A simple evolutionary game with feedback between perception and reality

TL;DR

This paper explores a simple evolutionary game where perception influences reality through a feedback loop, analyzing how different degrees of subjectivity affect outcomes, efficiency, and strategic advantages over time.

Contribution

It introduces a novel probabilistic game model with a feedback mechanism between wagers and outcome probabilities, bridging objective and subjective perceptions.

Findings

In self-reinforcing games, early advantages tend to persist and grow.

The inefficiency of the game decreases over time following a power law.

The degree of subjectivity influences the rate at which inefficiency diminishes.

Abstract

We study an evolutionary game of chance in which the probabilities for different outcomes (e.g., heads or tails) depend on the amount wagered on those outcomes. The game is perhaps the simplest possible probabilistic game in which perception affects reality. By varying the `reality map', which relates the amount wagered to the probability of the outcome, it is possible to move continuously from a purely objective game in which probabilities have no dependence on wagers, to a purely subjective game in which probabilities equal the amount wagered. The reality map can reflect self-reinforcing strategies or self-defeating strategies. In self-reinforcing games, rational players can achieve increasing returns and manipulate the outcome probabilities to their advantage; consequently, an early lead in the game, whether acquired by chance or by strategy, typically gives a persistent advantage. We investigate the game both in and out of equilibrium and with and without rational players. We introduce a method of measuring the inefficiency of the game and show that in the large time limit the inefficiency decreases slowly in its approach to equilibrium as a power law with an exponent between zero and one, depending on the subjectivity of the game.