Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Randomness Requirements and Asymmetries in Nash Equilibria

Edan Orzech, Martin Rinard

TL;DR

The paper analyzes the resources required to play Nash equilibria in finite two-player normal-form games by introducing a formal measure $C(x)$ for the complexity of finite rational distributions and a bounded-randomness game where players' strategies are constrained by their storage capacity. It proves strong upper bounds on the complexity of equilibrium strategies in terms of payoff magnitudes and size $n$, and constructs explicit examples that achieve exponential lower bounds, including a notable exponential vs. linear gap between the two players. The results rely on specialized constructions such as imitation games and carefully designed block matrices, yielding unique fully mixed Nash equilibria with disparate storage requirements. This work highlights nontrivial randomness assumptions in equilibrium play and reveals potentially large asymmetries in the resources required to represent and sample NE strategies, with implications for sampling algorithms and the computational complexity of exact NE computation.

Abstract

In general, Nash equilibria in normal-form games may require players to play (probabilistically) mixed strategies. We define a measure of the complexity of finite probability distributions and study the complexity required to play Nash equilibria in finite two player $n\times n$ games with rational payoffs. Our central results show that there exist games in which there is an exponential vs. linear gap in the complexity of the mixed distributions that the two players play in the (unique) Nash equilibrium of these games. This gap induces asymmetries in the amounts of space required by the players to represent and sample from the corresponding distributions using known state-of-the-art sampling algorithms. We also establish exponential upper and lower bounds on the complexity of Nash equilibria in normal-form games. These results highlight (i) the nontriviality of the assumption that players can play any mixed strategy and (ii) the disparity in resources that players may require to play Nash equilibria in normal-form games.

Randomness Requirements and Asymmetries in Nash Equilibria

TL;DR

The paper analyzes the resources required to play Nash equilibria in finite two-player normal-form games by introducing a formal measure for the complexity of finite rational distributions and a bounded-randomness game where players' strategies are constrained by their storage capacity. It proves strong upper bounds on the complexity of equilibrium strategies in terms of payoff magnitudes and size , and constructs explicit examples that achieve exponential lower bounds, including a notable exponential vs. linear gap between the two players. The results rely on specialized constructions such as imitation games and carefully designed block matrices, yielding unique fully mixed Nash equilibria with disparate storage requirements. This work highlights nontrivial randomness assumptions in equilibrium play and reveals potentially large asymmetries in the resources required to represent and sample NE strategies, with implications for sampling algorithms and the computational complexity of exact NE computation.

Abstract

In general, Nash equilibria in normal-form games may require players to play (probabilistically) mixed strategies. We define a measure of the complexity of finite probability distributions and study the complexity required to play Nash equilibria in finite two player games with rational payoffs. Our central results show that there exist games in which there is an exponential vs. linear gap in the complexity of the mixed distributions that the two players play in the (unique) Nash equilibrium of these games. This gap induces asymmetries in the amounts of space required by the players to represent and sample from the corresponding distributions using known state-of-the-art sampling algorithms. We also establish exponential upper and lower bounds on the complexity of Nash equilibria in normal-form games. These results highlight (i) the nontriviality of the assumption that players can play any mixed strategy and (ii) the disparity in resources that players may require to play Nash equilibria in normal-form games.
Paper Structure (3 sections, 3 theorems, 1 equation)

This paper contains 3 sections, 3 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Model and motivation
  3. Our results

Key Result

theorem thmcountertheorem

For every $A,B\in\mathbb{Z}^{n\times n}$ there holds In particular, if $A,B$ have entries which are $poly(n)$, then $C_i(A,B)=2^{O(n\log n)}$.

Theorems & Definitions (6)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem