# Strategies that enforce linear payoff relationships under observation errors in Repeated Prisoner's Dilemma game

## TL;DR

This paper analytically investigates strategies in the repeated prisoner's dilemma with observation errors, revealing that only zero-determinant and unconditional strategies can enforce linear payoff relationships despite errors.

## Contribution

It extends the understanding of linear payoff enforcement strategies to scenarios with observation errors, identifying the limited strategy sets capable of such enforcement.

## Key findings

- Only ZD and unconditional strategies enforce linear payoffs with errors.
- Numerical confirmation supports the analytical results.
- Observation errors do not enable new strategies to enforce linear relationships.

## Abstract

The theory of repeated games analyzes the long-term relationship of interacting players and mathematically reveals the condition of how cooperation is achieved, which is not achieved in a one-shot game. In the repeated prisoner's dilemma (RPD) game with no errors, zero-determinant (ZD) strategies allow a player to unilaterally set a linear relationship between the player's own payoff and the opponent's payoff regardless of the strategy that the opponent implements. In contrast, unconditional strategies such as ALLD and ALLC also unilaterally set a linear payoff relationship. Errors often happen between players in the real world. However, little is known about the existence of such strategies in the RPD game with errors. Here, we analytically search for all strategies that enforce a linear payoff relationship under observation errors in the RPD game. As a result, we found that, even in the case with observation errors, the only strategy sets that enforce a linear payoff relationship are either ZD strategies or unconditional strategies and that no other strategies can enforce it, which were numerically confirmed.