Autocratic Strategies of Multi-State Games

TL;DR

This paper extends the concept of autocratic strategies from single-state to multi-state deterministic games, providing methods to identify enforceable payoff intervals and construct strategies with finite memory.

Contribution

It introduces a framework for analyzing autocratic strategies in multi-state games, including fixed point equations and algorithms for approximating enforceable values.

Findings

Enforceable payoff intervals satisfy fixed point equations.

Finite memory strategies can enforce specific values.

An algorithm approximates enforceable values in each state.

Abstract

In a single-state repeated game, zero-determinant strategies can unilaterally force functions of the payoffs to take values in particular closed intervals. When the explicit use of a determinant is absent from the analysis, they are instead called autocratic. We extend their study to the setting of finite state games with deterministic transitions. For a given game we find that the endpoints of the intervals of enforceable values must satisfy fixed point equations. From these extreme enforceable values we show it is always possible to construct finite memory strategies to enforce a particular value. An algorithm is presented which will approximate the enforceable values in each state. Finally, we present formulas from which the exact solutions can be recovered from the approximate solution.