How Was the Cooperative Relationship in Trench Warfare Brought about? a Sociological Analysis of Equilibrium in Repeated Game

The cooperative relationship called "the Live-and-Let-Live System" which existed in trench warfare in World War I has been understood, especially by Axelrod, as Nash equilibrium of TFT strategy pair in repeated PD game. But, the actual interaction described by Axelrod himself contains truth inconvenient to this popular understanding. That is, there had been occasional attacks by snipers and artillery as well as retaliations, and, nevertheless, these retaliations had been effectively controlled and the escalation did not driven further. This contradicts the property of TFT Nash equilibrium, because, under the equilibrium, no defection (attack) should occur and, if it ever occurs, an unlimited chain of retaliations should arise.

This problem has been investigated as "noise problem" in evolutionary game. Axelrod has shown that revised strategies "Generosity TFT" and "Contrition TFT" which incorporate flexible response to noise (mistaken defection) are advantageous. However, analysis by evolutionary games is not appropriate for the cooperative relationship in trench warfare, since it was born by players' deliberate choice,

In this presentation, we investigate this cooperative relationship analytically focusing on player's subjective choice logic in repeated PD game. As a result, we show; (1) When disturbing noise occurs under a sufficiently high discount rate, the expected payoff of simple TFT strategy decreases and becomes lower than that of revised TFT strategies incorporating tolerance or contrition; (2) The superiority of those revised strategies is based, not on a mechanical application of Nash equilibrium, but on the subjective reason for keeping on those strategies; (3) And exchanges of "message" through individual choice of "moves" by both sides during the interaction play an important function for keeping the cooperative relationship.

These show that a certain cooperative relationship in trench warfare should be understood as a sociological relationship, not just as a mechanical application of game theory.