Distributed Computation of Equilibria in Misspecified Convex Stochastic Nash Games

Abstract

The distributed computation of Nash equilibria is assuming growing relevance in engineering where such problems emerge in the context of distributed control. Accordingly, we present schemes for computing equilibria of two classes of static stochastic convex games complicated by a parametric misspecification, a natural concern in the control of large-scale networked engineered system. In both schemes, players learn the equilibrium strategy while resolving the misspecification: (1) Monotone stochastic Nash games: We present a set of coupled stochastic approximation schemes distributed across agents in which the first scheme updates each agent's strategy via a projected (stochastic) gradient step while the second scheme updates every agent's belief regarding its misspecified parameter using an independently specified learning problem. We proceed to show that the produced sequences converge in an almost-sure sense to the true equilibrium strategy and the true parameter , respectively. Surprisingly, convergence in the equilibrium strategy achieves the optimal rate of convergence in a mean-squared sense with a quantifiable degradation in the rate constant; (2) Stochastic Nash-Cournot games with unobservable aggregate output: We refine (1) to a Cournot setting where we assume that the tuple of strategies is unobservable while payoff functions and strategy sets are public knowledge through a common knowledge assumption. By utilizing observations of noise-corrupted prices, iterative fixedpoint schemes are developed, allowing for simultaneously learning the equilibrium strategies and the misspecified parameter in an almost-sure sense.

Modeling and solution based on stochastic games for development of COA under uncertainty#br#

“…The uncertainty of a war situation transformation makes it difficult for a commander to predict the transformation paths after he/she executes operational actions [38,39], and therefore the final outcome of the war is unknown and so is the decision-making effect of operational actions. According to the game theory, the uncertain situation of a confrontation outcome is actually the uncertainty of the system transformation for people inside the game.…”

Distributed Variable Sample-Size Gradient-response and Best-response Schemes for Stochastic Nash Equilibrium Problems over Graphs

“…To overcome the fact that the communication network is sparse, we assume that to compute v i,k+1 , players communicate τ k rounds rather than once at major iteration k + 1. The strategy of each player is updated by a variable sample-size proximal stochastic gradient scheme characterized by (18) dependent on the constant step size α > 0 and S k , the number of sampled gradient are used at time k. We specify the scheme in Algorithm 1.…”

“…Merely monotone problems have been addressed via iterative regularization in deterministic [20,43] and stochastic [21] regimes. Extensions have been developed to contend with misspecification [18] and the lack of Lipschitzian properties [44].…”

Games with distributionally robust joint chance constraints

“…The robust optimization framework is used to handle distribution free uncertainties in the model [1,34]. For the uncertainties involving random variables, the expected payoff criterion is used in case of risk neutral players [15,16,19,26,35,36] and the risk measures CVaR and variance are used in the risk averse case [10,18,26]. For finite strategic games with random payoffs, Singh et al [29,30,31] introduced a chance constraint programming based payoff criterion.…”