Concave Pro-rata Games

TL;DR

This paper introduces concave pro-rata games, analyzing their equilibrium properties, efficiency bounds, and convergence behavior, with applications to decentralized exchanges and practical scenarios.

Contribution

It defines a new class of games, proves the uniqueness of symmetric pure equilibria, and analyzes their efficiency and convergence properties.

Findings

Unique symmetric pure equilibrium exists and is the only equilibrium.

Price of anarchy is at least proportional to the number of players.

Players tend to quickly converge to equilibrium in iterative play.

Abstract

In this paper, we introduce a family of games called concave pro-rata games. In such a game, players place their assets into a pool, and the pool pays out some concave function of all assets placed into it. Each player then receives a pro-rata share of the payout; i.e., each player receives an amount proportional to how much they placed in the pool. Such games appear in a number of practical scenarios, including as a simplified version of batched decentralized exchanges, such as those proposed by Penumbra. We show that this game has a number of interesting properties, including a symmetric pure equilibrium that is the unique equilibrium of this game, and we prove that its price of anarchy is $\Omega(n)$ in the number of players. We also show some numerical results in the iterated setting which suggest that players quickly converge to an equilibrium in iterated play.