Large tournament games

Abstract

We consider a stochastic tournament game in which each player is rewarded based on her rank in terms of the completion time of her own task and is subject to cost of effort. When players are homogeneous and the rewards are purely rank dependent, the equilibrium has a surprisingly explicit characterization, which allows us to conduct comparative statics and obtain explicit solution to several optimal reward design problems. In the general case when the players are heterogenous and payoffs are not purely rank dependent, we prove the existence, uniqueness and stability of the Nash equilibrium of the associated mean field game, and the existence of an approximate Nash equilibrium of the finite-player game. Our results have some potential economic implications; e.g., they lend support to government subsidies for R&D, assuming the profits to be made are substantial.

Many-player games of optimal consumption and investment under relative performance criteria

“…Mean field games, introduced in [17,22], are rarely explicitly solvable outside of linearquadratic examples. See [4,6,16,21,29] for some notable exceptions and the book [7] for further background on the active area of mean field games. From a mean field game perspective, our model is rather complex: It involves common noise, degenerate volatility coefficients, singular objective functions, and a mean field interaction through both the states and controls (i.e., an extended mean field game [7,Chapter I.4.6]).…”

Hodge allocation for cooperative rewards: a generalization of Shapley's cooperative value allocation theory via Hodge theory on graphs

“…Another important recent development is the mean field game theory, the study of strategic decision making by interacting agents in large populations; see, e.g., Cardaliaguet et al [7], Acciaio et al [1], Bayraktar et al [4,5], Possamaï et al [31], Lacker and Soret [21]. Carmona and Delarue [8,9] provides a detailed account.…”

Mean Field Contest with Singularity

“…This result is due to to [24]. 3 We say that F ∈ F is a mean field equilibrium if no player is incentivized to deviate from F ; that is, ξ F dF ≥ ξ F d F for all F ∈ F. The associated value function u(x) is defined as the supremum expected reward achievable for a player starting at level x (instead of x 0 ) if all others use F . Denote the average reward by R = 1 0 R(r)dr and set…”

“…There, the optimal designs are more similar between the two settings; part (a) of the above guess is always correct-the optimal reward has a sharp cut-off exactly at the target rank-though the shape over the ranks above the target is concave rather than being flat as in (b). A related mean field game is considered in [3], with diffusion instead of Poissonian dynamics. Both (a) and (b) turn out to be correct in the mean field setting.…”