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Efficient Method for Finding Optimal Strategies in Chopstick Auctions with Uniform Objects Values

Stanisław Kaźmierowski, Marcin Dziubiński

TL;DR

An algorithm for computing Nash equilibria (NE) in a class of conflicts with multiple battlefields with uniform battlefields values and a non-linear aggregation function is proposed and achieves a significant speed-up as compared to the standard, LP-based, approach.

Abstract

We propose an algorithm for computing Nash equilibria (NE) in a class of conflicts with multiple battlefields with uniform battlefield values and a non-linear aggregation function. By expanding the symmetrization idea of Hart [9], proposed for the Colonel Blotto game, to the wider class of symmetric conflicts with multiple battlefields, we reduce the number of strategies of the players by an exponential factor. We propose a clash matrix algorithm which allows for computing the payoffs in the symmetrized model in polynomial time. Combining symmetrization and clash matrix algorithm with the double oracle algorithm we obtain an algorithm for computing NE in the models in question that achieves a significant speed-up as compared to the standard, LP-based, approach. We also introduce a heuristic to further speed up the process. Overall, our approach offers an efficient and novel method for computing NE in a specific class of conflicts, with potential practical applications in various fields.

Efficient Method for Finding Optimal Strategies in Chopstick Auctions with Uniform Objects Values

TL;DR

An algorithm for computing Nash equilibria (NE) in a class of conflicts with multiple battlefields with uniform battlefields values and a non-linear aggregation function is proposed and achieves a significant speed-up as compared to the standard, LP-based, approach.

Abstract

We propose an algorithm for computing Nash equilibria (NE) in a class of conflicts with multiple battlefields with uniform battlefield values and a non-linear aggregation function. By expanding the symmetrization idea of Hart [9], proposed for the Colonel Blotto game, to the wider class of symmetric conflicts with multiple battlefields, we reduce the number of strategies of the players by an exponential factor. We propose a clash matrix algorithm which allows for computing the payoffs in the symmetrized model in polynomial time. Combining symmetrization and clash matrix algorithm with the double oracle algorithm we obtain an algorithm for computing NE in the models in question that achieves a significant speed-up as compared to the standard, LP-based, approach. We also introduce a heuristic to further speed up the process. Overall, our approach offers an efficient and novel method for computing NE in a specific class of conflicts, with potential practical applications in various fields.
Paper Structure (7 sections, 2 theorems, 12 equations, 1 figure)

This paper contains 7 sections, 2 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related literature
  3. Model
  4. Complexity of finding the best pure response
  5. Symmetrized model
  6. Computing payoffs from symmetric strategies
  7. Clash matrix

Key Result

proposition 1

For aggregation functions $f$ given by chopstick_agg and majoritarian_agg, there is no polynomial time algorithm to solve $BR_f$ unless $P=NP$.

Figures (1)

  • Figure 1: Comparison of number of pure and pure symmetric strategies

Theorems & Definitions (5)

  • definition 1: Aggregation function
  • definition 2: Conflict with multiple battlefields
  • proposition 1
  • definition 3: Symmetrized conflict with multiple battlefields
  • proposition 2