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Emergence of Exploitation as Symmetry Breaking in Iterated Prisoner's Dilemma
In society, mutual cooperation, defection, and asymmetric exploitative
relationships are common. Whereas cooperation and defection are studied
extensively in the literature on game theory, asymmetric exploitative
relationships between players are little explored. In a recent study, Press and
Dyson demonstrate that if only one player can learn about the other, asymmetric
exploitation is achieved in the prisoner's dilemma game. In contrast, however,
it is unknown whether such one-way exploitation is stably established when both
players learn about each other symmetrically and try to optimize their payoffs.
Here, we first formulate a dynamical system that describes the change in a
player's probabilistic strategy with reinforcement learning to obtain greater
payoffs, based on the recognition of the other player. By applying this
formulation to the standard prisoner's dilemma game, we numerically and
analytically demonstrate that an exploitative relationship can be achieved
despite symmetric strategy dynamics and symmetric rule of games. This
exploitative relationship is stable, even though the exploited player, who
receives a lower payoff than the exploiting player, has optimized the own
strategy. Whether the final equilibrium state is mutual cooperation, defection,
or exploitation, crucially depends on the initial conditions: Punishment
against a defector oscillates between the players, and thus a complicated basin
structure to the final equilibrium appears. In other words, slight differences
in the initial state may lead to drastic changes in the final state.
Considering the generality of the result, this study provides a new perspective
on the origin of exploitation in society.Comment: 19 pages, 7 figures, + supplement(8 pages, 2 figures
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