Long-term cooperation, competition, or exploitation among individuals can be modeled through repeated games. In repeated games, Press and Dyson discovered zero-determinant (ZD) strategies that enforce a special relationship between two players. This special relationship implies that a ZD player can unilaterally impose a linear payoff relationship to the opponent regardless of the opponent's strategies. A ZD player also has a property that can lead the opponent to an unconditional cooperation if the opponent tries to improve its payoff. This property has been mathematically confirmed by Chen and Zinger. Humans often underestimate a payoff obtained in the future. However, such discounting was not considered in their analysis. Here, we mathematically explored whether a ZD player can lead the opponent to an unconditional cooperation even if a discount factor is incorporated. Specifically, we represented the expected payoff with a discount factor as the form of determinants and calculated whether the values obtained by partially differentiating each factor in the strategy vector become positive. As a result, we proved that the strategy vector ends up as an unconditional cooperation even when starting from any initial strategy. This result was confirmed through numerical calculations. We extended the applicability of ZD strategies to real world problems.
翻译:个人之间的长期合作、竞争或剥削可以通过重复游戏来模拟。 在重复游戏中, Press 和 Dyson 发现了执行两个玩家之间特殊关系的零确定性策略。 这种特殊关系意味着ZD玩家可以单方面对对手强加线性补偿关系,而不管对手的战略如何。 ZD玩家还有一种财产,如果对手试图提高回报率,则可以引导对手进行无条件合作。这个财产已经得到了陈氏和津杰的数学确认。人类经常低估未来获得的回报。然而,这种折扣在他们的分析中并没有被考虑。在这里,我们用数学来探讨ZD玩家是否可以使对手获得无条件合作,即使采纳了折扣因素。具体地说,我们以一个折扣系数代表预期的收益,作为决定因素的形式,并计算出通过部分区分战略矢量获得的价值是否正值。结果证明,战略矢量最终是一种无条件合作,即使从任何初始战略开始,也是无条件合作。这个结果通过数字计算得到证实。我们把ZD战略的应用性扩大到现实世界。