Relative Performance

Mean-Field Price Formation on Trees with a Network of Relative Performance Concerns ArXiv ID: 2512.21621 “View on arXiv” Authors: Masaaki Fujii Abstract Financial firms and institutional investors are routinely evaluated based on their performance relative to their peers. These relative performance concerns significantly influence risk-taking behavior and market dynamics. While the literature studying Nash equilibrium under such relative performance competitions is extensive, its effect on asset price formation remains largely unexplored. This paper investigates mean-field equilibrium price formation of a single risky stock in a discrete-time market where agents exhibit exponential utility and relative performance concerns. Unlike existing literature that typically treats asset prices as exogenous, we impose a market-clearing condition to determine the price dynamics endogenously within a relative performance equilibrium. Using a binomial tree framework, we establish the existence and uniqueness of the market-clearing mean-field equilibrium in both single- and multi-population settings. Finally, we provide illustrative numerical examples demonstrating the equilibrium price distributions and agents’ optimal position sizes. ...

N-player and mean field games among fund managers considering excess logarithmic returns ArXiv ID: 2503.02722 “View on arXiv” Authors: Unknown Abstract This paper studies the competition among multiple fund managers with relative performance over the excess logarithmic return. Fund managers compete with each other and have expected utility or mean-variance criteria for excess logarithmic return. Each fund manager possesses a unique risky asset, and all fund managers can also invest in a public risk-free asset and a public risk asset. We construct both an $n$-player game and a mean field game (MFG) to address the competition problem under these two criteria. We explicitly define and rigorously solve the equilibrium and mean field equilibrium (MFE) for each criteria. In the four models, the excess logarithmic return as the evaluation criterion of the fund leads to the {" allocation fractions"} being constant. The introduction of the public risky asset yields different outcomes, with competition primarily affecting the investment in public assets, particularly evident in the MFG. We demonstrate that the MFE of the MFG represents the limit of the $n$-player game’s equilibrium as the competitive scale $n$ approaches infinity. Finally, the sensitivity analyses of the equilibrium are given. ...