The Reality Game

TL;DR

The paper introduces the 'reality game', an evolutionary game where perception influences reality, exploring how feedback mechanisms affect outcomes and convergence rates in subjective versus objective scenarios.

Contribution

It presents a novel framework for analyzing feedback between perception and reality in games, including methods to measure inefficiency and convergence behavior.

Findings

Convergence to equilibrium follows a power law with slow rates.

Subjectivity in the game slows down convergence.

Self-reinforcing and self-defeating dynamics are characterized.

Abstract

We introduce an evolutionary game with feedback between perception and reality, which we call the reality game. It is a game of chance in which the probabilities for different objective outcomes (e.g., heads or tails in a coin toss) depend on the amount wagered on those outcomes. 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. We study self-reinforcing games, in which betting more on an outcome increases its odds, and self-defeating games, in which the opposite is true. This is investigated in and out of equilibrium, with and without rational players, and both numerically and analytically. We introduce a method of measuring the inefficiency of the game, similar to measuring the magnitude of the arbitrage opportunities in a financial market. We prove that convergence to equilibrium is is a power law with an extremely slow rate of convergence: The more subjective the game, the slower the convergence.