646:. Consider a game consisting of an employer considering whether to hire a job applicant. The job applicant's ability might be one of two things: high or low. Their ability level is random; they either have low ability with probability 1/3 or high ability with probability 2/3. In this case, it is convenient to model nature as another player of sorts who chooses the applicant's ability according to those probabilities. Nature however does not have any payoffs. Nature's choice is represented in the game tree by a non-filled node. Edges coming from a nature's choice node are labelled with the probability of the event it represents occurring.
658:
probability the type of player 1 (which in this game is tantamount to selecting the payoffs in the subgame played), either t1 or t2. Player 1 has distinct information sets for these; i.e. player 1 knows what type they are (this need not be the case). However, player 2 does not observe nature's choice. They do not know the type of player 1; however, in this game they do observe player 1's actions; i.e. there is perfect information. Indeed, it is now appropriate to alter the above definition of complete information: at every stage in the game, every player knows what has been played
381:
1970:
292:
650:
1966:
delimiting numbers are placed at the bottom and top of the arc respectively, usually with a variable that is used to express the payoffs. The infinite number of decision nodes that could result are represented by a single node placed in the centre of the arc. A similar device is used to represent action spaces that, whilst not infinite, are large enough to prove impractical to represent with an edge for each action.
299:
The game on the right has two players: 1 and 2. The numbers by every non-terminal node indicate to which player that decision node belongs. The numbers by every terminal node represent the payoffs to the players (e.g. 2,1 represents a payoff of 2 to player 1 and a payoff of 1 to player 2). The labels
405:
The game on the right is the same as the above game except that player 2 does not know what player 1 does when they come to play. The first game described has perfect information; the game on the right does not. If both players are rational and both know that both players are rational and everything
74:
with payoffs (no imperfect or incomplete information), and add the other elements in subsequent chapters as refinements. Whereas the rest of this article follows this gentle approach with motivating examples, we present upfront the finite extensive-form games as (ultimately) constructed here. This
1965:
It may be that a player has an infinite number of possible actions to choose from at a particular decision node. The device used to represent this is an arc joining two edges protruding from the decision node in question. If the action space is a continuum between two numbers, the lower and upper
203:
The above presentation, while precisely defining the mathematical structure over which the game is played, elides however the more technical discussion of formalizing statements about how the game is played like "a player cannot distinguish between nodes in the same information set when making a
171:
A play is thus a path through the tree from the root to a terminal node. At any given non-terminal node belonging to Chance, an outgoing branch is chosen according to the probability distribution. At any rational player's node, the player must choose one of the equivalence classes for the edges,
358:
An advantage of representing the game in this way is that it is clear what the order of play is. The tree shows clearly that player 1 moves first and player 2 observes this move. However, in some games play does not occur like this. One player does not always observe the choice of another (for
657:
The game on the left is one of complete information (all the players and payoffs are known to everyone) but of imperfect information (the employer doesn't know what nature's move was.) The initial node is in the centre and it is not filled, so nature moves first. Nature selects with the same
372:
When the game reaches the information set, the player who is about to move cannot differentiate between nodes within the information set; i.e. if the information set contains more than one node, the player to whom that set belongs does not know which node in the set has been
134:+1 subsets, one for each (rational) player, and with a special subset for a fictitious player called Chance (or Nature). Each player's subset of nodes is referred to as the "nodes of the player". (A game of complete information thus has an empty set of Chance nodes.)
602:
In games with infinite action spaces and imperfect information, non-singleton information sets are represented, if necessary, by inserting a dotted line connecting the (non-nodal) endpoints behind the arc described above or by dashing the arc itself. In the
319:. The payoffs are as specified in the tree. There are four outcomes represented by the four terminal nodes of the tree: (U,U'), (U,D'), (D,U') and (D,D'). The payoffs associated with each outcome respectively are as follows (0,0), (2,1), (1,2) and (3,1).
899:
665:
In this game, if nature selects t1 as player 1's type, the game played will be like the very first game described, except that player 2 does not know it (and the very fact that this cuts through their information sets disqualify it from
1954:
2242:. The same process can be done for the leader except that in calculating its profit, it knows that firm 2 will play the above response and so this can be substituted into its maximisation problem. It can then solve for
1873:
1684:
473:. In this equilibrium, every strategy is rational given the beliefs held and every belief is consistent with the strategies played. Notice how the imperfection of information changes the outcome of the game.
1290:
598:
These preferences can be marked within the matrix, and any box where both players have a preference provides a nash equilibrium. This particular game has a single solution of (D,U’) with a payoff of (1,2).
684:, player 2 can only form the belief that they are on either node in the information set with probability 1/2 (because this is the chance of seeing either type). Player 2 maximises their payoff by playing
2240:
172:
which determines precisely one outgoing edge except (in general) the player doesn't know which one is being followed. (An outside observer knowing every other player's choices up to that point, and the
2430:
963:
581:
We will have a two by two matrix with a unique payoff for each combination of moves. Using the normal form game, it is now possible to solve the game and identify dominant strategies for both players.
2349:
1388:
50:) information each player has about the other player's moves when they make a decision, and their payoffs for all possible game outcomes. Extensive-form games also allow for the representation of
2881:, 6.1, "Disasters in Game Theory" and 7.2 "Measurability (The Axiom of Determinateness)", discusses problems in extending the finite-case definition to infinite number of options (or moves)
151:
there is a one-to-one correspondence between outgoing edges of any two nodes of the same information set—thus the set of all outgoing edges of an information set is partitioned in
1600:
1445:
1985:
between 0 and 5000). This would be specified elsewhere. Here, it will be supposed that it is the former and, for concreteness, it will be supposed it represents two firms engaged in
2486:
1750:
762:
1209:
1106:
1233:
1074:
1706:
1130:
1042:
1016:
1776:
2269:
2148:
2109:
2078:
2047:
2016:
377:
In extensive form, an information set is indicated by a dotted line connecting all nodes in that set or sometimes by a loop drawn around all the nodes in that set.
1523:
1494:
1339:
1181:
1620:
1543:
1465:
1310:
1150:
983:
1880:
406:
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.
662:. In the case of private information, every player knows what has been played by nature. Information sets are represented as before by broken lines.
184:—choosing precisely one class of outgoing edges for every information set (of his). In a game of perfect information, the information sets are
445:
In the second game it is less clear: player 2 cannot observe player 1's move. Player 1 would like to fool player 2 into thinking they have played
42:
allowing (as the name suggests) for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their
426:(because to player 2 a payoff of 2 is better than a payoff of 1) and player 1 will receive 1. Hence, in the first game, the equilibrium will be (
1783:
607:
described above, if the second player had not observed the first player's move the game would no longer fit the
Stackelberg model; it would be
303:
The initial node belongs to player 1, indicating that player 1 moves first. Play according to the tree is as follows: player 1 chooses between
62:
in that they provide a more complete description of the game in question, whereas normal-form simply boils down the game into a payoff matrix.
398:
is such that at any stage of the game, every player knows exactly what has taken place earlier in the game; i.e. every information set is a
189:
1625:
1240:
2787:
2159:
2354:
907:
2549:
2276:
418:(because for player 2 a payoff of 1 is preferable to a payoff of 0) and so player 1 will receive 2. However, if player 1 plays
1347:
2964:
2932:
2874:
2844:
2777:
2729:
2707:
2668:
2635:
2602:
3868:
2804:(1957). Games and decisions: introduction and critical survey. (Ch3: Extensive and Normal Forms, pp39–55). Wiley New York.
2741:(1961). The mathematics of games of strategy: theory and applications (Ch4: Games in extensive form, pp74–78). Rand Corp.
259:
has both moves of chance (the cards being dealt) and imperfect information (the cards secretly held by other players). (
3685:
3215:
3013:
188:. It's less evident how payoffs should be interpreted in games with Chance nodes. It is assumed that each player has a
2939:
contains Kuhn's lectures at
Princeton from 1952 (officially unpublished previously, but in circulation as photocopies)
4031:
3504:
3323:
2823:
2809:
2760:
2746:
3120:
2114:
3594:
700:, player 2 again forms the belief that they are at either node with probability 1/2. In this case player 2 plays
1551:
1393:
3464:
3130:
70:
Some authors, particularly in introductory textbooks, initially define the extensive-form game as being just a
17:
3303:
158:
every (directed) path in the tree from the root to a terminal node can cross each information set at most once
3645:
3058:
3033:
361:
145:
43:
629:. In extensive form it is represented as a game with complete but imperfect information using the so-called
3995:
3421:
3170:
3160:
3095:
2435:
894:{\displaystyle \Gamma =\langle {\mathcal {K}},\mathbf {H} ,,\{A(H)\}_{H\in \mathbf {H} },a,\rho ,u\rangle }
674:
458:
255:, has no imperfect information (all information sets are singletons) but has moves of chance. For example,
2784:
3210:
3190:
2818:
1994. A course in game theory (Ch6 Extensive game with perfect information, pp. 89–115). MIT press.
2512:
1711:
347:
3929:
3680:
3650:
3308:
3145:
3140:
2836:
1186:
1079:
3965:
3888:
3624:
3175:
3100:
2957:
2507:
224:
173:
148:, which make certain choices indistinguishable for the player when making a move, in the sense that:
1214:
1047:
192:
defined for every game outcome; this assumption entails that every rational player will evaluate an
3980:
3713:
3599:
3396:
3185:
3003:
619:
It may be the case that a player does not know exactly what the payoffs of the game are or of what
164:
138:
3783:
1689:
1113:
3985:
3584:
3554:
3205:
2993:
1986:
399:
228:
185:
1021:
988:
4010:
3990:
3970:
3919:
3589:
3494:
3353:
3298:
3225:
3195:
3115:
3043:
680:
If both types play the same action (pooling), an equilibrium cannot be sustained. If both play
625:
500:
51:
3469:
3454:
3023:
2894:
390:
205:
47:
1755:
3803:
3788:
3675:
3670:
3574:
3559:
3524:
3489:
3083:
3028:
2950:
2497:
2247:
2126:
2087:
2056:
2025:
1994:
193:
1499:
1470:
1315:
1157:
8:
3960:
3579:
3529:
3366:
3293:
3268:
3125:
2502:
608:
330:
to maximise their payoff and so player 1 will only receive 1. However, if player 1 plays
216:
155:, each class representing a possible choice for a player's move at some point—, and
3619:
1949:{\displaystyle u=(u_{i})_{i\in {\mathcal {I}}}:T\rightarrow \mathbb {R} ^{\mathcal {I}}}
3939:
3798:
3629:
3609:
3459:
3338:
3238:
3165:
3110:
2911:
2576:
2568:
2118:
1605:
1528:
1450:
1295:
1135:
968:
380:
248:
127:
3924:
3893:
3848:
3743:
3614:
3569:
3544:
3474:
3348:
3273:
3263:
3155:
3105:
3053:
2928:
2915:
2870:
2840:
2819:
2805:
2773:
2756:
2742:
2725:
2703:
2674:
2664:
2641:
2631:
2608:
2598:
1977:
The tree on the left represents such a game, either with infinite action spaces (any
490:
388:
If a game has an information set with more than one member that game is said to have
152:
4005:
4000:
3934:
3898:
3878:
3838:
3808:
3763:
3718:
3703:
3660:
3514:
3288:
3150:
3087:
3073:
3038:
2903:
2815:
2527:
2121:
of each payoff function with respect to the follower's (firm 2) strategy variable (
604:
484:
478:
80:
59:
3903:
3863:
3818:
3733:
3728:
3449:
3401:
3283:
3048:
3018:
2988:
2791:
2717:
2522:
2517:
748:
494:
231:) can be represented as an extensive form game with outcomes (i.e. win, lose, or
181:
55:
3768:
2567:
PBS Infinite Series. Perfect information defined at 0:25, with academic sources
3843:
3833:
3823:
3758:
3748:
3738:
3723:
3519:
3499:
3484:
3479:
3439:
3406:
3391:
3386:
3376:
3180:
2797:
2738:
638:
410:), play in the first game will be as follows: player 1 knows that if they play
244:
197:
123:, meaning there is one payoff for each player at the end of every possible play
76:
2851:
2678:
2645:
2612:
2564:
653:
A game with incomplete and imperfect information represented in extensive form
4025:
3883:
3873:
3828:
3813:
3793:
3564:
3539:
3411:
3381:
3371:
3358:
3258:
3200:
3135:
3068:
2850:. A comprehensive reference from a computational perspective; see Chapter 5.
2801:
2695:
2153:
631:
177:
591:
If player 2 plays Up (U’), player 1 prefers to play Down (D) (Payoff 1>0)
588:
If player 1 plays Down (D), player 2 prefers to play Up (U’) (Payoff 2>1)
585:
If player 1 plays Up (U), player 2 prefers to play Down (D’) (Payoff 1>0)
300:
by every edge of the graph are the name of the action that edge represents.
287:
the payoffs received by every player for every possible combination of moves
3858:
3853:
3708:
3278:
232:
1868:{\displaystyle \rho =\{\rho _{H}:A(H)\rightarrow |H\in \mathbf {H} _{0}\}}
649:
163:
the complete description of the game specified by the above parameters is
3975:
3778:
3773:
3753:
3549:
3534:
3343:
3313:
3243:
3233:
3063:
2998:
2974:
2752:
2691:
1978:
236:
103:
31:
2942:
2832:
Multiagent
Systems: Algorithmic, Game-Theoretic, and Logical Foundations
1969:
3604:
3253:
2907:
594:
If player 2 plays Down (D’), player 1 prefers to play Down (D) (3>2)
252:
291:
3509:
3429:
3248:
402:
set. Any game without perfect information has imperfect information.
220:
71:
2769:
Essentials of Game Theory: A Concise, Multidisciplinary
Introduction
2755:(1991) Game theory (Ch3 Extensive form games, pp67–106). MIT press.
677:; i.e. an equilibrium in which different types do different things.
3944:
3444:
2580:
457:
and player 1 will receive 3. In fact in the second game there is a
2572:
2550:
https://www.math.uni-hamburg/Infinite Games, Yurii
Khomskii (2010)
1981:
between 0 and 5000) or with very large action spaces (perhaps any
1679:{\displaystyle (\mathbf {H} _{i})_{i\in {\mathcal {I}}\cup \{0\}}}
1076:
be a set of decision nodes) and an immediate predecessor function
338:
and player 1 receives 2. Player 1 prefers 2 to 1 and so will play
3665:
3655:
3333:
1982:
667:
469:
and player 2 holds the belief that player 1 will definitely play
144:
Each set of nodes of a rational player is further partitioned in
2830:
1973:
A game with infinite action spaces represented in extensive form
1285:{\displaystyle a:V\setminus \{v^{0}\}\rightarrow {\mathcal {A}}}
359:
example, moves may be simultaneous or a move may be hidden). An
384:
A game with imperfect information represented in extensive form
311:; player 2 observes player 1's choice and then chooses between
2783:. An 88-page mathematical introduction; see Chapters 4 and 5.
2767:
434:) because player 1 prefers to receive 2 to 1 and so will play
3434:
1989:. The payoffs to the firms are represented on the left, with
256:
240:
113:
91:-player extensive-form game thus consists of the following:
2892:
Neumann, J. (1928). "Zur
Theorie der Gesellschaftsspiele".
2235:{\displaystyle q_{2}(q_{1})={\tfrac {5000-q_{1}-c_{2}}{2}}}
636:. This transformation introduces to the game the notion of
39:
2425:{\displaystyle q_{2}^{*}={\tfrac {5000+2c_{1}-3c_{2}}{4}}}
2113:
as some constants (here marginal costs to each firm). The
1875:
is a family of probabilities of the actions of nature, and
958:{\displaystyle {\mathcal {K}}=\langle V,v^{0},T,p\rangle }
2344:{\displaystyle q_{1}^{*}={\tfrac {5000+c_{2}-2c_{1}}{2}}}
759:
Formally, a finite game in extensive form is a structure
1383:{\displaystyle \forall H\in \mathbf {H} ,\forall v\in H}
176:
of Nature's moves, can determine the edge precisely.) A
1183:
is a set of actions available for each information set
2377:
2299:
2193:
723:
whatever action they observe, but then type 1 prefers
2694:(1992). "Games in extensive and strategic forms". In
2438:
2357:
2351:. Feeding this into firm 2's best response function,
2279:
2250:
2162:
2129:
2090:
2059:
2028:
1997:
1883:
1786:
1758:
1714:
1692:
1628:
1608:
1554:
1531:
1502:
1473:
1453:
1396:
1350:
1318:
1298:
1243:
1217:
1189:
1160:
1138:
1116:
1082:
1050:
1024:
991:
971:
910:
765:
2772:, San Rafael, CA: Morgan & Claypool Publishers,
727:. The only equilibrium hence is with type 1 playing
696:. This cannot be an equilibrium. If both types play
278:
for every player every opportunity they have to move
271:
A complete extensive-form representation specifies:
27:
Wide-ranging representation of a game in game theory
2922:
2552:
Infinite Games (section 1.1), Yurii
Khomskii (2010)
266:
2828:
2765:
2700:Handbook of Game Theory with Economic Applications
2480:
2424:
2343:
2263:
2234:
2142:
2103:
2072:
2041:
2010:
1948:
1867:
1770:
1744:
1700:
1678:
1614:
1594:
1537:
1517:
1488:
1459:
1439:
1382:
1333:
1304:
1284:
1227:
1211:which forms a partition on the set of all actions
1203:
1175:
1144:
1124:
1100:
1068:
1036:
1010:
977:
957:
893:
209:
112:Each terminal (leaf) node of the game tree has an
4023:
2560:
2558:
2545:
2543:
1108:on which the rules of the game are represented,
79:in 1953, who extended an earlier definition of
2864:
2661:Strategy : an introduction to game theory
2628:Strategy : an introduction to game theory
2595:Strategy : an introduction to game theory
498:game, player one and player two each have two
281:what each player can do at each of their moves
130:of the non-terminal nodes of the game tree in
65:
58:". Extensive-form representations differ from
2958:
2658:
2625:
2592:
2555:
2540:
1752:be a single player that makes a move at node
1292:is an action partition associating each node
334:, player 2 maximises their payoff by playing
1862:
1793:
1671:
1665:
1589:
1565:
1269:
1256:
952:
921:
888:
853:
837:
772:
369:Every node in the set belongs to one player.
623:their opponents are. This sort of game has
204:decision". These can be made precise using
83:from 1928. Following the presentation from
2965:
2951:
2829:Shoham, Yoav; Leyton-Brown, Kevin (2009),
2766:Leyton-Brown, Kevin; Shoham, Yoav (2008),
1595:{\displaystyle {\mathcal {I}}=\{1,...,I\}}
1440:{\displaystyle a_{v}:s(v)\rightarrow A(H)}
2972:
2488:is the subgame perfect Nash equilibrium.
2273:by taking the first derivative, yielding
1934:
1686:is a player partition of information set
1622:is (a special player called) nature, and
614:
54:in the form of chance events modeled as "
2117:of this game can be found by taking the
1968:
1960:
648:
379:
353:
290:
190:von Neumann–Morgenstern utility function
2891:
2722:Playing for real: a text on game theory
2716:
476:To more easily solve this game for the
260:
14:
4024:
747:. Through their actions, player 1 has
365:is a set of decision nodes such that:
2946:
2565:"Infinite Chess, PBS Infinite Series"
2481:{\displaystyle (q_{1}^{*},q_{2}^{*})}
965:is a finite tree with a set of nodes
284:what each player knows for every move
137:Each node of the Chance player has a
75:general definition was introduced by
2690:
754:
295:A game represented in extensive form
84:
1745:{\displaystyle \iota (v)=\iota (H)}
24:
3014:First-player and second-player win
2858:
1940:
1916:
1657:
1557:
1368:
1351:
1277:
1220:
913:
824:
777:
766:
704:, but then type 1 prefers to play
235:). Examples of such games include
25:
4043:
1253:
1204:{\displaystyle H\in \mathbf {H} }
1060:
3121:Coalition-proof Nash equilibrium
1852:
1694:
1634:
1361:
1197:
1152:called an information partition,
1118:
1101:{\displaystyle p:V\rightarrow D}
864:
801:
786:
743:and randomising if they observe
511:Player 2's Strategies: {U’ , D’}
267:Perfect and complete information
180:for a player thus consists of a
2925:Lectures on the theory of games
2115:subgame perfect Nash equilibria
2051:as the strategy they adopt and
449:when they have actually played
210:Shoham & Leyton-Brown (2009
44:choices at every decision point
3131:Evolutionarily stable strategy
2927:. Princeton University Press.
2724:. Oxford University Press US.
2652:
2619:
2586:
2475:
2439:
2186:
2173:
1929:
1904:
1890:
1840:
1836:
1824:
1821:
1818:
1812:
1739:
1733:
1724:
1718:
1645:
1629:
1525:the set of successor nodes of
1512:
1506:
1483:
1477:
1434:
1428:
1422:
1419:
1413:
1328:
1322:
1272:
1228:{\displaystyle {\mathcal {A}}}
1170:
1164:
1092:
1069:{\displaystyle D=V\setminus T}
849:
843:
831:
812:
796:
793:
692:, type 2 would prefer to play
508:Player 1's Strategies: {U , D}
13:
1:
3059:Simultaneous action selection
2533:
1956:is a payoff profile function.
482:, it can be converted to the
3996:List of games in game theory
3171:Quantal response equilibrium
3161:Perfect Bayesian equilibrium
3096:Bayes correlated equilibrium
2923:Harold William Kuhn (2003).
2659:Watson, Joel. (2013-05-09).
2626:Watson, Joel. (2013-05-09).
2593:Watson, Joel. (2013-05-09).
1701:{\displaystyle \mathbf {H} }
1602:is a finite set of players,
1125:{\displaystyle \mathbf {H} }
675:perfect Bayesian equilibrium
459:perfect Bayesian equilibrium
7:
3465:Optional prisoner's dilemma
3191:Self-confirming equilibrium
2513:Self-confirming equilibrium
2491:
453:so that player 2 will play
348:subgame perfect equilibrium
66:Finite extensive-form games
10:
4048:
3930:Principal variation search
3646:Aumann's agreement theorem
3309:Strategy-stealing argument
3216:Trembling hand equilibrium
3146:Markov perfect equilibrium
3141:Mertens-stable equilibrium
2837:Cambridge University Press
1037:{\displaystyle T\subset V}
1018:, a set of terminal nodes
1011:{\displaystyle v^{0}\in V}
438:and so player 2 will play
3966:Combinatorial game theory
3953:
3912:
3694:
3638:
3625:Princess and monster game
3420:
3322:
3224:
3176:Quasi-perfect equilibrium
3101:Bayesian Nash equilibrium
3082:
2981:
2702:. Vol. 1. Elsevier.
2508:Combinatorial game theory
225:combinatorial game theory
212:, chpt. 13) for details.
4032:Game theory game classes
3981:Evolutionary game theory
3714:Antoine Augustin Cournot
3600:Guess 2/3 of the average
3397:Strictly determined game
3186:Satisfaction equilibrium
3004:Escalation of commitment
2852:Downloadable free online
2119:first partial derivative
985:, a unique initial node
751:their type to player 2.
688:. However, if they play
141:over its outgoing edges.
139:probability distribution
38:is a specification of a
3986:Glossary of game theory
3585:Stackelberg competition
3206:Strong Nash equilibrium
2865:Horst Herrlich (2006).
2698:; Hart, Sergiu (eds.).
1987:Stackelberg competition
605:Stackelberg competition
342:and player 2 will play
229:artificial intelligence
219:two-player game over a
4011:Tragedy of the commons
3991:List of game theorists
3971:Confrontation analysis
3681:Sprague–Grundy theorem
3196:Sequential equilibrium
3116:Correlated equilibrium
2482:
2426:
2345:
2265:
2236:
2144:
2105:
2074:
2043:
2012:
1974:
1950:
1869:
1772:
1771:{\displaystyle v\in H}
1746:
1702:
1680:
1616:
1596:
1539:
1519:
1490:
1461:
1441:
1384:
1335:
1306:
1286:
1229:
1205:
1177:
1146:
1126:
1102:
1070:
1038:
1012:
979:
959:
895:
670:status). There is one
654:
626:incomplete information
615:Incomplete information
385:
296:
196:random outcome by its
52:incomplete information
3784:Jean-François Mertens
2895:Mathematische Annalen
2794:at many universities.
2483:
2427:
2346:
2266:
2264:{\displaystyle q_{1}}
2237:
2145:
2143:{\displaystyle q_{2}}
2106:
2104:{\displaystyle c_{2}}
2075:
2073:{\displaystyle c_{1}}
2044:
2042:{\displaystyle q_{2}}
2013:
2011:{\displaystyle q_{1}}
1972:
1961:Infinite action space
1951:
1870:
1773:
1747:
1703:
1681:
1617:
1597:
1540:
1520:
1496:is a bijection, with
1491:
1462:
1442:
1385:
1336:
1307:
1287:
1230:
1206:
1178:
1147:
1127:
1103:
1071:
1039:
1013:
980:
960:
896:
735:and player 2 playing
719:, player 2 will play
652:
461:where player 1 plays
422:, player 2 will play
414:, player 2 will play
391:imperfect information
383:
354:Imperfect information
326:, player 2 will play
294:
275:the players of a game
206:epistemic modal logic
3913:Search optimizations
3789:Jennifer Tour Chayes
3676:Revelation principle
3671:Purification theorem
3610:Nash bargaining game
3575:Bertrand competition
3560:El Farol Bar problem
3525:Electronic mail game
3490:Lewis signaling game
3029:Hierarchy of beliefs
2498:Axiom of determinacy
2436:
2355:
2277:
2248:
2160:
2127:
2088:
2057:
2026:
1995:
1881:
1784:
1756:
1712:
1690:
1626:
1606:
1552:
1529:
1518:{\displaystyle s(v)}
1500:
1489:{\displaystyle s(v)}
1471:
1451:
1394:
1348:
1334:{\displaystyle a(v)}
1316:
1296:
1241:
1215:
1187:
1176:{\displaystyle A(H)}
1158:
1136:
1114:
1080:
1048:
1022:
989:
969:
908:
763:
660:by the other players
3961:Bounded rationality
3580:Cournot competition
3530:Rock paper scissors
3505:Battle of the sexes
3495:Volunteer's dilemma
3367:Perfect information
3294:Dominant strategies
3126:Epsilon-equilibrium
3009:Extensive-form game
2597:. pp. 97–100.
2503:Perfect information
2474:
2456:
2372:
2294:
1312:to a single action
609:Cournot competition
516:
465:and player 2 plays
396:perfect information
217:perfect information
153:equivalence classes
36:extensive-form game
3940:Paranoid algorithm
3920:Alpha–beta pruning
3799:John Maynard Smith
3630:Rendezvous problem
3470:Traveler's dilemma
3460:Gift-exchange game
3455:Prisoner's dilemma
3372:Large Poisson game
3339:Bargaining problem
3239:Backward induction
3211:Subgame perfection
3166:Proper equilibrium
2908:10.1007/BF01448847
2790:2000-08-15 at the
2663:. pp. 22–26.
2630:. pp. 26–28.
2478:
2460:
2442:
2422:
2420:
2358:
2341:
2339:
2280:
2261:
2232:
2230:
2152:) and finding its
2140:
2101:
2070:
2039:
2008:
1975:
1946:
1865:
1768:
1742:
1698:
1676:
1612:
1592:
1535:
1515:
1486:
1457:
1437:
1390:, the restriction
1380:
1331:
1302:
1282:
1225:
1201:
1173:
1142:
1132:is a partition of
1122:
1098:
1066:
1034:
1008:
975:
955:
891:
655:
515:
488:. Given this is a
386:
322:If player 1 plays
297:
249:expectminimax tree
99:(rational) players
4019:
4018:
3925:Aspiration window
3894:Suzanne Scotchmer
3849:Oskar Morgenstern
3744:Donald B. Gillies
3686:Zermelo's theorem
3615:Induction puzzles
3570:Fair cake-cutting
3545:Public goods game
3475:Coordination game
3349:Intransitive game
3274:Forward induction
3156:Pareto efficiency
3136:Gibbs equilibrium
3106:Berge equilibrium
3054:Simultaneous game
2934:978-0-691-02772-2
2886:Historical papers
2876:978-3-540-30989-5
2846:978-0-521-89943-7
2779:978-1-59829-593-1
2731:978-0-19-530057-4
2709:978-0-444-88098-7
2670:978-0-393-91838-0
2637:978-0-393-91838-0
2604:978-0-393-91838-0
2419:
2338:
2229:
1615:{\displaystyle 0}
1538:{\displaystyle v}
1460:{\displaystyle a}
1305:{\displaystyle v}
1145:{\displaystyle D}
978:{\displaystyle V}
755:Formal definition
731:, type 2 playing
715:and type 2 plays
579:
578:
247:. A game over an
167:among the players
16:(Redirected from
4039:
4006:Topological game
4001:No-win situation
3899:Thomas Schelling
3879:Robert B. Wilson
3839:Merrill M. Flood
3809:John von Neumann
3719:Ariel Rubinstein
3704:Albert W. Tucker
3555:War of attrition
3515:Matching pennies
3289:Pairing strategy
3151:Nash equilibrium
3074:Mechanism design
3039:Normal-form game
2994:Cooperative game
2967:
2960:
2953:
2944:
2943:
2938:
2919:
2880:
2849:
2782:
2751:Fudenberg D and
2735:
2718:Binmore, Kenneth
2713:
2683:
2682:
2656:
2650:
2649:
2623:
2617:
2616:
2590:
2584:
2562:
2553:
2547:
2528:Solution concept
2487:
2485:
2484:
2479:
2473:
2468:
2455:
2450:
2431:
2429:
2428:
2423:
2421:
2415:
2414:
2413:
2398:
2397:
2378:
2371:
2366:
2350:
2348:
2347:
2342:
2340:
2334:
2333:
2332:
2317:
2316:
2300:
2293:
2288:
2272:
2270:
2268:
2267:
2262:
2260:
2259:
2241:
2239:
2238:
2233:
2231:
2225:
2224:
2223:
2211:
2210:
2194:
2185:
2184:
2172:
2171:
2151:
2149:
2147:
2146:
2141:
2139:
2138:
2112:
2110:
2108:
2107:
2102:
2100:
2099:
2081:
2079:
2077:
2076:
2071:
2069:
2068:
2050:
2048:
2046:
2045:
2040:
2038:
2037:
2019:
2017:
2015:
2014:
2009:
2007:
2006:
1955:
1953:
1952:
1947:
1945:
1944:
1943:
1937:
1922:
1921:
1920:
1919:
1902:
1901:
1874:
1872:
1871:
1866:
1861:
1860:
1855:
1843:
1805:
1804:
1777:
1775:
1774:
1769:
1751:
1749:
1748:
1743:
1707:
1705:
1704:
1699:
1697:
1685:
1683:
1682:
1677:
1675:
1674:
1661:
1660:
1643:
1642:
1637:
1621:
1619:
1618:
1613:
1601:
1599:
1598:
1593:
1561:
1560:
1544:
1542:
1541:
1536:
1524:
1522:
1521:
1516:
1495:
1493:
1492:
1487:
1466:
1464:
1463:
1458:
1446:
1444:
1443:
1438:
1406:
1405:
1389:
1387:
1386:
1381:
1364:
1340:
1338:
1337:
1332:
1311:
1309:
1308:
1303:
1291:
1289:
1288:
1283:
1281:
1280:
1268:
1267:
1234:
1232:
1231:
1226:
1224:
1223:
1210:
1208:
1207:
1202:
1200:
1182:
1180:
1179:
1174:
1151:
1149:
1148:
1143:
1131:
1129:
1128:
1123:
1121:
1107:
1105:
1104:
1099:
1075:
1073:
1072:
1067:
1043:
1041:
1040:
1035:
1017:
1015:
1014:
1009:
1001:
1000:
984:
982:
981:
976:
964:
962:
961:
956:
939:
938:
917:
916:
900:
898:
897:
892:
869:
868:
867:
830:
829:
828:
827:
810:
809:
804:
789:
781:
780:
739:if they observe
711:If type 1 plays
517:
514:
479:Nash equilibrium
165:common knowledge
146:information sets
95:A finite set of
46:, the (possibly
21:
4047:
4046:
4042:
4041:
4040:
4038:
4037:
4036:
4022:
4021:
4020:
4015:
3949:
3935:max^n algorithm
3908:
3904:William Vickrey
3864:Reinhard Selten
3819:Kenneth Binmore
3734:David K. Levine
3729:Daniel Kahneman
3696:
3690:
3666:Negamax theorem
3656:Minimax theorem
3634:
3595:Diner's dilemma
3450:All-pay auction
3416:
3402:Stochastic game
3354:Mean-field game
3325:
3318:
3284:Markov strategy
3220:
3086:
3078:
3049:Sequential game
3034:Information set
3019:Game complexity
2989:Congestion game
2977:
2971:
2935:
2877:
2867:Axiom of choice
2861:
2859:Further reading
2847:
2814:Osborne MJ and
2792:Wayback Machine
2780:
2732:
2710:
2687:
2686:
2671:
2657:
2653:
2638:
2624:
2620:
2605:
2591:
2587:
2563:
2556:
2548:
2541:
2536:
2518:Sequential game
2494:
2469:
2464:
2451:
2446:
2437:
2434:
2433:
2409:
2405:
2393:
2389:
2379:
2376:
2367:
2362:
2356:
2353:
2352:
2328:
2324:
2312:
2308:
2301:
2298:
2289:
2284:
2278:
2275:
2274:
2255:
2251:
2249:
2246:
2245:
2243:
2219:
2215:
2206:
2202:
2195:
2192:
2180:
2176:
2167:
2163:
2161:
2158:
2157:
2134:
2130:
2128:
2125:
2124:
2122:
2095:
2091:
2089:
2086:
2085:
2083:
2064:
2060:
2058:
2055:
2054:
2052:
2033:
2029:
2027:
2024:
2023:
2021:
2002:
1998:
1996:
1993:
1992:
1990:
1963:
1939:
1938:
1933:
1932:
1915:
1914:
1907:
1903:
1897:
1893:
1882:
1879:
1878:
1856:
1851:
1850:
1839:
1800:
1796:
1785:
1782:
1781:
1757:
1754:
1753:
1713:
1710:
1709:
1693:
1691:
1688:
1687:
1656:
1655:
1648:
1644:
1638:
1633:
1632:
1627:
1624:
1623:
1607:
1604:
1603:
1556:
1555:
1553:
1550:
1549:
1530:
1527:
1526:
1501:
1498:
1497:
1472:
1469:
1468:
1452:
1449:
1448:
1401:
1397:
1395:
1392:
1391:
1360:
1349:
1346:
1345:
1317:
1314:
1313:
1297:
1294:
1293:
1276:
1275:
1263:
1259:
1242:
1239:
1238:
1219:
1218:
1216:
1213:
1212:
1196:
1188:
1185:
1184:
1159:
1156:
1155:
1137:
1134:
1133:
1117:
1115:
1112:
1111:
1081:
1078:
1077:
1049:
1046:
1045:
1023:
1020:
1019:
996:
992:
990:
987:
986:
970:
967:
966:
934:
930:
912:
911:
909:
906:
905:
863:
856:
852:
823:
822:
815:
811:
805:
800:
799:
785:
776:
775:
764:
761:
760:
757:
639:nature's choice
617:
525:
522:
362:information set
356:
269:
251:, like that of
223:(as defined in
68:
56:moves by nature
28:
23:
22:
15:
12:
11:
5:
4045:
4035:
4034:
4017:
4016:
4014:
4013:
4008:
4003:
3998:
3993:
3988:
3983:
3978:
3973:
3968:
3963:
3957:
3955:
3951:
3950:
3948:
3947:
3942:
3937:
3932:
3927:
3922:
3916:
3914:
3910:
3909:
3907:
3906:
3901:
3896:
3891:
3886:
3881:
3876:
3871:
3869:Robert Axelrod
3866:
3861:
3856:
3851:
3846:
3844:Olga Bondareva
3841:
3836:
3834:Melvin Dresher
3831:
3826:
3824:Leonid Hurwicz
3821:
3816:
3811:
3806:
3801:
3796:
3791:
3786:
3781:
3776:
3771:
3766:
3761:
3759:Harold W. Kuhn
3756:
3751:
3749:Drew Fudenberg
3746:
3741:
3739:David M. Kreps
3736:
3731:
3726:
3724:Claude Shannon
3721:
3716:
3711:
3706:
3700:
3698:
3692:
3691:
3689:
3688:
3683:
3678:
3673:
3668:
3663:
3661:Nash's theorem
3658:
3653:
3648:
3642:
3640:
3636:
3635:
3633:
3632:
3627:
3622:
3617:
3612:
3607:
3602:
3597:
3592:
3587:
3582:
3577:
3572:
3567:
3562:
3557:
3552:
3547:
3542:
3537:
3532:
3527:
3522:
3520:Ultimatum game
3517:
3512:
3507:
3502:
3500:Dollar auction
3497:
3492:
3487:
3485:Centipede game
3482:
3477:
3472:
3467:
3462:
3457:
3452:
3447:
3442:
3440:Infinite chess
3437:
3432:
3426:
3424:
3418:
3417:
3415:
3414:
3409:
3407:Symmetric game
3404:
3399:
3394:
3392:Signaling game
3389:
3387:Screening game
3384:
3379:
3377:Potential game
3374:
3369:
3364:
3356:
3351:
3346:
3341:
3336:
3330:
3328:
3320:
3319:
3317:
3316:
3311:
3306:
3304:Mixed strategy
3301:
3296:
3291:
3286:
3281:
3276:
3271:
3266:
3261:
3256:
3251:
3246:
3241:
3236:
3230:
3228:
3222:
3221:
3219:
3218:
3213:
3208:
3203:
3198:
3193:
3188:
3183:
3181:Risk dominance
3178:
3173:
3168:
3163:
3158:
3153:
3148:
3143:
3138:
3133:
3128:
3123:
3118:
3113:
3108:
3103:
3098:
3092:
3090:
3080:
3079:
3077:
3076:
3071:
3066:
3061:
3056:
3051:
3046:
3041:
3036:
3031:
3026:
3024:Graphical game
3021:
3016:
3011:
3006:
3001:
2996:
2991:
2985:
2983:
2979:
2978:
2970:
2969:
2962:
2955:
2947:
2941:
2940:
2933:
2920:
2883:
2882:
2875:
2860:
2857:
2856:
2855:
2845:
2826:
2812:
2795:
2778:
2763:
2749:
2736:
2730:
2714:
2708:
2696:Aumann, Robert
2685:
2684:
2669:
2651:
2636:
2618:
2603:
2585:
2554:
2538:
2537:
2535:
2532:
2531:
2530:
2525:
2520:
2515:
2510:
2505:
2500:
2493:
2490:
2477:
2472:
2467:
2463:
2459:
2454:
2449:
2445:
2441:
2418:
2412:
2408:
2404:
2401:
2396:
2392:
2388:
2385:
2382:
2375:
2370:
2365:
2361:
2337:
2331:
2327:
2323:
2320:
2315:
2311:
2307:
2304:
2297:
2292:
2287:
2283:
2258:
2254:
2228:
2222:
2218:
2214:
2209:
2205:
2201:
2198:
2191:
2188:
2183:
2179:
2175:
2170:
2166:
2137:
2133:
2098:
2094:
2067:
2063:
2036:
2032:
2005:
2001:
1962:
1959:
1958:
1957:
1942:
1936:
1931:
1928:
1925:
1918:
1913:
1910:
1906:
1900:
1896:
1892:
1889:
1886:
1876:
1864:
1859:
1854:
1849:
1846:
1842:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1808:
1803:
1799:
1795:
1792:
1789:
1779:
1767:
1764:
1761:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1696:
1673:
1670:
1667:
1664:
1659:
1654:
1651:
1647:
1641:
1636:
1631:
1611:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1559:
1534:
1514:
1511:
1508:
1505:
1485:
1482:
1479:
1476:
1456:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1404:
1400:
1379:
1376:
1373:
1370:
1367:
1363:
1359:
1356:
1353:
1343:
1342:
1330:
1327:
1324:
1321:
1301:
1279:
1274:
1271:
1266:
1262:
1258:
1255:
1252:
1249:
1246:
1236:
1222:
1199:
1195:
1192:
1172:
1169:
1166:
1163:
1153:
1141:
1120:
1109:
1097:
1094:
1091:
1088:
1085:
1065:
1062:
1059:
1056:
1053:
1033:
1030:
1027:
1007:
1004:
999:
995:
974:
954:
951:
948:
945:
942:
937:
933:
929:
926:
923:
920:
915:
890:
887:
884:
881:
878:
875:
872:
866:
862:
859:
855:
851:
848:
845:
842:
839:
836:
833:
826:
821:
818:
814:
808:
803:
798:
795:
792:
788:
784:
779:
774:
771:
768:
756:
753:
634:transformation
616:
613:
596:
595:
592:
589:
586:
577:
576:
567:
554:
550:
549:
540:
537:
533:
532:
529:
526:
523:
520:
513:
512:
509:
394:. A game with
375:
374:
370:
355:
352:
346:. This is the
289:
288:
285:
282:
279:
276:
268:
265:
245:infinite chess
169:
168:
161:
160:
159:
156:
142:
135:
124:
110:
100:
77:Harold W. Kuhn
67:
64:
26:
18:Extensive form
9:
6:
4:
3:
2:
4044:
4033:
4030:
4029:
4027:
4012:
4009:
4007:
4004:
4002:
3999:
3997:
3994:
3992:
3989:
3987:
3984:
3982:
3979:
3977:
3974:
3972:
3969:
3967:
3964:
3962:
3959:
3958:
3956:
3954:Miscellaneous
3952:
3946:
3943:
3941:
3938:
3936:
3933:
3931:
3928:
3926:
3923:
3921:
3918:
3917:
3915:
3911:
3905:
3902:
3900:
3897:
3895:
3892:
3890:
3889:Samuel Bowles
3887:
3885:
3884:Roger Myerson
3882:
3880:
3877:
3875:
3874:Robert Aumann
3872:
3870:
3867:
3865:
3862:
3860:
3857:
3855:
3852:
3850:
3847:
3845:
3842:
3840:
3837:
3835:
3832:
3830:
3829:Lloyd Shapley
3827:
3825:
3822:
3820:
3817:
3815:
3814:Kenneth Arrow
3812:
3810:
3807:
3805:
3802:
3800:
3797:
3795:
3794:John Harsanyi
3792:
3790:
3787:
3785:
3782:
3780:
3777:
3775:
3772:
3770:
3767:
3765:
3764:Herbert Simon
3762:
3760:
3757:
3755:
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3732:
3730:
3727:
3725:
3722:
3720:
3717:
3715:
3712:
3710:
3707:
3705:
3702:
3701:
3699:
3693:
3687:
3684:
3682:
3679:
3677:
3674:
3672:
3669:
3667:
3664:
3662:
3659:
3657:
3654:
3652:
3649:
3647:
3644:
3643:
3641:
3637:
3631:
3628:
3626:
3623:
3621:
3618:
3616:
3613:
3611:
3608:
3606:
3603:
3601:
3598:
3596:
3593:
3591:
3588:
3586:
3583:
3581:
3578:
3576:
3573:
3571:
3568:
3566:
3565:Fair division
3563:
3561:
3558:
3556:
3553:
3551:
3548:
3546:
3543:
3541:
3540:Dictator game
3538:
3536:
3533:
3531:
3528:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3508:
3506:
3503:
3501:
3498:
3496:
3493:
3491:
3488:
3486:
3483:
3481:
3478:
3476:
3473:
3471:
3468:
3466:
3463:
3461:
3458:
3456:
3453:
3451:
3448:
3446:
3443:
3441:
3438:
3436:
3433:
3431:
3428:
3427:
3425:
3423:
3419:
3413:
3412:Zero-sum game
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3388:
3385:
3383:
3382:Repeated game
3380:
3378:
3375:
3373:
3370:
3368:
3365:
3363:
3361:
3357:
3355:
3352:
3350:
3347:
3345:
3342:
3340:
3337:
3335:
3332:
3331:
3329:
3327:
3321:
3315:
3312:
3310:
3307:
3305:
3302:
3300:
3299:Pure strategy
3297:
3295:
3292:
3290:
3287:
3285:
3282:
3280:
3277:
3275:
3272:
3270:
3267:
3265:
3262:
3260:
3259:De-escalation
3257:
3255:
3252:
3250:
3247:
3245:
3242:
3240:
3237:
3235:
3232:
3231:
3229:
3227:
3223:
3217:
3214:
3212:
3209:
3207:
3204:
3202:
3201:Shapley value
3199:
3197:
3194:
3192:
3189:
3187:
3184:
3182:
3179:
3177:
3174:
3172:
3169:
3167:
3164:
3162:
3159:
3157:
3154:
3152:
3149:
3147:
3144:
3142:
3139:
3137:
3134:
3132:
3129:
3127:
3124:
3122:
3119:
3117:
3114:
3112:
3109:
3107:
3104:
3102:
3099:
3097:
3094:
3093:
3091:
3089:
3085:
3081:
3075:
3072:
3070:
3069:Succinct game
3067:
3065:
3062:
3060:
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3037:
3035:
3032:
3030:
3027:
3025:
3022:
3020:
3017:
3015:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2995:
2992:
2990:
2987:
2986:
2984:
2980:
2976:
2968:
2963:
2961:
2956:
2954:
2949:
2948:
2945:
2936:
2930:
2926:
2921:
2917:
2913:
2909:
2905:
2901:
2897:
2896:
2890:
2889:
2888:
2887:
2878:
2872:
2868:
2863:
2862:
2853:
2848:
2842:
2838:
2834:
2833:
2827:
2825:
2824:0-262-65040-1
2821:
2817:
2816:Rubinstein A.
2813:
2811:
2810:0-486-65943-7
2807:
2803:
2799:
2796:
2793:
2789:
2786:
2781:
2775:
2771:
2770:
2764:
2762:
2761:0-262-06141-4
2758:
2754:
2750:
2748:
2747:0-486-64216-X
2744:
2740:
2737:
2733:
2727:
2723:
2719:
2715:
2711:
2705:
2701:
2697:
2693:
2689:
2688:
2680:
2676:
2672:
2666:
2662:
2655:
2647:
2643:
2639:
2633:
2629:
2622:
2614:
2610:
2606:
2600:
2596:
2589:
2582:
2578:
2574:
2570:
2566:
2561:
2559:
2551:
2546:
2544:
2539:
2529:
2526:
2524:
2521:
2519:
2516:
2514:
2511:
2509:
2506:
2504:
2501:
2499:
2496:
2495:
2489:
2470:
2465:
2461:
2457:
2452:
2447:
2443:
2416:
2410:
2406:
2402:
2399:
2394:
2390:
2386:
2383:
2380:
2373:
2368:
2363:
2359:
2335:
2329:
2325:
2321:
2318:
2313:
2309:
2305:
2302:
2295:
2290:
2285:
2281:
2256:
2252:
2226:
2220:
2216:
2212:
2207:
2203:
2199:
2196:
2189:
2181:
2177:
2168:
2164:
2155:
2154:best response
2135:
2131:
2120:
2116:
2096:
2092:
2065:
2061:
2034:
2030:
2003:
1999:
1988:
1984:
1980:
1971:
1967:
1926:
1923:
1911:
1908:
1898:
1894:
1887:
1884:
1877:
1857:
1847:
1844:
1833:
1830:
1827:
1815:
1809:
1806:
1801:
1797:
1790:
1787:
1780:
1765:
1762:
1759:
1736:
1730:
1727:
1721:
1715:
1668:
1662:
1652:
1649:
1639:
1609:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1562:
1548:
1547:
1546:
1532:
1509:
1503:
1480:
1474:
1454:
1431:
1425:
1416:
1410:
1407:
1402:
1398:
1377:
1374:
1371:
1365:
1357:
1354:
1341:, fulfilling:
1325:
1319:
1299:
1264:
1260:
1250:
1247:
1244:
1237:
1193:
1190:
1167:
1161:
1154:
1139:
1110:
1095:
1089:
1086:
1083:
1063:
1057:
1054:
1051:
1031:
1028:
1025:
1005:
1002:
997:
993:
972:
949:
946:
943:
940:
935:
931:
927:
924:
918:
904:
903:
902:
885:
882:
879:
876:
873:
870:
860:
857:
846:
840:
834:
819:
816:
806:
790:
782:
769:
752:
750:
746:
742:
738:
734:
730:
726:
722:
718:
714:
709:
707:
703:
699:
695:
691:
687:
683:
678:
676:
673:
669:
663:
661:
651:
647:
645:
641:
640:
635:
633:
628:
627:
622:
612:
610:
606:
600:
593:
590:
587:
584:
583:
582:
574:
573:
568:
566:
564:
560:
555:
552:
551:
547:
546:
541:
538:
535:
534:
530:
527:
519:
518:
510:
507:
506:
505:
503:
502:
497:
496:
492:
487:
486:
481:
480:
474:
472:
468:
464:
460:
456:
452:
448:
443:
441:
437:
433:
429:
425:
421:
417:
413:
409:
403:
401:
397:
393:
392:
382:
378:
371:
368:
367:
366:
364:
363:
351:
349:
345:
341:
337:
333:
329:
325:
320:
318:
314:
310:
306:
301:
293:
286:
283:
280:
277:
274:
273:
272:
264:
262:
258:
254:
250:
246:
242:
238:
234:
230:
226:
222:
218:
213:
211:
207:
201:
199:
195:
191:
187:
183:
179:
178:pure strategy
175:
166:
162:
157:
154:
150:
149:
147:
143:
140:
136:
133:
129:
125:
122:
118:
116:
111:
109:
106:, called the
105:
101:
98:
94:
93:
92:
90:
86:
82:
78:
73:
63:
61:
57:
53:
49:
45:
41:
37:
33:
19:
3859:Peyton Young
3854:Paul Milgrom
3769:Hervé Moulin
3709:Amos Tversky
3651:Folk theorem
3362:-player game
3359:
3279:Grim trigger
3008:
2924:
2899:
2893:
2885:
2884:
2869:. Springer.
2866:
2835:, New York:
2831:
2768:
2721:
2699:
2692:Hart, Sergiu
2660:
2654:
2627:
2621:
2594:
2588:
1976:
1964:
1344:
758:
744:
740:
736:
732:
728:
724:
720:
716:
712:
710:
705:
701:
697:
693:
689:
685:
681:
679:
671:
664:
659:
656:
644:God's choice
643:
637:
630:
624:
620:
618:
601:
597:
580:
571:
570:
562:
558:
556:
544:
543:
499:
491:simultaneous
489:
483:
477:
475:
470:
466:
462:
454:
450:
446:
444:
439:
435:
431:
427:
423:
419:
415:
411:
408:ad infinitum
407:
404:
395:
389:
387:
376:
360:
357:
343:
339:
335:
331:
327:
323:
321:
316:
312:
308:
304:
302:
298:
270:
261:Binmore 2007
214:
202:
170:
131:
120:
114:
107:
96:
88:
69:
35:
29:
3976:Coopetition
3779:Jean Tirole
3774:John Conway
3754:Eric Maskin
3550:Blotto game
3535:Pirate game
3344:Global game
3314:Tit for tat
3244:Bid shading
3234:Appeasement
3084:Equilibrium
3064:Solved game
2999:Determinacy
2982:Definitions
2975:game theory
2902:: 295–320.
2785:Free online
1979:real number
531:Down' (D')
485:normal form
263:, chpt. 2)
237:tic-tac-toe
174:realization
104:rooted tree
85:Hart (1992)
81:von Neumann
60:normal-form
32:game theory
3620:Trust game
3605:Kuhn poker
3269:Escalation
3264:Deterrence
3254:Cheap talk
3226:Strategies
3044:Preference
2973:Topics of
2739:Dresher M.
2679:1123193808
2646:1123193808
2613:1123193808
2581:1510.08155
2534:References
2523:Signalling
2156:function,
672:separating
501:strategies
495:sequential
253:backgammon
186:singletons
3804:John Nash
3510:Stag hunt
3249:Collusion
2916:122961988
2802:Raiffa H.
2798:Luce R.D.
2753:Tirole J.
2573:1302.4377
2471:∗
2453:∗
2400:−
2369:∗
2319:−
2291:∗
2213:−
2200:−
1930:→
1912:∈
1848:∈
1822:→
1798:ρ
1788:ρ
1763:∈
1731:ι
1716:ι
1663:∪
1653:∈
1423:→
1375:∈
1369:∀
1358:∈
1352:∀
1273:→
1254:∖
1194:∈
1093:→
1061:∖
1029:⊂
1003:∈
953:⟩
922:⟨
889:⟩
880:ρ
861:∈
820:∈
773:⟨
767:Γ
749:signalled
553:Down (D)
528:Up' (U')
524:Player 1
400:singleton
221:game tree
200:utility.
182:selection
128:partition
108:game tree
72:game tree
48:imperfect
4026:Category
3945:Lazy SMP
3639:Theorems
3590:Deadlock
3445:Checkers
3326:of games
3088:concepts
2788:Archived
2720:(2007).
2492:See also
632:Harsanyi
521:Player 2
373:reached.
198:expected
194:a priori
3697:figures
3480:Chicken
3334:Auction
3324:Classes
2271:
2244:
2150:
2123:
2111:
2084:
2080:
2053:
2049:
2022:
2018:
1991:
1983:integer
901:where:
668:subgame
536:Up (U)
121:payoffs
2931:
2914:
2873:
2843:
2822:
2808:
2776:
2759:
2745:
2728:
2706:
2677:
2667:
2644:
2634:
2611:
2601:
1708:. Let
539:(0,0)
243:, and
208:; see
117:-tuple
3435:Chess
3422:Games
2912:S2CID
2577:arXiv
2569:arXiv
1044:(let
257:poker
241:chess
87:, an
34:, an
3111:Core
2929:ISBN
2871:ISBN
2841:ISBN
2820:ISBN
2806:ISBN
2800:and
2774:ISBN
2757:ISBN
2743:ISBN
2726:ISBN
2704:ISBN
2675:OCLC
2665:ISBN
2642:OCLC
2632:ISBN
2609:OCLC
2599:ISBN
2575:and
2432:and
2381:5000
2303:5000
2197:5000
2082:and
2020:and
621:type
575:,1)
315:and
307:and
233:draw
227:and
40:game
3695:Key
2904:doi
2900:100
1467:on
1447:of
737:U'
721:D'
702:D'
690:D'
686:D'
642:or
542:(2,
467:U'
455:D'
440:D'
432:D'
424:U'
416:D'
344:D'
336:D'
328:U'
317:D'
313:U'
119:of
30:In
4028::
3430:Go
2910:.
2898:.
2839:,
2673:.
2640:.
2607:.
2557:^
2542:^
1545:.
708:.
611:.
548:)
504:.
442:.
430:,
350:.
239:,
215:A
126:A
102:A
3360:n
2966:e
2959:t
2952:v
2937:.
2918:.
2906::
2879:.
2854:.
2734:.
2712:.
2681:.
2648:.
2615:.
2583:.
2579::
2571::
2476:)
2466:2
2462:q
2458:,
2448:1
2444:q
2440:(
2417:4
2411:2
2407:c
2403:3
2395:1
2391:c
2387:2
2384:+
2374:=
2364:2
2360:q
2336:2
2330:1
2326:c
2322:2
2314:2
2310:c
2306:+
2296:=
2286:1
2282:q
2257:1
2253:q
2227:2
2221:2
2217:c
2208:1
2204:q
2190:=
2187:)
2182:1
2178:q
2174:(
2169:2
2165:q
2136:2
2132:q
2097:2
2093:c
2066:1
2062:c
2035:2
2031:q
2004:1
2000:q
1941:I
1935:R
1927:T
1924::
1917:I
1909:i
1905:)
1899:i
1895:u
1891:(
1888:=
1885:u
1863:}
1858:0
1853:H
1845:H
1841:|
1837:]
1834:1
1831:,
1828:0
1825:[
1819:)
1816:H
1813:(
1810:A
1807::
1802:H
1794:{
1791:=
1778:.
1766:H
1760:v
1740:)
1737:H
1734:(
1728:=
1725:)
1722:v
1719:(
1695:H
1672:}
1669:0
1666:{
1658:I
1650:i
1646:)
1640:i
1635:H
1630:(
1610:0
1590:}
1587:I
1584:,
1581:.
1578:.
1575:.
1572:,
1569:1
1566:{
1563:=
1558:I
1533:v
1513:)
1510:v
1507:(
1504:s
1484:)
1481:v
1478:(
1475:s
1455:a
1435:)
1432:H
1429:(
1426:A
1420:)
1417:v
1414:(
1411:s
1408::
1403:v
1399:a
1378:H
1372:v
1366:,
1362:H
1355:H
1329:)
1326:v
1323:(
1320:a
1300:v
1278:A
1270:}
1265:0
1261:v
1257:{
1251:V
1248::
1245:a
1235:.
1221:A
1198:H
1191:H
1171:)
1168:H
1165:(
1162:A
1140:D
1119:H
1096:D
1090:V
1087::
1084:p
1064:T
1058:V
1055:=
1052:D
1032:V
1026:T
1006:V
998:0
994:v
973:V
950:p
947:,
944:T
941:,
936:0
932:v
928:,
925:V
919:=
914:K
886:u
883:,
877:,
874:a
871:,
865:H
858:H
854:}
850:)
847:H
844:(
841:A
838:{
835:,
832:]
825:I
817:i
813:)
807:i
802:H
797:(
794:[
791:,
787:H
783:,
778:K
770:=
745:U
741:D
733:U
729:D
725:D
717:D
713:U
706:D
698:U
694:U
682:D
572:3
569:(
565:)
563:2
561:,
559:1
557:(
545:1
493:/
471:D
463:D
451:D
447:U
436:U
428:U
420:D
412:U
340:U
332:U
324:D
309:D
305:U
132:n
115:n
97:n
89:n
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.