Linear algebraic structure of zero-determinant strategies in repeated games
Linear algebraic structure of zero-determinant strategies in repeated games
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DOI：
10.1371/journal.pone.0230973
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发表时间：
2020-04-02
期刊：
影响因子：
3.7
通讯作者：
Tanaka, Toshiyuki
中科院分区：
文献类型：
--
作者：
Ueda, Masahiko;Tanaka, Toshiyuki
零决定性策略是近年来在重复博弈中发现的一类新的策略，在进化博弈论中引起了广泛的关注。ZD策略单方面强制玩家平均收益之间的线性关系。虽然ZD策略的存在性和进化稳定性在简单博弈中已经得到了研究，但它们的数学性质还不清楚。例如，当不止一个参与者采用ZD策略时会发生什么情况尚未澄清。在本文中，我们提供了一个一般的框架，调查的情况下，多个球员采用ZD策略的线性代数。首先，我们从理论上证明了ZD策略下平均收益的线性关系总是有解的，这意味着不相容的线性关系是不可能的。其次，我们证明了线性支付关系在一定条件下是相互独立的。这些结果适用于公众监督的一般游戏，包括完美监督游戏。此外，我们提供了一个简单的例子，一个玩家可以同时执行两个线性关系，也就是说，同时控制她和她的对手的平均收益的两个玩家的游戏。所有这些结果阐明了ZD策略的一般数学性质。
Zero-determinant (ZD) strategies, a recently found novel class of strategies in repeated games, has attracted much attention in evolutionary game theory. A ZD strategy unilaterally enforces a linear relation between average payoffs of players. Although existence and evolutional stability of ZD strategies have been studied in simple games, their mathematical properties have not been well-known yet. For example, what happens when more than one players employ ZD strategies have not been clarified. In this paper, we provide a general framework for investigating situations where more than one players employ ZD strategies in terms of linear algebra. First, we theoretically prove that a set of linear relations of average payoffs enforced by ZD strategies always has solutions, which implies that incompatible linear relations are impossible. Second, we prove that linear payoff relations are independent of each other under some conditions. These results hold for general games with public monitoring including perfect-monitoring games. Furthermore, we provide a simple example of a two-player game in which one player can simultaneously enforce two linear relations, that is, simultaneously control her and her opponent's average payoffs. All of these results elucidate general mathematical properties of ZD strategies.