A Mean Field Competition

Nutz, Marcel; Zhang, Yuchong

doi:10.1287/moor.2018.0966

Cited by 26 publications

(29 citation statements)

References 50 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Next, we consider the quality of the mean field proxy from the point of view of the principal: we fix the optimal design R * from the mean field setting (Theorem 4.1) and compare the resulting expected performance in the n-player game with the performance (5.10) of the exact optimizer given by k * n . For comparison, we mention that the analogous question was considered in the Poissonian model of [31], for the same performance functional of the principal, and there the mean field proxy was shown to be O(1/n)-optimal for the n-player design problem.…”

Section: Convergence Of the Optimal Reward Designmentioning

confidence: 99%

“…On the other hand, knowing only the mean field limit may suggest an over-simplified picture for the finite player game. One previous model where the optimal reward design problem was solved completely for both n-player and mean field setting, is the Poissonian game of [31] where players control the jump intensity and are ranked according to their jump times. 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).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Mean Field Contest with Singularity

Nutz

Zhang

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We formulate a mean field game where each player stops a privately observed Brownian motion with absorption. Players are ranked according to their level of stopping and rewarded as a function of their relative rank. There is a unique mean field equilibrium and it is shown to be the limit of associated n-player games. Conversely, the mean field strategy induces n-player ε-Nash equilibria for any continuous reward functionbut not for discontinuous ones. In a second part, we study the problem of a principal who can choose how to distribute a reward budget over the ranks and aims to maximize the performance of the median player. The optimal reward design (contract) is found in closed form, complementing the merely partial results available in the n-player case. We then analyze the quality of the mean field design when used as a proxy for the optimizer in the n-player game. Surprisingly, the quality deteriorates dramatically as n grows. We explain this with an asymptotic singularity in the induced n-player equilibrium distributions.

show abstract

Section: Convergence Of the Optimal Reward Designmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mean Field Contest with Singularity

Nutz

Zhang

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Moreover, Fubini's theorem for iterated integrals holds in the Fubini extension. Some recent gametheoretical models considering a continuum of agents in a Fubini extension are the rank-based reward models [35,36,37,45], the static graphon game in [8], and the model of [40].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Graphon Games: II. The Linear-Quadratic Case

Aurell¹,

Carmona²,

Laurière³

2021

Preprint

View full text Add to dashboard Cite

In this paper, we analyze linear-quadratic stochastic differential games with a continuum of players interacting through graphon aggregates, each state being subject to idiosyncratic Brownian shocks. The major technical issue is the joint measurability of the player state trajectories with respect to samples and player labels, which is required to compute for example costs involving the graphon aggregate. To resolve this issue we set the game in a Fubini extension of a product probability space. We provide conditions under which the graphon aggregates are deterministic and the linear state equation is uniquely solvable for all players in the continuum. The Pontryagin maximum principle yields equilibrium conditions for the graphon game in the form of a forward-backward stochastic differential equation, for which we establish existence and uniqueness. We then study how graphon games approximate games with finitely many players over graphs with random weights. We illustrate some of the results with a numerical example.

show abstract

“…In these works continuity with respect to the rank was assumed. Related tournament games where the players are ranked according to their completion times has been considered by [1] for controlled Brownian motion dynamics and by [21] for one-stage Poisson dynamics with controlled jump intensity. In the Appendix we are going to construct an extension of Schrödinger bridges from space to time which can then be applied to construct the equilibrium in [1] as well.…”

Section: Introductionmentioning

confidence: 99%