2951:(terminal nodes and nodes at the maximum search depth). Non-leaf nodes inherit their value from a descendant leaf node. The heuristic value is a score measuring the favorability of the node for the maximizing player. Hence nodes resulting in a favorable outcome, such as a win, for the maximizing player have higher scores than nodes more favorable for the minimizing player. The heuristic value for terminal (game ending) leaf nodes are scores corresponding to win, loss, or draw, for the maximizing player. For non terminal leaf nodes at the maximum search depth, an evaluation function estimates a heuristic value for the node. The quality of this estimate and the search depth determine the quality and accuracy of the final minimax result.
3041:
3049:
3149:
Minimax theory has been extended to decisions where there is no other player, but where the consequences of decisions depend on unknown facts. For example, deciding to prospect for minerals entails a cost, which will be wasted if the minerals are not present, but will bring major rewards if they are.
2688:
win in one move, while another move will lead to the situation where player A can, at best, draw, then player B's best move is the one leading to a draw. Late in the game, it's easy to see what the "best" move is. The minimax algorithm helps find the best move, by working backwards from the
3650:
strategy where voters, when faced with two or more candidates, choose the one they perceive as the least harmful or the "lesser evil." To do so, "voting should not be viewed as a form of personal self-expression or moral judgement directed in retaliation towards major party candidates who fail to
3618:
measurements (that outcomes include "how much better or worse"), and returns ordinal data, using only the modeled outcomes: the conclusion of a minimax analysis is: "this strategy is minimax, as the worst case is (outcome), which is less bad than any other strategy". Compare to expected value
1363:
540:
2785:. The above algorithm will assign a value of positive or negative infinity to any position since the value of every position will be the value of some final winning or losing position. Often this is generally only possible at the very end of complicated games such as
2800:
evaluation function which gives values to non-final game states without considering all possible following complete sequences. We can then limit the minimax algorithm to look only at a certain number of moves ahead. This number is called the "look-ahead", measured in
3456:
2793:, since it is not computationally feasible to look ahead as far as the completion of the game, except towards the end, and instead, positions are given finite values as estimates of the degree of belief that they will lead to a win for one player or another.
112:
is the highest value that the player can be sure to get without knowing the actions of the other players; equivalently, it is the lowest value the other players can force the player to receive when they know the player's action. Its formal definition is:
375:
Calculating the maximin value of a player is done in a worst-case approach: for each possible action of the player, we check all possible actions of the other players and determine the worst possible combination of actions – the one that gives player
2633:"Maximin" is a term commonly used for non-zero-sum games to describe the strategy which maximizes one's own minimum payoff. In non-zero-sum games, this is not generally the same as minimizing the opponent's maximum gain, nor the same as the
3103:
values, and assign it to that same node (e.g. the node on the left will choose the minimum between "10" and "+∞", therefore assigning the value "10" to itself). The next step, in level 2, consists of choosing for each node the
3573:
232:
909:
2721:
and it indicates how good it would be for a player to reach that position. The player then makes the move that maximizes the minimum value of the position resulting from the opponent's possible following moves. If it is
1152:
95:, covering both the cases where players take alternate moves and those where they make simultaneous moves, it has also been extended to more complex games and to general decision-making in the presence of uncertainty.
413:
1948:
1875:
3674:. Rawls defined this principle as the rule which states that social and economic inequalities should be arranged so that "they are to be of the greatest benefit to the least-advantaged members of society".
785:
of a player is the smallest value that the other players can force the player to receive, without knowing the player's actions; equivalently, it is the largest value the player can be sure to get when they
1126:
2208:
arises because each player minimizes the maximum payoff possible for the other – since the game is zero-sum, they also minimize their own maximum loss (i.e., maximize their minimum payoff). See also
2845:
or repeated positions). The number of nodes to be explored for the analysis of a game is therefore approximately the branching factor raised to the power of the number of plies. It is therefore
3354:
3026:
4190:
1062:
984:
1721:
708:
624:
3303:
3256:
1431:
3342:
3947:
During the 1997 match, the software search extended the search to about 40 plies along the forcing lines, even though the non-extended search reached only about 12 plies.
1598:
1762:
1507:
1800:
3780:
3218:
3052:
An animated pedagogical example that attempts to be human-friendly by substituting initial infinite (or arbitrarily large) values for emptiness and by avoiding using the
1989:
1633:
1466:
2101:
2139:
2027:
1665:
3492:
2062:
1539:
336:
3651:
reflect our values, or of a corrupt system designed to limit choices to those acceptable to corporate elites," but rather as an opportunity to reduce harm or loss.
776:
744:
365:
300:
662:
267:
5698:
4183:
4156:
3506:
3124:, where it chooses the move with the largest value (represented in the figure with a blue arrow). This is the move that the player should make in order to
119:
796:
5622:
3896:
5713:
5523:
1358:{\displaystyle {\overline {v_{i}}}=\min _{a_{-i}}\max _{a_{i}}{v_{i}(a_{i},a_{-i})}=\min _{a_{-i}}{\Big (}\max _{a_{i}}{v_{i}(a_{i},a_{-i})}{\Big )}}
3599:
to changes in the assumptions, in contrast to these other decision techniques. Various extensions of this non-probabilistic approach exist, notably
914:
The definition is very similar to that of the maximin value – only the order of the maximum and minimum operators is inverse. In the above example:
535:{\displaystyle {\begin{array}{c|cc}\hline &L&R\\\hline T&3,1&2,-20\\M&5,0&-10,1\\B&-100,2&4,4\\\hline \end{array}}}
4176:
4135:
5688:
4449:
419:
5348:
2684:
win in one move, their best move is that winning move. If player B knows that one move will lead to the situation where player A
1880:
1135:
tries to maximize their value before knowing what the others will do; in minimax the maximization comes before the minimization, so player
1808:
545:(where the first number in each of the cell is the pay-out of the row player and the second number is the pay-out of the column player).
5723:
387:
For example, consider the following game for two players, where the first player ("row player") may choose any of three moves, labelled
2954:
Minimax treats the two players (the maximizing player and the minimizing player) separately in its code. Based on the observation that
2856:. Other heuristic pruning methods can also be used, but not all of them are guaranteed to give the same result as the unpruned search.
5165:
4700:
4498:
4984:
4803:
3905:
3871:
3842:
2209:
88:. When dealing with gains, it is referred to as "maximin" – to maximize the minimum gain. Originally formulated for several-player
1077:
4605:
2717:, usually a two-player game. A value is associated with each position or state of the game. This value is computed by means of a
2353:
will choose B2; and eventually both players will realize the difficulty of making a choice. So a more stable strategy is needed.
5074:
2846:
4615:
3166:, take an approach which minimizes the maximum expected loss, using the same techniques as in the two-person zero-sum games.
4944:
5516:
5125:
4543:
4518:
4114:
3709:
3060:
Suppose the game being played only has a maximum of two possible moves per player each turn. The algorithm generates the
2852:
The performance of the naïve minimax algorithm may be improved dramatically, without affecting the result, by the use of
5475:
4901:
4655:
4645:
4580:
4368:
4338:
4323:
3118:. The algorithm continues evaluating the maximum and minimum values of the child nodes alternately until it reaches the
5546:
4695:
4675:
4109:
5160:
2626:
is distinct from minimax. Minimax is used in zero-sum games to denote minimizing the opponent's maximum payoff. In a
3451:{\displaystyle \sup _{\theta }R(\theta ,{\tilde {\delta }})=\inf _{\delta }\ \sup _{\theta }\ R(\theta ,\delta )\ .}
5703:
5662:
5637:
5409:
5130:
4788:
4630:
4625:
4393:
4267:
4257:
4237:
3834:
2650:
2837:
of each node (i.e., the average number of legal moves in a position). The number of nodes to be explored usually
5693:
5683:
5597:
5445:
5368:
5104:
4660:
4585:
4442:
4272:
3739:
2957:
2742:
2662:
1015:
937:
5718:
5509:
5460:
5193:
5079:
4876:
4670:
4488:
3719:
3604:
31:
5263:
667:
583:
3264:
3226:
1371:
5657:
5465:
5064:
5034:
4690:
4478:
4411:
4373:
3759:
3754:
3724:
3312:
2817:
53:
5399:
3684:
2853:
5652:
5617:
5567:
5490:
5470:
5450:
5069:
4974:
4833:
4783:
4778:
4710:
4680:
4600:
4528:
4217:
2325:
is B2 since the worst possible result is then no payment. However, this solution is not stable, since if
2200:
regardless of Player 2's strategy, and similarly Player 2 can guarantee themselves a payoff of −
549:
1670:
1544:
5632:
4508:
4277:
4247:
2860:
2321:
is A2 since the worst possible result is then having to pay 1, while the simple maximin choice for
4949:
4934:
4104:
4006:
5708:
5647:
5551:
5283:
5268:
5155:
5150:
5054:
5039:
5004:
4969:
4568:
4513:
4435:
4388:
4363:
4308:
4122:
3671:
3583:
A key feature of minimax decision making is being non-probabilistic: in contrast to decisions using
3093:
are assigned with negative infinity. At level 3, the algorithm will choose, for each node, the
3072:). Because of the limitation of computation resources, as explained above, the tree is limited to a
2630:, this is identical to minimizing one's own maximum loss, and to maximizing one's own minimum gain.
5582:
5440:
5059:
5009:
4846:
4773:
4753:
4610:
4493:
4398:
4383:
4328:
4143:
3986:
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3665:
3611:
3170:
2718:
1726:
1471:
81:
1767:
5627:
5419:
5278:
5109:
5089:
4939:
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4723:
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4318:
4313:
4227:
4083:
4075:
4030:
3466:
3197:
2838:
2813:
at that time) looked ahead at least 12 plies, then applied a heuristic evaluation function.
2746:
2372:
will not choose B3 since some mixtures of B1 and B2 will produce a better result, no matter what
1953:
1603:
1436:
2067:
3962:
2106:
1994:
5404:
5373:
5328:
5223:
5094:
5049:
5024:
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4828:
4758:
4748:
4640:
4590:
4538:
4378:
4222:
3911:
3901:
3887:
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3838:
3714:
3592:
3182:
1638:
3471:
2032:
1515:
5612:
5577:
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5414:
5378:
5358:
5318:
5288:
5243:
5198:
5183:
5140:
4994:
4635:
4572:
4558:
4523:
4358:
4343:
4067:
4053:"Can the maximin principle serve as a basis for morality? a critique of John Rawls's theory"
4022:
3938:
3859:
3588:
2829:
2634:
2154:
311:
3173:
have been developed, for two-player games in which chance (for example, dice) is a factor.
749:
717:
343:
278:
5572:
5532:
5383:
5343:
5298:
5213:
5208:
4929:
4881:
4768:
4533:
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3798:
3462:
3188:
3163:
2790:
2161:
57:
5248:
3159:
3064:
on the right, where the circles represent the moves of the player running the algorithm (
2305:
displayed on the table ("Payoff matrix for player A"). Assume the payoff matrix for
644:
249:
5607:
5323:
5313:
5303:
5238:
5228:
5218:
5203:
4999:
4979:
4964:
4959:
4919:
4886:
4871:
4866:
4856:
4665:
4199:
3704:
3699:
3694:
3689:
3600:
3584:
3155:
3085:
using a heuristic evaluation function, obtaining the values shown. The moves where the
2810:
2646:
2611:
2749:. An alternative is using a rule that if the result of a move is an immediate win for
2364:
will not choose A3 since either A1 or A2 will produce a better result, no matter what
1139:
is in a much better position – they maximize their value knowing what the others did.
5677:
5363:
5353:
5308:
5293:
5273:
5099:
5044:
5019:
4891:
4861:
4851:
4838:
4743:
4685:
4620:
4553:
4348:
4303:
4087:
4048:
4002:
3568:{\displaystyle \int _{\Theta }R(\theta ,\delta )\ \operatorname {d} \Pi (\theta )\ .}
3306:
3259:
3133:
2859:
A naïve minimax algorithm may be trivially modified to additionally return an entire
2806:
2627:
2298:
2150:
227:{\displaystyle {\underline {v_{i}}}=\max _{a_{i}}\min _{a_{-i}}{v_{i}(a_{i},a_{-i})}}
77:
3089:
wins are assigned with positive infinity, while the moves that lead to a win of the
2309:
is the same matrix with the signs reversed (i.e., if the choices are A1 and B1 then
904:{\displaystyle {\overline {v_{i}}}=\min _{a_{-i}}\max _{a_{i}}{v_{i}(a_{i},a_{-i})}}
5338:
5333:
5188:
4763:
4298:
4262:
4232:
4052:
3958:
3891:
2180:(a) Given Player 2's strategy, the best payoff possible for Player 1 is
4168:
4151:
1131:
Intuitively, in maximin the maximization comes after the minimization, so player
384:
can take in order to make sure that this smallest value is the highest possible.
17:
5642:
5455:
5258:
5253:
5233:
5029:
5014:
4823:
4793:
4728:
4718:
4548:
4483:
4459:
4252:
3826:
3744:
3643:
3114:
2842:
2802:
2714:
2677:
1950:
the payoff vector resulting from both players playing their minimax strategies,
92:
61:
5084:
4738:
4242:
3982:
3660:
3099:
2948:
2872:
2834:
2187:(b) Given Player 1's strategy, the best payoff possible for Player 2 is −
407:. The result of the combination of both moves is expressed in a payoff table:
69:
65:
27:
Decision rule used for minimizing the possible loss for a worst case scenario
4989:
4909:
4733:
4203:
3734:
3591:, it makes no assumptions about the probabilities of various outcomes, just
3221:
3192:
3120:
3081:
3061:
2822:
2797:
2710:
399:, and the second player ("column player") may choose either of two moves,
5424:
4924:
2697:
the chances of A winning (i.e., to maximize B's own chances of winning).
2689:
end of the game. At each step it assumes that player A is trying to
2169:
89:
4427:
3596:
2578:
chooses, by using a randomized strategy of choosing B1 with probability
5592:
5145:
4813:
4079:
4034:
3808:
3729:
3634:
Minimax thus can be used on ordinal data, and can be more transparent.
3496:
3053:
3029:
2849:
to completely analyze games such as chess using the minimax algorithm.
2693:
the chances of A winning, while on the next turn player B is trying to
30:
This article is about the decision theory concept. For other uses, see
3942:
2761:
of any other move is the maximum of the values resulting from each of
2753:, it is assigned positive infinity and if it is an immediate win for
2733:
A possible allocation method consists in assigning a certain win for
1943:{\displaystyle \ {\underline {v_{col}}}\leq {\overline {v_{col}}}\,,}
4071:
4026:
3929:
Hsu, Feng-Hsiung (1999). "IBM's Deep Blue chess grandmaster chips".
3461:
An alternative criterion in the decision theoretic framework is the
3048:
2841:
with the number of plies (it is less than exponential if evaluating
1870:{\displaystyle \ {\underline {v_{row}}}\leq {\overline {v_{row}}}\ }
3040:
2915:
value := max(value, minimax(child, depth − 1, FALSE))
2196:
Equivalently, Player 1's strategy guarantees them a payoff of
5501:
4914:
3659:
In philosophy, the term "maximin" is often used in the context of
2934:
value := min(value, minimax(child, depth − 1, TRUE))
2786:
2297:
solutions. Suppose each player has three choices and consider the
3619:
analysis, whose conclusion is of the form: "This strategy yields
5602:
4282:
4161:
3915:
5505:
4431:
4172:
2680:, where each player can win, lose, or draw. If player A
3967:
2141:
resulting from both players playing their maximin strategy.
1121:{\displaystyle {\underline {v_{i}}}\leq {\overline {v_{i}}}}
790:
the actions of the other players. Its formal definition is:
380:
the smallest value. Then, we determine which action player
3937:(2). Los Alamitos, CA, USA: IEEE Computer Society: 70–81.
2614:
minimax strategies cannot be improved and are now stable.
2383:
can avoid having to make an expected payment of more than
2172:
game with finitely many strategies, there exists a value
714:
If both players play their respective maximin strategies
2875:
for the depth-limited minimax algorithm is given below.
2645:
The minimax values are very important in the theory of
3509:
3474:
3357:
3315:
3267:
3229:
3200:
2960:
2109:
2103:
cannot similarly be ranked against the payoff vector
2070:
2035:
1997:
1956:
1883:
1811:
1770:
1729:
1673:
1641:
1606:
1547:
1518:
1474:
1439:
1374:
1155:
1080:
1018:
940:
799:
752:
720:
670:
647:
586:
416:
346:
314:
281:
252:
122:
3900:(4th ed.). Hoboken: Pearson. pp. 149–150.
3112:
values. Once again, the values are assigned to each
3068:), and squares represent the moves of the opponent (
5560:
5539:
5433:
5392:
5174:
5118:
4900:
4802:
4709:
4567:
4466:
4291:
4210:
2947:The minimax function returns a heuristic value for
2665:, there is a minimax algorithm for game solutions.
3567:
3486:
3450:
3336:
3297:
3250:
3212:
3020:
2809:(the first one to beat a reigning world champion,
2649:. One of the central theorems in this theory, the
2133:
2095:
2056:
2021:
1983:
1942:
1869:
1794:
1756:
1715:
1659:
1627:
1592:
1533:
1501:
1460:
1425:
1357:
1120:
1056:
978:
903:
770:
738:
702:
656:
618:
534:
359:
330:
294:
261:
226:
3963:An Eight Point Brief for LEV (Lesser Evil Voting)
2816:The algorithm can be thought of as exploring the
1350:
1282:
3412:
3399:
3359:
3177:Minimax criterion in statistical decision theory
3150:One approach is to treat this as a game against
2988:
2964:
2285:The following example of a zero-sum game, where
2176:and a mixed strategy for each player, such that
1288:
1261:
1197:
1177:
1146:is by reading from right to left: When we write
841:
821:
161:
144:
1802:over these outcomes. (Conversely for maximin.)
338:denotes the actions taken by all other players.
4007:"Some ordinalist-utilitarian notes on Rawls's
3595:of what the possible outcomes are. It is thus
5517:
4443:
4184:
988:The column player can get a maximum value of
560:, which guarantees them a payoff of at least
8:
4157:Dictionary of Algorithms and Data Structures
3866:(print ed.). Cambridge, MA: MIT Press.
3670:where he refers to it in the context of The
3646:" voting (LEV) can be seen as a form of the
3614:(that outcomes be compared and ranked), not
2730:gives a value to each of their legal moves.
4140:Dictionary of Philosophical Terms and Names
3021:{\displaystyle \ \max(a,b)=-\min(-a,-b)\ ,}
5623:Heuristics in judgment and decision-making
5524:
5510:
5502:
4450:
4436:
4428:
4191:
4177:
4169:
3897:Artificial Intelligence: A Modern Approach
3494:An estimator is Bayes if it minimizes the
2713:for choosing the next move in an n-player
2153:, the minimax solution is the same as the
1057:{\displaystyle {\overline {v_{col}}}=1\ .}
979:{\displaystyle {\overline {v_{row}}}=4\ .}
918:The row player can get a maximum value of
548:For the sake of example, we consider only
3820:
3818:
3514:
3508:
3473:
3415:
3402:
3381:
3380:
3362:
3356:
3320:
3319:
3314:
3266:
3228:
3199:
3028:minimax may often be simplified into the
2959:
2676:, stated below, deals with games such as
2108:
2089:
2069:
2034:
1996:
1955:
1936:
1919:
1913:
1893:
1887:
1882:
1847:
1841:
1821:
1815:
1810:
1779:
1774:
1769:
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1733:
1728:
1709:
1697:
1681:
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1647:
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1605:
1578:
1565:
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1524:
1519:
1517:
1483:
1478:
1473:
1448:
1443:
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1408:
1395:
1382:
1373:
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1348:
1335:
1322:
1309:
1304:
1296:
1291:
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1280:
1269:
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1200:
1185:
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1162:
1156:
1154:
1107:
1101:
1087:
1081:
1079:
1025:
1019:
1017:
947:
941:
939:
888:
875:
862:
857:
849:
844:
829:
824:
806:
800:
798:
751:
719:
677:
671:
669:
646:
593:
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418:
417:
415:
351:
345:
319:
313:
286:
280:
251:
211:
198:
185:
180:
169:
164:
152:
147:
129:
123:
121:
3047:
3039:
2796:This can be extended if we can supply a
2558:can ensure an expected gain of at least
2349:will choose A1 to gain 3; and then
2219:
703:{\displaystyle {\underline {v_{col}}}=0}
619:{\displaystyle {\underline {v_{row}}}=2}
269:denotes all other players except player
3786:(Report). Fraser Institute. p. 25.
3771:
3305:usually specified as the integral of a
3298:{\displaystyle \ R(\theta ,\delta )\ .}
3251:{\displaystyle \ \theta \in \Theta \ .}
2881:minimax(node, depth, maximizingPlayer)
2337:will choose B1 to gain 1; then if
1426:{\displaystyle \ v_{i}(a_{i},a_{-i})\ }
244:is the index of the player of interest.
3337:{\displaystyle \ {\tilde {\delta }}\ }
2765:'s possible replies. For this reason,
2701:Minimax algorithm with alternate moves
2160:In the context of zero-sum games, the
1667:) to yield a set of marginal outcomes
1071:, the maximin is at most the minimax:
2833:of the tree is the average number of
2293:make simultaneous moves, illustrates
568:is risky since it can lead to payoff
7:
1805:Although it is always the case that
5699:Optimization algorithms and methods
2805:". For example, the chess computer
1716:{\displaystyle \ v'_{i}(a_{-i})\,,}
1593:{\displaystyle v_{i}(a_{i},a_{-i})}
4499:First-player and second-player win
3547:
3541:
3515:
3475:
3239:
3158:), and using a similar mindset as
3145:Minimax in the face of uncertainty
2757:, negative infinity. The value to
25:
4060:American Political Science Review
3579:Non-probabilistic decision theory
2360:by others and can be eliminated:
2220:
2210:example of a game without a value
641:puts them in the risk of getting
5714:Theorems in discrete mathematics
4606:Coalition-proof Nash equilibrium
3781:Provincial Healthcare Index 2013
3140:Minimax for individual decisions
2899:the heuristic value of node
2653:, relies on the minimax values.
2399:by choosing A1 with probability
2317:). Then, the maximin choice for
633:and secure a payoff of at least
367:is the value function of player
3779:Bacchus, Barua (January 2013).
3610:Further, minimax only requires
4616:Evolutionarily stable strategy
3556:
3550:
3535:
3523:
3439:
3427:
3392:
3386:
3371:
3325:
3286:
3274:
3009:
2991:
2979:
2967:
2125:
2113:
2086:
2074:
2051:
2036:
2013:
2001:
1975:
1960:
1706:
1690:
1587:
1558:
1417:
1388:
1344:
1315:
1253:
1224:
1142:Another way to understand the
897:
868:
765:
753:
733:
721:
302:is the action taken by player
220:
191:
1:
4544:Simultaneous action selection
3710:Lesser of two evils principle
3079:The algorithm evaluates each
2944:minimax(origin, depth, TRUE)
1757:{\displaystyle \ {a_{-i}}\ .}
1635:(for every possible value of
1502:{\displaystyle \ {a_{-i}}\ .}
552:. Check each player in turn:
52:) is a decision rule used in
5689:Game artificial intelligence
5476:List of games in game theory
4656:Quantal response equilibrium
4646:Perfect Bayesian equilibrium
4581:Bayes correlated equilibrium
2863:along with a minimax score.
2719:position evaluation function
2622:Frequently, in game theory,
1931:
1859:
1795:{\displaystyle \ {a_{-i}}\ }
1368:the initial set of outcomes
1168:
1113:
1037:
959:
812:
5547:Expected utility hypothesis
4945:Optional prisoner's dilemma
4676:Self-confirming equilibrium
4110:Encyclopedia of Mathematics
3800:Turing and von Neumann
3220:that is used to estimate a
3213:{\displaystyle \ \delta \ }
2221:Payoff matrix for player A
1984:{\displaystyle \ (2,-20)\ }
1628:{\displaystyle \ {a_{i}}\ }
1461:{\displaystyle \ {a_{i}}\ }
992:(if the other player plays
930:(if the other player plays
922:(if the other player plays
629:The column player can play
5740:
5724:Fixed points (mathematics)
5663:Evidential decision theory
5410:Principal variation search
5126:Aumann's agreement theorem
4789:Strategy-stealing argument
4701:Trembling hand equilibrium
4631:Markov perfect equilibrium
4626:Mertens-stable equilibrium
3835:Cambridge University Press
3180:
2096:{\displaystyle \ (M,R)\,,}
576:can result in a payoff of
29:
5598:Principle of indifference
5446:Combinatorial game theory
5105:Princess and monster game
4661:Quasi-perfect equilibrium
4586:Bayesian Nash equilibrium
4407:
3797:Professor Raymond Flood.
3187:In classical statistical
2926:value := +∞
2907:value := −∞
2743:combinatorial game theory
2663:combinatorial game theory
2657:Combinatorial game theory
2134:{\displaystyle \ (3,1)\ }
2022:{\displaystyle \ (T,R)\ }
5461:Evolutionary game theory
5194:Antoine Augustin Cournot
5080:Guess 2/3 of the average
4877:Strictly determined game
4671:Satisfaction equilibrium
4489:Escalation of commitment
4131:— A visualization applet
3829:; Zamir, Shmuel (2013).
3605:Info-gap decision theory
2892:node is a terminal node
2594:and B2 with probability
2434:The expected payoff for
2415:and A2 with probability
1660:{\displaystyle {a_{-i}}}
556:The row player can play
32:Minimax (disambiguation)
5658:Emotional choice theory
5466:Glossary of game theory
5065:Stackelberg competition
4691:Strong Nash equilibrium
4412:List of data structures
3864:A Course in Game Theory
3760:Gamma-minimax inference
3725:Monte Carlo tree search
3487:{\displaystyle \Pi \ .}
3056:coding simplifications.
2924:(* minimizing player *)
2839:increases exponentially
2672:version of the minimax
2057:{\displaystyle (-10,1)}
1534:{\displaystyle {a_{i}}}
746:, the payoff vector is
54:artificial intelligence
5653:Causal decision theory
5618:St. Petersburg paradox
5568:Decision-matrix method
5491:Tragedy of the commons
5471:List of game theorists
5451:Confrontation analysis
5161:Sprague–Grundy theorem
4681:Sequential equilibrium
4601:Correlated equilibrium
3740:Sion's minimax theorem
3569:
3488:
3452:
3338:
3299:
3252:
3214:
3057:
3045:
3044:A minimax tree example
3022:
2194:
2135:
2097:
2058:
2023:
1985:
1944:
1871:
1796:
1764:We then minimize over
1758:
1723:which depends only on
1717:
1661:
1629:
1594:
1535:
1503:
1462:
1427:
1359:
1122:
1058:
980:
905:
772:
740:
704:
658:
620:
536:
361:
332:
331:{\displaystyle a_{-i}}
296:
263:
228:
5633:Bayesian epistemology
5264:Jean-François Mertens
4015:Journal of Philosophy
3655:Maximin in philosophy
3570:
3489:
3465:in the presence of a
3453:
3339:
3309:. In this framework,
3300:
3253:
3215:
3051:
3043:
3023:
2741:as −1. This leads to
2554:chose B2. Similarly,
2168:For every two-person
2166:
2136:
2098:
2059:
2024:
1986:
1945:
1872:
1797:
1759:
1718:
1662:
1630:
1600:, by maximizing over
1595:
1536:
1504:
1463:
1428:
1360:
1123:
1059:
981:
906:
773:
771:{\displaystyle (3,1)}
741:
739:{\displaystyle (T,L)}
705:
659:
621:
537:
362:
360:{\displaystyle v_{i}}
333:
297:
295:{\displaystyle a_{i}}
264:
229:
5648:Social choice theory
5552:Intertemporal choice
5393:Search optimizations
5269:Jennifer Tour Chayes
5156:Revelation principle
5151:Purification theorem
5090:Nash bargaining game
5055:Bertrand competition
5040:El Farol Bar problem
5005:Electronic mail game
4970:Lewis signaling game
4514:Hierarchy of beliefs
4309:Breadth-first search
4129:. Curriculum: Games.
3858:Osborne, Martin J.;
3837:. pp. 176–180.
3755:Wald's maximin model
3672:Difference Principle
3507:
3472:
3355:
3313:
3265:
3227:
3198:
3171:expectiminimax trees
2958:
2345:will choose B1 then
2333:will choose A2 then
2107:
2068:
2033:
1995:
1954:
1881:
1809:
1768:
1727:
1671:
1639:
1604:
1545:
1516:
1472:
1437:
1372:
1153:
1078:
1016:
938:
797:
750:
718:
668:
645:
584:
530:
414:
344:
312:
279:
250:
120:
5583:Strategic dominance
5441:Bounded rationality
5060:Cournot competition
5010:Rock paper scissors
4985:Battle of the sexes
4975:Volunteer's dilemma
4847:Perfect information
4774:Dominant strategies
4611:Epsilon-equilibrium
4494:Extensive-form game
4399:Topological sorting
4329:Dynamic programming
4136:"Maximin principle"
4105:"Minimax principle"
3988:A Theory of Justice
3825:Maschler, Michael;
3750:Transposition table
3666:A Theory of Justice
3638:Minimax in politics
3612:ordinal measurement
2861:Principal Variation
2222:
1689:
657:{\displaystyle -20}
86:imum loss) scenario
5628:Probability theory
5420:Paranoid algorithm
5400:Alpha–beta pruning
5279:John Maynard Smith
5110:Rendezvous problem
4950:Traveler's dilemma
4940:Gift-exchange game
4935:Prisoner's dilemma
4852:Large Poisson game
4819:Bargaining problem
4724:Backward induction
4696:Subgame perfection
4651:Proper equilibrium
4417:List of algorithms
4324:Divide and conquer
4319:Depth-first search
4314:Brute-force search
4228:Binary search tree
4123:"Mixed strategies"
3888:Russell, Stuart J.
3685:Alpha–beta pruning
3565:
3484:
3467:prior distribution
3448:
3420:
3407:
3367:
3334:
3295:
3248:
3210:
3058:
3046:
3018:
2942:(* Initial call *)
2854:alpha–beta pruning
2164:is equivalent to:
2131:
2093:
2054:
2019:
1981:
1940:
1908:
1867:
1836:
1792:
1754:
1713:
1677:
1657:
1625:
1590:
1531:
1499:
1458:
1423:
1355:
1303:
1279:
1212:
1195:
1118:
1096:
1054:
976:
901:
856:
839:
768:
736:
700:
692:
654:
616:
608:
532:
529:
357:
328:
292:
262:{\displaystyle -i}
259:
224:
179:
159:
138:
5704:Search algorithms
5671:
5670:
5499:
5498:
5405:Aspiration window
5374:Suzanne Scotchmer
5329:Oskar Morgenstern
5224:Donald B. Gillies
5166:Zermelo's theorem
5095:Induction puzzles
5050:Fair cake-cutting
5025:Public goods game
4955:Coordination game
4829:Intransitive game
4759:Forward induction
4641:Pareto efficiency
4621:Gibbs equilibrium
4591:Berge equilibrium
4539:Simultaneous game
4425:
4424:
4223:Associative array
4009:Theory of Justice
3961:and John Halle, "
3943:10.1109/40.755469
3715:Minimax Condorcet
3593:scenario analysis
3561:
3540:
3480:
3444:
3423:
3411:
3410:
3398:
3389:
3358:
3333:
3328:
3318:
3291:
3270:
3258:We also assume a
3244:
3232:
3209:
3203:
3183:Minimax estimator
3091:minimizing player
3087:maximizing player
3076:of 4 moves.
3070:minimizing player
3066:maximizing player
3014:
2963:
2903:maximizingPlayer
2783:minimax algorithm
2781:, hence the name
2779:minimizing player
2771:maximizing player
2726:'s turn to move,
2707:minimax algorithm
2641:In repeated games
2574:, no matter what
2356:Some choices are
2283:
2282:
2145:In zero-sum games
2130:
2112:
2073:
2018:
2000:
1980:
1959:
1934:
1888:
1886:
1866:
1862:
1816:
1814:
1791:
1773:
1750:
1732:
1676:
1624:
1609:
1495:
1477:
1457:
1442:
1422:
1377:
1287:
1260:
1196:
1176:
1171:
1116:
1082:
1067:For every player
1050:
1040:
972:
962:
840:
820:
815:
672:
588:
160:
143:
124:
18:Minimax algorithm
16:(Redirected from
5731:
5694:Graph algorithms
5684:Detection theory
5613:Ellsberg paradox
5578:Expected utility
5526:
5519:
5512:
5503:
5486:Topological game
5481:No-win situation
5379:Thomas Schelling
5359:Robert B. Wilson
5319:Merrill M. Flood
5289:John von Neumann
5199:Ariel Rubinstein
5184:Albert W. Tucker
5035:War of attrition
4995:Matching pennies
4636:Nash equilibrium
4559:Mechanism design
4524:Normal-form game
4479:Cooperative game
4452:
4445:
4438:
4429:
4394:String-searching
4193:
4186:
4179:
4170:
4165:
4147:
4142:. Archived from
4130:
4127:cut-the-knot.org
4118:
4092:
4091:
4057:
4045:
4039:
4038:
3999:
3993:
3992:
3979:
3973:
3956:
3950:
3949:
3926:
3920:
3919:
3884:
3878:
3877:
3855:
3849:
3848:
3822:
3813:
3812:
3794:
3788:
3787:
3785:
3776:
3642:The concept of "
3633:
3631:
3627:
3623:
3589:expected utility
3574:
3572:
3571:
3566:
3559:
3538:
3519:
3518:
3493:
3491:
3490:
3485:
3478:
3457:
3455:
3454:
3449:
3442:
3421:
3419:
3408:
3406:
3391:
3390:
3382:
3366:
3348:if it satisfies
3343:
3341:
3340:
3335:
3331:
3330:
3329:
3321:
3316:
3304:
3302:
3301:
3296:
3289:
3268:
3257:
3255:
3254:
3249:
3242:
3230:
3219:
3217:
3216:
3211:
3207:
3201:
3027:
3025:
3024:
3019:
3012:
2961:
2830:branching factor
2745:as developed by
2635:Nash equilibrium
2609:
2607:
2606:
2603:
2600:
2593:
2591:
2590:
2587:
2584:
2573:
2571:
2570:
2567:
2564:
2549:
2548:
2546:
2545:
2542:
2539:
2535:
2529:
2527:
2526:
2523:
2520:
2513:
2511:
2510:
2507:
2504:
2491:
2490:
2488:
2487:
2484:
2481:
2477:
2471:
2469:
2468:
2465:
2462:
2455:
2453:
2452:
2449:
2446:
2433:
2431:
2429:
2428:
2425:
2422:
2414:
2412:
2411:
2408:
2405:
2398:
2396:
2395:
2392:
2389:
2261:
2223:
2203:
2199:
2190:
2183:
2175:
2155:Nash equilibrium
2140:
2138:
2137:
2132:
2128:
2110:
2102:
2100:
2099:
2094:
2071:
2063:
2061:
2060:
2055:
2028:
2026:
2025:
2020:
2016:
1998:
1990:
1988:
1987:
1982:
1978:
1957:
1949:
1947:
1946:
1941:
1935:
1930:
1929:
1914:
1909:
1904:
1903:
1884:
1876:
1874:
1873:
1868:
1864:
1863:
1858:
1857:
1842:
1837:
1832:
1831:
1812:
1801:
1799:
1798:
1793:
1789:
1788:
1787:
1786:
1771:
1763:
1761:
1760:
1755:
1748:
1747:
1746:
1745:
1730:
1722:
1720:
1719:
1714:
1705:
1704:
1685:
1674:
1666:
1664:
1663:
1658:
1656:
1655:
1654:
1634:
1632:
1631:
1626:
1622:
1621:
1620:
1619:
1607:
1599:
1597:
1596:
1591:
1586:
1585:
1570:
1569:
1557:
1556:
1540:
1538:
1537:
1532:
1530:
1529:
1528:
1511:marginalize away
1508:
1506:
1505:
1500:
1493:
1492:
1491:
1490:
1475:
1467:
1465:
1464:
1459:
1455:
1454:
1453:
1452:
1440:
1433:depends on both
1432:
1430:
1429:
1424:
1420:
1416:
1415:
1400:
1399:
1387:
1386:
1375:
1364:
1362:
1361:
1356:
1354:
1353:
1347:
1343:
1342:
1327:
1326:
1314:
1313:
1302:
1301:
1300:
1286:
1285:
1278:
1277:
1276:
1256:
1252:
1251:
1236:
1235:
1223:
1222:
1211:
1210:
1209:
1194:
1193:
1192:
1172:
1167:
1166:
1157:
1138:
1134:
1127:
1125:
1124:
1119:
1117:
1112:
1111:
1102:
1097:
1092:
1091:
1070:
1063:
1061:
1060:
1055:
1048:
1041:
1036:
1035:
1020:
1011:
1007:
1003:
999:
995:
991:
985:
983:
982:
977:
970:
963:
958:
957:
942:
933:
929:
925:
921:
910:
908:
907:
902:
900:
896:
895:
880:
879:
867:
866:
855:
854:
853:
838:
837:
836:
816:
811:
810:
801:
777:
775:
774:
769:
745:
743:
742:
737:
709:
707:
706:
701:
693:
688:
687:
663:
661:
660:
655:
640:
636:
632:
625:
623:
622:
617:
609:
604:
603:
579:
575:
571:
567:
563:
559:
541:
539:
538:
533:
531:
421:
406:
402:
398:
394:
390:
383:
379:
370:
366:
364:
363:
358:
356:
355:
337:
335:
334:
329:
327:
326:
305:
301:
299:
298:
293:
291:
290:
272:
268:
266:
265:
260:
243:
233:
231:
230:
225:
223:
219:
218:
203:
202:
190:
189:
178:
177:
176:
158:
157:
156:
139:
134:
133:
104:In general games
21:
5739:
5738:
5734:
5733:
5732:
5730:
5729:
5728:
5719:Decision theory
5674:
5673:
5672:
5667:
5573:Decision matrix
5556:
5535:
5533:Decision theory
5530:
5500:
5495:
5429:
5415:max^n algorithm
5388:
5384:William Vickrey
5344:Reinhard Selten
5299:Kenneth Binmore
5214:David K. Levine
5209:Daniel Kahneman
5176:
5170:
5146:Negamax theorem
5136:Minimax theorem
5114:
5075:Diner's dilemma
4930:All-pay auction
4896:
4882:Stochastic game
4834:Mean-field game
4805:
4798:
4769:Markov strategy
4705:
4571:
4563:
4534:Sequential game
4519:Information set
4504:Game complexity
4474:Congestion game
4462:
4456:
4426:
4421:
4403:
4334:Graph traversal
4287:
4211:Data structures
4206:
4200:Data structures
4197:
4150:
4134:
4121:
4103:
4100:
4095:
4072:10.2307/1959090
4055:
4047:
4046:
4042:
4027:10.2307/2025006
4001:
4000:
3996:
3981:
3980:
3976:
3957:
3953:
3928:
3927:
3923:
3908:
3886:
3885:
3881:
3874:
3857:
3856:
3852:
3845:
3824:
3823:
3816:
3805:Gresham College
3796:
3795:
3791:
3783:
3778:
3777:
3773:
3769:
3764:
3680:
3657:
3640:
3629:
3625:
3621:
3620:
3581:
3510:
3505:
3504:
3470:
3469:
3463:Bayes estimator
3353:
3352:
3311:
3310:
3263:
3262:
3225:
3224:
3196:
3195:
3189:decision theory
3185:
3179:
3164:resistentialism
3147:
3142:
3038:
2956:
2955:
2945:
2939:
2869:
2709:is a recursive
2703:
2659:
2643:
2620:
2604:
2601:
2598:
2597:
2595:
2588:
2585:
2582:
2581:
2579:
2568:
2565:
2562:
2561:
2559:
2543:
2540:
2537:
2536:
2533:
2531:
2524:
2521:
2518:
2517:
2515:
2508:
2505:
2502:
2501:
2499:
2497:
2485:
2482:
2479:
2478:
2475:
2473:
2466:
2463:
2460:
2459:
2457:
2450:
2447:
2444:
2443:
2441:
2439:
2426:
2423:
2420:
2419:
2417:
2416:
2409:
2406:
2403:
2402:
2400:
2393:
2390:
2387:
2386:
2384:
2313:pays 3 to
2259:
2218:
2201:
2197:
2188:
2181:
2173:
2162:minimax theorem
2147:
2105:
2104:
2066:
2065:
2064:in the case of
2031:
2030:
1993:
1992:
1991:in the case of
1952:
1951:
1915:
1889:
1879:
1878:
1843:
1817:
1807:
1806:
1775:
1766:
1765:
1734:
1725:
1724:
1693:
1669:
1668:
1643:
1637:
1636:
1611:
1602:
1601:
1574:
1561:
1548:
1543:
1542:
1520:
1514:
1513:
1479:
1470:
1469:
1444:
1435:
1434:
1404:
1391:
1378:
1370:
1369:
1331:
1318:
1305:
1292:
1265:
1240:
1227:
1214:
1201:
1181:
1158:
1151:
1150:
1136:
1132:
1103:
1083:
1076:
1075:
1068:
1021:
1014:
1013:
1009:
1005:
1001:
997:
993:
989:
943:
936:
935:
931:
927:
923:
919:
884:
871:
858:
845:
825:
802:
795:
794:
748:
747:
716:
715:
673:
666:
665:
643:
642:
638:
634:
630:
589:
582:
581:
577:
573:
569:
565:
561:
557:
550:pure strategies
528:
527:
516:
502:
496:
495:
481:
470:
464:
463:
449:
438:
432:
431:
426:
412:
411:
404:
400:
396:
392:
388:
381:
377:
368:
347:
342:
341:
315:
310:
309:
303:
282:
277:
276:
270:
248:
247:
241:
207:
194:
181:
165:
148:
125:
118:
117:
106:
101:
58:decision theory
35:
28:
23:
22:
15:
12:
11:
5:
5737:
5735:
5727:
5726:
5721:
5716:
5711:
5706:
5701:
5696:
5691:
5686:
5676:
5675:
5669:
5668:
5666:
5665:
5660:
5655:
5650:
5645:
5640:
5635:
5630:
5625:
5620:
5615:
5610:
5608:Allais paradox
5605:
5600:
5595:
5590:
5585:
5580:
5575:
5570:
5564:
5562:
5558:
5557:
5555:
5554:
5549:
5543:
5541:
5537:
5536:
5531:
5529:
5528:
5521:
5514:
5506:
5497:
5496:
5494:
5493:
5488:
5483:
5478:
5473:
5468:
5463:
5458:
5453:
5448:
5443:
5437:
5435:
5431:
5430:
5428:
5427:
5422:
5417:
5412:
5407:
5402:
5396:
5394:
5390:
5389:
5387:
5386:
5381:
5376:
5371:
5366:
5361:
5356:
5351:
5349:Robert Axelrod
5346:
5341:
5336:
5331:
5326:
5324:Olga Bondareva
5321:
5316:
5314:Melvin Dresher
5311:
5306:
5304:Leonid Hurwicz
5301:
5296:
5291:
5286:
5281:
5276:
5271:
5266:
5261:
5256:
5251:
5246:
5241:
5239:Harold W. Kuhn
5236:
5231:
5229:Drew Fudenberg
5226:
5221:
5219:David M. Kreps
5216:
5211:
5206:
5204:Claude Shannon
5201:
5196:
5191:
5186:
5180:
5178:
5172:
5171:
5169:
5168:
5163:
5158:
5153:
5148:
5143:
5141:Nash's theorem
5138:
5133:
5128:
5122:
5120:
5116:
5115:
5113:
5112:
5107:
5102:
5097:
5092:
5087:
5082:
5077:
5072:
5067:
5062:
5057:
5052:
5047:
5042:
5037:
5032:
5027:
5022:
5017:
5012:
5007:
5002:
5000:Ultimatum game
4997:
4992:
4987:
4982:
4980:Dollar auction
4977:
4972:
4967:
4965:Centipede game
4962:
4957:
4952:
4947:
4942:
4937:
4932:
4927:
4922:
4920:Infinite chess
4917:
4912:
4906:
4904:
4898:
4897:
4895:
4894:
4889:
4887:Symmetric game
4884:
4879:
4874:
4872:Signaling game
4869:
4867:Screening game
4864:
4859:
4857:Potential game
4854:
4849:
4844:
4836:
4831:
4826:
4821:
4816:
4810:
4808:
4800:
4799:
4797:
4796:
4791:
4786:
4784:Mixed strategy
4781:
4776:
4771:
4766:
4761:
4756:
4751:
4746:
4741:
4736:
4731:
4726:
4721:
4715:
4713:
4707:
4706:
4704:
4703:
4698:
4693:
4688:
4683:
4678:
4673:
4668:
4666:Risk dominance
4663:
4658:
4653:
4648:
4643:
4638:
4633:
4628:
4623:
4618:
4613:
4608:
4603:
4598:
4593:
4588:
4583:
4577:
4575:
4565:
4564:
4562:
4561:
4556:
4551:
4546:
4541:
4536:
4531:
4526:
4521:
4516:
4511:
4509:Graphical game
4506:
4501:
4496:
4491:
4486:
4481:
4476:
4470:
4468:
4464:
4463:
4457:
4455:
4454:
4447:
4440:
4432:
4423:
4422:
4420:
4419:
4414:
4408:
4405:
4404:
4402:
4401:
4396:
4391:
4386:
4381:
4376:
4371:
4366:
4361:
4356:
4351:
4346:
4341:
4336:
4331:
4326:
4321:
4316:
4311:
4306:
4301:
4295:
4293:
4289:
4288:
4286:
4285:
4280:
4275:
4270:
4265:
4260:
4255:
4250:
4245:
4240:
4235:
4230:
4225:
4220:
4214:
4212:
4208:
4207:
4198:
4196:
4195:
4188:
4181:
4173:
4167:
4166:
4148:
4146:on 2006-03-07.
4132:
4119:
4099:
4098:External links
4096:
4094:
4093:
4066:(2): 594–606.
4040:
4021:(9): 245–263.
3994:
3991:. p. 152.
3974:
3972:June 15, 2016.
3951:
3921:
3906:
3892:Norvig, Peter.
3879:
3872:
3860:Rubinstein, A.
3850:
3843:
3814:
3789:
3770:
3768:
3765:
3763:
3762:
3757:
3752:
3747:
3742:
3737:
3732:
3727:
3722:
3720:Minimax regret
3717:
3712:
3707:
3705:Horizon effect
3702:
3700:Computer chess
3697:
3695:Maxn algorithm
3692:
3690:Expectiminimax
3687:
3681:
3679:
3676:
3656:
3653:
3639:
3636:
3601:minimax regret
3585:expected value
3580:
3577:
3576:
3575:
3564:
3558:
3555:
3552:
3549:
3546:
3543:
3537:
3534:
3531:
3528:
3525:
3522:
3517:
3513:
3483:
3477:
3459:
3458:
3447:
3441:
3438:
3435:
3432:
3429:
3426:
3418:
3414:
3405:
3401:
3397:
3394:
3388:
3385:
3379:
3376:
3373:
3370:
3365:
3361:
3327:
3324:
3294:
3288:
3285:
3282:
3279:
3276:
3273:
3247:
3241:
3238:
3235:
3206:
3181:Main article:
3178:
3175:
3156:move by nature
3146:
3143:
3141:
3138:
3037:
3034:
3017:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2940:
2930:child of node
2911:child of node
2877:
2868:
2865:
2811:Garry Kasparov
2777:is called the
2769:is called the
2747:John H. Conway
2737:as +1 and for
2702:
2699:
2658:
2655:
2647:repeated games
2642:
2639:
2619:
2616:
2281:
2280:
2277:
2274:
2271:
2267:
2266:
2263:
2257:
2254:
2250:
2249:
2246:
2243:
2240:
2236:
2235:
2232:
2229:
2226:
2217:
2214:
2193:
2192:
2185:
2151:zero-sum games
2149:In two-player
2146:
2143:
2127:
2124:
2121:
2118:
2115:
2092:
2088:
2085:
2082:
2079:
2076:
2053:
2050:
2047:
2044:
2041:
2038:
2015:
2012:
2009:
2006:
2003:
1977:
1974:
1971:
1968:
1965:
1962:
1939:
1933:
1928:
1925:
1922:
1918:
1912:
1907:
1902:
1899:
1896:
1892:
1861:
1856:
1853:
1850:
1846:
1840:
1835:
1830:
1827:
1824:
1820:
1785:
1782:
1778:
1753:
1744:
1741:
1737:
1712:
1708:
1703:
1700:
1696:
1692:
1688:
1684:
1680:
1653:
1650:
1646:
1618:
1614:
1589:
1584:
1581:
1577:
1573:
1568:
1564:
1560:
1555:
1551:
1527:
1523:
1498:
1489:
1486:
1482:
1451:
1447:
1419:
1414:
1411:
1407:
1403:
1398:
1394:
1390:
1385:
1381:
1366:
1365:
1352:
1346:
1341:
1338:
1334:
1330:
1325:
1321:
1317:
1312:
1308:
1299:
1295:
1290:
1284:
1275:
1272:
1268:
1263:
1259:
1255:
1250:
1247:
1243:
1239:
1234:
1230:
1226:
1221:
1217:
1208:
1204:
1199:
1191:
1188:
1184:
1179:
1175:
1170:
1165:
1161:
1129:
1128:
1115:
1110:
1106:
1100:
1095:
1090:
1086:
1065:
1064:
1053:
1047:
1044:
1039:
1034:
1031:
1028:
1024:
986:
975:
969:
966:
961:
956:
953:
950:
946:
912:
911:
899:
894:
891:
887:
883:
878:
874:
870:
865:
861:
852:
848:
843:
835:
832:
828:
823:
819:
814:
809:
805:
767:
764:
761:
758:
755:
735:
732:
729:
726:
723:
712:
711:
699:
696:
691:
686:
683:
680:
676:
653:
650:
627:
615:
612:
607:
602:
599:
596:
592:
572:, and playing
543:
542:
526:
523:
520:
517:
515:
512:
509:
506:
503:
501:
498:
497:
494:
491:
488:
485:
482:
480:
477:
474:
471:
469:
466:
465:
462:
459:
456:
453:
450:
448:
445:
442:
439:
437:
434:
433:
430:
427:
425:
422:
420:
373:
372:
354:
350:
339:
325:
322:
318:
307:
289:
285:
274:
258:
255:
245:
235:
234:
222:
217:
214:
210:
206:
201:
197:
193:
188:
184:
175:
172:
168:
163:
155:
151:
146:
142:
137:
132:
128:
105:
102:
100:
97:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5736:
5725:
5722:
5720:
5717:
5715:
5712:
5710:
5707:
5705:
5702:
5700:
5697:
5695:
5692:
5690:
5687:
5685:
5682:
5681:
5679:
5664:
5661:
5659:
5656:
5654:
5651:
5649:
5646:
5644:
5641:
5639:
5638:Risk aversion
5636:
5634:
5631:
5629:
5626:
5624:
5621:
5619:
5616:
5614:
5611:
5609:
5606:
5604:
5601:
5599:
5596:
5594:
5591:
5589:
5586:
5584:
5581:
5579:
5576:
5574:
5571:
5569:
5566:
5565:
5563:
5559:
5553:
5550:
5548:
5545:
5544:
5542:
5538:
5534:
5527:
5522:
5520:
5515:
5513:
5508:
5507:
5504:
5492:
5489:
5487:
5484:
5482:
5479:
5477:
5474:
5472:
5469:
5467:
5464:
5462:
5459:
5457:
5454:
5452:
5449:
5447:
5444:
5442:
5439:
5438:
5436:
5434:Miscellaneous
5432:
5426:
5423:
5421:
5418:
5416:
5413:
5411:
5408:
5406:
5403:
5401:
5398:
5397:
5395:
5391:
5385:
5382:
5380:
5377:
5375:
5372:
5370:
5369:Samuel Bowles
5367:
5365:
5364:Roger Myerson
5362:
5360:
5357:
5355:
5354:Robert Aumann
5352:
5350:
5347:
5345:
5342:
5340:
5337:
5335:
5332:
5330:
5327:
5325:
5322:
5320:
5317:
5315:
5312:
5310:
5309:Lloyd Shapley
5307:
5305:
5302:
5300:
5297:
5295:
5294:Kenneth Arrow
5292:
5290:
5287:
5285:
5282:
5280:
5277:
5275:
5274:John Harsanyi
5272:
5270:
5267:
5265:
5262:
5260:
5257:
5255:
5252:
5250:
5247:
5245:
5244:Herbert Simon
5242:
5240:
5237:
5235:
5232:
5230:
5227:
5225:
5222:
5220:
5217:
5215:
5212:
5210:
5207:
5205:
5202:
5200:
5197:
5195:
5192:
5190:
5187:
5185:
5182:
5181:
5179:
5173:
5167:
5164:
5162:
5159:
5157:
5154:
5152:
5149:
5147:
5144:
5142:
5139:
5137:
5134:
5132:
5129:
5127:
5124:
5123:
5121:
5117:
5111:
5108:
5106:
5103:
5101:
5098:
5096:
5093:
5091:
5088:
5086:
5083:
5081:
5078:
5076:
5073:
5071:
5068:
5066:
5063:
5061:
5058:
5056:
5053:
5051:
5048:
5046:
5045:Fair division
5043:
5041:
5038:
5036:
5033:
5031:
5028:
5026:
5023:
5021:
5020:Dictator game
5018:
5016:
5013:
5011:
5008:
5006:
5003:
5001:
4998:
4996:
4993:
4991:
4988:
4986:
4983:
4981:
4978:
4976:
4973:
4971:
4968:
4966:
4963:
4961:
4958:
4956:
4953:
4951:
4948:
4946:
4943:
4941:
4938:
4936:
4933:
4931:
4928:
4926:
4923:
4921:
4918:
4916:
4913:
4911:
4908:
4907:
4905:
4903:
4899:
4893:
4892:Zero-sum game
4890:
4888:
4885:
4883:
4880:
4878:
4875:
4873:
4870:
4868:
4865:
4863:
4862:Repeated game
4860:
4858:
4855:
4853:
4850:
4848:
4845:
4843:
4841:
4837:
4835:
4832:
4830:
4827:
4825:
4822:
4820:
4817:
4815:
4812:
4811:
4809:
4807:
4801:
4795:
4792:
4790:
4787:
4785:
4782:
4780:
4779:Pure strategy
4777:
4775:
4772:
4770:
4767:
4765:
4762:
4760:
4757:
4755:
4752:
4750:
4747:
4745:
4744:De-escalation
4742:
4740:
4737:
4735:
4732:
4730:
4727:
4725:
4722:
4720:
4717:
4716:
4714:
4712:
4708:
4702:
4699:
4697:
4694:
4692:
4689:
4687:
4686:Shapley value
4684:
4682:
4679:
4677:
4674:
4672:
4669:
4667:
4664:
4662:
4659:
4657:
4654:
4652:
4649:
4647:
4644:
4642:
4639:
4637:
4634:
4632:
4629:
4627:
4624:
4622:
4619:
4617:
4614:
4612:
4609:
4607:
4604:
4602:
4599:
4597:
4594:
4592:
4589:
4587:
4584:
4582:
4579:
4578:
4576:
4574:
4570:
4566:
4560:
4557:
4555:
4554:Succinct game
4552:
4550:
4547:
4545:
4542:
4540:
4537:
4535:
4532:
4530:
4527:
4525:
4522:
4520:
4517:
4515:
4512:
4510:
4507:
4505:
4502:
4500:
4497:
4495:
4492:
4490:
4487:
4485:
4482:
4480:
4477:
4475:
4472:
4471:
4469:
4465:
4461:
4453:
4448:
4446:
4441:
4439:
4434:
4433:
4430:
4418:
4415:
4413:
4410:
4409:
4406:
4400:
4397:
4395:
4392:
4390:
4387:
4385:
4382:
4380:
4377:
4375:
4372:
4370:
4367:
4365:
4362:
4360:
4357:
4355:
4352:
4350:
4349:Hash function
4347:
4345:
4342:
4340:
4337:
4335:
4332:
4330:
4327:
4325:
4322:
4320:
4317:
4315:
4312:
4310:
4307:
4305:
4304:Binary search
4302:
4300:
4297:
4296:
4294:
4290:
4284:
4281:
4279:
4276:
4274:
4271:
4269:
4266:
4264:
4261:
4259:
4256:
4254:
4251:
4249:
4246:
4244:
4241:
4239:
4236:
4234:
4231:
4229:
4226:
4224:
4221:
4219:
4216:
4215:
4213:
4209:
4205:
4201:
4194:
4189:
4187:
4182:
4180:
4175:
4174:
4171:
4163:
4159:
4158:
4153:
4149:
4145:
4141:
4137:
4133:
4128:
4124:
4120:
4116:
4112:
4111:
4106:
4102:
4101:
4097:
4089:
4085:
4081:
4077:
4073:
4069:
4065:
4061:
4054:
4051:(June 1975).
4050:
4044:
4041:
4036:
4032:
4028:
4024:
4020:
4016:
4012:
4010:
4004:
3998:
3995:
3990:
3989:
3984:
3978:
3975:
3971:
3969:
3964:
3960:
3955:
3952:
3948:
3944:
3940:
3936:
3932:
3925:
3922:
3917:
3913:
3909:
3907:9780134610993
3903:
3899:
3898:
3893:
3889:
3883:
3880:
3875:
3873:9780262150415
3869:
3865:
3861:
3854:
3851:
3846:
3844:9781107005488
3840:
3836:
3832:
3828:
3821:
3819:
3815:
3810:
3806:
3802:
3801:
3793:
3790:
3782:
3775:
3772:
3766:
3761:
3758:
3756:
3753:
3751:
3748:
3746:
3743:
3741:
3738:
3736:
3733:
3731:
3728:
3726:
3723:
3721:
3718:
3716:
3713:
3711:
3708:
3706:
3703:
3701:
3698:
3696:
3693:
3691:
3688:
3686:
3683:
3682:
3677:
3675:
3673:
3669:
3667:
3662:
3654:
3652:
3649:
3645:
3637:
3635:
3617:
3613:
3608:
3606:
3602:
3598:
3594:
3590:
3586:
3578:
3562:
3553:
3544:
3532:
3529:
3526:
3520:
3511:
3503:
3502:
3501:
3499:
3498:
3481:
3468:
3464:
3445:
3436:
3433:
3430:
3424:
3416:
3403:
3395:
3383:
3377:
3374:
3368:
3363:
3351:
3350:
3349:
3347:
3322:
3308:
3307:loss function
3292:
3283:
3280:
3277:
3271:
3261:
3260:risk function
3245:
3236:
3233:
3223:
3204:
3194:
3191:, we have an
3190:
3184:
3176:
3174:
3172:
3169:In addition,
3167:
3165:
3161:
3157:
3153:
3144:
3139:
3137:
3135:
3131:
3127:
3123:
3122:
3117:
3116:
3111:
3107:
3102:
3101:
3096:
3092:
3088:
3084:
3083:
3077:
3075:
3071:
3067:
3063:
3055:
3050:
3042:
3035:
3033:
3031:
3015:
3006:
3003:
3000:
2997:
2994:
2985:
2982:
2976:
2973:
2970:
2952:
2950:
2943:
2937:
2933:
2929:
2925:
2922:
2918:
2914:
2910:
2906:
2902:
2898:
2895:
2891:
2887:
2884:
2880:
2876:
2874:
2866:
2864:
2862:
2857:
2855:
2850:
2848:
2844:
2840:
2836:
2832:
2831:
2825:
2824:
2819:
2814:
2812:
2808:
2804:
2799:
2794:
2792:
2788:
2784:
2780:
2776:
2772:
2768:
2764:
2760:
2756:
2752:
2748:
2744:
2740:
2736:
2731:
2729:
2725:
2720:
2716:
2712:
2708:
2700:
2698:
2696:
2692:
2687:
2683:
2679:
2675:
2671:
2666:
2664:
2656:
2654:
2652:
2648:
2640:
2638:
2636:
2631:
2629:
2628:zero-sum game
2625:
2617:
2615:
2613:
2577:
2557:
2553:
2496:chose B1 and
2495:
2437:
2382:
2377:
2375:
2371:
2367:
2363:
2359:
2354:
2352:
2348:
2344:
2340:
2336:
2332:
2328:
2324:
2320:
2316:
2312:
2308:
2304:
2300:
2299:payoff matrix
2296:
2292:
2288:
2278:
2275:
2272:
2270:A chooses A3
2269:
2268:
2264:
2258:
2255:
2253:A chooses A2
2252:
2251:
2247:
2244:
2241:
2239:A chooses A1
2238:
2237:
2234:B chooses B3
2233:
2231:B chooses B2
2230:
2228:B chooses B1
2227:
2225:
2224:
2215:
2213:
2211:
2207:
2186:
2179:
2178:
2177:
2171:
2165:
2163:
2158:
2156:
2152:
2144:
2142:
2122:
2119:
2116:
2090:
2083:
2080:
2077:
2048:
2045:
2042:
2039:
2010:
2007:
2004:
1972:
1969:
1966:
1963:
1937:
1926:
1923:
1920:
1916:
1910:
1905:
1900:
1897:
1894:
1890:
1854:
1851:
1848:
1844:
1838:
1833:
1828:
1825:
1822:
1818:
1803:
1783:
1780:
1776:
1751:
1742:
1739:
1735:
1710:
1701:
1698:
1694:
1686:
1682:
1678:
1651:
1648:
1644:
1616:
1612:
1582:
1579:
1575:
1571:
1566:
1562:
1553:
1549:
1525:
1521:
1512:
1496:
1487:
1484:
1480:
1449:
1445:
1412:
1409:
1405:
1401:
1396:
1392:
1383:
1379:
1339:
1336:
1332:
1328:
1323:
1319:
1310:
1306:
1297:
1293:
1273:
1270:
1266:
1257:
1248:
1245:
1241:
1237:
1232:
1228:
1219:
1215:
1206:
1202:
1189:
1186:
1182:
1173:
1163:
1159:
1149:
1148:
1147:
1145:
1140:
1108:
1104:
1098:
1093:
1088:
1084:
1074:
1073:
1072:
1051:
1045:
1042:
1032:
1029:
1026:
1022:
987:
973:
967:
964:
954:
951:
948:
944:
917:
916:
915:
892:
889:
885:
881:
876:
872:
863:
859:
850:
846:
833:
830:
826:
817:
807:
803:
793:
792:
791:
789:
784:
783:minimax value
779:
762:
759:
756:
730:
727:
724:
697:
694:
689:
684:
681:
678:
674:
651:
648:
628:
613:
610:
605:
600:
597:
594:
590:
555:
554:
553:
551:
546:
524:
521:
518:
513:
510:
507:
504:
499:
492:
489:
486:
483:
478:
475:
472:
467:
460:
457:
454:
451:
446:
443:
440:
435:
428:
423:
410:
409:
408:
385:
352:
348:
340:
323:
320:
316:
308:
287:
283:
275:
256:
253:
246:
240:
239:
238:
215:
212:
208:
204:
199:
195:
186:
182:
173:
170:
166:
153:
149:
140:
135:
130:
126:
116:
115:
114:
111:
110:maximin value
103:
98:
96:
94:
91:
87:
85:
79:
76:the possible
75:
71:
67:
63:
59:
55:
51:
47:
43:
39:
33:
19:
5587:
5339:Peyton Young
5334:Paul Milgrom
5249:Hervé Moulin
5189:Amos Tversky
5135:
5131:Folk theorem
4842:-player game
4839:
4764:Grim trigger
4374:Root-finding
4353:
4299:Backtracking
4263:Segment tree
4233:Fenwick tree
4155:
4144:the original
4139:
4126:
4108:
4063:
4059:
4049:Harsanyi, J.
4043:
4018:
4014:
4008:
4005:(May 1973).
3997:
3987:
3977:
3968:New Politics
3966:
3959:Noam Chomsky
3954:
3946:
3934:
3930:
3924:
3895:
3882:
3863:
3853:
3830:
3827:Solan, Eilon
3807:– via
3799:
3792:
3774:
3664:
3658:
3647:
3641:
3615:
3609:
3582:
3495:
3460:
3345:
3186:
3168:
3160:Murphy's law
3151:
3148:
3129:
3125:
3119:
3113:
3109:
3105:
3098:
3094:
3090:
3086:
3080:
3078:
3073:
3069:
3065:
3059:
2953:
2946:
2941:
2935:
2931:
2927:
2923:
2920:
2916:
2912:
2908:
2904:
2900:
2896:
2893:
2889:
2885:
2882:
2878:
2870:
2858:
2851:
2843:forced moves
2827:
2821:
2815:
2795:
2782:
2778:
2774:
2770:
2766:
2762:
2758:
2754:
2750:
2738:
2734:
2732:
2727:
2723:
2706:
2704:
2694:
2690:
2685:
2681:
2673:
2669:
2667:
2660:
2651:folk theorem
2644:
2632:
2623:
2621:
2575:
2555:
2551:
2493:
2435:
2380:
2378:
2373:
2369:
2365:
2361:
2357:
2355:
2350:
2346:
2342:
2338:
2334:
2330:
2326:
2322:
2318:
2314:
2310:
2306:
2302:
2294:
2290:
2286:
2284:
2205:
2195:
2167:
2159:
2148:
1804:
1510:
1367:
1143:
1141:
1130:
1066:
913:
787:
782:
780:
713:
547:
544:
386:
374:
236:
109:
107:
83:
82:worst case (
73:
50:saddle point
49:
45:
41:
37:
36:
5709:Game theory
5643:Game theory
5456:Coopetition
5259:Jean Tirole
5254:John Conway
5234:Eric Maskin
5030:Blotto game
5015:Pirate game
4824:Global game
4794:Tit for tat
4729:Bid shading
4719:Appeasement
4569:Equilibrium
4549:Solved game
4484:Determinacy
4467:Definitions
4460:game theory
4253:Linked list
3831:Game Theory
3745:Tit for Tat
3644:lesser evil
3115:parent node
3032:algorithm.
2847:impractical
2678:tic-tac-toe
2204:. The name
99:Game theory
93:game theory
62:game theory
40:(sometimes
5678:Categories
5100:Trust game
5085:Kuhn poker
4754:Escalation
4749:Deterrence
4739:Cheap talk
4711:Strategies
4529:Preference
4458:Topics of
4389:Sweep line
4364:Randomized
4292:Algorithms
4243:Hash table
4204:algorithms
3931:IEEE Micro
3767:References
3661:John Rawls
3344:is called
3110:child node
3100:child node
3074:look-ahead
2949:leaf nodes
2919:value
2888:depth = 0
2873:pseudocode
2867:Pseudocode
2828:effective
2637:strategy.
1012:). Hence:
664:). Hence:
580:). Hence:
74:minimizing
70:philosophy
66:statistics
5540:Decisions
5284:John Nash
4990:Stag hunt
4734:Collusion
4384:Streaming
4369:Recursion
4152:"Minimax"
4115:EMS Press
4088:118261543
4003:Arrow, K.
3983:Rawls, J.
3803:(video).
3735:Negascout
3554:θ
3548:Π
3545:
3533:δ
3527:θ
3516:Θ
3512:∫
3476:Π
3437:δ
3431:θ
3417:θ
3404:δ
3387:~
3384:δ
3375:θ
3364:θ
3326:~
3323:δ
3284:δ
3278:θ
3240:Θ
3237:∈
3234:θ
3222:parameter
3205:δ
3193:estimator
3132:possible
3121:root node
3082:leaf node
3004:−
2995:−
2986:−
2823:game tree
2807:Deep Blue
2798:heuristic
2711:algorithm
2674:algorithm
2438:would be
2376:chooses.
2368:chooses;
2358:dominated
2341:believes
2329:believes
2040:−
1970:−
1932:¯
1911:≤
1906:_
1860:¯
1839:≤
1834:_
1781:−
1740:−
1699:−
1649:−
1580:−
1509:We first
1485:−
1410:−
1337:−
1271:−
1246:−
1187:−
1169:¯
1114:¯
1099:≤
1094:_
1038:¯
960:¯
890:−
831:−
813:¯
690:_
649:−
637:(playing
606:_
564:(playing
505:−
484:−
458:−
321:−
254:−
213:−
171:−
136:_
5561:Concepts
5425:Lazy SMP
5119:Theorems
5070:Deadlock
4925:Checkers
4806:of games
4573:concepts
3985:(1971).
3916:20190474
3894:(2021).
3862:(1994).
3678:See also
3616:interval
3126:minimize
3095:smallest
2928:for each
2909:for each
2879:function
2835:children
2695:minimize
2691:maximize
2610:. These
2550:in case
2532:−
2492:in case
2474:−
2170:zero-sum
1687:′
1144:notation
90:zero-sum
5593:Leximin
5588:Minimax
5177:figures
4960:Chicken
4814:Auction
4804:Classes
4379:Sorting
4354:Minimax
4117:, 2001
4080:1959090
4035:2025006
3809:YouTube
3730:Negamax
3648:minimax
3497:average
3346:minimax
3130:maximum
3108:of the
3106:largest
3097:of the
3054:negamax
3036:Example
3030:negamax
2624:maximin
2618:Maximin
2608:
2596:
2592:
2580:
2572:
2560:
2547:
2528:
2516:
2512:
2500:
2489:
2470:
2458:
2454:
2442:
2430:
2418:
2413:
2401:
2397:
2385:
2379:Player
2295:maximin
2216:Example
2206:minimax
934:), so:
237:Where:
38:Minimax
4359:Online
4344:Greedy
4273:String
4086:
4078:
4033:
3914:
3904:
3870:
3841:
3597:robust
3560:
3539:
3479:
3443:
3422:
3409:
3332:
3317:
3290:
3269:
3243:
3231:
3208:
3202:
3152:nature
3013:
2962:
2938:value
2936:return
2917:return
2897:return
2826:. The
2670:simple
2514:+ 0 ×
2456:− 1 ×
2129:
2111:
2072:
2017:
1999:
1979:
1958:
1885:
1865:
1813:
1790:
1772:
1749:
1731:
1675:
1623:
1608:
1494:
1476:
1456:
1441:
1421:
1376:
1049:
971:
80:for a
68:, and
42:Minmax
4915:Chess
4902:Games
4268:Stack
4258:Queue
4238:Graph
4218:Array
4160:. US
4084:S2CID
4076:JSTOR
4056:(PDF)
4031:JSTOR
3784:(PDF)
3500:risk
3154:(see
2820:of a
2818:nodes
2803:plies
2787:chess
2612:mixed
2498:−2 ×
2184:, and
1541:from
1004:) or
926:) or
395:, or
5603:Risk
4596:Core
4339:Fold
4283:Trie
4278:Tree
4248:Heap
4202:and
4162:NIST
3912:LCCN
3902:ISBN
3868:ISBN
3839:ISBN
3628:) =
3603:and
3134:loss
3128:the
3062:tree
2921:else
2905:then
2894:then
2871:The
2773:and
2715:game
2440:3 ×
2301:for
2289:and
1877:and
1468:and
1008:(if
1000:(if
788:know
781:The
570:−100
108:The
78:loss
72:for
5175:Key
4068:doi
4023:doi
3965:,"
3939:doi
3663:'s
3587:or
3413:sup
3400:inf
3360:sup
3162:or
2989:min
2965:max
2789:or
2686:can
2682:can
2661:In
2279:+1
2276:−3
2273:−4
2265:+4
2256:−1
2248:+2
2245:−2
2242:+3
2029:or
1289:max
1262:min
1198:max
1178:min
996:),
842:max
822:min
578:−10
508:100
403:or
162:min
145:max
84:max
48:or
5680::
4910:Go
4154:.
4138:.
4125:.
4113:,
4107:,
4082:.
4074:.
4064:69
4062:.
4058:.
4029:.
4019:70
4017:.
4013:.
3945:.
3935:19
3933:.
3910:.
3890:;
3833:.
3817:^
3632:."
3607:.
3136:.
2932:do
2913:do
2901:if
2890:or
2886:if
2883:is
2791:go
2705:A
2668:A
2605:3
2589:3
2569:3
2544:3
2530:=
2525:6
2509:6
2486:3
2472:=
2467:6
2451:6
2427:6
2410:6
2394:3
2262:0
2212:.
2157:.
2043:10
1973:20
778:.
652:20
487:10
461:20
391:,
64:,
60:,
56:,
46:MM
44:,
5525:e
5518:t
5511:v
4840:n
4451:e
4444:t
4437:v
4192:e
4185:t
4178:v
4164:.
4090:.
4070::
4037:.
4025::
4011:"
3970:,
3941::
3918:.
3876:.
3847:.
3811:.
3668:,
3630:n
3626:X
3624:(
3622:ℰ
3563:.
3557:)
3551:(
3542:d
3536:)
3530:,
3524:(
3521:R
3482:.
3446:.
3440:)
3434:,
3428:(
3425:R
3396:=
3393:)
3378:,
3372:(
3369:R
3293:.
3287:)
3281:,
3275:(
3272:R
3246:.
3016:,
3010:)
3007:b
3001:,
2998:a
2992:(
2983:=
2980:)
2977:b
2974:,
2971:a
2968:(
2801:"
2775:B
2767:A
2763:B
2759:A
2755:B
2751:A
2739:B
2735:A
2728:A
2724:A
2602:/
2599:2
2586:/
2583:1
2576:A
2566:/
2563:1
2556:B
2552:B
2541:/
2538:1
2534:+
2522:/
2519:5
2506:/
2503:1
2494:B
2483:/
2480:1
2476:+
2464:/
2461:5
2448:/
2445:1
2436:A
2432::
2424:/
2421:5
2407:/
2404:1
2391:/
2388:1
2381:A
2374:A
2370:B
2366:B
2362:A
2351:B
2347:A
2343:B
2339:A
2335:B
2331:A
2327:B
2323:B
2319:A
2315:A
2311:B
2307:B
2303:A
2291:B
2287:A
2260:+
2202:V
2198:V
2191:.
2189:V
2182:V
2174:V
2126:)
2123:1
2120:,
2117:3
2114:(
2091:,
2087:)
2084:R
2081:,
2078:M
2075:(
2052:)
2049:1
2046:,
2037:(
2014:)
2011:R
2008:,
2005:T
2002:(
1976:)
1967:,
1964:2
1961:(
1938:,
1927:l
1924:o
1921:c
1917:v
1901:l
1898:o
1895:c
1891:v
1855:w
1852:o
1849:r
1845:v
1829:w
1826:o
1823:r
1819:v
1784:i
1777:a
1752:.
1743:i
1736:a
1711:,
1707:)
1702:i
1695:a
1691:(
1683:i
1679:v
1652:i
1645:a
1617:i
1613:a
1588:)
1583:i
1576:a
1572:,
1567:i
1563:a
1559:(
1554:i
1550:v
1526:i
1522:a
1497:.
1488:i
1481:a
1450:i
1446:a
1418:)
1413:i
1406:a
1402:,
1397:i
1393:a
1389:(
1384:i
1380:v
1351:)
1345:)
1340:i
1333:a
1329:,
1324:i
1320:a
1316:(
1311:i
1307:v
1298:i
1294:a
1283:(
1274:i
1267:a
1258:=
1254:)
1249:i
1242:a
1238:,
1233:i
1229:a
1225:(
1220:i
1216:v
1207:i
1203:a
1190:i
1183:a
1174:=
1164:i
1160:v
1137:i
1133:i
1109:i
1105:v
1089:i
1085:v
1069:i
1052:.
1046:1
1043:=
1033:l
1030:o
1027:c
1023:v
1010:B
1006:4
1002:M
998:1
994:T
990:1
974:.
968:4
965:=
955:w
952:o
949:r
945:v
932:L
928:5
924:R
920:4
898:)
893:i
886:a
882:,
877:i
873:a
869:(
864:i
860:v
851:i
847:a
834:i
827:a
818:=
808:i
804:v
766:)
763:1
760:,
757:3
754:(
734:)
731:L
728:,
725:T
722:(
710:.
698:0
695:=
685:l
682:o
679:c
675:v
639:R
635:0
631:L
626:.
614:2
611:=
601:w
598:o
595:r
591:v
574:M
566:B
562:2
558:T
525:4
522:,
519:4
514:2
511:,
500:B
493:1
490:,
479:0
476:,
473:5
468:M
455:,
452:2
447:1
444:,
441:3
436:T
429:R
424:L
405:R
401:L
397:B
393:M
389:T
382:i
378:i
371:.
369:i
353:i
349:v
324:i
317:a
306:.
304:i
288:i
284:a
273:.
271:i
257:i
242:i
221:)
216:i
209:a
205:,
200:i
196:a
192:(
187:i
183:v
174:i
167:a
154:i
150:a
141:=
131:i
127:v
34:.
20:)
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