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Compromise by "multimatum"

Federico Echenique, Matías Núñez

TL;DR

The paper tackles two-agent social choice over a continuum of outcomes and proposes a normative, transfers-free compromise rule implemented via the multimatum mechanism. By introducing a common cardinal yardstick $ u$ on the outcome space, it enables meaningful comparisons of two agents' preferences despite potentially incommensurate scales, and defines a compromise as the outcome maximizing the worst-off agent’s lower-contour measure. The authors prove that, in regular convex environments with thin indifference curves, the multimatum mechanism fully implements the compromise in subgame-perfect equilibrium, and every compromise outcome can be sustained by an SPNE. The work links theory to applications in political economy, inequality-averse preferences, and spatial settings, and distinguishes its results from finite-settings Nash implementation, highlighting the role of cardinalization and the no-transfer constraint. It also opens avenues for experimental validation and extensions to outside options and multi-agent settings, while illustrating the mechanism’s normative appeal and potential real-world relevance.

Abstract

We propose a solution and a mechanism for two-agent social choice problems with large (infinite) policy spaces. Our solution is an efficient compromise rule between the two agents, built on a common cardinalization of their preferences. Our mechanism, the \emph{multimatum} has the two players alternate in proposing sets of alternatives from which the other must choose. Our main result shows that the multimatum fully implements our compromise solution in subgame perfect Nash equilibrium. \\ We demonstrate the power and versatility of this approach through applications to political economy, other-regarding preferences, and facility location.

Compromise by "multimatum"

TL;DR

The paper tackles two-agent social choice over a continuum of outcomes and proposes a normative, transfers-free compromise rule implemented via the multimatum mechanism. By introducing a common cardinal yardstick on the outcome space, it enables meaningful comparisons of two agents' preferences despite potentially incommensurate scales, and defines a compromise as the outcome maximizing the worst-off agent’s lower-contour measure. The authors prove that, in regular convex environments with thin indifference curves, the multimatum mechanism fully implements the compromise in subgame-perfect equilibrium, and every compromise outcome can be sustained by an SPNE. The work links theory to applications in political economy, inequality-averse preferences, and spatial settings, and distinguishes its results from finite-settings Nash implementation, highlighting the role of cardinalization and the no-transfer constraint. It also opens avenues for experimental validation and extensions to outside options and multi-agent settings, while illustrating the mechanism’s normative appeal and potential real-world relevance.

Abstract

We propose a solution and a mechanism for two-agent social choice problems with large (infinite) policy spaces. Our solution is an efficient compromise rule between the two agents, built on a common cardinalization of their preferences. Our mechanism, the \emph{multimatum} has the two players alternate in proposing sets of alternatives from which the other must choose. Our main result shows that the multimatum fully implements our compromise solution in subgame perfect Nash equilibrium. \\ We demonstrate the power and versatility of this approach through applications to political economy, other-regarding preferences, and facility location.
Paper Structure (32 sections, 16 theorems, 53 equations, 9 figures, 1 table)

This paper contains 32 sections, 16 theorems, 53 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model
  3. Preferences in metric spaces
  4. Convex preferences
  5. The model
  6. Main Result
  7. Subgame-perfect implementation of a compromise
  8. Applications
  9. Political economy
  10. Private consumption with other-regarding preferences
  11. Pareto Efficient allocations
  12. Indifference curves
  13. Players' indifference curves.
  14. Compromise and Inequality aversion
  15. Spatial voting and facility location.
  16. ...and 17 more sections

Key Result

Proposition 1

Suppose that $X$ is a full-dimensional, convex, and compact subset of $\mathbf{R}^d$ endowed with the usual Euclidean norm, and that the probability measure $\nu$ is absolutely continuous with respect to the Lebesgue measure on $\mathbf{R}^d$. Let $\succeq$ be an explicitly convex and locally strict

Figures (9)

  • Figure 1: Compromise in a regular problem .
  • Figure 2: Compromise in a non-regular problem
  • Figure 3: Two Compromises in a regular problem
  • Figure 4: Utility of Party 1 and 2 over private consumption with $\theta_1=1/2$ and $\theta_2=1/3$. Lower contour sets are colored.
  • Figure 5: Facility location example.
  • ...and 4 more figures

Theorems & Definitions (36)

  • Proposition 1
  • Definition 1
  • Remark 1
  • Definition 2
  • Lemma 1
  • proof
  • Example 1
  • Example 2
  • Example 3
  • Definition 3
  • ...and 26 more