Compromise by "multimatum"

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 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.

💡 Research Summary

The paper addresses a two‑agent social choice problem in which the set of feasible policies is a compact, connected metric space that may be infinite or continuous. A full‑support, non‑atomic probability measure ν on this space is taken as a common “size” or “generosity” metric that both agents agree to use. Each agent i has a continuous preference relation ⪰i with thin indifference curves; the lower contour set Li(x) = {y : x ⪰i y} is therefore measurable under ν. By evaluating an outcome x through the ν‑measure of its lower contour set, ν(Li(x)), the authors obtain a cardinal representation of each agent’s ordinal preferences that is comparable across agents because the same ν is used for both.

The authors define a “compromise solution” x* as the maximizer of the minimum of the two ν‑measures:
 maxx∈X min{ν(L1(x)), ν(L2(x))}.
They prove (Lemma 1) that, under a regularity condition (local non‑satiation at the solution), any compromise solution is Pareto‑optimal, gives each agent the same ν‑measure (at least ½), and all agents are indifferent among all compromise solutions. Thus the rule selects outcomes that are simultaneously fair (equal ν‑measure) and efficient.

To implement this rule without eliciting the agents’ underlying utilities, the paper proposes the “multimatum” mechanism. The game proceeds as follows:

1. Player 1 (the proposer) announces a measurable set C ⊆ X; ν(C) is interpreted as the generosity of the offer.
2. Player 2 (the responder) either (i) picks any element of C and ends the game, or (ii) makes a counter‑offer by announcing a set B with ν(B) ≥ ν(C).
3. If a counter‑offer occurs, Player 1 must then select an element from B, after which the game terminates.

The crucial restriction ν(B) ≥ ν(C) prevents the responder from extracting all surplus and forces the proposer to be sufficiently generous. Conversely, because the responder can always counter‑offer, the proposer cannot be overly greedy. The authors show (Theorem 1) that every subgame‑perfect Nash equilibrium (SPNE) of this game yields exactly a compromise solution, and for every compromise solution there exists an SPNE that implements it. Hence the mechanism fully implements the compromise rule in SPNE, requiring only the common measure ν and no knowledge of the agents’ utility functions.

The paper illustrates the approach with several applications:

• Political economy – extending the Bowden et al. (2014) model of two parties bargaining over a public good and private consumption, deriving comparative statics of the compromise outcome as party preferences change.
• Other‑regarding preferences – applying the framework to Fehr‑Schmidt (1999) inequality‑averse utilities, showing that the same compromise rule applies when agents care about each other’s allocations.
• Facility location – demonstrating that with single‑peaked preferences the compromise solution coincides with the median location, and the multimatum mechanism implements it.
• Additional examples include random social choice and lottery agreement, underscoring the versatility of the method.

The discussion contrasts the multimatum with classic ultimatum or Nash‑implementation approaches, emphasizing that the former does not require an equilibrium selection criterion and works even when agents’ utilities are not cardinally comparable. The authors acknowledge limitations: the need for a common, ex‑ogenously given ν; the reliance on thin indifference curves and local non‑satiation (which may fail in discrete or non‑continuous settings); and the practical difficulty of measuring ν(C) in real negotiations. Nevertheless, the paper contributes a novel, analytically tractable solution concept and a simple, robust mechanism for implementing it in infinite‑dimensional bargaining environments. Future work could extend the model to more than two agents, explore endogenous determination of ν, or relax the continuity assumptions.