Unexploitable Games and Unbeatable Strategies

Abstract

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 zerodeterminant 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.INDEX TERMS Imitation strategies, repeated games, unbeatable strategies, zero-determinant strategies.

Necessary and Sufficient Condition for the Existence of Zero-Determinant Strategies in Repeated Games

“…Because it has been known that TFT is unbeatable in two-player symmetric potential games [43], relation between unbeatability and ZD strategies was focused on, and it was indeed shown that unbeatable TFT is a ZD strategy [39]. Furthermore, relation between the existence of unbeatable imitation [44] and that of unbeatable ZD strategies in two-player symmetric games [23] and in multi-player totally symmetric games [40] has also been shown. These application studies show that the concept of ZD strategies can be useful as methods to control multi-agent systems and to analyze human behavior.…”

“…In the field of information and communications technology, ZD strategies have been applied in resource sharing in wireless networks [27,28], smart grid systems [29], mining in blockchain [30,31], crowdsourcing [32,33], mobile crowdsensing [34], the Internet of Things [35], cloud computing [36], federated learning [37], and data trading [38], in order to unilaterally realize some favorable states. Another field of application is analysis of imitation in behavioral science [39,40]. Originally, in the prisoner's dilemma game, it was pointed out that the Tit-for-Tat (TFT) strategy [41,42], which imitates the previous action of the opponent, is a ZD strategy which unilaterally equalizes the payoffs of two players [1].…”