps D.M.
8778:References
5299:such that
5020:, call it
3964:on action
3407:. Because
3356:is convex.
3130:such that
2800:David Gale
2601:Occurrence
2438:interest.
2016:See also:
1885:Hunt stag
1877:Hunt stag
1430:such that
80:newspapers
10896:John Nash
10602:Stag hunt
10346:Collusion
10016:EMS Press
9875:MIT Press
9728:MIT Press
9670:122603890
9662:1432-1270
9623:0036-1399
9500:0012-9682
9169:: 18–73.
9140:0018-9286
9070:CiteSeerX
8984:143728467
8582:Strategy
8466:⟹
8456:−
8444:−
8434:⟹
8384:−
8369:−
8336:−
8303:−
8285:−
8270:−
8205:⟹
8198:−
8180:−
8173:⟹
8126:−
8108:−
8093:−
8039:−
8024:−
7991:−
7935:−
7883:−
7814:Strategy
7589:∗
7585:σ
7562:∗
7553:σ
7501:∗
7492:σ
7480:−
7457:∈
7450:∑
7417:∗
7413:σ
7383:∗
7374:σ
7358:∈
7351:∑
7329:∗
7321:−
7317:σ
7308:∗
7299:σ
7282:−
7274:∗
7266:−
7262:σ
7218:∗
7209:σ
7193:∈
7186:∑
7167:∗
7159:−
7155:σ
7146:∗
7137:σ
7120:−
7107:∗
7099:−
7095:σ
7054:∗
7045:σ
7029:∈
7022:∑
6998:∗
6990:−
6986:σ
6977:∗
6968:σ
6951:−
6943:∗
6935:−
6931:σ
6922:∗
6913:σ
6819:∗
6815:σ
6786:−
6756:∗
6747:σ
6706:∗
6702:σ
6649:∗
6645:σ
6595:∗
6591:σ
6561:∗
6552:σ
6537:∗
6529:−
6525:σ
6516:∗
6507:σ
6490:−
6482:∗
6474:−
6470:σ
6426:∗
6417:σ
6399:∈
6393:∀
6367:⋅
6359:∗
6355:σ
6317:∗
6308:σ
6248:⋅
6240:∗
6236:σ
6207:−
6186:∗
6177:σ
6173:⇒
6160:⋅
6152:∗
6148:σ
6124:∗
6115:σ
6103:−
6092:⇒
6079:⋅
6071:∗
6067:σ
6043:∗
6034:σ
6025:∗
6016:σ
6009:⇒
5991:⋅
5983:∗
5979:σ
5955:∗
5946:σ
5922:∗
5913:σ
5909:⇒
5882:∗
5878:σ
5849:∈
5842:∑
5831:∗
5827:σ
5802:∗
5793:σ
5789:⇒
5774:∗
5770:σ
5748:∗
5739:σ
5735:⇒
5723:∗
5719:σ
5704:∗
5700:σ
5670:∗
5666:σ
5615:⋅
5564:∗
5560:σ
5531:∈
5524:∑
5474:∗
5470:σ
5439:∈
5432:∑
5408:∗
5404:σ
5375:∈
5368:∑
5327:∗
5323:σ
5277:∈
5242:⋯
5230:∈
5224:∃
5181:∗
5177:σ
5144:∈
5138:∀
5123:⋯
5111:∈
5105:∀
5060:∗
5056:σ
5033:∗
5029:σ
5008:Δ
4968:Δ
4928:Δ
4888:σ
4835:Δ
4767:∈
4744:σ
4716:∈
4709:∑
4691:σ
4657:σ
4637:Δ
4634:→
4631:Δ
4609:⋯
4538:σ
4508:∈
4501:∑
4479:σ
4443:σ
4427:∈
4420:∑
4401:σ
4373:∈
4366:∑
4332:∈
4323:Δ
4320:∈
4317:σ
4285:σ
4249:σ
4230:σ
4178:…
4119:−
4115:σ
4102:σ
4085:−
4074:−
4070:σ
4023:σ
3975:∈
3932:Δ
3929:∈
3926:σ
3903:Δ
3850:Δ
3846:×
3843:⋯
3840:×
3831:Δ
3824:Δ
3767:×
3764:⋯
3761:×
3611:−
3607:σ
3597:∈
3585:σ
3578:λ
3575:−
3557:σ
3553:λ
3525:−
3521:σ
3511:∈
3499:σ
3486:σ
3449:σ
3388:Σ
3368:Σ
3341:σ
3307:σ
3276:σ
3248:Σ
3193:σ
3181:−
3173:×
3162:−
3158:σ
3116:Σ
3108:→
3105:Σ
3102::
3044:−
3040:Σ
3036:×
3027:Σ
3020:Σ
3000:Σ
2997:∈
2994:σ
2963:−
2959:σ
2946:σ
2929:
2918:σ
2880:−
2876:σ
2831:−
2827:σ
2749:stability
2741:economics
2684:arms race
2628:There is
2541:stability
2535:Stability
2502:Option C
2486:Option B
2470:Option A
2465:Option C
2462:Option B
2459:Option A
2184:, 50 via
1930:stag hunt
1545:Existence
1201:∗
1188:≠
1162:∈
1117:∗
1109:−
1067:∗
1059:−
1046:∗
963:∈
918:∗
910:−
876:≥
868:∗
860:−
847:∗
791:∗
761:∗
753:−
679:−
654:∗
646:−
616:∗
608:−
595:∗
574:∗
543:…
435:proposed
374:oligopoly
322:bank runs
314:stag hunt
257:oligopoly
156:Subset of
110:June 2023
11037:Lazy SMP
10731:Theorems
10682:Deadlock
10537:Checkers
10418:of games
10185:concepts
10031:Archived
9976:(1998),
9844:(2010),
9796:(1997),
9555:Archived
9193:16055840
9148:21850096
8949:57072280
8881:16588946
8633:See also
7804:Examples
5657:. Since
3593:′
3507:′
3012:, where
2682:" or an
2593:context
2148:, where
2093:(one of
1853:strategy
1751:Examples
1456:Variants
523:, where
445:repeated
429:credible
409:zero-sum
346:football
264:strategy
219:Used for
10789:figures
10572:Chicken
10426:Auction
10416:Classes
10018:, 2001
9785:(1947)
9748:(1991)
9508:1911749
8872:1063129
8849:Bibcode
7842:−1, +1
7839:+1, −1
7831:+1, −1
7828:−1, +1
5506:So let
5356:. Then
3545:, then
2781:arise.
2680:chicken
2589:. In a
2200:edge).
1718:simplex
1679:concave
1600:Denote
1575:compact
360:History
249:Cournot
94:scholar
9906:
9881:
9856:
9830:
9808:
9768:
9734:
9704:
9668:
9660:
9621:
9580:
9506:
9498:
9390:
9271:882466
9269:
9191:
9146:
9138:
9072:
8982:
8947:
8903:
8879:
8869:
8802:
6273:Since
4206:where
3717:where
2986:Here,
2699:(...)
2607:unique
2512:10, 10
2508:15, 5
2505:10, 5
2497:5, 15
2490:40, 25
2481:5, 10
2477:25, 40
1665:convex
714:. Let
561:. Let
255:in an
235:, the
96:
89:
82:
75:
67:
10527:Chess
10514:Games
9666:S2CID
9504:JSTOR
9412:(PDF)
9303:(PDF)
9292:(PDF)
9235:(PDF)
9189:S2CID
9171:arXiv
9144:S2CID
9099:arXiv
9058:(PDF)
8980:S2CID
8945:S2CID
8758:Notes
8610:0, 0
8607:0, 0
8599:0, 0
8596:1, 1
7742:. If
2494:0, 0
2473:0, 0
2101:, or
1716:is a
1663:T is
1637:,...,
1611:+...+
1469:or a
1439:Price
1414:Price
348:(see
328:(see
101:JSTOR
87:books
10208:Core
9904:ISBN
9879:ISBN
9854:ISBN
9828:ISBN
9806:ISBN
9781:and
9766:ISBN
9732:ISBN
9702:ISBN
9658:ISSN
9619:ISSN
9578:ISBN
9496:ISSN
9388:ISBN
9267:SSRN
9136:ISSN
8901:ISBN
8877:PMID
8841:PNAS
8800:ISBN
8620:The
8535:and
7522:>
7399:Gain
6801:Gain
6688:Gain
6663:>
6631:Gain
6577:Gain
6341:Gain
6284:>
6222:Gain
6134:Gain
6053:Gain
5965:Gain
5602:Gain
5488:>
5456:Gain
5341:>
5309:Gain
5266:and
5163:Gain
4960:and
4550:>
4525:Gain
4466:Gain
4309:for
4272:Gain
4010:Gain
2764:(U).
2743:and
2585:and
2215:and
2186:ABCD
2099:ABCD
2083:game
1847:The
1499:weak
1370:High
1075:>
998:weak
399:and
324:and
243:for
222:All
73:news
10787:Key
9650:doi
9611:doi
9535:10
9486:hdl
9478:doi
9432:hdl
9424:doi
9365:doi
9257:hdl
9249:doi
9245:144
9181:doi
9128:doi
9080:doi
9012:doi
8972:doi
8935:hdl
8927:doi
8867:PMC
8857:doi
4038:max
3997:be
3322:is
2633:on.
2209:ACD
2205:ABD
2190:ACD
2182:ABD
2176:to
2128:100
2107:ABD
2103:ACD
2095:ABD
2089:to
2074:to
1681:in
1501:or
1346:Low
1330:Yes
1278:Yes
352:),
231:In
143:in
56:by
11120::
10522:Go
10014:,
10008:,
9982:,
9966:,
9902:,
9877:,
9873:,
9852:,
9848:,
9826:,
9822:,
9804:,
9800:,
9760:,
9730:,
9726:,
9700:,
9696:,
9664:.
9656:.
9644:.
9640:.
9617:.
9607:21
9605:.
9601:.
9502:.
9494:.
9484:.
9474:33
9472:.
9468:.
9430:,
9420:17
9418:,
9414:,
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9359:,
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9326:.
9294:.
9279:^
9265:.
9255:.
9243:.
9237:.
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9078:.
9066:92
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8978:.
8968:44
8966:.
8943:.
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8875:.
8865:.
8855:.
8845:36
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8839:.
8568:.
5491:1.
5198:0.
4553:0.
3915:.
3474:.
2704:.
2415:,
2406:,
2397:,
2395:−1
2388:,
2386:−2
2374:,
2365:,
2356:,
2354:−1
2347:,
2345:−2
2335:−1
2333:,
2326:−1
2324:,
2315:,
2306:,
2304:−2
2294:−2
2292:,
2285:−2
2283:,
2276:−2
2274:,
2265:,
2223:.
2198:CD
2194:AB
2170:AB
2097:,
1997:10
1994:10
1968:10
1965:10
1736:.
1604::=
1541:.
1505:.
1493:.
1354:No
1286:No
259:.
196:,
192:,
188:,
184:,
180:,
166:,
162:,
10452:n
10063:e
10056:t
10049:v
9986:.
9970:.
9917:.
9837:.
9711:.
9672:.
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9586:.
9540:.
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9488::
9480::
9454:.
9440:.
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9426::
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9367::
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9251::
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9101::
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9082::
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8986:.
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8883:.
8859::
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8554:2
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8418:E
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8360:)
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8319:[
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7428:)
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6351:(
6346:i
6312:i
6287:1
6281:C
6254:.
6251:)
6245:,
6232:(
6227:i
6216:)
6210:1
6204:C
6200:1
6195:(
6191:=
6181:i
6163:)
6157:,
6144:(
6139:i
6129:=
6119:i
6110:)
6106:1
6100:C
6096:(
6082:)
6076:,
6063:(
6058:i
6048:+
6038:i
6030:=
6020:i
6012:C
5998:)
5994:)
5988:,
5975:(
5970:i
5960:+
5950:i
5941:(
5935:C
5932:1
5927:=
5917:i
5896:)
5893:a
5890:(
5887:)
5874:(
5869:i
5865:g
5857:i
5853:A
5846:a
5836:)
5823:(
5818:i
5814:g
5807:=
5797:i
5779:)
5766:(
5761:i
5757:f
5753:=
5743:i
5728:)
5715:(
5712:f
5709:=
5643:i
5639:A
5618:)
5612:,
5609:i
5606:(
5578:.
5575:)
5572:a
5569:,
5556:(
5551:i
5547:g
5539:i
5535:A
5528:a
5520:=
5517:C
5485:)
5482:a
5479:,
5466:(
5461:i
5447:i
5443:A
5436:a
5428:+
5425:1
5422:=
5419:)
5416:a
5413:,
5400:(
5395:i
5391:g
5383:i
5379:A
5372:a
5344:0
5338:)
5335:a
5332:,
5319:(
5314:i
5285:i
5281:A
5274:a
5254:,
5251:}
5248:N
5245:,
5239:,
5236:1
5233:{
5227:i
5195:=
5192:)
5189:a
5186:,
5173:(
5168:i
5157::
5152:i
5148:A
5141:a
5135:,
5132:}
5129:N
5126:,
5120:,
5117:1
5114:{
5108:i
5082:G
4988:f
4948:f
4908:f
4866:i
4862:f
4839:i
4812:i
4808:f
4775:i
4771:A
4764:a
4756:)
4753:b
4750:(
4747:)
4741:(
4736:i
4732:g
4724:i
4720:A
4713:b
4703:)
4700:a
4697:(
4694:)
4688:(
4683:i
4679:g
4672:=
4669:)
4666:a
4663:(
4660:)
4654:(
4649:i
4645:f
4628::
4625:)
4620:N
4616:f
4612:,
4606:,
4601:1
4597:f
4593:(
4590:=
4587:f
4581:{
4547:)
4544:a
4541:,
4535:(
4530:i
4516:i
4512:A
4505:a
4497:+
4494:1
4491:=
4488:)
4485:a
4482:,
4476:(
4471:i
4461:+
4458:)
4455:a
4452:(
4447:i
4435:i
4431:A
4424:a
4416:=
4413:)
4410:a
4407:(
4404:)
4398:(
4393:i
4389:g
4381:i
4377:A
4370:a
4340:i
4336:A
4329:a
4326:,
4294:)
4291:a
4288:,
4282:(
4277:i
4267:+
4264:)
4261:a
4258:(
4253:i
4245:=
4242:)
4239:a
4236:(
4233:)
4227:(
4222:i
4218:g
4194:)
4189:N
4185:g
4181:,
4175:,
4170:1
4166:g
4162:(
4159:=
4156:g
4133:.
4130:}
4127:)
4122:i
4111:,
4106:i
4098:(
4093:i
4089:u
4082:)
4077:i
4066:,
4063:a
4060:(
4055:i
4051:u
4047:,
4044:0
4041:{
4035:=
4032:)
4029:a
4026:,
4020:(
4015:i
3983:i
3979:A
3972:a
3952:i
3881:i
3877:A
3854:N
3835:1
3827:=
3802:i
3798:A
3775:N
3771:A
3756:1
3752:A
3748:=
3745:A
3725:N
3705:)
3702:u
3699:,
3696:A
3693:,
3690:N
3687:(
3684:=
3681:G
3642:r
3619:)
3614:i
3603:(
3600:r
3589:i
3581:)
3572:1
3569:(
3566:+
3561:i
3533:)
3528:i
3517:(
3514:r
3503:i
3495:,
3490:i
3458:)
3453:i
3445:(
3442:r
3420:i
3416:u
3344:)
3338:(
3335:r
3310:)
3304:(
3301:r
3279:)
3273:(
3270:r
3222:r
3202:)
3197:i
3189:(
3184:i
3177:r
3170:)
3165:i
3154:(
3149:i
3145:r
3141:=
3138:r
3112:2
3099:r
3074:i
3070:u
3047:i
3031:i
3023:=
2971:)
2966:i
2955:,
2950:i
2942:(
2937:i
2933:u
2922:i
2913:x
2910:a
2907:m
2903:g
2900:r
2897:a
2891:=
2888:)
2883:i
2872:(
2867:i
2863:r
2839:)
2834:i
2823:(
2818:i
2814:r
2678:"
2417:3
2413:3
2408:4
2404:0
2399:3
2390:2
2376:0
2372:4
2367:2
2363:2
2358:3
2349:2
2331:3
2322:3
2317:1
2313:1
2308:2
2290:2
2281:2
2272:2
2267:0
2263:0
2217:C
2213:B
2178:D
2174:A
2156:x
2136:2
2133:+
2125:x
2120:+
2117:1
2091:D
2087:A
2076:D
2072:A
2058:x
2034:x
1989:0
1986:0
1976:0
1973:0
1921:1
1918:1
1913:1
1910:0
1900:0
1897:1
1892:2
1889:2
1839:2
1834:2
1827:3
1822:1
1810:1
1805:3
1798:4
1793:4
1713:i
1711:S
1698:.
1695:i
1693:-
1690:s
1685:i
1683:s
1674:i
1672:u
1657:i
1652:i
1650:S
1646:i
1641:n
1639:S
1635:1
1632:S
1628:S
1624:S
1620:S
1615:n
1613:m
1609:1
1606:m
1602:m
1598:.
1593:i
1591:s
1587:i
1418:}
1410:{
1407:=
1402:i
1398:S
1377:.
1374:}
1366:=
1363:p
1359:|
1350:,
1342:=
1339:p
1335:|
1326:{
1323:=
1318:i
1314:S
1293:.
1290:}
1282:,
1274:{
1271:=
1266:i
1262:S
1239:i
1235:S
1196:i
1192:s
1183:i
1179:s
1175:,
1170:i
1166:S
1157:i
1153:s
1145:l
1142:l
1139:a
1135:r
1132:o
1129:f
1122:)
1112:i
1105:s
1101:,
1096:i
1092:s
1088:(
1083:i
1079:u
1072:)
1062:i
1055:s
1051:,
1041:i
1037:s
1033:(
1028:i
1024:u
971:i
967:S
958:i
954:s
946:l
943:l
940:a
936:r
933:o
930:f
923:)
913:i
906:s
902:,
897:i
893:s
889:(
884:i
880:u
873:)
863:i
856:s
852:,
842:i
838:s
834:(
829:i
825:u
787:s
766:)
756:i
749:s
745:,
740:i
736:s
732:(
727:i
723:u
702:i
682:1
676:N
649:i
642:s
621:)
611:i
604:s
600:,
590:i
586:s
582:(
579:=
570:s
549:N
546:,
540:,
537:1
534:=
531:i
511:i
489:i
485:S
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
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