Nash equilibrium - Knowledge

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. 6268: 7539: 5690: 6880: 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}}} 2760: 8629:

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. 8497: 412:

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 " 7959: 36: 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 2777:

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%). 2429:

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. 4794: 6616: 2023: 4563: 2561:

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. 7799:

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 5501: 4574: 1496:

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

5208: 6846: 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. 4360: 3629: 1485:

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

9451: 5588: 2857: 1018: 3128: 3059: 6676: 6380: 5354: 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 7964: 6885: 1303: 6733: 2546:

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:

5264: 4352: 631: 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 ' 2566:

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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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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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.

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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.

4003: 2146: 819: 2849: 3942: 3010: 8566: 8533: 5100: 7574: 6329: 3468: 1450: 559: 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 7601: 5682: 5072: 5045: 4851: 5297: 3995: 3354: 3320: 3289: 666: 465:

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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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

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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 9841: 3548: 1513:

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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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 1732:

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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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

11123: 3133: 2751:" theory above. In these situations the assumption that the strategy observed is actually a NE has often been borne out by research. 2004:

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}^{*}} 9554: 5512: 2570:

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.

1555: 1256: 6681: 1706:. Rosen also proves that, under certain technical conditions which include strict concavity, the equilibrium is unique. 79: 10777: 10312: 10110: 9857: 9705: 3740: 189: 11133: 10596: 10415: 8904: 8803: 8723: 8625: 1000:: a player might be indifferent among several strategies given the other players' choices. It is unique and called a 309: 119: 53: 10030: 10217: 10005: 9032: 5219: 4312: 2105:). The "payoff" of each strategy is the travel time of each route. In the graph on the right, a car travelling via 1744: 1526: 17: 9054: 564: 86: 10686: 9323: 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. 2726:

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:

9967: 5596: 4151: 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. 1743:

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}} 717: 68: 9408: 2203:

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.

2112: 10307: 10287: 10010: 8729: 5203:{\displaystyle \forall i\in \{1,\cdots ,N\},\forall a\in A_{i}:\quad {\text{Gain}}_{i}(\sigma ^{*},a)=0.} 2808: 436: 417: 181: 9216: 3921: 2989: 11021: 10772: 10742: 10400: 10242: 10237: 9899: 9761: 9239: 9055:"Testing Mixed-Strategy Equilibria when Players Are Heterogeneous: The Case of Penalty Kicks in Soccer" 8712: 8502:

Thus, a mixed-strategy Nash equilibrium in this game is for each player to randomly choose H or T with

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Or, the strategy set might be a finite set of conditional strategies responding to other players, e.g.

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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

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in 1949, von Neumann famously dismissed it with the words, "That's trivial, you know. That's just a

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in crowds, energy systems, transportation systems, evacuation problems and wireless communications.

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Ward, H. (1996). "Game Theory and the Politics of Global Warming: The State of Play and Beyond".

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The players all will do their utmost to maximize their expected payoff as described by the game.

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Nash equilibria need not exist if the set of choices is infinite and non-compact. For example:

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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)

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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

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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: 18:Nash Equilibrium 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: 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: 6923: 6918: 6914: 6910: 6905: 6901: 6897: 6894: 6892: 6890: 6887: 6886: 6860: 6849: 6848: 6837: 6834: 6831: 6828: 6825: 6820: 6816: 6812: 6807: 6796: 6790: 6787: 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: 6567: 6562: 6557: 6553: 6549: 6546: 6543: 6538: 6533: 6530: 6526: 6522: 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:, 9361:42 9359:, 9347:^ 9326:. 9294:. 9279:^ 9265:. 9255:. 9243:. 9237:. 9187:. 9179:. 9165:. 9142:. 9134:. 9124:62 9122:. 9078:. 9066:92 9064:. 9060:. 9008:74 9006:. 9002:. 8978:. 8968:44 8966:. 8943:. 8933:. 8923:92 8921:. 8875:. 8865:. 8855:. 8845:36 8843:. 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:. 9652:: 9646:2 9625:. 9613:: 9586:. 9540:. 9510:. 9488:: 9480:: 9454:. 9440:. 9434:: 9426:: 9396:. 9371:. 9367:: 9341:. 9312:. 9273:. 9259:: 9251:: 9195:. 9183:: 9173:: 9167:1 9150:. 9130:: 9107:. 9101:: 9086:. 9082:: 9042:. 9022:, 9020:. 9014:: 8988:, 8986:. 8974:: 8951:. 8937:: 8929:: 8907:. 8883:. 8859:: 8851:: 8808:. 8554:2 8551:1 8546:= 8543:q 8521:2 8518:1 8513:= 8510:p 8481:2 8478:1 8473:= 8470:p 8462:p 8459:2 8453:1 8450:= 8447:1 8441:p 8438:2 8430:] 8422:[ 8418:E 8414:= 8411:] 8403:[ 8399:E 8390:p 8387:2 8381:1 8378:= 8375:) 8372:p 8366:1 8363:( 8360:) 8357:1 8354:+ 8351:( 8348:+ 8345:p 8342:) 8339:1 8333:( 8330:= 8327:] 8319:[ 8315:E 8306:1 8300:p 8297:2 8294:= 8291:) 8288:p 8282:1 8279:( 8276:) 8273:1 8267:( 8264:+ 8261:p 8258:) 8255:1 8252:+ 8249:( 8246:= 8243:] 8235:[ 8231:E 8220:2 8217:1 8212:= 8209:q 8201:1 8195:q 8192:2 8189:= 8186:q 8183:2 8177:1 8169:] 8161:[ 8157:E 8153:= 8150:] 8142:[ 8138:E 8129:1 8123:q 8120:2 8117:= 8114:) 8111:q 8105:1 8102:( 8099:) 8096:1 8090:( 8087:+ 8084:q 8081:) 8078:1 8075:+ 8072:( 8069:= 8066:] 8058:[ 8054:E 8045:q 8042:2 8036:1 8033:= 8030:) 8027:q 8021:1 8018:( 8015:) 8012:1 8009:+ 8006:( 8003:+ 8000:q 7997:) 7994:1 7988:( 7985:= 7982:] 7974:[ 7970:E 7941:) 7938:q 7932:1 7929:( 7909:q 7889:) 7886:p 7880:1 7877:( 7857:p 7782:A 7778:s 7755:A 7751:s 7728:A 7724:s 7701:A 7697:s 7674:A 7670:s 7647:A 7643:s 7611:G 7557:i 7525:0 7517:2 7513:) 7509:a 7506:( 7496:i 7487:) 7483:1 7477:C 7473:( 7465:i 7461:A 7454:a 7446:= 7428:) 7425:a 7422:, 7409:( 7404:i 7394:) 7391:a 7388:( 7378:i 7366:i 7362:A 7355:a 7347:= 7337:) 7334:) 7324:i 7313:, 7303:i 7295:( 7290:i 7286:u 7279:) 7269:i 7258:, 7253:i 7249:a 7245:( 7240:i 7236:u 7232:( 7229:) 7226:a 7223:( 7213:i 7201:i 7197:A 7190:a 7182:= 7172:) 7162:i 7151:, 7141:i 7133:( 7128:i 7124:u 7116:) 7112:) 7102:i 7091:, 7086:i 7082:a 7078:( 7073:i 7069:u 7065:) 7062:a 7059:( 7049:i 7037:i 7033:A 7026:a 7017:( 7013:= 7003:) 6993:i 6982:, 6972:i 6964:( 6959:i 6955:u 6948:) 6938:i 6927:, 6917:i 6909:( 6904:i 6900:u 6896:= 6889:0 6859:0 6836:0 6833:= 6830:) 6827:a 6824:, 6811:( 6806:i 6795:) 6789:1 6783:C 6779:1 6774:( 6770:= 6767:) 6764:a 6761:( 6751:i 6723:0 6720:= 6717:) 6714:a 6711:, 6698:( 6693:i 6666:0 6660:) 6657:a 6654:, 6641:( 6636:i 6606:) 6603:a 6600:, 6587:( 6582:i 6572:) 6569:a 6566:( 6556:i 6548:= 6545:) 6542:) 6532:i 6521:, 6511:i 6503:( 6498:i 6494:u 6487:) 6477:i 6466:, 6461:i 6457:a 6453:( 6448:i 6444:u 6440:( 6437:) 6434:a 6431:( 6421:i 6412:: 6407:i 6403:A 6396:a 6370:) 6364:, 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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Index