869:. One approach is to continue to treat individual players as reasoning in isolation, but to allow them, with some probability, to reason from the perspective of a collective. Another approach is to assume that players within any collective agent know that the agent exists, but that other players do not know this, although they suspect it with some probability. For example, Alice and Bob may sometimes optimize as individuals and sometimes collude as a team, depending on the state of nature, but other players may not know which of these is the case.
269:. Bayesian games are also useful in that they do not require infinite sequential calculations. Infinite sequential calculations would arise where players (essentially) try to "get into each other's heads". For example, one may ask questions and decide "If I expect some action from player B, then player B will anticipate that I expect that action, so then I should anticipate that anticipation"
1228:
type2 appearing from whether the previous firm entering the market was blocked, it is a
Bayesian game. The reason for these judgements is that there are blocking costs for player2, which may need to make significant price cuts to prevent player1 from entering the market, so it will block player1 when the profit it steals from entering the market is greater than the blocking costs.
63:
to these characteristics and by calculating the outcome of the game using
Bayesian probability, the result is a game whose solution is, for technical reasons, far easier to calculate than a similar game in a non-Bayesian context. For those technical reasons, see the Specification of games section in
264:
There are two important and novel aspects to
Bayesian games that were themselves specified by Harsanyi. The first is that Bayesian games should be considered and structured identically to complete information games. Except, by attaching probability to the game, the final game functions as though it
1227:
A new company (player1) that wants to enter a market that is monopolised by a large company will encounter two types of monopolist (player2), type1 is prevented and type2 is allowed. Player1 will never have complete information about player2, but may be able to infer the probability of type1 and
1151:
There is a used car. Player 1 is a potential buyer who is interested in the car. Player 2 is the owner of the car and knows the value v of the car (how good it is, etc.). Player 1 does not and believes that the value v of the car to the owner (Player 2) is distributed uniformly between 0 and 100
905:
The sheriff would rather defend himself and shoot if the suspect shoots, or not shoot if the suspect does not (even if the suspect is a criminal). The suspect would rather shoot if he is a criminal, even if the sheriff does not shoot, but would rather not shoot if he is a civilian, even if the
299:
An analogous concept can be defined for a
Bayesian game, the difference being that every player's strategy maximizes their expected payoff given their beliefs about the state of nature. A player's beliefs about the state of nature are formed by conditioning the prior probabilities
273:. Bayesian games allows for the calculation of these outcomes in one move by simultaneously assigning different probability weights to different outcomes. The effect of this is that Bayesian games allow for the modeling of a number of games that in a non-Bayesian setting would be
853:, to represent environment states (e.g. physical world states) with stochastic transitions between states as well as uncertainty about the types of different players in each state. The resulting model is solved via a recursive combination of the Bayesian Nash equilibrium and the
857:. Stochastic Bayesian games have been used to address diverse problems, including defense and security planning, cybersecurity of power plants, autonomous driving, mobile edge computing, self-stabilization in dynamic systems, and misbehavior treating in crowdsourcing IoT.
38:
is a strategic decision-making model which assumes players have incomplete information. Players hold private information relevant to the game, meaning that the payoffs are not common knowledge. Bayesian games model the outcome of player interactions using aspects of
1092:
for the suspect is to shoot, and when the type is "civilian", the dominant strategy for the suspect is not to shoot; alternative strictly dominated strategy can thus be removed. Given this, if the sheriff shoots, he will have a payoff of 0 with probability
420:
For finite
Bayesian games, i.e., both the action and the type space are finite, there are two equivalent representations. The first is called the agent-form game (see Theorem 9.51 of the Game Theory book) which expands the number of players from
1076:
If both players are rational and both know that both players are rational and everything that is known by any player is known to be known by every player (i.e. player 1 knows player 2 knows that player 1 is rational and player 2 knows this, etc.
58:
for these and other contributions to game theory in 1994. Roughly speaking, Harsanyi defined
Bayesian games in the following way: players are assigned by nature at the start of the game a set of characteristics. By mapping
647:, i.e., the pure policy is a combination of actions the player should take for different types. Nash Equilibrium (NE) can be computed in these two equivalent representations, and the BNE can be recovered from the NE.
885:
The suspect can either be of type "criminal" or type "civilian". The sheriff has only one type. The suspect knows its type and the
Sheriff's type, but the Sheriff does not know the suspect's type. Thus, there is
696:
Nature's node is usually denoted by an unfilled circle. Its strategy is always specified and always completely mixed. Usually, Nature is at the root of the tree, however Nature can move at other points as well.
840:
requires that, starting from any information set, subsequent play be optimal. It requires that beliefs be updated consistently with Bayes' rule on every path of play that occurs with positive probability.
1322:
Harsanyi, John C., 1967/1968. "Games with
Incomplete Information Played by Bayesian Players, I-III." Management Science 14 (3): 159-183 (Part I), 14 (5): 320-334 (Part II), 14 (7): 486-502 (Part III).
829:
Bayesian Nash equilibrium can result in implausible equilibria in dynamic games, where players move sequentially rather than simultaneously. As in games of complete information, these can arise via
517:
327:
is defined as a strategy profile that maximizes the expected payoff for each player given their beliefs and given the strategies played by the other players. That is, a strategy profile
296:
to every other strategy in the profile; i.e., there is no strategy that a player could play that would yield a higher payoff, given all the strategies played by the other players.
645:
265:
were an incomplete information game. Therefore, players can be essentially modelled as having incomplete information and the probability space of the game still follows the
249:
Players know their own type, but only a probability distribution of other players. A player studies the expected value of the other player's type when considering payoffs.
395:
584:
345:
898:
that the suspect is a civilian; both players are aware of this probability (common prior assumption, which can be converted into a complete-information game with
547:
449:
243:
Players do not know their own types or those of other players. A player recognises payoffs as expected values based on a prior distribution of all possible types.
368:
415:
318:
1375:
Harsanyi, John C. (1968). "Games with
Incomplete Information Played by "Bayesian" Players, I-III. Part III. The Basic Probability Distribution of the Game".
519:, i.e., every type of each player becomes a player. The second is called the induced normal form (see Section 6.3.3 of Multiagent Systems) which still has
818:
in an extensive form game is a combination of strategies and a specification of beliefs such that the following two conditions are satisfied:
833:
strategies off the equilibrium path. In games of incomplete information there is also the additional possibility of non-credible beliefs.
1332:
Harsanyi, John C. (1968). "Games with
Incomplete Information Played by "Bayesian" Players, I-III. Part II. Bayesian Equilibrium Points".
1924:"Addressing Inherent Uncertainty: Risk-Sensitive Behavior Generation for Automated Driving using Distributional Reinforcement Learning"
55:
17:
43:. They are notable because they allowed, for the first time in game theory, for the specification of the solutions to games with
1155:
Player 1 can make a bid p between 0 and 100 (inclusive) I Player 2 can then accept or reject the offer. The payoffs as follows:
2344:
2071:
1953:
1797:
1623:
1590:
1299:
1528:
Harsanyi, John C. (2004). "Games with Incomplete Information Played by "Bayesian" Players, I-III: Part I. The Basic Model".
3243:
712:
is at one of her decision nodes in an information set, she does not know which node within the information set she is at.
3060:
2595:
2393:
3411:
2879:
2698:
1815:
1724:
1813:
Albrecht, Stefano; Crandall, Jacob; Ramamoorthy, Subramanian (2016). "Belief and Truth in Hypothesised Behaviours".
1639:
Ponssard, J. -P.; Sorin, S. (June 1980). "The LP formulation of finite zero-sum games with incomplete information".
2500:
2969:
760:
738:
In Bayesian games, player's beliefs about the game are denoted by a probability distribution over various types.
285:
A Bayesian-Nash Equilibrium of a Bayesian game is a Nash equilibrium of its associated ex-ante normal form game.
454:
2839:
2510:
2294:
2024:
Ramtin, Amir Reza; Towsley, Don (2021). "A Game-Theoretic Approach to Self-Stabilization with Selfish Agents".
2678:
3020:
2438:
2413:
716:
3406:
3370:
2796:
2550:
2540:
2475:
815:
809:
236:
There are three stages of Bayesian games, each describing the players' knowledge of types within the game.
2590:
2570:
837:
589:
3304:
3055:
3025:
2683:
2525:
2520:
1237:
651:
Consider two players with a zero-sum objective function. A linear program can be formed to compute BNE.
3340:
3263:
2999:
2555:
2480:
2337:
882:
A sheriff faces an armed suspect. Both must simultaneously decide whether to shoot the other or not.
865:
The definition of Bayesian games and Bayesian equilibrium has been extended to deal with collective
3355:
3088:
2974:
2771:
2565:
2383:
1242:
1082:
266:
60:
3158:
2241:
741:
If players do not have private information, the probability distribution over types is known as a
3360:
2959:
2929:
2585:
2373:
2087:
Bacharach, M. (1999). "Interactive team reasoning: A contribution to the theory of cooperation".
890:(because the suspect has private information), making it a Bayesian game. There is a probability
373:
274:
2052:
2023 20th Annual IEEE International Conference on Sensing, Communication, and Networking (SECON)
54:
introduced the concept of Bayesian games in three papers from 1967 and 1968: He was awarded the
3385:
3365:
3345:
3294:
2964:
2869:
2728:
2673:
2605:
2575:
2495:
2423:
1472:
1216:
2284:
1276:
2844:
2829:
2403:
2045:
Su, Runbo; Sfar, Arbia Riahi; Natalizio, Enrico; Moyal, Pascal; Song, Ye-Qiong (2023-09-11).
1134:
552:
330:
2305:
1683:
3178:
3163:
3050:
3045:
2949:
2934:
2899:
2864:
2463:
2408:
2330:
2050:
1494:
1247:
887:
44:
40:
255:
Players know their own types and those of other players. The payoffs are known to players.
8:
3335:
2954:
2904:
2741:
2668:
2648:
2505:
2388:
1089:
899:
830:
822:
Bayesian consistency: the beliefs are consistent with the strategies under consideration;
665:
522:
424:
2994:
2157:
1861:
1498:
1473:"Ex ante versus ex post equilibria in classical Bayesian games with a nonlocal resource"
1421:
Kajii, A.; Morris, S. (1997). "The Robustness of Equilibria to Incomplete Information".
350:
160:
Assign a payoff to a player given their type and the action profile. A payoff function,
3314:
3173:
3004:
2984:
2834:
2713:
2618:
2545:
2490:
2261:
2217:
2182:
2025:
2006:
1959:
1931:
1889:
1862:"Defense and security planning under resource uncertainty and multi-period commitments"
1842:
1824:
1664:
1553:
1510:
1484:
1438:
1400:
1357:
1305:
1252:
866:
730:
Information sets are denoted by dotted lines, which is the most common notation today.
400:
303:
1572:
3299:
3268:
3223:
3118:
2989:
2944:
2919:
2849:
2723:
2653:
2643:
2535:
2485:
2433:
2290:
2222:
2204:
2067:
2063:
2010:
1998:
1963:
1949:
1893:
1881:
1793:
1762:
1720:
1668:
1656:
1619:
1586:
1545:
1514:
1392:
1349:
1295:
1198:
Only "lemons" (used cars in bad conditions, specifically with value at most equal to
1140:
775:
1979:"Fast and Secure Computational Offloading With Lagrange Coded Mobile Edge Computing"
1309:
1205:
Player 1 can guarantee herself a payoff of zero by bidding 0, hence in equilibrium,
228:
is a player's choice of action at each point where the player must make a decision.
3380:
3375:
3309:
3273:
3253:
3213:
3183:
3138:
3093:
3078:
3035:
2889:
2530:
2467:
2453:
2418:
2253:
2212:
2194:
2137:
2127:
2096:
2059:
1990:
1941:
1873:
1846:
1834:
1785:
1754:
1712:
1648:
1611:
1578:
1537:
1502:
1430:
1384:
1341:
1287:
907:
854:
289:
51:
1704:
1212:
Since only "lemons" (used cars in bad conditions) are traded, the market collapses
1085:), play in the game will be as follows according to perfect Bayesian equilibrium:
3278:
3238:
3193:
3108:
3103:
2824:
2776:
2663:
2428:
2398:
2368:
1838:
1716:
1291:
850:
3143:
2046:
1506:
1152:(i.e., each of two value sub-intervals of of equal length are equally likely).
3218:
3208:
3198:
3133:
3123:
3113:
3098:
2894:
2874:
2859:
2854:
2814:
2781:
2766:
2761:
2751:
2560:
1978:
1923:
1789:
1105:; if the sheriff does not shoot, he will have a payoff of -2 with probability
3400:
3258:
3248:
3203:
3188:
3168:
2939:
2914:
2786:
2756:
2746:
2733:
2638:
2580:
2515:
2448:
2208:
2002:
1994:
1945:
1885:
1766:
1660:
1615:
1582:
1549:
1396:
1353:
910:
for both players depends on the type of the suspect. This game is defined by
293:
2183:"A Generalized Quantum-Inspired Decision Making Model for Intelligent Agent"
138:. "Types" capture the private information a player can have. A type profile
3233:
3228:
3083:
2658:
2226:
2100:
1758:
1541:
2199:
1784:. Springer Texts in Business and Economics. Berlin: Springer. p. 60.
1388:
1345:
3350:
3153:
3148:
3128:
2924:
2909:
2718:
2688:
2623:
2613:
2443:
2378:
2354:
825:
Sequential rationality: the players choose optimally given their beliefs.
708:
s decision nodes that she cannot distinguish between. That is, if player
31:
2322:
2142:
1557:
849:
Stochastic Bayesian games combine the definitions of Bayesian games and
2979:
2633:
2265:
2242:"The Market for "Lemons": Quality Uncertainty and the Market Mechanism"
1909:
Stochastic Bayesian Games for the Cybersecurity of Nuclear Power Plants
1652:
1456:
Grüne-Yanoff, Till; Lehtinen, Aki (2012). "Philosophy of Game Theory".
1442:
1404:
1361:
2132:
2115:
2047:"A Game Theoretical Model addressing Misbehavior in Crowdsourcing IoT"
1877:
836:
To deal with these issues, Perfect Bayesian equilibrium, according to
2884:
2804:
2628:
2257:
1434:
668:
with perfect or imperfect information, have the following elements:
3319:
2819:
2030:
1936:
1829:
1489:
655:
1471:
Koniorczyk, Mátyás; Bodor, András; Pintér, Miklós (29 June 2020).
185:
A probability distribution over all possible type profiles, where
3040:
3030:
2708:
860:
67:
549:
players yet expands the number of each player i's actions from
347:
is a Bayesian Nash equilibrium if and only if for every player
2809:
1745:"Bayes' rule: a tutorial introduction to Bayesian analysis".
1182:
Player 2's strategy: Accept all bids above a certain cut-off
370:
keeping the strategies of every other player fixed, strategy
1860:
Caballero, William N.; Banks, David; Wu, Keru (2022-08-08).
681:
Set of actions for each player at each of her decision nodes
1812:
1711:, Cambridge University Press, pp. 75–143, 2013-03-21,
793:
is reached with strictly positive probability according to
678:
A player function assigning a player to each decision node
2080:
1922:
Bernhard, Julian; Pollok, Stefan; Knoll, Alois (2019).
1571:
Maschler, Michael; Solan, Eilon; Zamir, Shmuel (2013).
1139:
The Market for Lemons is related to a concept known as
259:
592:
457:
1470:
1416:
1414:
555:
525:
427:
403:
376:
353:
333:
306:
2044:
1455:
1921:
1277:"Bayesian Games: Games with Incomplete Information"
660:
320:on the player's own type according to Bayes' rule.
1605:
1570:
1411:
894:that the suspect is a criminal, and a probability
753:An assessment of an extensive form game is a pair
639:
578:
541:
511:
443:
409:
389:
362:
339:
312:
1976:
3398:
2107:
1859:
906:sheriff shoots. Thus, the payoff matrix of this
656:Extensive form games with incomplete information
288:In a non-Bayesian game, a strategy profile is a
81:, where it consists of the following elements:
2289:. Princeton University Press. pp. 144–52.
1222:
803:
1284:Encyclopedia of Complexity and Systems Science
1215:No trade is possible even when trade would be
691:
2338:
2023:
1928:2019 IEEE Intelligent Vehicles Symposium (IV)
1911:. PhD Dissertation, University of Pittsburgh.
1691:Department of Computer Science and Automation
1638:
861:Incomplete information over collective agency
68:Normal form games with incomplete information
844:
280:
1930:. Paris, France: IEEE. pp. 2148–2155.
1420:
512:{\textstyle \sum _{i=1}^{|N|}|\Theta _{i}|}
2345:
2331:
2181:Hu, Yuhuang; Loo, Chu Kiong (2014-03-17).
1606:Shoham, Yoav; Leyton-Brown, Kevin (2008).
197:is the probability that Player 1 has type
2352:
2216:
2198:
2141:
2131:
2086:
2029:
1983:IEEE Transactions on Vehicular Technology
1977:Asheralieva, Alia; Niyato, Dusit (2021).
1935:
1906:
1828:
1610:. Cambridge: Cambridge University Press.
1577:. Cambridge: Cambridge University Press.
1488:
1117:. Thus, the Sheriff will always shoot if
715:For two decision nodes to be in the same
121:is a list of actions, one for each player
56:Nobel Memorial Prize in Economic Sciences
1681:
1527:
1374:
1331:
1190:, is known as a cut-off strategy, where
1128:
397:maximizes the expected payoff of player
2282:
2239:
1270:
1268:
292:if every strategy in that profile is a
150:is a list of types, one for each player
105:The set of actions available to Player
14:
3399:
2113:
1779:
733:
2326:
2303:
2180:
1274:
1033:
986:
1641:International Journal of Game Theory
1265:
1097:and a payoff of -1 with probability
877:
640:{\textstyle |A_{i}|^{|\Theta _{i}|}}
417:according to that player's beliefs.
260:Improvements over non-Bayesian games
2306:"Games with Incomplete Information"
1166:Player 2's payoff: Bid Accepted is
1159:Player 1's payoff: Bid Accepted is
1109:and a payoff of 0 with probability
24:
2394:First-player and second-player win
2286:Game Theory for Applied Economists
2276:
2246:The Quarterly Journal of Economics
2240:Akerlof, George A. (August 1970).
1753:(6): 51–3301–51-3301. 2014-01-21.
621:
495:
219:
92:The set of players within the game
25:
3423:
1088:When the type is "criminal", the
687:A payoff function for each player
2501:Coalition-proof Nash equilibrium
2064:10.1109/SECON58729.2023.10287527
974:It is assumed that the payoffs,
661:Elements of extensive form games
172:denotes the utilities of player
2233:
2174:
2150:
2038:
2017:
1970:
1915:
1900:
1853:
1806:
1773:
1738:
1697:
1675:
1632:
1599:
325:Bayesian Nash equilibrium (BNE)
231:
2511:Evolutionarily stable strategy
1866:Naval Research Logistics (NRL)
1564:
1521:
1464:
1449:
1368:
1325:
1316:
748:
723:Belong to the same player; and
631:
616:
610:
594:
572:
557:
535:
527:
505:
490:
483:
475:
437:
429:
77:A Bayesian game is defined by
13:
1:
2439:Simultaneous action selection
1907:Maccarone, Lee Tylor (2021).
1258:
1113:, i.e. an expected payoff of
1101:, i.e. an expected payoff of
700:An information set of player
3371:List of games in game theory
2551:Quantal response equilibrium
2541:Perfect Bayesian equilibrium
2476:Bayes correlated equilibrium
2187:The Scientific World Journal
1839:10.1016/j.artint.2016.02.004
1717:10.1017/cbo9780511794216.005
1292:10.1007/978-0-387-30440-3_29
1223:Enter the monopolized market
1178:Side point: cut-off strategy
816:perfect Bayesian equilibrium
810:Perfect Bayesian equilibrium
804:Perfect Bayesian equilibrium
726:Have the same set of actions
134:The set of types of players
7:
2840:Optional prisoner's dilemma
2571:Self-confirming equilibrium
1507:10.1103/PhysRevA.101.062115
1231:
1186:, and Reject and bid below
855:Bellman optimality equation
838:subgame perfect equilibrium
692:Nature and information sets
390:{\displaystyle \sigma _{i}}
72:
10:
3428:
3305:Principal variation search
3021:Aumann's agreement theorem
2684:Strategy-stealing argument
2596:Trembling hand equilibrium
2526:Markov perfect equilibrium
2521:Mertens-stable equilibrium
2058:. IEEE. pp. 195–203.
1238:Bayesian-optimal mechanism
1132:
872:
807:
3341:Combinatorial game theory
3328:
3287:
3069:
3013:
3000:Princess and monster game
2795:
2697:
2604:
2556:Quasi-perfect equilibrium
2481:Bayesian Nash equilibrium
2462:
2361:
1790:10.1007/978-3-662-46950-7
1682:Narahari, Y (July 2012).
1049:
1036:
1002:
989:
845:Stochastic Bayesian games
281:Bayesian Nash equilibrium
61:probability distributions
18:Bayesian Nash equilibrium
3412:Game theory game classes
3356:Evolutionary game theory
3089:Antoine Augustin Cournot
2975:Guess 2/3 of the average
2772:Strictly determined game
2566:Satisfaction equilibrium
2384:Escalation of commitment
2304:Levin, Jonathan (2002).
2283:Gibbons, Robert (1992).
1995:10.1109/TVT.2021.3070723
1946:10.1109/IVS.2019.8813791
1616:10.1017/cbo9780511811654
1583:10.1017/cbo9780511794216
1243:Bayesian-optimal pricing
267:law of total probability
3361:Glossary of game theory
2960:Stackelberg competition
2586:Strong Nash equilibrium
1816:Artificial Intelligence
1458:Philosophy of Economics
1194:is called the cut-off.
978:, are given as follows:
579:{\displaystyle |A_{i}|}
340:{\displaystyle \sigma }
224:In a strategic game, a
3386:Tragedy of the commons
3366:List of game theorists
3346:Confrontation analysis
3056:Sprague–Grundy theorem
2576:Sequential equilibrium
2496:Correlated equilibrium
2101:10.1006/reec.1999.0188
1759:10.5860/choice.51-3301
1705:"Strategic-form games"
1684:"Extensive Form Games"
1542:10.1287/mnsc.1040.0270
1275:Zamir, Shmuel (2009).
1217:economically efficient
944:= {Criminal, Civilian}
919:N = {Suspect, Sheriff}
888:incomplete information
704:is a subset of player
641:
580:
543:
513:
488:
445:
411:
391:
364:
341:
314:
45:incomplete information
3159:Jean-François Mertens
2089:Research in Economics
1780:Peters, Hans (2015).
1747:Choice Reviews Online
1389:10.1287/mnsc.14.7.486
1346:10.1287/mnsc.14.5.320
1135:The Market for Lemons
1129:The market for lemons
900:imperfect information
684:Set of terminal nodes
675:Set of decision nodes
642:
581:
544:
514:
458:
446:
412:
392:
365:
342:
315:
3288:Search optimizations
3164:Jennifer Tour Chayes
3051:Revelation principle
3046:Purification theorem
2985:Nash bargaining game
2950:Bertrand competition
2935:El Farol Bar problem
2900:Electronic mail game
2865:Lewis signaling game
2409:Hierarchy of beliefs
2116:"Agency equilibrium"
1248:Bayesian programming
666:Extensive form games
590:
553:
523:
455:
425:
401:
374:
351:
331:
304:
109:. An action profile
50:Hungarian economist
41:Bayesian probability
3407:Bayesian statistics
3336:Bounded rationality
2955:Cournot competition
2905:Rock paper scissors
2880:Battle of the sexes
2870:Volunteer's dilemma
2742:Perfect information
2669:Dominant strategies
2506:Epsilon-equilibrium
2389:Extensive-form game
2200:10.1155/2014/240983
2114:Newton, J. (2019).
1499:2020PhRvA.101f2115K
1163:, Bid Rejected is 0
1030:
983:
734:The role of beliefs
542:{\displaystyle |N|}
444:{\displaystyle |N|}
253:Ex-post stage game.
247:Interim stage game.
241:Ex-ante stage game.
27:Game theory concept
3315:Paranoid algorithm
3295:Alpha–beta pruning
3174:John Maynard Smith
3005:Rendezvous problem
2845:Traveler's dilemma
2835:Gift-exchange game
2830:Prisoner's dilemma
2747:Large Poisson game
2714:Bargaining problem
2619:Backward induction
2591:Subgame perfection
2546:Proper equilibrium
1653:10.1007/bf01769767
1608:Multiagent Systems
1530:Management Science
1377:Management Science
1334:Management Science
1253:Bayesian inference
1170:, Bid Rejected is
1034:Type = "Civilian"
1029:
987:Type = "Criminal"
982:
637:
576:
539:
509:
441:
407:
387:
363:{\displaystyle i,}
360:
337:
310:
154:Payoff functions,
3394:
3393:
3300:Aspiration window
3269:Suzanne Scotchmer
3224:Oskar Morgenstern
3119:Donald B. Gillies
3061:Zermelo's theorem
2990:Induction puzzles
2945:Fair cake-cutting
2920:Public goods game
2850:Coordination game
2724:Intransitive game
2654:Forward induction
2536:Pareto efficiency
2516:Gibbs equilibrium
2486:Berge equilibrium
2434:Simultaneous game
2133:10.3390/g10010014
2073:979-8-3503-0052-9
1955:978-1-7281-0560-4
1878:10.1002/nav.22071
1799:978-3-662-46949-1
1625:978-0-511-81165-4
1592:978-0-511-79421-6
1536:(12): 1804–1817.
1477:Physical Review A
1301:978-0-387-75888-6
1141:adverse selection
1090:dominant strategy
1074:
1073:
1050:Suspect's action
1037:Sheriff's action
1027:
1026:
1003:Suspect's action
990:Sheriff's action
878:Sheriff's dilemma
761:Behavior Strategy
410:{\displaystyle i}
313:{\displaystyle p}
16:(Redirected from
3419:
3381:Topological game
3376:No-win situation
3274:Thomas Schelling
3254:Robert B. Wilson
3214:Merrill M. Flood
3184:John von Neumann
3094:Ariel Rubinstein
3079:Albert W. Tucker
2930:War of attrition
2890:Matching pennies
2531:Nash equilibrium
2454:Mechanism design
2419:Normal-form game
2374:Cooperative game
2347:
2340:
2333:
2324:
2323:
2319:
2317:
2315:
2310:
2300:
2270:
2269:
2237:
2231:
2230:
2220:
2202:
2178:
2172:
2171:
2169:
2168:
2154:
2148:
2147:
2145:
2135:
2111:
2105:
2104:
2084:
2078:
2077:
2057:
2042:
2036:
2035:
2033:
2021:
2015:
2014:
1989:(5): 4924–4942.
1974:
1968:
1967:
1939:
1919:
1913:
1912:
1904:
1898:
1897:
1872:(7): 1009–1026.
1857:
1851:
1850:
1832:
1810:
1804:
1803:
1777:
1771:
1770:
1742:
1736:
1735:
1734:
1733:
1701:
1695:
1694:
1688:
1679:
1673:
1672:
1636:
1630:
1629:
1603:
1597:
1596:
1568:
1562:
1561:
1525:
1519:
1518:
1492:
1468:
1462:
1461:
1453:
1447:
1446:
1429:(6): 1283–1309.
1418:
1409:
1408:
1372:
1366:
1365:
1329:
1323:
1320:
1314:
1313:
1281:
1272:
1083:common knowledge
1031:
1028:
984:
981:
908:Normal-form game
851:stochastic games
646:
644:
643:
638:
636:
635:
634:
629:
628:
619:
613:
607:
606:
597:
585:
583:
582:
577:
575:
570:
569:
560:
548:
546:
545:
540:
538:
530:
518:
516:
515:
510:
508:
503:
502:
493:
487:
486:
478:
472:
450:
448:
447:
442:
440:
432:
416:
414:
413:
408:
396:
394:
393:
388:
386:
385:
369:
367:
366:
361:
346:
344:
343:
338:
319:
317:
316:
311:
290:Nash equilibrium
86:Set of players,
52:John C. Harsanyi
21:
3427:
3426:
3422:
3421:
3420:
3418:
3417:
3416:
3397:
3396:
3395:
3390:
3324:
3310:max^n algorithm
3283:
3279:William Vickrey
3239:Reinhard Selten
3194:Kenneth Binmore
3109:David K. Levine
3104:Daniel Kahneman
3071:
3065:
3041:Negamax theorem
3031:Minimax theorem
3009:
2970:Diner's dilemma
2825:All-pay auction
2791:
2777:Stochastic game
2729:Mean-field game
2700:
2693:
2664:Markov strategy
2600:
2466:
2458:
2429:Sequential game
2414:Information set
2399:Game complexity
2369:Congestion game
2357:
2351:
2313:
2311:
2308:
2297:
2279:
2277:Further reading
2274:
2273:
2258:10.2307/1879431
2238:
2234:
2179:
2175:
2166:
2164:
2156:
2155:
2151:
2112:
2108:
2085:
2081:
2074:
2055:
2043:
2039:
2022:
2018:
1975:
1971:
1956:
1920:
1916:
1905:
1901:
1858:
1854:
1811:
1807:
1800:
1778:
1774:
1744:
1743:
1739:
1731:
1729:
1727:
1703:
1702:
1698:
1686:
1680:
1676:
1637:
1633:
1626:
1604:
1600:
1593:
1569:
1565:
1526:
1522:
1469:
1465:
1454:
1450:
1435:10.2307/2171737
1419:
1412:
1373:
1369:
1330:
1326:
1321:
1317:
1302:
1286:. p. 426.
1279:
1273:
1266:
1261:
1234:
1225:
1137:
1131:
969:
961:
951:
943:
935:
927:
880:
875:
863:
847:
812:
806:
798:
791:
783:
751:
736:
717:information set
694:
663:
658:
630:
624:
620:
615:
614:
609:
608:
602:
598:
593:
591:
588:
587:
571:
565:
561:
556:
554:
551:
550:
534:
526:
524:
521:
520:
504:
498:
494:
489:
482:
474:
473:
462:
456:
453:
452:
436:
428:
426:
423:
422:
402:
399:
398:
381:
377:
375:
372:
371:
352:
349:
348:
332:
329:
328:
305:
302:
301:
283:
262:
234:
222:
220:Pure strategies
213:
202:
194:
190:
169:
165:
147:
143:
130:
118:
114:
101:
75:
70:
28:
23:
22:
15:
12:
11:
5:
3425:
3415:
3414:
3409:
3392:
3391:
3389:
3388:
3383:
3378:
3373:
3368:
3363:
3358:
3353:
3348:
3343:
3338:
3332:
3330:
3326:
3325:
3323:
3322:
3317:
3312:
3307:
3302:
3297:
3291:
3289:
3285:
3284:
3282:
3281:
3276:
3271:
3266:
3261:
3256:
3251:
3246:
3244:Robert Axelrod
3241:
3236:
3231:
3226:
3221:
3219:Olga Bondareva
3216:
3211:
3209:Melvin Dresher
3206:
3201:
3199:Leonid Hurwicz
3196:
3191:
3186:
3181:
3176:
3171:
3166:
3161:
3156:
3151:
3146:
3141:
3136:
3134:Harold W. Kuhn
3131:
3126:
3124:Drew Fudenberg
3121:
3116:
3114:David M. Kreps
3111:
3106:
3101:
3099:Claude Shannon
3096:
3091:
3086:
3081:
3075:
3073:
3067:
3066:
3064:
3063:
3058:
3053:
3048:
3043:
3038:
3036:Nash's theorem
3033:
3028:
3023:
3017:
3015:
3011:
3010:
3008:
3007:
3002:
2997:
2992:
2987:
2982:
2977:
2972:
2967:
2962:
2957:
2952:
2947:
2942:
2937:
2932:
2927:
2922:
2917:
2912:
2907:
2902:
2897:
2895:Ultimatum game
2892:
2887:
2882:
2877:
2875:Dollar auction
2872:
2867:
2862:
2860:Centipede game
2857:
2852:
2847:
2842:
2837:
2832:
2827:
2822:
2817:
2815:Infinite chess
2812:
2807:
2801:
2799:
2793:
2792:
2790:
2789:
2784:
2782:Symmetric game
2779:
2774:
2769:
2767:Signaling game
2764:
2762:Screening game
2759:
2754:
2752:Potential game
2749:
2744:
2739:
2731:
2726:
2721:
2716:
2711:
2705:
2703:
2695:
2694:
2692:
2691:
2686:
2681:
2679:Mixed strategy
2676:
2671:
2666:
2661:
2656:
2651:
2646:
2641:
2636:
2631:
2626:
2621:
2616:
2610:
2608:
2602:
2601:
2599:
2598:
2593:
2588:
2583:
2578:
2573:
2568:
2563:
2561:Risk dominance
2558:
2553:
2548:
2543:
2538:
2533:
2528:
2523:
2518:
2513:
2508:
2503:
2498:
2493:
2488:
2483:
2478:
2472:
2470:
2460:
2459:
2457:
2456:
2451:
2446:
2441:
2436:
2431:
2426:
2421:
2416:
2411:
2406:
2404:Graphical game
2401:
2396:
2391:
2386:
2381:
2376:
2371:
2365:
2363:
2359:
2358:
2350:
2349:
2342:
2335:
2327:
2321:
2320:
2301:
2295:
2278:
2275:
2272:
2271:
2252:(3): 488–500.
2232:
2173:
2149:
2106:
2079:
2072:
2037:
2016:
1969:
1954:
1914:
1899:
1852:
1805:
1798:
1772:
1737:
1725:
1696:
1674:
1631:
1624:
1598:
1591:
1563:
1520:
1463:
1448:
1410:
1383:(7): 486–502.
1367:
1340:(5): 320–334.
1324:
1315:
1300:
1263:
1262:
1260:
1257:
1256:
1255:
1250:
1245:
1240:
1233:
1230:
1224:
1221:
1220:
1219:
1213:
1210:
1203:
1175:
1174:
1164:
1133:Main article:
1130:
1127:
1072:
1071:
1068:
1065:
1061:
1060:
1057:
1054:
1051:
1047:
1046:
1043:
1039:
1038:
1035:
1025:
1024:
1021:
1018:
1014:
1013:
1010:
1007:
1004:
1000:
999:
996:
992:
991:
988:
980:
979:
972:
967:
959:
955:
949:
941:
938:
936:= {Shoot, Not}
933:
928:= {Shoot, Not}
925:
921:
879:
876:
874:
871:
862:
859:
846:
843:
827:
826:
823:
808:Main article:
805:
802:
796:
789:
781:
770:An assessment
768:
767:
764:
750:
747:
735:
732:
728:
727:
724:
693:
690:
689:
688:
685:
682:
679:
676:
673:
672:Set of players
662:
659:
657:
654:
653:
652:
633:
627:
623:
618:
612:
605:
601:
596:
574:
568:
564:
559:
537:
533:
529:
507:
501:
497:
492:
485:
481:
477:
471:
468:
465:
461:
439:
435:
431:
406:
384:
380:
359:
356:
336:
309:
282:
279:
261:
258:
257:
256:
250:
244:
233:
230:
221:
218:
217:
216:
211:
200:
192:
188:
176:
167:
163:
151:
145:
141:
128:
122:
116:
112:
99:
93:
74:
71:
69:
66:
64:this article.
26:
9:
6:
4:
3:
2:
3424:
3413:
3410:
3408:
3405:
3404:
3402:
3387:
3384:
3382:
3379:
3377:
3374:
3372:
3369:
3367:
3364:
3362:
3359:
3357:
3354:
3352:
3349:
3347:
3344:
3342:
3339:
3337:
3334:
3333:
3331:
3329:Miscellaneous
3327:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3298:
3296:
3293:
3292:
3290:
3286:
3280:
3277:
3275:
3272:
3270:
3267:
3265:
3264:Samuel Bowles
3262:
3260:
3259:Roger Myerson
3257:
3255:
3252:
3250:
3249:Robert Aumann
3247:
3245:
3242:
3240:
3237:
3235:
3232:
3230:
3227:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3205:
3204:Lloyd Shapley
3202:
3200:
3197:
3195:
3192:
3190:
3189:Kenneth Arrow
3187:
3185:
3182:
3180:
3177:
3175:
3172:
3170:
3169:John Harsanyi
3167:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3139:Herbert Simon
3137:
3135:
3132:
3130:
3127:
3125:
3122:
3120:
3117:
3115:
3112:
3110:
3107:
3105:
3102:
3100:
3097:
3095:
3092:
3090:
3087:
3085:
3082:
3080:
3077:
3076:
3074:
3068:
3062:
3059:
3057:
3054:
3052:
3049:
3047:
3044:
3042:
3039:
3037:
3034:
3032:
3029:
3027:
3024:
3022:
3019:
3018:
3016:
3012:
3006:
3003:
3001:
2998:
2996:
2993:
2991:
2988:
2986:
2983:
2981:
2978:
2976:
2973:
2971:
2968:
2966:
2963:
2961:
2958:
2956:
2953:
2951:
2948:
2946:
2943:
2941:
2940:Fair division
2938:
2936:
2933:
2931:
2928:
2926:
2923:
2921:
2918:
2916:
2915:Dictator game
2913:
2911:
2908:
2906:
2903:
2901:
2898:
2896:
2893:
2891:
2888:
2886:
2883:
2881:
2878:
2876:
2873:
2871:
2868:
2866:
2863:
2861:
2858:
2856:
2853:
2851:
2848:
2846:
2843:
2841:
2838:
2836:
2833:
2831:
2828:
2826:
2823:
2821:
2818:
2816:
2813:
2811:
2808:
2806:
2803:
2802:
2800:
2798:
2794:
2788:
2787:Zero-sum game
2785:
2783:
2780:
2778:
2775:
2773:
2770:
2768:
2765:
2763:
2760:
2758:
2757:Repeated game
2755:
2753:
2750:
2748:
2745:
2743:
2740:
2738:
2736:
2732:
2730:
2727:
2725:
2722:
2720:
2717:
2715:
2712:
2710:
2707:
2706:
2704:
2702:
2696:
2690:
2687:
2685:
2682:
2680:
2677:
2675:
2674:Pure strategy
2672:
2670:
2667:
2665:
2662:
2660:
2657:
2655:
2652:
2650:
2647:
2645:
2642:
2640:
2639:De-escalation
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2620:
2617:
2615:
2612:
2611:
2609:
2607:
2603:
2597:
2594:
2592:
2589:
2587:
2584:
2582:
2581:Shapley value
2579:
2577:
2574:
2572:
2569:
2567:
2564:
2562:
2559:
2557:
2554:
2552:
2549:
2547:
2544:
2542:
2539:
2537:
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2517:
2514:
2512:
2509:
2507:
2504:
2502:
2499:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2479:
2477:
2474:
2473:
2471:
2469:
2465:
2461:
2455:
2452:
2450:
2449:Succinct game
2447:
2445:
2442:
2440:
2437:
2435:
2432:
2430:
2427:
2425:
2422:
2420:
2417:
2415:
2412:
2410:
2407:
2405:
2402:
2400:
2397:
2395:
2392:
2390:
2387:
2385:
2382:
2380:
2377:
2375:
2372:
2370:
2367:
2366:
2364:
2360:
2356:
2348:
2343:
2341:
2336:
2334:
2329:
2328:
2325:
2307:
2302:
2298:
2292:
2288:
2287:
2281:
2280:
2267:
2263:
2259:
2255:
2251:
2247:
2243:
2236:
2228:
2224:
2219:
2214:
2210:
2206:
2201:
2196:
2192:
2188:
2184:
2177:
2163:
2159:
2153:
2144:
2139:
2134:
2129:
2125:
2121:
2117:
2110:
2102:
2098:
2095:(2): 117–47.
2094:
2090:
2083:
2075:
2069:
2065:
2061:
2054:
2053:
2048:
2041:
2032:
2027:
2020:
2012:
2008:
2004:
2000:
1996:
1992:
1988:
1984:
1980:
1973:
1965:
1961:
1957:
1951:
1947:
1943:
1938:
1933:
1929:
1925:
1918:
1910:
1903:
1895:
1891:
1887:
1883:
1879:
1875:
1871:
1867:
1863:
1856:
1848:
1844:
1840:
1836:
1831:
1826:
1822:
1818:
1817:
1809:
1801:
1795:
1791:
1787:
1783:
1776:
1768:
1764:
1760:
1756:
1752:
1748:
1741:
1728:
1726:9780511794216
1722:
1718:
1714:
1710:
1706:
1700:
1692:
1685:
1678:
1670:
1666:
1662:
1658:
1654:
1650:
1647:(2): 99–105.
1646:
1642:
1635:
1627:
1621:
1617:
1613:
1609:
1602:
1594:
1588:
1584:
1580:
1576:
1575:
1567:
1559:
1555:
1551:
1547:
1543:
1539:
1535:
1531:
1524:
1516:
1512:
1508:
1504:
1500:
1496:
1491:
1486:
1482:
1478:
1474:
1467:
1459:
1452:
1444:
1440:
1436:
1432:
1428:
1424:
1417:
1415:
1406:
1402:
1398:
1394:
1390:
1386:
1382:
1378:
1371:
1363:
1359:
1355:
1351:
1347:
1343:
1339:
1335:
1328:
1319:
1311:
1307:
1303:
1297:
1293:
1289:
1285:
1278:
1271:
1269:
1264:
1254:
1251:
1249:
1246:
1244:
1241:
1239:
1236:
1235:
1229:
1218:
1214:
1211:
1208:
1204:
1201:
1197:
1196:
1195:
1193:
1189:
1185:
1180:
1179:
1173:
1169:
1165:
1162:
1158:
1157:
1156:
1153:
1149:
1148:
1144:
1142:
1136:
1126:
1124:
1120:
1116:
1112:
1108:
1104:
1100:
1096:
1091:
1086:
1084:
1080:
1069:
1066:
1063:
1062:
1058:
1055:
1052:
1048:
1044:
1041:
1040:
1032:
1022:
1019:
1016:
1015:
1011:
1008:
1005:
1001:
997:
994:
993:
985:
977:
973:
971:
963:
956:
954:
947:
939:
937:
929:
922:
920:
917:
916:
915:
913:
909:
903:
901:
897:
893:
889:
883:
870:
868:
858:
856:
852:
842:
839:
834:
832:
824:
821:
820:
819:
817:
811:
801:
799:
792:
785:
784:) = Pr / Σ Pr
777:
773:
766:Belief system
765:
762:
759:
758:
757:
756:
746:
744:
739:
731:
725:
722:
721:
720:
718:
713:
711:
707:
703:
698:
686:
683:
680:
677:
674:
671:
670:
669:
667:
650:
649:
648:
625:
603:
599:
566:
562:
531:
499:
479:
469:
466:
463:
459:
433:
418:
404:
382:
378:
357:
354:
334:
326:
321:
307:
297:
295:
294:best response
291:
286:
278:
276:
272:
268:
254:
251:
248:
245:
242:
239:
238:
237:
229:
227:
226:pure strategy
214:
207:
203:
196:
184:
182:
177:
175:
171:
159:
157:
152:
149:
137:
133:
131:
123:
120:
108:
104:
102:
96:Action sets,
94:
91:
89:
84:
83:
82:
80:
65:
62:
57:
53:
48:
46:
42:
37:
36:Bayesian game
33:
19:
3234:Peyton Young
3229:Paul Milgrom
3144:Hervé Moulin
3084:Amos Tversky
3026:Folk theorem
2737:-player game
2734:
2659:Grim trigger
2312:. Retrieved
2285:
2249:
2245:
2235:
2190:
2186:
2176:
2165:. Retrieved
2161:
2152:
2143:10419/219237
2123:
2119:
2109:
2092:
2088:
2082:
2051:
2040:
2019:
1986:
1982:
1972:
1927:
1917:
1908:
1902:
1869:
1865:
1855:
1820:
1814:
1808:
1781:
1775:
1750:
1746:
1740:
1730:, retrieved
1708:
1699:
1690:
1677:
1644:
1640:
1634:
1607:
1601:
1573:
1566:
1533:
1529:
1523:
1480:
1476:
1466:
1457:
1451:
1426:
1423:Econometrica
1422:
1380:
1376:
1370:
1337:
1333:
1327:
1318:
1283:
1226:
1206:
1202:) are traded
1199:
1191:
1187:
1183:
1181:
1177:
1176:
1171:
1167:
1160:
1154:
1150:
1146:
1145:
1138:
1122:
1121:, i.e. when
1119:p-1 > -2p
1118:
1114:
1110:
1106:
1102:
1098:
1094:
1087:
1079:ad infinitum
1078:
1075:
975:
965:
957:
953:
945:
931:
923:
918:
911:
904:
895:
891:
884:
881:
864:
848:
835:
831:non-credible
828:
813:
794:
787:
779:
772:<b, μ>
771:
769:
763:profile; and
755:<b, μ>
754:
752:
743:common prior
742:
740:
737:
729:
719:, they must
714:
709:
705:
701:
699:
695:
664:
419:
324:
322:
298:
287:
284:
277:to compute.
271:ad infinitum
270:
263:
252:
246:
240:
235:
232:Three stages
225:
223:
209:
205:
198:
186:
180:
178:
173:
161:
155:
153:
139:
135:
126:
124:
110:
106:
97:
95:
87:
85:
78:
76:
49:
35:
29:
3351:Coopetition
3154:Jean Tirole
3149:John Conway
3129:Eric Maskin
2925:Blotto game
2910:Pirate game
2719:Global game
2689:Tit for tat
2624:Bid shading
2614:Appeasement
2464:Equilibrium
2444:Solved game
2379:Determinacy
2362:Definitions
2355:game theory
1782:Game Theory
1709:Game Theory
1574:Game Theory
912:(N,A,T,p,u)
776:Bayes' rule
749:Bayes' rule
204:and Player
166:, . . . , u
144:, . . . , t
125:Type sets,
115:, . . . , a
79:(N,A,T,p,u)
32:game theory
3401:Categories
2995:Trust game
2980:Kuhn poker
2649:Escalation
2644:Deterrence
2634:Cheap talk
2606:Strategies
2424:Preference
2353:Topics of
2296:1400835887
2193:: 240983.
2167:2016-06-16
2158:"Coursera"
2031:2108.07362
1937:2102.03119
1830:1507.07688
1732:2023-04-23
1490:2005.12727
1483:(6): 2–3.
1259:References
1123:p > 1/3
774:satisfies
275:irrational
191:, . . . ,t
187:p(t) = p(t
3179:John Nash
2885:Stag hunt
2629:Collusion
2314:25 August
2209:1537-744X
2126:(1): 14.
2011:234331661
2003:0018-9545
1964:201811314
1894:251461541
1886:0894-069X
1823:: 63–94.
1767:0009-4978
1669:120632621
1661:0020-7276
1550:0025-1909
1515:218889282
1397:0025-1909
1354:0025-1909
970:= (1 - p)
914:, where:
786:whenever
622:Θ
496:Θ
460:∑
379:σ
335:σ
208:has type
3320:Lazy SMP
3014:Theorems
2965:Deadlock
2820:Checkers
2701:of games
2468:concepts
2227:24778580
2162:Coursera
1558:30046151
1310:14218591
1232:See also
968:Civilian
960:Criminal
73:Elements
3072:figures
2855:Chicken
2709:Auction
2699:Classes
2266:1879431
2218:3977121
1847:2599762
1495:Bibcode
1443:2171737
1405:2628894
1362:2628673
1067:-2, -1
1059:-1, -2
1056:-3, -1
1020:-2, -1
950:Sheriff
942:Suspect
934:Sheriff
926:Suspect
873:Example
179:Prior,
2293:
2264:
2225:
2215:
2207:
2070:
2009:
2001:
1962:
1952:
1892:
1884:
1845:
1796:
1765:
1723:
1667:
1659:
1622:
1589:
1556:
1548:
1513:
1460:: 532.
1441:
1403:
1395:
1360:
1352:
1308:
1298:
1161:3/2v-p
1147:Set up
1053:Shoot
1042:Shoot
1012:2, -2
1006:Shoot
995:Shoot
867:agency
140:t = (t
111:a = (a
2810:Chess
2797:Games
2309:(PDF)
2262:JSTOR
2120:Games
2056:(PDF)
2026:arXiv
2007:S2CID
1960:S2CID
1932:arXiv
1890:S2CID
1843:S2CID
1825:arXiv
1687:(PDF)
1665:S2CID
1554:JSTOR
1511:S2CID
1485:arXiv
1439:JSTOR
1401:JSTOR
1358:JSTOR
1306:S2CID
1280:(PDF)
1070:0, 0
1023:-1,1
1009:0, 0
952:= {*}
780:μ(x|h
162:u= (u
2491:Core
2316:2016
2291:ISBN
2223:PMID
2205:ISSN
2191:2014
2068:ISBN
1999:ISSN
1950:ISBN
1882:ISSN
1794:ISBN
1763:ISSN
1721:ISBN
1693:: 1.
1657:ISSN
1620:ISBN
1587:ISBN
1546:ISSN
1393:ISSN
1350:ISSN
1296:ISBN
1064:Not
1045:Not
1017:Not
998:Not
34:, a
3070:Key
2254:doi
2213:PMC
2195:doi
2138:hdl
2128:doi
2097:doi
2060:doi
1991:doi
1942:doi
1874:doi
1835:doi
1821:235
1786:doi
1755:doi
1713:doi
1649:doi
1612:doi
1579:doi
1538:doi
1503:doi
1431:doi
1385:doi
1342:doi
1288:doi
1209:= 0
1115:-2p
1111:1-p
1103:p-1
1099:1-p
962:= p
902:).
896:1-p
778:if
586:to
451:to
30:In
3403::
2805:Go
2260:.
2250:84
2248:.
2244:.
2221:.
2211:.
2203:.
2189:.
2185:.
2160:.
2136:.
2124:10
2122:.
2118:.
2093:53
2091:.
2066:.
2049:.
2005:.
1997:.
1987:70
1985:.
1981:.
1958:.
1948:.
1940:.
1926:.
1888:.
1880:.
1870:69
1868:.
1864:.
1841:.
1833:.
1819:.
1792:.
1761:.
1751:51
1749:.
1719:,
1707:,
1689:.
1663:.
1655:.
1643:.
1618:.
1585:.
1552:.
1544:.
1534:50
1532:.
1509:.
1501:.
1493:.
1479:.
1475:.
1437:.
1427:65
1425:.
1413:^
1399:.
1391:.
1381:14
1379:.
1356:.
1348:.
1338:14
1336:.
1304:.
1294:.
1282:.
1267:^
1192:P*
1188:P*
1184:P*
1143:.
1125:.
1081:–
964:,
946:,
930:,
814:A
800:.
797:−i
745:.
706:i'
323:A
47:.
2735:n
2346:e
2339:t
2332:v
2318:.
2299:.
2268:.
2256::
2229:.
2197::
2170:.
2146:.
2140::
2130::
2103:.
2099::
2076:.
2062::
2034:.
2028::
2013:.
1993::
1966:.
1944::
1934::
1896:.
1876::
1849:.
1837::
1827::
1802:.
1788::
1769:.
1757::
1715::
1671:.
1651::
1645:9
1628:.
1614::
1595:.
1581::
1560:.
1540::
1517:.
1505::
1497::
1487::
1481:1
1445:.
1433::
1407:.
1387::
1364:.
1344::
1312:.
1290::
1207:p
1200:p
1172:v
1168:p
1107:p
1095:p
976:u
966:p
958:p
948:T
940:T
932:A
924:A
892:p
795:b
790:i
788:h
782:i
710:i
702:i
632:|
626:i
617:|
611:|
604:i
600:A
595:|
573:|
567:i
563:A
558:|
536:|
532:N
528:|
506:|
500:i
491:|
484:|
480:N
476:|
470:1
467:=
464:i
438:|
434:N
430:|
405:i
383:i
358:,
355:i
308:p
215:.
212:N
210:t
206:N
201:1
199:t
195:)
193:N
189:1
183::
181:p
174:i
170:)
168:N
164:1
158::
156:u
148:)
146:N
142:1
136:i
132::
129:i
127:t
119:)
117:N
113:1
107:i
103::
100:i
98:a
90::
88:N
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.