# Terminal Ranking Games

## TL;DR

This paper studies a mean field game where agents are rewarded based on project rankings, using Schrödinger bridges to explicitly find equilibria and analyze the impact of reward structures on welfare and efficiency.

## Contribution

It introduces a novel application of Schrödinger bridges to explicitly compute equilibria in ranking-based mean field games and addresses mechanism design and welfare analysis.

## Key findings

- Explicit equilibrium calculations using Schrödinger bridges.
- Identification of reward functions for desired equilibria.
- Analysis of reward inequality effects on welfare and efficiency.

## Abstract

We analyze a mean field tournament: a mean field game in which the agents receive rewards according to the ranking of the terminal value of their projects and are subject to cost of effort. Using Schr\"{o}dinger bridges we are able to explicitly calculate the equilibrium. This allows us to identify the reward functions which would yield a desired equilibrium and solve several related mechanism design problems. We are also able to identify the effect of reward inequality on the players' welfare as well as calculate the price of anarchy.