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Mechanisms that play a game, not toss a coin

Toby Walsh

TL;DR

This work replaces stochastic randomness in mechanisms with a deterministic framework built on modular arithmetic games. By eliciting uniform or quasi-uniform strategy equilibria, the authors derandomize mechanisms in voting, location, task allocation, peer selection, school choice, and resource allocation, retaining many normative properties such as efficiency, fairness, and incentive compatibility in a weakened, quasi-strategy-proof sense. The three proposed derandomization paradigms—game-first, game-last, and game-interleaved—enable audits and deterministic analysis while preserving performance (e.g., $7/4$-approximation in makespan, Pareto efficiency in RP) and, in at least one domain, introduce a new desirable property (responsiveness). The results offer practical avenues for deploying auditable, deterministic mechanisms without sacrificing core normative benefits.

Abstract

Randomized mechanisms can have good normative properties compared to their deterministic counterparts. However, randomized mechanisms are problematic in several ways such as in their verifiability. We propose here to derandomize such mechanisms by having agents play a game instead of tossing a coin. The game is designed so an agent's best action is to play randomly, and this play then injects ``randomness'' into the mechanism. This derandomization retains many of the good normative properties of the original randomized mechanism but gives a mechanism that is deterministic and easy, for instance, to audit. We consider three related methods to derandomize randomized mechanism in six different domains: voting, facility location, task allocation, school choice, peer selection, and resource allocation. We propose a number of novel derandomized mechanisms for these six domains with good normative properties. Each mechanism has a mixed Nash equilibrium in which agents play a modular arithmetic game with an uniform mixed strategy. In all but one mixed Nash equilibrium, agents report their preferences over the original problem sincerely. The derandomized methods are thus ``quasi-strategy proof''. In one domain, we additionally show that a new and desirable normative property emerges as a result of derandomization.

Mechanisms that play a game, not toss a coin

TL;DR

This work replaces stochastic randomness in mechanisms with a deterministic framework built on modular arithmetic games. By eliciting uniform or quasi-uniform strategy equilibria, the authors derandomize mechanisms in voting, location, task allocation, peer selection, school choice, and resource allocation, retaining many normative properties such as efficiency, fairness, and incentive compatibility in a weakened, quasi-strategy-proof sense. The three proposed derandomization paradigms—game-first, game-last, and game-interleaved—enable audits and deterministic analysis while preserving performance (e.g., -approximation in makespan, Pareto efficiency in RP) and, in at least one domain, introduce a new desirable property (responsiveness). The results offer practical avenues for deploying auditable, deterministic mechanisms without sacrificing core normative benefits.

Abstract

Randomized mechanisms can have good normative properties compared to their deterministic counterparts. However, randomized mechanisms are problematic in several ways such as in their verifiability. We propose here to derandomize such mechanisms by having agents play a game instead of tossing a coin. The game is designed so an agent's best action is to play randomly, and this play then injects ``randomness'' into the mechanism. This derandomization retains many of the good normative properties of the original randomized mechanism but gives a mechanism that is deterministic and easy, for instance, to audit. We consider three related methods to derandomize randomized mechanism in six different domains: voting, facility location, task allocation, school choice, peer selection, and resource allocation. We propose a number of novel derandomized mechanisms for these six domains with good normative properties. Each mechanism has a mixed Nash equilibrium in which agents play a modular arithmetic game with an uniform mixed strategy. In all but one mixed Nash equilibrium, agents report their preferences over the original problem sincerely. The derandomized methods are thus ``quasi-strategy proof''. In one domain, we additionally show that a new and desirable normative property emerges as a result of derandomization.
Paper Structure (14 sections, 11 theorems, 1 algorithm)

This paper contains 14 sections, 11 theorems, 1 algorithm.

Table of Contents

  1. Introduction
  2. Modular Arithmetic Game
  3. Two Simple Examples
  4. Random Dictator
  5. Left-Right-Middle Mechanism
  6. Task Allocation
  7. Peer Selection
  8. Sequential Elimination
  9. Random Sequential Elimination
  10. School Choice
  11. Resource Allocation
  12. Probabilistic Serial
  13. Random Priority
  14. Conclusions

Key Result

Theorem 1

The parity game has an unique Nash equilibrium in which the even and odd agent both play uniform random bits. The outcome is even/odd with equal probability.

Theorems & Definitions (19)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Example 1
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 9 more