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$α$-Rank-Collections: Analyzing Expected Strategic Behavior with Uncertain Utilities

Fabian R. Pieroth, Martin Bichler

TL;DR

The paper addresses strategic analysis under uncertain cardinal utilities by modeling opponents' preferences as a Bayesian game and extending $\alpha$-Rank to form $\alpha$-Rank-collections, which yield the long-run distribution over strategy profiles averaged over a prior on utilities. It leverages MCCs and replicator dynamics to predict ex-ante behavior without committing to a single Bayes-Nash equilibrium, and proves invariance to positive affine transformations and smoothness properties to ensure robust predictions. The approach is demonstrated on Hawk-Dove and Boston-style matching experiments, showing more nuanced predictions than standard Bayes-Nash equilibria and highlighting the framework's practical applicability to non-strategyproof mechanisms. Overall, $\alpha$-Rank-collections offer a scalable, affine-invariant tool for analyzing strategic behavior under uncertainty, with promising implications for the design and evaluation of matching markets and other ordinal settings.

Abstract

Game theory relies heavily on the availability of cardinal utility functions, but in fields such as matching markets, only ordinal preferences are typically elicited. The literature focuses on mechanisms with simple dominant strategies, but many real-world applications lack dominant strategies, making the intensity of preferences between outcomes important for determining strategies. Even though precise information about cardinal utilities is not available, some data about the likelihood of utility functions is often accessible. We propose to use Bayesian games to formalize uncertainty about the decision-makers' utilities by viewing them as a collection of normal-form games. Instead of searching for the Bayes-Nash equilibrium, we study how uncertainty in utilities is reflected in uncertainty of strategic play. To do this, we introduce a novel solution concept called $α$-Rank-collections, which extends $α$-Rank to Bayesian games. This allows us to analyze strategic play in, for example, non-strategyproof matching markets, for which appropriate solution concepts are currently lacking. $α$-Rank-collections characterize the expected probability of encountering a certain strategy profile under replicator dynamics in the long run, rather than predicting a specific equilibrium strategy profile. We experimentally evaluate $α$-Rank-collections using instances of the Boston mechanism, finding that our solution concept provides more nuanced predictions compared to Bayes-Nash equilibria. Additionally, we prove that $α$-Rank-collections are invariant to positive affine transformations, a standard property for a solution concept, and are efficient to approximate.

$α$-Rank-Collections: Analyzing Expected Strategic Behavior with Uncertain Utilities

TL;DR

The paper addresses strategic analysis under uncertain cardinal utilities by modeling opponents' preferences as a Bayesian game and extending -Rank to form -Rank-collections, which yield the long-run distribution over strategy profiles averaged over a prior on utilities. It leverages MCCs and replicator dynamics to predict ex-ante behavior without committing to a single Bayes-Nash equilibrium, and proves invariance to positive affine transformations and smoothness properties to ensure robust predictions. The approach is demonstrated on Hawk-Dove and Boston-style matching experiments, showing more nuanced predictions than standard Bayes-Nash equilibria and highlighting the framework's practical applicability to non-strategyproof mechanisms. Overall, -Rank-collections offer a scalable, affine-invariant tool for analyzing strategic behavior under uncertainty, with promising implications for the design and evaluation of matching markets and other ordinal settings.

Abstract

Game theory relies heavily on the availability of cardinal utility functions, but in fields such as matching markets, only ordinal preferences are typically elicited. The literature focuses on mechanisms with simple dominant strategies, but many real-world applications lack dominant strategies, making the intensity of preferences between outcomes important for determining strategies. Even though precise information about cardinal utilities is not available, some data about the likelihood of utility functions is often accessible. We propose to use Bayesian games to formalize uncertainty about the decision-makers' utilities by viewing them as a collection of normal-form games. Instead of searching for the Bayes-Nash equilibrium, we study how uncertainty in utilities is reflected in uncertainty of strategic play. To do this, we introduce a novel solution concept called -Rank-collections, which extends -Rank to Bayesian games. This allows us to analyze strategic play in, for example, non-strategyproof matching markets, for which appropriate solution concepts are currently lacking. -Rank-collections characterize the expected probability of encountering a certain strategy profile under replicator dynamics in the long run, rather than predicting a specific equilibrium strategy profile. We experimentally evaluate -Rank-collections using instances of the Boston mechanism, finding that our solution concept provides more nuanced predictions compared to Bayes-Nash equilibria. Additionally, we prove that -Rank-collections are invariant to positive affine transformations, a standard property for a solution concept, and are efficient to approximate.
Paper Structure (18 sections, 7 theorems, 14 equations, 4 figures, 7 tables, 2 algorithms)

This paper contains 18 sections, 7 theorems, 14 equations, 4 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Preliminaries
  4. Ordinal and Cardinal Utility Theory
  5. Markov-Conley-Chains and alpha-Rank
  6. alpha-Rank-Collections
  7. Capturing Uncertainty in Strategic Behavior
  8. Stability and Smoothness
  9. Approximation and Computational Costs
  10. Experimental Evaluation
  11. The General Hawk-Dove Interaction
  12. Two-Sided Matching Market
  13. Experimental Setting.
  14. Equilibrium Strategies.
  15. Results.
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\text{BG} = (\mathcal{N}, \mathcal{S}, \mathcal{V}, F, u)$ be a Bayesian game. Furthermore, let every utility function $u_i$ be linear in the types, then $\alpha$-Rank-collections are invariant to positive affine transformations of the type space $\mathcal{V}$. If $\text{BG}$ additionally has p

Figures (4)

  • Figure 1: Values for stationary distributions for different values of $\alpha$ over the strategy space. Action "0" corresponds to "Hawk" and action "1" to "Dove".
  • Figure 2: Game graph of the Hawk-Dove game graph with $p=0.75$. The nodes' sizes and colors represent the $\alpha$-Rank-collection's mass. Nodes with high mass are large and red, whereas low mass nodes are brighter and small. The edges' size represents the improvement in the deviating agent's utility weighted by the prior distribution over the four possible instances. Action "0" corresponds to "Hawk" and action "1" to "Dove".
  • Figure 3: The $\alpha$-Rank distribution's mass on the strategies profiles $\text{NE}_{\text{DA}}$ under DA and $\text{NE}_{\text{Bo}}$ under Boston for different values of the vNM function for Silver. The values of Gold and Bronze are fixed to $100$ and $25$ respectively.
  • Figure 4: Graph visualizations with a linear-repulsion linear-attraction model of the aligned environment's game graph under Boston with different varying values for $v_S$, and fixed values $v_G=100$, $v_B=25$ for all students. The nodes' sizes and colors represent the $\alpha$-Rank distribution's mass. The edges are green, and their brightness and size depict the improvement in the deviating player's utility.

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Definition 3: $\alpha$-Rank-collection
  • Theorem 1
  • Theorem 2
  • proof
  • Definition 4
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • ...and 5 more