Ueda Masahiko

Imitation is simple behavior which uses successful actions of others in order to deal with one’s own problems. Because success of imitation generally depends on whether profit of an imitating agent coincides with those of other agents or not, game theory is suitable for specifying situations where imitation can be successful. One of the concepts describing successfulness of imitation in repeated two-player symmetric games is unbeatability. For infinitely repeated two-player symmetric games, a necessary and sufficient condition for some imitation strategy to be unbeatable was specified. However, situations where imitation can be unbeatable in multi-player games are still not clear. In order to analyze successfulness of imitation in multi-player situations, here we introduce a class of totally symmetric games called unexploitable games, which is a natural extension of two-player symmetric games without exploitation cycles. We then prove that, for infinitely repeated unexploitable games, there exist unbeatable imitation strategies. Furthermore, we also prove that, for infinitely repeated non-trivial unexploitable games, there exist unbeatable zero-determinant strategies, which unilaterally enforce some relationships on payoffs of players. These claims are demonstrated in the public goods game, which is the simplest unexploitable game. These results show that there are situations where imitation is unbeatable even in multi-player games.

We investigate symmetric equilibria of mutual reinforcement learning when both players alternately learn the optimal memory-two strategies against the oppo-nent in the repeated prisoners' dilemma game. We provide a necessary condi-tion for memory-two deterministic strategies to form symmetric equilibria. We then provide three examples of memory-two deterministic strategies which form symmetric mutual reinforcement learning equilibria. We also prove that mu-tual reinforcement learning equilibria formed by memory-two strategies are also mutual reinforcement learning equilibria when both players use reinforcement learning of memory-n strategies with n > 2.

Zero-determinant strategies are memory-one strategies in repeated　games which unilaterally enforce linear relations between expected payoffs of　players. Recently, the concept of zero-determinant strategies was extended　to　the class of memory-n strategies with n _ 1, which enables more compli-cated control of payoffs by one player. However, what we can do by　memory-n　zero-determinant strategies is still not clear. Here, we show that memory-n　zero-determinant strategies in repeated games can be used to control condi-tional expectations of payoffs. Equivalently, they can be used to control ex-pected payoffs in biased ensembles, where a history of action pro_les with large　value of bias function is more weighted. Controlling conditional expectations　of payoffs is useful for strengthening zero-determinant strategies, because play-ers can choose conditions in such a way that only unfavorable action pro_les　to one player are contained in the conditions. We provide several examples　of memory-n zero-determinant strategies in the repeated prisoner's dilemma　game. We also explain that a deformed version of zero-determinant strategies　is easily extended to the memory-n case.

Zero-determinant strategies are a class of memory-one strategies in repeated games which unilaterally enforce linear relationships between payoffs. It has long been unclear for what stage games zero-determinant strategies exist. We provide a necessary and sufficient condition for the existence of zero-determinant strategies. This condition can be interpreted as the existence of two different actions which unilaterally adjust the total value of a linear combination of payoffs. A relation between the class of stage games where zero-determinant strategies exist and other class of stage games is also provided.

We study the diffusion of Brownian particles in a Gaussian random velocity field with short memory. By extending the derivation of an effective Fokker–Planck equation for the Lanvegin equation with weakly colored noise to a random velocity-field problem, we find that the effect of thermal noise on particles is suppressed by the existence of memory. We also find that the renormalization effect for the relative diffusion of two particles is stronger than that for single-particle diffusion. The results are compared with those of molecular dynamics simulations.