Eizo Akiyama

Chaos in learning a simple two-person game

We investigate the problem of learning to play the game of rock–paper–scissors. Each player attempts to improve her/his average score by adjusting the frequency of the three possible responses, using reinforcement learning. For the zero sum game the learning process displays Hamiltonian chaos. Thus, the learning trajectory can be simple or complex, depending on initial conditions. We also investigate the non-zero sum case and show that it can give rise to chaotic transients. This is, to our knowledge, the first demonstration of Hamiltonian chaos in learning a basic two-person game, extending earlier findings of chaotic attractors in dissipative systems. As we argue here, chaos provides an important self-consistency condition for determining when players will learn to behave as though they were fully rational. That chaos can occur in learning a simple game indicates one should use caution in assuming real people will learn to play a game according to a Nash equilibrium strategy.

Reputation and the evolution of cooperation in sizable groups

The evolution of cooperation in social dilemmas has been of considerable concern in various fields such as sociobiology, economics and sociology. It might be that, in the real world, reputation plays an important role in the evolution of cooperation. Recently, studies that have addressed indirect reciprocity have revealed that cooperation can evolve through reputation, even though pairs of individuals interact only a few times. To our knowledge, most indirect reciprocity models have presumed dyadic interaction; no studies have attempted analysis of the evolution of cooperation in large communities where the effect of reputation is included. We investigate the evolution of cooperation in sizable groups in which the reputation of individuals affects the decision-making process. This paper presents the following: (i) cooperation can evolve in a four-person case, (ii) the evolution of cooperation becomes difficult as group size increases, even if the effect of reputation is included, and (iii) three kinds of final social states exist. In medium-sized communities, cooperative species can coexist in a stable manner with betrayal species.

Neural basis of conditional cooperation

Cooperation among genetically unrelated individuals is a fundamental aspect of society, but it has been a longstanding puzzle in biological and social sciences. Recently, theoretical studies in biology and economics showed that conditional cooperation-cooperating only with those who have exhibited cooperative behavior-can spread over a society. Furthermore, experimental studies in psychology demonstrated that people are actually conditional cooperators. In this study, we used functional magnetic resonance imaging to investigate the neural system underlying conditional cooperation by scanning participants during interaction with cooperative, neutral and non-cooperative opponents in prisoners dilemma games. The results showed that: (i) participants cooperated more frequently with both cooperative and neutral opponents than with non-cooperative opponents; and (ii) a brain area related to cognitive inhibition of pre-potent responses (right dorsolateral prefrontal cortex) showed greater activation, especially when participants confronted non-cooperative opponents. Consequently, we suggest that cognitive inhibition of the motivation to cooperate with non-cooperators drives the conditional behavior.

Dynamical systems game theory and dynamics of games

A theoretical framework we call dynamical systems game is presented, in which the game itself can change due to the influence of players’ behaviors and states. That is, the nature of the game itself is described as a dynamical system. The relation between game dynamics and the evolution of strategies is discussed by applying this framework. Computer experiments are carried out for simple one-person games to demonstrate the evolution of dynamical systems with the effective use of dynamical resources.

Evolutionary stability of first-order-information indirect reciprocity in sizable groups

Indirect reciprocity is considered as a key mechanism for explaining the evolution of cooperation in situations where the same individuals interact only a few times. Under indirect reciprocity, an individual who helps others gets returns indirectly from others who know her good reputation. Recently, many studies have discussed the effect of reputation criteria based only on the former actions of the others (first-order information) and of those based also on the former reputation of opponents of the others (second-order information) on the evolution of indirect reciprocity. In this study, we investigate the evolutionary stability of the indirectly reciprocal strategy (discriminating strategy: DIS), which cooperates only with opponents who have good reputations, in n(>2)-person games where more than two individuals take part in a single group (interaction). We show that in n-person games, DIS is an evolutionarily stable strategy (ESS) even under the image-scoring reputation criterion, which is based only on first-order information and where cooperations (defections) are judged to be good (bad). This result is in contrast to that of 2-person games where DIS is not an ESS under reputation criteria based only on first-order information.

Dynamical systems game theory II: A new approach to the problem of the social dilemma

The “social dilemma” is a problem inherent in forming and maintaining cooperation among selfish individuals, and is of fundamental importance in the biological and social sciences. From the viewpoint of traditional game theory, the existence of the social dilemma necessarily implies degeneration into selfish behavior as the numbers of members in a community increases, unless there exists some external power. In the real world, however, cooperation is often formed and maintained merely through mutual interactions, without the influence of an external power. To answer questions concerning appearance and maintenance of cooperative behavior in societies, we study what we call the “Lumberjacks’ Dilemma (LD) game”, as an application of the dynamical systems (DS) game theory presented in [Physica D 147 (2000) 221], which can naturally deal with the dynamic aspects of games. Dynamical processes that lead to the formation and maintenance of cooperation, which is often observed in the real communities, are realized in our model. The mechanism underlying this formation and maintenance is explained from the DS game point of view, by analyzing the functional dependence of the attractor of the game dynamics on a parameter characterizing the strategy. It is demonstrated that norms for cooperation are formed as strategies that are manifested as specific attractors of game dynamics. The change in the stability of this cooperative behavior as the number of members increases is also discussed. Finally, the relevance of our study to cooperation seen in the real world is discussed.

It is Not Just Confusion! Strategic Uncertainty in an Experimental Asset Market

To what extent is the observed mis-pricing in experimental asset markets caused by strategic uncertainty (SU) and by individual bounded rationality (IBR)? We address this question by comparing subjects initial price forecasts in two market environments - one with six human traders, and the other with one human and five computer traders. We find that both SU and IBR account equally for the median initial forecasts deviation from the fundamental values. The effect of SU is greater for subjects with a perfect score in the Cognitive Reflection Test, and it is not significant for those with low scores.

Chaos, oscillation and the evolution of indirect reciprocity in n-person games

Evolution of cooperation among genetically unrelated individuals has been of considerable concern in various fields such as biology, economics, and psychology. The evolution of cooperation is often explained by reciprocity. Under reciprocity, cooperation can prevail in a society because a donor of cooperation receives reciprocation from the recipient of the cooperation, called direct reciprocity, or from someone else in the community, called indirect reciprocity. Nowak and Sigmund [1993. Chaos and the evolution of cooperation. Proc. Natl. Acad. Sci. USA 90, 5091-5094] have demonstrated that directly reciprocal cooperation in two-person prisoners dilemma games with mutation of strategies can be maintained dynamically as periodic or chaotic oscillation. Furthermore, Eriksson and Lindgren [2005. Cooperation driven by mutations in multi-person Prisoners Dilemma. J. Theor. Biol. 232, 399-409] have reported that directly reciprocal cooperation in n-person prisoners dilemma games (n>2) can be maintained as periodic oscillation. Is dynamic cooperation observed only in direct reciprocity? Results of this study show that indirectly reciprocal cooperation in n-person prisoners dilemma games can be maintained dynamically as periodic or chaotic oscillation. This is, to our knowledge, the first demonstration of chaos in indirect reciprocity. Furthermore, the results show that oscillatory dynamics are observed in common in the evolution of reciprocal cooperation whether for direct or indirect.

Heterogeneity of link weight and the evolution of cooperation

In this paper, we investigate the effect of heterogeneity of link weight, heterogeneity of the frequency or amount of interactions among individuals, on the evolution of cooperation. Based on an analysis of the evolutionary prisoner’s dilemma game on a weighted one-dimensional lattice network with intra-individual heterogeneity, we confirm that moderate level of link-weight heterogeneity can facilitate cooperation. Furthermore, we identify two key mechanisms by which link-weight heterogeneity promotes the evolution of cooperation: mechanisms for spread and maintenance of cooperation. We also derive the corresponding conditions under which the mechanisms can work through evolutionary dynamics.

Selection of opponents in the prisoner׳s dilemma in dynamic networks: an experimental approach.

To investigate how a human subject selects her neighbors (opponents) to play the Prisoner׳s Dilemma within a social network, we conducted a human-subject experiment. The results are as follows: (1) A subject is more likely to dismiss the links to her neighbors most frequently when the subject chooses C and when the neighbor chooses D; (2) a subject who has more neighbors is less likely to dismiss links than a subject who has fewer neighbors; and (3) a subject is more likely to create links to (=select) opponents who have more neighbors than to opponents who have fewer neighbors.