Extensive Form Games

The two-player Iterated Prisoner's Dilemma game is a model for both sentient and evolutionary behaviors, especially including the emergence of cooperation. It is generally assumed that there exists no simple ultimatum strategy whereby one player can enforce a unilateral claim to an unfair share of rewards. Here, we show that such strategies unexpectedly do exist. In particular, a player X who is witting of these strategies can (i) deterministically set her opponent Y's score, independently of his strategy or response, or (ii) enforce an extortionate linear relation between her and his scores. Against such a player, an evolutionary player's best response is to accede to the extortion. Only a player with a theory of mind about his opponent can do better, in which case Iterated Prisoner's Dilemma is an Ultimatum Game. evolution of cooperation | game theory | tit for tat I terated 2 × 2 games, with Iterated Prisoner's Dilemma (IPD) as the notable example, have long been touchstone models for elucidating both sentient human behaviors, such as cartel pricing, and Darwinian phenomena, such as the evolution of cooperation (1-6). Well-known popular treatments (7-9) have further established IPD as foundational lore in fields as diverse as political science and evolutionary biology. It would be surprising if any significant mathematical feature of IPD has remained undescribed, but that appears to be the case, as we show in this paper. shows the setup for a single play of Prisoner's Dilemma (PD). If X and Y cooperate (c), then each earns a reward R. If one defects (d), the defector gets an even larger payment T, and the naive cooperator gets S, usually zero. However, if both defect, then both get a meager payment P. To be interesting, the game must satisfy two inequalities: T > R > P > S guarantees that the Nash equilibrium of the game is mutual defection, whereas 2R > T þ S makes mutual cooperation the globally best outcome. The "conventional values" ðT; R; P; SÞ ¼ ð5; 3; 1; 0Þ occur most often in the literature. We derive most results in the general case, and indicate when there is a specialization to the conventional values. shows an iterated IPD game consisting of multiple, successive plays by the same opponents. Opponents may now condition their play on their opponent's strategy insofar as each can deduce it from the previous play. However, we give each player only a finite memory of previous play (10). One might have thought that a player with longer memory always has the advantage over a more forgetful player. In the game of bridge, for example, a player who remembers all of the cards played has the advantage over a player who remembers only the last trick; however, that is not the case when the same game (same allowed moves and same payoff matrices) is indefinitely repeated. In fact, it is easy to prove (Appendix A) that, for any strategy of the longer-memory player Y, shorter-memory X's score is exactly the same as if Y had played a certain shorter-memory strategy (roughly, the marginalization of Y's long-memory strategy: its average over states remembered by Y but not by X), disregarding any history in excess of that shared with X. This fact is important. We derive strategies for X assuming that both players have memory of only a single previous move, and the above theorem shows that this involves no loss of generality. Longer memory will not give Y any advantage. , then, shows the most general memory-one game. The four outcomes of the previous move are labeled 1; . . . ; 4 for the respective outcomes xy ∈ ðcc; cd; dc; ddÞ, where c and d denote cooperation and defection. X's strategy is p ¼ ðp 1 ; p 2 ; p 3 ; p 4 Þ, her probabilities for cooperating under each of the previous outcomes. Y's strategy is analogously q ¼ ðq 1 ; q 2 ; q 3 ; q 4 Þ for outcomes seen from his perspective, that is, in the order of yx ∈ ðcc; cd; dc; ddÞ. The outcome of this play is determined by a product of probabilities, as shown in

International Journal of Game Theory, 1971

For a class of repeated two-person zero-sum games with incomplete reformation it was proved by AUMANN and MASCHLER that lim v. exists, v. being the value of the game with n repetitions. n~o0 As for the speed of convergence AUMANN and MASCHLER showed that the error term 6. = I vn-lira v.I is bounded from above by c/Vn for some positive constant c. Both results have been generalized by M~RTENS and ZAMIR. It is shown in this paper that the above mentioned theorem about the speed of convergence is sharp in the sense that there are games in which 3. > c'/V~-for some positive constant c'. However there are games for which 6. is of a lower order of magnitude, for instance c'(log n)/n <<_ 3. < c (log n)/n or c'/n < 6. < c/n. Sufficient conditions are given here for games to belong to one of these categories as well as examples of games from each category.

International Journal of Game Theory, 1973

For a class of repeated two-person zero-sum games with incomplete information it was proved by AUMANN and MASCnLER that lim v. exists, v. being the value of the game with n repetitions. If the players know at each stage the moves done by both players at all previous stages, AUMANN and MASCHLER could prove that the error term 6. = I v.-lim v.[ satisfies 6. < c/V~ for some c > 0. It was then shown by ZAMm that this bound is the lowest possible. In this paper it is shown that if previous moves are not always announced, 6. may be of higher order of magnitude e.g. 6. >_ c/n a/3 for some c > 0. New upper bounds for 6. are given for two classes of games.

1982

1997

From the $ex$ ante point of view,.we consider decision making in partially interactive games. In a partially interactive game, players may not be required to predict all the others' decisions such as in Prisoner's Dilemma. In some games, some players can ignore some other players' decision making but need to predict decisions of some different ones. In this case, the common knowledge of the entire structure of a game may be unnecessary, but some different form of the knowledge of the payoff functions of some players as well as of their behavioral postulates may be required. We develop some generalizations of Nash equilibrium and dominant strategies in Part I, and then investigate epistemic aspects of decision making in games in the game logic framework in Parts II and III. One new concept, an interaction structure, plays an essential role in both developments. We give a full characterization of the knowledge structure necessary for decision making in the games whose Nash equilibria satisfies a certain weakening of interchangeability. This characterization tells when the solution necessarily involves the common knowledge or not. In Part I, we develop game theoretical notions, and in Parts II and III, we will investigate the knowledge structure required for decision making in games in the game logic framework.