Repeated Matching Pennies with Limited Randomness

TL;DR

This paper investigates how limited randomness affects the existence of Nash equilibria in repeated Matching Pennies games, revealing trade-offs between randomness, computational constraints, and equilibrium existence.

Contribution

It provides a full characterization of approximate equilibria under randomness constraints and links the existence of equilibria to computational assumptions like one-way functions.

Findings

Approximate equilibria exist with less randomness when players are computationally bounded.

Existence of Nash equilibria with limited randomness depends on computational assumptions such as one-way functions.

Trade-offs between randomness and computational time enable equilibrium existence under constraints.

Abstract

We consider a repeated Matching Pennies game in which players have limited access to randomness. Playing the (unique) Nash equilibrium in this n-stage game requires n random bits. Can there be Nash equilibria that use less than n random coins? Our main results are as follows: We give a full characterization of approximate equilibria, showing that, for any e in [0, 1], the game has a e-Nash equilibrium if and only if both players have (1 - e)n random coins. When players are bound to run in polynomial time, Nash equilibria can exist if and only if one-way functions exist. It is possible to trade-off randomness for running time. In particular, under reasonable assumptions, if we give one player only O(log n) random coins but allow him to run in arbitrary polynomial time and we restrict his opponent to run in time n^k, for some fixed k, then we can sustain an Nash equilibrium. When the game is played for an infinite amount of rounds with time discounted utilities, under reasonable assumptions, we can reduce the amount of randomness required to achieve a e-Nash equilibrium to n, where n is the number of random coins necessary to achieve an approximate Nash equilibrium in the general case.