635:. 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.
647:
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
370:
1955:
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.
1959:
281:
639:
288:
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
394:
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
63:
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
1954:
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
192:
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
160:
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,
347:
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
646:
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
361:
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
123:+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.)
591:
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
308:. 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).
888:
654:
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
1943:
2231:. 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
1862:
1673:
462:. 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.
1279:
587:
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).
673:, 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
2229:
161:
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
2419:
952:
570:
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.
2338:
1377:
39:) 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
2870:, 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)
140:
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
1589:
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1974:
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
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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.
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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.
651:. In the case of private information, every player knows what has been played by nature. Information sets are represented as before by broken lines.
173:—choosing precisely one class of outgoing edges for every information set (of his). In a game of perfect information, the information sets are
434:
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
31:
allowing (as the name suggests) for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their
415:(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 (
1772:
596:
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
292:
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
51:
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.
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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
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407:(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
1336:
3119:
3448:
2793:(1957). Games and decisions: introduction and critical survey. (Ch3: Extensive and Normal Forms, pp39–55). Wiley New York.
3629:
3047:
3022:
2730:(1961). The mathematics of games of strategy: theory and applications (Ch4: Games in extensive form, pp74–78). Rand Corp.
350:
134:
32:
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has both moves of chance (the cards being dealt) and imperfect information (the cards secretly held by other players). (
177:. It's less evident how payoffs should be interpreted in games with Chance nodes. It is assumed that each player has a
3199:
3179:
2928:
contains Kuhn's lectures at
Princeton from 1952 (officially unpublished previously, but in circulation as photocopies)
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689:, player 2 again forms the belief that they are at either node with probability 1/2. In this case player 2 plays
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127:
59:
Some authors, particularly in introductory textbooks, initially define the extensive-form game as being just a
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every (directed) path in the tree from the root to a terminal node can cross each information set at most once
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3194:
2982:
1975:
618:. In extensive form it is represented as a game with complete but imperfect information using the so-called
388:
217:
174:
3903:
2424:
883:{\displaystyle \Gamma =\langle {\mathcal {K}},\mathbf {H} ,,\{A(H)\}_{H\in \mathbf {H} },a,\rho ,u\rangle }
244:, has no imperfect information (all information sets are singletons) but has moves of chance. For example,
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40:
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1994. A course in game theory (Ch6 Extensive game with perfect information, pp. 89–115). MIT press.
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137:, which make certain choices indistinguishable for the player when making a move, in the sense that:
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defined for every game outcome; this assumption entails that every rational player will evaluate an
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It may be the case that a player does not know exactly what the payoffs of the game are or of what
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If both types play the same action (pooling), an equilibrium cannot be sustained. If both play
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to maximise their payoff and so player 1 will only receive 1. However, if player 1 plays
144:, each class representing a possible choice for a player's move at some point—, and
1938:{\displaystyle u=(u_{i})_{i\in {\mathcal {I}}}:T\rightarrow \mathbb {R} ^{\mathcal {I}}}
638:
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1966:
The tree on the left represents such a game, either with infinite action spaces (any
620:
377:
If a game has an information set with more than one member that game is said to have
166:
3842:
3837:
3692:
3267:
2110:
of each payoff function with respect to the follower's (firm 2) strategy variable (
280:
221:
220:) can be represented as an extensive form game with outcomes (i.e. win, lose, or
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3762:
3757:
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PBS Infinite Series. Perfect information defined at 0:25, with academic sources
1967:
225:
92:
20:
399:), play in the first game will be as follows: player 1 knows that if they play
112:, meaning there is one payoff for each player at the end of every possible play
3588:
3242:
2840:
2667:
2634:
2601:
2553:
642:
A game with incomplete and imperfect information represented in extensive form
241:
2839:. A comprehensive reference from a computational perspective; see Chapter 5.
3493:
3413:
3237:
580:
If player 2 plays Up (U’), player 1 prefers to play Down (D) (Payoff 1>0)
577:
If player 1 plays Down (D), player 2 prefers to play Up (U’) (Payoff 2>1)
574:
If player 1 plays Up (U), player 2 prefers to play Down (D’) (Payoff 1>0)
289:
by every edge of the graph are the name of the action that edge represents.
276:
the payoffs received by every player for every possible combination of moves
209:
60:
1857:{\displaystyle \rho =\{\rho _{H}:A(H)\rightarrow |H\in \mathbf {H} _{0}\}}
152:
the complete description of the game specified by the above parameters is
3928:
3428:
2931:
2821:
Multiagent
Systems: Algorithmic, Game-Theoretic, and Logical Foundations
3649:
3639:
3317:
2896:
1971:
656:
583:
If player 2 plays Down (D’), player 1 prefers to play Down (D) (3>2)
391:
set. Any game without perfect information has imperfect information.
2758:
Essentials of Game Theory: A Concise, Multidisciplinary
Introduction
2744:(1991) Game theory (Ch3 Extensive form games, pp67–106). MIT press.
666:; i.e. an equilibrium in which different types do different things.
2569:
446:
and player 1 will receive 3. In fact in the second game there is a
3418:
2561:
2539:
https://www.math.uni-hamburg/Infinite Games, Yurii
Khomskii (2010)
1970:
between 0 and 5000) or with very large action spaces (perhaps any
1957:
1668:{\displaystyle (\mathbf {H} _{i})_{i\in {\mathcal {I}}\cup \{0\}}}
1065:
be a set of decision nodes) and an immediate predecessor function
637:
368:
327:
and player 1 receives 2. Player 1 prefers 2 to 1 and so will play
279:
245:
229:
102:
458:
and player 2 holds the belief that player 1 will definitely play
133:
Each set of nodes of a rational player is further partitioned in
2819:
1962:
A game with infinite action spaces represented in extensive form
1274:{\displaystyle a:V\setminus \{v^{0}\}\rightarrow {\mathcal {A}}}
348:
example, moves may be simultaneous or a move may be hidden). An
28:
2935:
373:
A game with imperfect information represented in extensive form
300:; player 2 observes player 1's choice and then chooses between
2772:. An 88-page mathematical introduction; see Chapters 4 and 5.
2756:
423:) because player 1 prefers to receive 2 to 1 and so will play
1978:. The payoffs to the firms are represented on the left, with
1929:
1905:
1646:
1546:
1266:
1209:
902:
813:
766:
80:-player extensive-form game thus consists of the following:
2881:
Neumann, J. (1928). "Zur
Theorie der Gesellschaftsspiele".
2224:{\displaystyle q_{2}(q_{1})={\tfrac {5000-q_{1}-c_{2}}{2}}}
625:. This transformation introduces to the game the notion of
2414:{\displaystyle q_{2}^{*}={\tfrac {5000+2c_{1}-3c_{2}}{4}}}
2102:
as some constants (here marginal costs to each firm). The
1864:
is a family of probabilities of the actions of nature, and
947:{\displaystyle {\mathcal {K}}=\langle V,v^{0},T,p\rangle }
2333:{\displaystyle q_{1}^{*}={\tfrac {5000+c_{2}-2c_{1}}{2}}}
748:
Formally, a finite game in extensive form is a structure
1372:{\displaystyle \forall H\in \mathbf {H} ,\forall v\in H}
165:
of Nature's moves, can determine the edge precisely.) A
1172:
is a set of actions available for each information set
2366:
2288:
2182:
712:
whatever action they observe, but then type 1 prefers
2683:(1992). "Games in extensive and strategic forms". In
2427:
2346:
2340:. Feeding this into firm 2's best response function,
2268:
2239:
2151:
2118:
2079:
2048:
2017:
1986:
1872:
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1681:
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1149:
1127:
1105:
1071:
1039:
1013:
980:
960:
899:
754:
2761:, San Rafael, CA: Morgan & Claypool Publishers,
716:. The only equilibrium hence is with type 1 playing
685:. This cannot be an equilibrium. If both types play
267:
for every player every opportunity they have to move
260:
A complete extensive-form representation specifies:
16:
Wide-ranging representation of a game in game theory
3937:
3896:
3678:
3622:
3404:
3306:
3213:
3071:
2970:
2541:
Infinite Games (section 1.1), Yurii
Khomskii (2010)
2689:Handbook of Game Theory with Economic Applications
2469:
2413:
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2131:
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1999:
1937:
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1200:which forms a partition on the set of all actions
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1113:
1089:
1057:
1025:
999:
966:
946:
882:
198:
101:Each terminal (leaf) node of the game tree has an
2549:
2547:
2534:
2532:
1097:on which the rules of the game are represented,
68:in 1953, who extended an earlier definition of
2650:Strategy : an introduction to game theory
2617:Strategy : an introduction to game theory
2584:Strategy : an introduction to game theory
487:game, player one and player two each have two
270:what each player can do at each of their moves
119:of the non-terminal nodes of the game tree in
47:". Extensive-form representations differ from
2947:
1741:be a single player that makes a move at node
1281:is an action partition associating each node
323:, player 2 maximises their payoff by playing
8:
1851:
1782:
1660:
1654:
1578:
1554:
1258:
1245:
941:
910:
877:
842:
826:
761:
358:Every node in the set belongs to one player.
612:their opponents are. This sort of game has
193:decision". These can be made precise using
72:from 1928. Following the presentation from
2954:
2940:
2932:
2818:Shoham, Yoav; Leyton-Brown, Kevin (2009),
2755:Leyton-Brown, Kevin; Shoham, Yoav (2008),
1584:{\displaystyle {\mathcal {I}}=\{1,...,I\}}
1429:{\displaystyle a_{v}:s(v)\rightarrow A(H)}
2477:is the subgame perfect Nash equilibrium.
2458:
2453:
2440:
2435:
2426:
2398:
2382:
2365:
2356:
2351:
2345:
2317:
2301:
2287:
2278:
2273:
2267:
2262:by taking the first derivative, yielding
2244:
2238:
2208:
2195:
2181:
2169:
2156:
2150:
2123:
2117:
2084:
2078:
2053:
2047:
2022:
2016:
1991:
1985:
1928:
1927:
1923:
1922:
1904:
1903:
1896:
1886:
1871:
1845:
1840:
1828:
1789:
1774:
1746:
1702:
1682:
1680:
1675:is a player partition of information set
1645:
1644:
1637:
1627:
1622:
1616:
1611:is (a special player called) nature, and
1596:
1545:
1544:
1542:
1519:
1490:
1461:
1441:
1390:
1384:
1349:
1338:
1306:
1286:
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1208:
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1177:
1148:
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1070:
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1012:
985:
979:
959:
923:
901:
900:
898:
852:
845:
812:
811:
804:
794:
789:
774:
765:
764:
753:
43:in the form of chance events modeled as "
2106:of this game can be found by taking the
503:
179:von Neumann–Morgenstern utility function
2711:Playing for real: a text on game theory
2528:
1242:
1049:
465:To more easily solve this game for the
249:
736:. Through their actions, player 1 has
354:is a set of decision nodes such that:
2554:"Infinite Chess, PBS Infinite Series"
2470:{\displaystyle (q_{1}^{*},q_{2}^{*})}
954:is a finite tree with a set of nodes
273:what each player knows for every move
126:Each node of the Chance player has a
64:general definition was introduced by
7:
284:A game represented in extensive form
73:
1734:{\displaystyle \iota (v)=\iota (H)}
3003:First-player and second-player win
1357:
1340:
755:
693:, but then type 1 prefers to play
224:). Examples of such games include
14:
1193:{\displaystyle H\in \mathbf {H} }
3110:Coalition-proof Nash equilibrium
1841:
1683:
1623:
1350:
1186:
1141:called an information partition,
1107:
1090:{\displaystyle p:V\rightarrow D}
853:
790:
775:
732:and randomising if they observe
500:Player 2's Strategies: {U’ , D’}
256:Perfect and complete information
169:for a player thus consists of a
2914:Lectures on the theory of games
2104:subgame perfect Nash equilibria
2040:as the strategy they adopt and
438:when they have actually played
199:Shoham & Leyton-Brown (2009
33:choices at every decision point
3120:Evolutionarily stable strategy
2916:. Princeton University Press.
2713:. Oxford University Press US.
2464:
2428:
2175:
2162:
1918:
1893:
1879:
1829:
1825:
1813:
1810:
1807:
1801:
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1514:the set of successor nodes of
1501:
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1317:
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1261:
1217:{\displaystyle {\mathcal {A}}}
1159:
1153:
1081:
1058:{\displaystyle D=V\setminus T}
838:
832:
820:
801:
785:
782:
681:, type 2 would prefer to play
497:Player 1's Strategies: {U , D}
1:
3048:Simultaneous action selection
1945:is a payoff profile function.
471:, it can be converted to the
3980:List of games in game theory
3160:Quantal response equilibrium
3150:Perfect Bayesian equilibrium
3085:Bayes correlated equilibrium
2912:Harold William Kuhn (2003).
2648:Watson, Joel. (2013-05-09).
2615:Watson, Joel. (2013-05-09).
2582:Watson, Joel. (2013-05-09).
1690:{\displaystyle \mathbf {H} }
1591:is a finite set of players,
1114:{\displaystyle \mathbf {H} }
664:perfect Bayesian equilibrium
448:perfect Bayesian equilibrium
3449:Optional prisoner's dilemma
3180:Self-confirming equilibrium
2502:Self-confirming equilibrium
442:so that player 2 will play
337:subgame perfect equilibrium
55:Finite extensive-form games
4032:
3914:Principal variation search
3630:Aumann's agreement theorem
3293:Strategy-stealing argument
3205:Trembling hand equilibrium
3135:Markov perfect equilibrium
3130:Mertens-stable equilibrium
2826:Cambridge University Press
1026:{\displaystyle T\subset V}
1007:, a set of terminal nodes
1000:{\displaystyle v^{0}\in V}
427:and so player 2 will play
3950:Combinatorial game theory
3609:Princess and monster game
3165:Quasi-perfect equilibrium
3090:Bayesian Nash equilibrium
2691:. Vol. 1. Elsevier.
2497:Combinatorial game theory
214:combinatorial game theory
201:, chpt. 13) for details.
4016:Game theory game classes
3965:Evolutionary game theory
3698:Antoine Augustin Cournot
3584:Guess 2/3 of the average
3381:Strictly determined game
3175:Satisfaction equilibrium
2993:Escalation of commitment
2841:Downloadable free online
2108:first partial derivative
974:, a unique initial node
740:their type to player 2.
677:. However, if they play
130:over its outgoing edges.
128:probability distribution
27:is a specification of a
3970:Glossary of game theory
3569:Stackelberg competition
3195:Strong Nash equilibrium
2854:Horst Herrlich (2006).
2687:; Hart, Sergiu (eds.).
1976:Stackelberg competition
594:Stackelberg competition
331:and player 2 will play
218:artificial intelligence
208:two-player game over a
3995:Tragedy of the commons
3975:List of game theorists
3955:Confrontation analysis
3665:Sprague–Grundy theorem
3185:Sequential equilibrium
3105:Correlated equilibrium
2471:
2415:
2334:
2254:
2225:
2133:
2094:
2063:
2032:
2001:
1963:
1939:
1858:
1761:
1760:{\displaystyle v\in H}
1735:
1691:
1669:
1605:
1585:
1528:
1508:
1479:
1450:
1430:
1373:
1324:
1295:
1275:
1218:
1194:
1166:
1135:
1115:
1091:
1059:
1027:
1001:
968:
948:
884:
659:status). There is one
643:
615:incomplete information
604:Incomplete information
374:
285:
185:random outcome by its
41:incomplete information
3768:Jean-François Mertens
2884:Mathematische Annalen
2783:at many universities.
2472:
2416:
2335:
2255:
2253:{\displaystyle q_{1}}
2226:
2134:
2132:{\displaystyle q_{2}}
2095:
2093:{\displaystyle c_{2}}
2064:
2062:{\displaystyle c_{1}}
2033:
2031:{\displaystyle q_{2}}
2002:
2000:{\displaystyle q_{1}}
1961:
1950:Infinite action space
1940:
1859:
1762:
1736:
1692:
1670:
1606:
1586:
1529:
1509:
1485:is a bijection, with
1480:
1451:
1431:
1374:
1325:
1296:
1276:
1219:
1195:
1167:
1136:
1116:
1092:
1060:
1028:
1002:
969:
949:
885:
724:and player 2 playing
708:, player 2 will play
641:
450:where player 1 plays
411:, player 2 will play
403:, player 2 will play
380:imperfect information
372:
343:Imperfect information
315:, player 2 will play
283:
264:the players of a game
195:epistemic modal logic
3897:Search optimizations
3773:Jennifer Tour Chayes
3660:Revelation principle
3655:Purification theorem
3594:Nash bargaining game
3559:Bertrand competition
3544:El Farol Bar problem
3509:Electronic mail game
3474:Lewis signaling game
3018:Hierarchy of beliefs
2487:Axiom of determinacy
2425:
2344:
2266:
2237:
2149:
2116:
2077:
2046:
2015:
1984:
1870:
1773:
1745:
1701:
1679:
1615:
1595:
1541:
1518:
1507:{\displaystyle s(v)}
1489:
1478:{\displaystyle s(v)}
1460:
1440:
1383:
1337:
1323:{\displaystyle a(v)}
1305:
1285:
1230:
1204:
1176:
1165:{\displaystyle A(H)}
1147:
1125:
1103:
1069:
1037:
1011:
978:
958:
897:
752:
649:by the other players
3945:Bounded rationality
3564:Cournot competition
3514:Rock paper scissors
3489:Battle of the sexes
3479:Volunteer's dilemma
3351:Perfect information
3278:Dominant strategies
3115:Epsilon-equilibrium
2998:Extensive-form game
2586:. pp. 97–100.
2492:Perfect information
2463:
2445:
2361:
2283:
1301:to a single action
598:Cournot competition
505:
454:and player 2 plays
385:perfect information
206:perfect information
142:equivalence classes
25:extensive-form game
3924:Paranoid algorithm
3904:Alpha–beta pruning
3783:John Maynard Smith
3614:Rendezvous problem
3454:Traveler's dilemma
3444:Gift-exchange game
3439:Prisoner's dilemma
3356:Large Poisson game
3323:Bargaining problem
3228:Backward induction
3200:Subgame perfection
3155:Proper equilibrium
2897:10.1007/BF01448847
2779:2000-08-15 at the
2652:. pp. 22–26.
2619:. pp. 26–28.
2467:
2449:
2431:
2411:
2409:
2347:
2330:
2328:
2269:
2250:
2221:
2219:
2141:) and finding its
2129:
2090:
2059:
2028:
1997:
1964:
1935:
1854:
1757:
1731:
1687:
1665:
1601:
1581:
1524:
1504:
1475:
1446:
1426:
1379:, the restriction
1369:
1320:
1291:
1271:
1214:
1190:
1162:
1131:
1121:is a partition of
1111:
1087:
1055:
1023:
997:
964:
944:
880:
644:
504:
477:. Given this is a
375:
311:If player 1 plays
286:
238:expectminimax tree
88:(rational) players
4003:
4002:
3909:Aspiration window
3878:Suzanne Scotchmer
3833:Oskar Morgenstern
3728:Donald B. Gillies
3670:Zermelo's theorem
3599:Induction puzzles
3554:Fair cake-cutting
3529:Public goods game
3459:Coordination game
3333:Intransitive game
3263:Forward induction
3145:Pareto efficiency
3125:Gibbs equilibrium
3095:Berge equilibrium
3043:Simultaneous game
2923:978-0-691-02772-2
2875:Historical papers
2865:978-3-540-30989-5
2835:978-0-521-89943-7
2768:978-1-59829-593-1
2720:978-0-19-530057-4
2698:978-0-444-88098-7
2659:978-0-393-91838-0
2626:978-0-393-91838-0
2593:978-0-393-91838-0
2408:
2327:
2218:
1604:{\displaystyle 0}
1527:{\displaystyle v}
1449:{\displaystyle a}
1294:{\displaystyle v}
1134:{\displaystyle D}
967:{\displaystyle V}
744:Formal definition
720:, type 2 playing
704:and type 2 plays
568:
567:
236:. A game over an
156:among the players
4023:
3990:Topological game
3985:No-win situation
3883:Thomas Schelling
3863:Robert B. Wilson
3823:Merrill M. Flood
3793:John von Neumann
3703:Ariel Rubinstein
3688:Albert W. Tucker
3539:War of attrition
3499:Matching pennies
3140:Nash equilibrium
3063:Mechanism design
3028:Normal-form game
2983:Cooperative game
2956:
2949:
2942:
2933:
2927:
2908:
2869:
2838:
2771:
2740:Fudenberg D and
2724:
2707:Binmore, Kenneth
2702:
2672:
2671:
2645:
2639:
2638:
2612:
2606:
2605:
2579:
2573:
2551:
2542:
2536:
2517:Solution concept
2476:
2474:
2473:
2468:
2462:
2457:
2444:
2439:
2420:
2418:
2417:
2412:
2410:
2404:
2403:
2402:
2387:
2386:
2367:
2360:
2355:
2339:
2337:
2336:
2331:
2329:
2323:
2322:
2321:
2306:
2305:
2289:
2282:
2277:
2261:
2259:
2257:
2256:
2251:
2249:
2248:
2230:
2228:
2227:
2222:
2220:
2214:
2213:
2212:
2200:
2199:
2183:
2174:
2173:
2161:
2160:
2140:
2138:
2136:
2135:
2130:
2128:
2127:
2101:
2099:
2097:
2096:
2091:
2089:
2088:
2070:
2068:
2066:
2065:
2060:
2058:
2057:
2039:
2037:
2035:
2034:
2029:
2027:
2026:
2008:
2006:
2004:
2003:
1998:
1996:
1995:
1944:
1942:
1941:
1936:
1934:
1933:
1932:
1926:
1911:
1910:
1909:
1908:
1891:
1890:
1863:
1861:
1860:
1855:
1850:
1849:
1844:
1832:
1794:
1793:
1766:
1764:
1763:
1758:
1740:
1738:
1737:
1732:
1696:
1694:
1693:
1688:
1686:
1674:
1672:
1671:
1666:
1664:
1663:
1650:
1649:
1632:
1631:
1626:
1610:
1608:
1607:
1602:
1590:
1588:
1587:
1582:
1550:
1549:
1533:
1531:
1530:
1525:
1513:
1511:
1510:
1505:
1484:
1482:
1481:
1476:
1455:
1453:
1452:
1447:
1435:
1433:
1432:
1427:
1395:
1394:
1378:
1376:
1375:
1370:
1353:
1329:
1327:
1326:
1321:
1300:
1298:
1297:
1292:
1280:
1278:
1277:
1272:
1270:
1269:
1257:
1256:
1223:
1221:
1220:
1215:
1213:
1212:
1199:
1197:
1196:
1191:
1189:
1171:
1169:
1168:
1163:
1140:
1138:
1137:
1132:
1120:
1118:
1117:
1112:
1110:
1096:
1094:
1093:
1088:
1064:
1062:
1061:
1056:
1032:
1030:
1029:
1024:
1006:
1004:
1003:
998:
990:
989:
973:
971:
970:
965:
953:
951:
950:
945:
928:
927:
906:
905:
889:
887:
886:
881:
858:
857:
856:
819:
818:
817:
816:
799:
798:
793:
778:
770:
769:
728:if they observe
700:If type 1 plays
506:
468:Nash equilibrium
154:common knowledge
135:information sets
84:A finite set of
35:, the (possibly
4031:
4030:
4026:
4025:
4024:
4022:
4021:
4020:
4006:
4005:
4004:
3999:
3933:
3919:max^n algorithm
3892:
3888:William Vickrey
3848:Reinhard Selten
3803:Kenneth Binmore
3718:David K. Levine
3713:Daniel Kahneman
3680:
3674:
3650:Negamax theorem
3640:Minimax theorem
3618:
3579:Diner's dilemma
3434:All-pay auction
3400:
3386:Stochastic game
3338:Mean-field game
3309:
3302:
3273:Markov strategy
3209:
3075:
3067:
3038:Sequential game
3023:Information set
3008:Game complexity
2978:Congestion game
2966:
2960:
2924:
2911:
2880:
2866:
2856:Axiom of choice
2853:
2850:
2848:Further reading
2836:
2817:
2803:Osborne MJ and
2781:Wayback Machine
2769:
2754:
2721:
2705:
2699:
2679:
2676:
2675:
2660:
2647:
2646:
2642:
2627:
2614:
2613:
2609:
2594:
2581:
2580:
2576:
2552:
2545:
2537:
2530:
2525:
2507:Sequential game
2483:
2423:
2422:
2394:
2378:
2368:
2342:
2341:
2313:
2297:
2290:
2264:
2263:
2240:
2235:
2234:
2232:
2204:
2191:
2184:
2165:
2152:
2147:
2146:
2119:
2114:
2113:
2111:
2080:
2075:
2074:
2072:
2049:
2044:
2043:
2041:
2018:
2013:
2012:
2010:
1987:
1982:
1981:
1979:
1952:
1921:
1892:
1882:
1868:
1867:
1839:
1785:
1771:
1770:
1743:
1742:
1699:
1698:
1677:
1676:
1633:
1621:
1613:
1612:
1593:
1592:
1539:
1538:
1516:
1515:
1487:
1486:
1458:
1457:
1438:
1437:
1386:
1381:
1380:
1335:
1334:
1303:
1302:
1283:
1282:
1248:
1228:
1227:
1202:
1201:
1174:
1173:
1145:
1144:
1123:
1122:
1101:
1100:
1067:
1066:
1035:
1034:
1009:
1008:
981:
976:
975:
956:
955:
919:
895:
894:
841:
800:
788:
750:
749:
746:
628:nature's choice
606:
514:
511:
351:information set
345:
258:
240:, like that of
212:(as defined in
57:
45:moves by nature
17:
12:
11:
5:
4029:
4027:
4019:
4018:
4008:
4007:
4001:
4000:
3998:
3997:
3992:
3987:
3982:
3977:
3972:
3967:
3962:
3957:
3952:
3947:
3941:
3939:
3935:
3934:
3932:
3931:
3926:
3921:
3916:
3911:
3906:
3900:
3898:
3894:
3893:
3891:
3890:
3885:
3880:
3875:
3870:
3865:
3860:
3855:
3853:Robert Axelrod
3850:
3845:
3840:
3835:
3830:
3828:Olga Bondareva
3825:
3820:
3818:Melvin Dresher
3815:
3810:
3808:Leonid Hurwicz
3805:
3800:
3795:
3790:
3785:
3780:
3775:
3770:
3765:
3760:
3755:
3750:
3745:
3743:Harold W. Kuhn
3740:
3735:
3733:Drew Fudenberg
3730:
3725:
3723:David M. Kreps
3720:
3715:
3710:
3708:Claude Shannon
3705:
3700:
3695:
3690:
3684:
3682:
3676:
3675:
3673:
3672:
3667:
3662:
3657:
3652:
3647:
3645:Nash's theorem
3642:
3637:
3632:
3626:
3624:
3620:
3619:
3617:
3616:
3611:
3606:
3601:
3596:
3591:
3586:
3581:
3576:
3571:
3566:
3561:
3556:
3551:
3546:
3541:
3536:
3531:
3526:
3521:
3516:
3511:
3506:
3504:Ultimatum game
3501:
3496:
3491:
3486:
3484:Dollar auction
3481:
3476:
3471:
3469:Centipede game
3466:
3461:
3456:
3451:
3446:
3441:
3436:
3431:
3426:
3424:Infinite chess
3421:
3416:
3410:
3408:
3402:
3401:
3399:
3398:
3393:
3391:Symmetric game
3388:
3383:
3378:
3376:Signaling game
3373:
3371:Screening game
3368:
3363:
3361:Potential game
3358:
3353:
3348:
3340:
3335:
3330:
3325:
3320:
3314:
3312:
3304:
3303:
3301:
3300:
3295:
3290:
3288:Mixed strategy
3285:
3280:
3275:
3270:
3265:
3260:
3255:
3250:
3245:
3240:
3235:
3230:
3225:
3219:
3217:
3211:
3210:
3208:
3207:
3202:
3197:
3192:
3187:
3182:
3177:
3172:
3170:Risk dominance
3167:
3162:
3157:
3152:
3147:
3142:
3137:
3132:
3127:
3122:
3117:
3112:
3107:
3102:
3097:
3092:
3087:
3081:
3079:
3069:
3068:
3066:
3065:
3060:
3055:
3050:
3045:
3040:
3035:
3030:
3025:
3020:
3015:
3013:Graphical game
3010:
3005:
3000:
2995:
2990:
2985:
2980:
2974:
2972:
2968:
2967:
2961:
2959:
2958:
2951:
2944:
2936:
2930:
2929:
2922:
2909:
2872:
2871:
2864:
2849:
2846:
2845:
2844:
2834:
2815:
2801:
2784:
2767:
2752:
2738:
2725:
2719:
2703:
2697:
2685:Aumann, Robert
2674:
2673:
2658:
2640:
2625:
2607:
2592:
2574:
2543:
2527:
2526:
2524:
2521:
2520:
2519:
2514:
2509:
2504:
2499:
2494:
2489:
2482:
2479:
2466:
2461:
2456:
2452:
2448:
2443:
2438:
2434:
2430:
2407:
2401:
2397:
2393:
2390:
2385:
2381:
2377:
2374:
2371:
2364:
2359:
2354:
2350:
2326:
2320:
2316:
2312:
2309:
2304:
2300:
2296:
2293:
2286:
2281:
2276:
2272:
2247:
2243:
2217:
2211:
2207:
2203:
2198:
2194:
2190:
2187:
2180:
2177:
2172:
2168:
2164:
2159:
2155:
2126:
2122:
2087:
2083:
2056:
2052:
2025:
2021:
1994:
1990:
1951:
1948:
1947:
1946:
1931:
1925:
1920:
1917:
1914:
1907:
1902:
1899:
1895:
1889:
1885:
1881:
1878:
1875:
1865:
1853:
1848:
1843:
1838:
1835:
1831:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1792:
1788:
1784:
1781:
1778:
1768:
1756:
1753:
1750:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1685:
1662:
1659:
1656:
1653:
1648:
1643:
1640:
1636:
1630:
1625:
1620:
1600:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1548:
1523:
1503:
1500:
1497:
1494:
1474:
1471:
1468:
1465:
1445:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1393:
1389:
1368:
1365:
1362:
1359:
1356:
1352:
1348:
1345:
1342:
1332:
1331:
1319:
1316:
1313:
1310:
1290:
1268:
1263:
1260:
1255:
1251:
1247:
1244:
1241:
1238:
1235:
1225:
1211:
1188:
1184:
1181:
1161:
1158:
1155:
1152:
1142:
1130:
1109:
1098:
1086:
1083:
1080:
1077:
1074:
1054:
1051:
1048:
1045:
1042:
1022:
1019:
1016:
996:
993:
988:
984:
963:
943:
940:
937:
934:
931:
926:
922:
918:
915:
912:
909:
904:
879:
876:
873:
870:
867:
864:
861:
855:
851:
848:
844:
840:
837:
834:
831:
828:
825:
822:
815:
810:
807:
803:
797:
792:
787:
784:
781:
777:
773:
768:
763:
760:
757:
745:
742:
623:transformation
605:
602:
585:
584:
581:
578:
575:
566:
565:
556:
543:
539:
538:
529:
526:
522:
521:
518:
515:
512:
509:
502:
501:
498:
383:. A game with
364:
363:
359:
344:
341:
335:. This is the
278:
277:
274:
271:
268:
265:
257:
254:
234:infinite chess
158:
157:
150:
149:
148:
145:
131:
124:
113:
99:
89:
66:Harold W. Kuhn
56:
53:
15:
13:
10:
9:
6:
4:
3:
2:
4028:
4017:
4014:
4013:
4011:
3996:
3993:
3991:
3988:
3986:
3983:
3981:
3978:
3976:
3973:
3971:
3968:
3966:
3963:
3961:
3958:
3956:
3953:
3951:
3948:
3946:
3943:
3942:
3940:
3938:Miscellaneous
3936:
3930:
3927:
3925:
3922:
3920:
3917:
3915:
3912:
3910:
3907:
3905:
3902:
3901:
3899:
3895:
3889:
3886:
3884:
3881:
3879:
3876:
3874:
3873:Samuel Bowles
3871:
3869:
3868:Roger Myerson
3866:
3864:
3861:
3859:
3858:Robert Aumann
3856:
3854:
3851:
3849:
3846:
3844:
3841:
3839:
3836:
3834:
3831:
3829:
3826:
3824:
3821:
3819:
3816:
3814:
3813:Lloyd Shapley
3811:
3809:
3806:
3804:
3801:
3799:
3798:Kenneth Arrow
3796:
3794:
3791:
3789:
3786:
3784:
3781:
3779:
3778:John Harsanyi
3776:
3774:
3771:
3769:
3766:
3764:
3761:
3759:
3756:
3754:
3751:
3749:
3748:Herbert Simon
3746:
3744:
3741:
3739:
3736:
3734:
3731:
3729:
3726:
3724:
3721:
3719:
3716:
3714:
3711:
3709:
3706:
3704:
3701:
3699:
3696:
3694:
3691:
3689:
3686:
3685:
3683:
3677:
3671:
3668:
3666:
3663:
3661:
3658:
3656:
3653:
3651:
3648:
3646:
3643:
3641:
3638:
3636:
3633:
3631:
3628:
3627:
3625:
3621:
3615:
3612:
3610:
3607:
3605:
3602:
3600:
3597:
3595:
3592:
3590:
3587:
3585:
3582:
3580:
3577:
3575:
3572:
3570:
3567:
3565:
3562:
3560:
3557:
3555:
3552:
3550:
3549:Fair division
3547:
3545:
3542:
3540:
3537:
3535:
3532:
3530:
3527:
3525:
3524:Dictator game
3522:
3520:
3517:
3515:
3512:
3510:
3507:
3505:
3502:
3500:
3497:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3460:
3457:
3455:
3452:
3450:
3447:
3445:
3442:
3440:
3437:
3435:
3432:
3430:
3427:
3425:
3422:
3420:
3417:
3415:
3412:
3411:
3409:
3407:
3403:
3397:
3396:Zero-sum game
3394:
3392:
3389:
3387:
3384:
3382:
3379:
3377:
3374:
3372:
3369:
3367:
3366:Repeated game
3364:
3362:
3359:
3357:
3354:
3352:
3349:
3347:
3345:
3341:
3339:
3336:
3334:
3331:
3329:
3326:
3324:
3321:
3319:
3316:
3315:
3313:
3311:
3305:
3299:
3296:
3294:
3291:
3289:
3286:
3284:
3283:Pure strategy
3281:
3279:
3276:
3274:
3271:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3249:
3248:De-escalation
3246:
3244:
3241:
3239:
3236:
3234:
3231:
3229:
3226:
3224:
3221:
3220:
3218:
3216:
3212:
3206:
3203:
3201:
3198:
3196:
3193:
3191:
3190:Shapley value
3188:
3186:
3183:
3181:
3178:
3176:
3173:
3171:
3168:
3166:
3163:
3161:
3158:
3156:
3153:
3151:
3148:
3146:
3143:
3141:
3138:
3136:
3133:
3131:
3128:
3126:
3123:
3121:
3118:
3116:
3113:
3111:
3108:
3106:
3103:
3101:
3098:
3096:
3093:
3091:
3088:
3086:
3083:
3082:
3080:
3078:
3074:
3070:
3064:
3061:
3059:
3058:Succinct game
3056:
3054:
3051:
3049:
3046:
3044:
3041:
3039:
3036:
3034:
3031:
3029:
3026:
3024:
3021:
3019:
3016:
3014:
3011:
3009:
3006:
3004:
3001:
2999:
2996:
2994:
2991:
2989:
2986:
2984:
2981:
2979:
2976:
2975:
2973:
2969:
2965:
2957:
2952:
2950:
2945:
2943:
2938:
2937:
2934:
2925:
2919:
2915:
2910:
2906:
2902:
2898:
2894:
2890:
2886:
2885:
2879:
2878:
2877:
2876:
2867:
2861:
2857:
2852:
2851:
2847:
2842:
2837:
2831:
2827:
2823:
2822:
2816:
2814:
2813:0-262-65040-1
2810:
2806:
2805:Rubinstein A.
2802:
2800:
2799:0-486-65943-7
2796:
2792:
2788:
2785:
2782:
2778:
2775:
2770:
2764:
2760:
2759:
2753:
2751:
2750:0-262-06141-4
2747:
2743:
2739:
2737:
2736:0-486-64216-X
2733:
2729:
2726:
2722:
2716:
2712:
2708:
2704:
2700:
2694:
2690:
2686:
2682:
2678:
2677:
2669:
2665:
2661:
2655:
2651:
2644:
2641:
2636:
2632:
2628:
2622:
2618:
2611:
2608:
2603:
2599:
2595:
2589:
2585:
2578:
2575:
2571:
2567:
2563:
2559:
2555:
2550:
2548:
2544:
2540:
2535:
2533:
2529:
2522:
2518:
2515:
2513:
2510:
2508:
2505:
2503:
2500:
2498:
2495:
2493:
2490:
2488:
2485:
2484:
2480:
2478:
2459:
2454:
2450:
2446:
2441:
2436:
2432:
2405:
2399:
2395:
2391:
2388:
2383:
2379:
2375:
2372:
2369:
2362:
2357:
2352:
2348:
2324:
2318:
2314:
2310:
2307:
2302:
2298:
2294:
2291:
2284:
2279:
2274:
2270:
2245:
2241:
2215:
2209:
2205:
2201:
2196:
2192:
2188:
2185:
2178:
2170:
2166:
2157:
2153:
2144:
2143:best response
2124:
2120:
2109:
2105:
2085:
2081:
2054:
2050:
2023:
2019:
1992:
1988:
1977:
1973:
1969:
1960:
1956:
1949:
1915:
1912:
1900:
1897:
1887:
1883:
1876:
1873:
1866:
1846:
1836:
1833:
1822:
1819:
1816:
1804:
1798:
1795:
1790:
1786:
1779:
1776:
1769:
1754:
1751:
1748:
1725:
1719:
1716:
1710:
1704:
1657:
1651:
1641:
1638:
1628:
1598:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1551:
1537:
1536:
1535:
1521:
1498:
1492:
1469:
1463:
1443:
1420:
1414:
1405:
1399:
1396:
1391:
1387:
1366:
1363:
1360:
1354:
1346:
1343:
1330:, fulfilling:
1314:
1308:
1288:
1253:
1249:
1239:
1236:
1233:
1226:
1182:
1179:
1156:
1150:
1143:
1128:
1099:
1084:
1078:
1075:
1072:
1052:
1046:
1043:
1040:
1020:
1017:
1014:
994:
991:
986:
982:
961:
938:
935:
932:
929:
924:
920:
916:
913:
907:
893:
892:
891:
874:
871:
868:
865:
862:
859:
849:
846:
835:
829:
823:
808:
805:
795:
779:
771:
758:
743:
741:
739:
735:
731:
727:
723:
719:
715:
711:
707:
703:
698:
696:
692:
688:
684:
680:
676:
672:
667:
665:
662:
658:
652:
650:
640:
636:
634:
630:
629:
624:
622:
617:
616:
611:
603:
601:
599:
595:
589:
582:
579:
576:
573:
572:
571:
563:
562:
557:
555:
553:
549:
544:
541:
540:
536:
535:
530:
527:
524:
523:
519:
516:
508:
507:
499:
496:
495:
494:
492:
491:
486:
485:
481:
476:
475:
470:
469:
463:
461:
457:
453:
449:
445:
441:
437:
432:
430:
426:
422:
418:
414:
410:
406:
402:
398:
392:
390:
386:
382:
381:
371:
367:
360:
357:
356:
355:
353:
352:
342:
340:
338:
334:
330:
326:
322:
318:
314:
309:
307:
303:
299:
295:
290:
282:
275:
272:
269:
266:
263:
262:
261:
255:
253:
251:
247:
243:
239:
235:
231:
227:
223:
219:
215:
211:
207:
202:
200:
196:
190:
188:
184:
180:
176:
172:
168:
167:pure strategy
164:
155:
151:
146:
143:
139:
138:
136:
132:
129:
125:
122:
118:
114:
111:
107:
105:
100:
98:
95:, called the
94:
90:
87:
83:
82:
81:
79:
75:
71:
67:
62:
54:
52:
50:
46:
42:
38:
34:
30:
26:
22:
3843:Peyton Young
3838:Paul Milgrom
3753:Hervé Moulin
3693:Amos Tversky
3635:Folk theorem
3346:-player game
3343:
3268:Grim trigger
2997:
2913:
2888:
2882:
2874:
2873:
2858:. Springer.
2855:
2824:, New York:
2820:
2757:
2710:
2688:
2681:Hart, Sergiu
2649:
2643:
2616:
2610:
2583:
2577:
1965:
1953:
1333:
747:
733:
729:
725:
721:
717:
713:
709:
705:
701:
699:
694:
690:
686:
682:
678:
674:
670:
668:
660:
653:
648:
645:
633:God's choice
632:
626:
619:
613:
609:
607:
590:
586:
569:
560:
559:
551:
547:
545:
533:
532:
488:
480:simultaneous
478:
472:
466:
464:
459:
455:
451:
443:
439:
435:
433:
428:
424:
420:
416:
412:
408:
404:
400:
397:ad infinitum
396:
393:
384:
378:
376:
365:
349:
346:
332:
328:
324:
320:
316:
312:
310:
305:
301:
297:
293:
291:
287:
259:
250:Binmore 2007
203:
191:
159:
120:
109:
103:
96:
85:
77:
58:
24:
18:
3960:Coopetition
3763:Jean Tirole
3758:John Conway
3738:Eric Maskin
3534:Blotto game
3519:Pirate game
3328:Global game
3298:Tit for tat
3233:Bid shading
3223:Appeasement
3073:Equilibrium
3053:Solved game
2988:Determinacy
2971:Definitions
2964:game theory
2891:: 295–320.
2774:Free online
1968:real number
520:Down' (D')
474:normal form
252:, chpt. 2)
226:tic-tac-toe
163:realization
93:rooted tree
74:Hart (1992)
70:von Neumann
49:normal-form
21:game theory
3604:Trust game
3589:Kuhn poker
3258:Escalation
3253:Deterrence
3243:Cheap talk
3215:Strategies
3033:Preference
2962:Topics of
2728:Dresher M.
2668:1123193808
2635:1123193808
2602:1123193808
2570:1510.08155
2523:References
2512:Signalling
2145:function,
661:separating
490:strategies
484:sequential
242:backgammon
175:singletons
3788:John Nash
3494:Stag hunt
3238:Collusion
2905:122961988
2791:Raiffa H.
2787:Luce R.D.
2742:Tirole J.
2562:1302.4377
2460:∗
2442:∗
2389:−
2358:∗
2308:−
2280:∗
2202:−
2189:−
1919:→
1901:∈
1837:∈
1811:→
1787:ρ
1777:ρ
1752:∈
1720:ι
1705:ι
1652:∪
1642:∈
1412:→
1364:∈
1358:∀
1347:∈
1341:∀
1262:→
1243:∖
1183:∈
1082:→
1050:∖
1018:⊂
992:∈
942:⟩
911:⟨
878:⟩
869:ρ
850:∈
809:∈
762:⟨
756:Γ
738:signalled
542:Down (D)
517:Up' (U')
513:Player 1
389:singleton
210:game tree
189:utility.
171:selection
117:partition
97:game tree
61:game tree
37:imperfect
4010:Category
3929:Lazy SMP
3623:Theorems
3574:Deadlock
3429:Checkers
3310:of games
3077:concepts
2777:Archived
2709:(2007).
2481:See also
621:Harsanyi
510:Player 2
362:reached.
187:expected
183:a priori
3681:figures
3464:Chicken
3318:Auction
3308:Classes
2260:
2233:
2139:
2112:
2100:
2073:
2069:
2042:
2038:
2011:
2007:
1980:
1972:integer
890:where:
657:subgame
525:Up (U)
110:payoffs
2920:
2903:
2862:
2832:
2811:
2797:
2765:
2748:
2734:
2717:
2695:
2666:
2656:
2633:
2623:
2600:
2590:
1697:. Let
528:(0,0)
232:, and
197:; see
106:-tuple
3419:Chess
3406:Games
2901:S2CID
2566:arXiv
2558:arXiv
1033:(let
246:poker
230:chess
76:, an
23:, an
3100:Core
2918:ISBN
2860:ISBN
2830:ISBN
2809:ISBN
2795:ISBN
2789:and
2763:ISBN
2746:ISBN
2732:ISBN
2715:ISBN
2693:ISBN
2664:OCLC
2654:ISBN
2631:OCLC
2621:ISBN
2598:OCLC
2588:ISBN
2564:and
2421:and
2370:5000
2292:5000
2186:5000
2071:and
2009:and
610:type
564:,1)
304:and
296:and
222:draw
216:and
29:game
3679:Key
2893:doi
2889:100
1456:on
1436:of
726:U'
710:D'
691:D'
679:D'
675:D'
631:or
531:(2,
456:U'
444:D'
429:D'
421:D'
413:U'
405:D'
333:D'
325:D'
317:U'
306:D'
302:U'
108:of
19:In
4012::
3414:Go
2899:.
2887:.
2828:,
2662:.
2629:.
2596:.
2546:^
2531:^
1534:.
697:.
600:.
537:)
493:.
431:.
419:,
339:.
228:,
204:A
115:A
91:A
3344:n
2955:e
2948:t
2941:v
2926:.
2907:.
2895::
2868:.
2843:.
2723:.
2701:.
2670:.
2637:.
2604:.
2572:.
2568::
2560::
2465:)
2455:2
2451:q
2447:,
2437:1
2433:q
2429:(
2406:4
2400:2
2396:c
2392:3
2384:1
2380:c
2376:2
2373:+
2363:=
2353:2
2349:q
2325:2
2319:1
2315:c
2311:2
2303:2
2299:c
2295:+
2285:=
2275:1
2271:q
2246:1
2242:q
2216:2
2210:2
2206:c
2197:1
2193:q
2179:=
2176:)
2171:1
2167:q
2163:(
2158:2
2154:q
2125:2
2121:q
2086:2
2082:c
2055:1
2051:c
2024:2
2020:q
1993:1
1989:q
1930:I
1924:R
1916:T
1913::
1906:I
1898:i
1894:)
1888:i
1884:u
1880:(
1877:=
1874:u
1852:}
1847:0
1842:H
1834:H
1830:|
1826:]
1823:1
1820:,
1817:0
1814:[
1808:)
1805:H
1802:(
1799:A
1796::
1791:H
1783:{
1780:=
1767:.
1755:H
1749:v
1729:)
1726:H
1723:(
1717:=
1714:)
1711:v
1708:(
1684:H
1661:}
1658:0
1655:{
1647:I
1639:i
1635:)
1629:i
1624:H
1619:(
1599:0
1579:}
1576:I
1573:,
1570:.
1567:.
1564:.
1561:,
1558:1
1555:{
1552:=
1547:I
1522:v
1502:)
1499:v
1496:(
1493:s
1473:)
1470:v
1467:(
1464:s
1444:a
1424:)
1421:H
1418:(
1415:A
1409:)
1406:v
1403:(
1400:s
1397::
1392:v
1388:a
1367:H
1361:v
1355:,
1351:H
1344:H
1318:)
1315:v
1312:(
1309:a
1289:v
1267:A
1259:}
1254:0
1250:v
1246:{
1240:V
1237::
1234:a
1224:.
1210:A
1187:H
1180:H
1160:)
1157:H
1154:(
1151:A
1129:D
1108:H
1085:D
1079:V
1076::
1073:p
1053:T
1047:V
1044:=
1041:D
1021:V
1015:T
995:V
987:0
983:v
962:V
939:p
936:,
933:T
930:,
925:0
921:v
917:,
914:V
908:=
903:K
875:u
872:,
866:,
863:a
860:,
854:H
847:H
843:}
839:)
836:H
833:(
830:A
827:{
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814:I
806:i
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796:i
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783:[
780:,
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759:=
734:U
730:D
722:U
718:D
714:D
706:D
702:U
695:D
687:U
683:U
671:D
561:3
558:(
554:)
552:2
550:,
548:1
546:(
534:1
482:/
460:D
452:D
440:D
436:U
425:U
417:U
409:D
401:U
329:U
321:U
313:D
298:D
294:U
121:n
104:n
86:n
78:n
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