skip to main content

Explore Research Products in the NSF-PAR

This content will become publicly available on October 1, 2024

Title: A Two-Player Resource-Sharing Game with Asymmetric Information

This paper considers a two-player game where each player chooses a resource from a finite collection of options. Each resource brings a random reward. Both players have statistical information regarding the rewards of each resource. Additionally, there exists an information asymmetry where each player has knowledge of the reward realizations of different subsets of the resources. If both players choose the same resource, the reward is divided equally between them, whereas if they choose different resources, each player gains the full reward of the resource. We first implement the iterative best response algorithm to find an ϵ-approximate Nash equilibrium for this game. This method of finding a Nash equilibrium may not be desirable when players do not trust each other and place no assumptions on the incentives of the opponent. To handle this case, we solve the problem of maximizing the worst-case expected utility of the first player. The solution leads to counter-intuitive insights in certain special cases. To solve the general version of the problem, we develop an efficient algorithmic solution that combines online convex optimization and the drift-plus penalty technique.

  more » « less
Award ID(s):
1824418
NSF-PAR ID:
10494715
Author(s) / Creator(s):
;
Corporate Creator(s):
Editor(s):
Yllka Velaj and Ulrich Berger
Publisher / Repository:
Games
Date Published:
Journal Name:
Games
Volume:
14
Issue:
5
ISSN:
2073-4336
Page Range / eLocation ID:
61
Subject(s) / Keyword(s):
["resource-sharing games","congestion games","potential games","worst-case utilitymaximization","drift-plus penalty method"]
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ( , arXiv:2402.05300v2)
    arXiv:2402.05300v2 (Ed.)
    This paper considers a multi-player resource-sharing game with a fair reward allocation model. Multiple players choose from a collection of resources. Each resource brings a random reward equally divided among the players who choose it. We consider two settings. The first setting is a one-slot game where the mean rewards of the resources are known to all the players, and the objective of player 1 is to maximize their worst-case expected utility. Certain special cases of this setting have explicit solutions. These cases provide interesting yet non-intuitive insights into the problem. The second setting is an online setting, where the game is played over a finite time horizon, where the mean rewards are unknown to the first player. Instead, the first player receives, as feedback, the rewards of the resources they chose after the action. We develop a novel Upper Confidence Bound (UCB) algorithm that minimizes the worst-case regret of the first player using the feedback received. 
    more » « less
  2. ; ( , Mathematics of Operations Research)
    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 function—but 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. Funding: M. Nutz is supported by an Alfred P. Sloan Fellowship and the Division of Mathematical Sciences of the National Science Foundation [Grants DMS-1812661 and DMS-2106056]. Y. Zhang is supported in part by the Natural Sciences and Engineering Research Council of Canada [NSERC Discovery Grant RGPIN-2020-06290]. 
    more » « less
  3. ; ( , 60th IEEE Conference on Decision and Control (CDC))
    The aim of this paper is to find the distributed solution of the generalized Nash equilibrium problem (GNEP) for a group of players that can communicate with each other over a connected communication network. Each player tries to minimize a local objective function of its own that may depend on the other players’ decisions, and collectively all the players’ decisions are subject to some globally shared resource constraints. After reformulating the local optimization problems, we introduce the notion of network Lagrangian and recast the GNEP as the zero finding problem of a properly defined operator. Utilizing the Douglas-Rachford operator splitting method, a distributed algorithm is proposed that requires only local information exchanges between neighboring players in each iteration. The convergence of the proposed algorithm to an exact variational generalized Nash equilibrium is established under two different sets of assumptions. The effectiveness of the proposed algorithm is demonstrated using the example of a Nash-Cournot production game. 
    more » « less
  4. ; ; ( , Proceedings of Machine Learning Research)
    Francisco Ruiz, Jennifer Dy (Ed.)
    We study the sample complexity of identifying an approximate equilibrium for two-player zero-sum n × 2 matrix games. That is, in a sequence of repeated game plays, how many rounds must the two players play before reaching an approximate equilibrium (e.g., Nash)? We derive instance-dependent bounds that define an ordering over game matrices that captures the intuition that the dynamics of some games converge faster than others. Specifically, we consider a stochastic observation model such that when the two players choose actions i and j, respectively, they both observe each other’s played actions and a stochastic observation Xij such that E [Xij ] = Aij . To our knowledge, our work is the first case of instance-dependent lower bounds on the number of rounds the players must play before reaching an approximate equilibrium in the sense that the number of rounds depends on the specific properties of the game matrix A as well as the desired accuracy. We also prove a converse statement: there exist player strategies that achieve this lower bound. 
    more » « less
  5. ( , AAMAS Conference proceedings)
    null (Ed.)
    A two-player finite game is represented by two payoff matrices (A, B), one for each player. Imitation games are a subclass of two-player games in which B is the identity matrix, implying that the second player gets a positive payoff only if she "imitates" the first. Given that the problem of computing a Nash equilibrium (NE) is known to be provably hard, even to approximate, we ask if it is any easier for imitation games. We show that much like the general case, for any c > 0, computing a 1 over n^c -approximate NE of imitation games remains PPAD-hard, where n is the number of moves available to the players. On the other hand, we design a polynomial-time algorithm to find ε-approximate NE for any given constant ε > 0 (PTAS). The former result also rules out the smooth complexity being in P, unless PPAD ⊂ RP. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email