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The Complexity of Computing Robust Mediated Equilibria in Ordinal Games

Vincent Conitzer

TL;DR

This work addresses the problem of computing robust equilibria in ordinal (cardinality-unknown) games by embedding them in a pre-Bayesian framework with a mediator enforcing on-path punishment. It shows that folk-theorem-style constructions yield equilibria robust to any cardinal utilities consistent with ordinal constraints, and provides polynomial-time algorithms in several settings: finite-type spaces (via LPs), total orders (via stochastic dominance and LPs), and partial orders (via polynomial-time separation or max-flow-based methods). It also establishes hardness results when utilities obey richer CNF-encoded constraints, and discusses when robustness can be considered without loss of generality in relation to mediated and program equilibria. Together, the results offer a practical, theory-grounded approach to robust coordination in AI systems under ordinal uncertainty, and point to mechanism-design directions for cases lacking robust equilibria.

Abstract

Usually, to apply game-theoretic methods, we must specify utilities precisely, and we run the risk that the solutions we compute are not robust to errors in this specification. Ordinal games provide an attractive alternative: they require specifying only which outcomes are preferred to which other ones. Unfortunately, they provide little guidance for how to play unless there are pure Nash equilibria; evaluating mixed strategies appears to fundamentally require cardinal utilities. In this paper, we observe that we can in fact make good use of mixed strategies in ordinal games if we consider settings that allow for folk theorems. These allow us to find equilibria that are robust, in the sense that they remain equilibria no matter which cardinal utilities are the correct ones -- as long as they are consistent with the specified ordinal preferences. We analyze this concept and study the computational complexity of finding such equilibria in a range of settings.

The Complexity of Computing Robust Mediated Equilibria in Ordinal Games

TL;DR

This work addresses the problem of computing robust equilibria in ordinal (cardinality-unknown) games by embedding them in a pre-Bayesian framework with a mediator enforcing on-path punishment. It shows that folk-theorem-style constructions yield equilibria robust to any cardinal utilities consistent with ordinal constraints, and provides polynomial-time algorithms in several settings: finite-type spaces (via LPs), total orders (via stochastic dominance and LPs), and partial orders (via polynomial-time separation or max-flow-based methods). It also establishes hardness results when utilities obey richer CNF-encoded constraints, and discusses when robustness can be considered without loss of generality in relation to mediated and program equilibria. Together, the results offer a practical, theory-grounded approach to robust coordination in AI systems under ordinal uncertainty, and point to mechanism-design directions for cases lacking robust equilibria.

Abstract

Usually, to apply game-theoretic methods, we must specify utilities precisely, and we run the risk that the solutions we compute are not robust to errors in this specification. Ordinal games provide an attractive alternative: they require specifying only which outcomes are preferred to which other ones. Unfortunately, they provide little guidance for how to play unless there are pure Nash equilibria; evaluating mixed strategies appears to fundamentally require cardinal utilities. In this paper, we observe that we can in fact make good use of mixed strategies in ordinal games if we consider settings that allow for folk theorems. These allow us to find equilibria that are robust, in the sense that they remain equilibria no matter which cardinal utilities are the correct ones -- as long as they are consistent with the specified ordinal preferences. We analyze this concept and study the computational complexity of finding such equilibria in a range of settings.
Paper Structure (10 sections, 13 theorems, 6 equations, 6 figures)

This paper contains 10 sections, 13 theorems, 6 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Definitions and Notation
  3. Examples and Basic Results
  4. Finite Sets of Types
  5. Total Orders
  6. Partial Orders
  7. Partial Orders over Outcome Distributions
  8. Richer Constraints on Utilities
  9. When Are the Robust Equilibria Considered Here without Loss of Generality?
  10. Other Future Research

Key Result

Proposition 1

Figure fi:MP1's game has no robust equilibrium.

Figures (6)

  • Figure 1: Left: game with an equilibrium with utilities $8.5,9$. Right: only the equilibrium with utilities $7,7$ survives.
  • Figure 2: Left: together with the orders $o_{11} \succeq_1 o_{22} \succeq_1 o_{12} \succeq_1 o_{21}$ and $o_{21} \succeq_2 o_{12} \succeq_2 o_{22} \succeq_2 o_{11}$, this is an ordinal game. Right: example utilities satisfying these constraints.
  • Figure 3: Left: together with the orders $o_{1} \succeq_1 o_{2}$ and $o_{2} \succeq_2 o_{1}$, this is an ordinal game. Right: example utilities satisfying these constraints.
  • Figure 4: Left: together with the orders $o_{dc} \succeq_1 o_{cc1} \succeq_1 o_{dd11} \succeq_1 o_{dd22} \succeq_1 o_{cc2} \succeq_1 o_{dd12} \succeq_1 o_{dd21} \succeq_1 o_{cd}$ and $o_{cd} \succeq_2 o_{cc2} \succeq_2 o_{dd21} \succeq_2 o_{dd12} \succeq_2 o_{cc1} \succeq_2 o_{dd22} \succeq_2 o_{dd11} \succeq_2 o_{dc}$, this is an ordinal game. Right: example utilities satisfying these constraints.
  • Figure 5: Left: together with the ordering constraints $o_{11} \succeq_1 o_{22}$ and $o_{11} \succeq_1 (0.6o_{22} + 0.4o_{23})$, this is an ordinal game. Right: example utilities satisfying these constraints.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Definition 1
  • Proposition 1
  • proof
  • Definition 2
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Theorem 1
  • Definition 3
  • ...and 16 more