Continuous-time integral dynamics for Aggregative Game equilibrium seeking

In this paper, we consider continuous-time semi-decentralized dynamics for the equilibrium computation in a class of aggregative games. Specifically, we propose a scheme where decentralized projected-gradient dynamics are driven by an integral control law. To prove global exponential convergence of the proposed dynamics to an aggregative equilibrium, we adopt a quadratic Lyapunov function argument. We derive a sufficient condition for global convergence that we position within the recent literature on aggregative games, and in particular we show that it improves on established results

Incentive and stability in the Rock-Paper-Scissors game: an experimental investigation

In a two-person Rock-Paper-Scissors (RPS) game, if we set a loss worth nothing and a tie worth 1, and the payoff of winning (the incentive a) as a variable, this game is called as generalized RPS game. The generalized RPS game is a representative mathematical model to illustrate the game dynamics, appearing widely in textbook. However, how actual motions in these games depend on the incentive has never been reported quantitatively. Using the data from 7 games with different incentives, including 84 groups of 6 subjects playing the game in 300-round, with random-pair tournaments and local information recorded, we find that, both on social and individual level, the actual motions are changing continuously with the incentive. More expressively, some representative findings are, (1) in social collective strategy transit views, the forward transition vector field is more and more centripetal as the stability of the system increasing; (2) In the individual behavior of strategy transit view, there exists a phase transformation as the stability of the systems increasing, and the phase transformation point being near the standard RPS; (3) Conditional response behaviors are structurally changing accompanied by the controlled incentive. As a whole, the best response behavior increases and the win-stay lose-shift (WSLS) behavior declines with the incentive. Further, the outcome of win, tie, and lose influence the best response behavior and WSLS behavior. Both as the best response behavior, the win-stay behavior declines with the incentive while the lose-left-shift behavior increase with the incentive. And both as the WSLS behavior, the lose-left-shift behavior increase with the incentive, but the lose-right-shift behaviors declines with the incentive. We hope to learn which one in tens of learning models can interpret the empirical observation above.Comment: 19 pages, 14 figures, Keywords: experimental economics, conditional response, best response, win-stay-lose-shift, evolutionary game theory, behavior economic

A Douglas-Rachford splitting for semi-decentralized equilibrium seeking in generalized aggregative games

We address the generalized aggregative equilibrium seeking problem for noncooperative agents playing average aggregative games with affine coupling constraints. First, we use operator theory to characterize the generalized aggregative equilibria of the game as the zeros of a monotone set-valued operator. Then, we massage the Douglas-Rachford splitting to solve the monotone inclusion problem and derive a single layer, semi-decentralized algorithm whose global convergence is guaranteed under mild assumptions. The potential of the proposed Douglas-Rachford algorithm is shown on a simplified resource allocation game, where we observe faster convergence with respect to forward-backward algorithms.Comment: arXiv admin note: text overlap with arXiv:1803.1044

Projected-gradient algorithms for generalized equilibrium seeking in Aggregative Games are preconditioned Forward-Backward methods

We show that projected-gradient methods for the distributed computation of generalized Nash equilibria in aggregative games are preconditioned forward-backward splitting methods applied to the KKT operator of the game. Specifically, we adopt the preconditioned forward-backward design, recently conceived by Yi and Pavel in the manuscript "A distributed primal-dual algorithm for computation of generalized Nash equilibria via operator splitting methods" for generalized Nash equilibrium seeking in aggregative games. Consequently, we notice that two projected-gradient methods recently proposed in the literature are preconditioned forward-backward methods. More generally, we provide a unifying operator-theoretic ground to design projected-gradient methods for generalized equilibrium seeking in aggregative games

Cournot tatonnement and Nash equilibrium in binary status games

We study a rather simplified game model of competition for status. Each player chooses a scalar variable (say, the level of conspicuous consumption), and then those who chose the highest level obtain the "high" status, while everybody else remains with the "low" status. Each player strictly prefers the high status, but they also have intrinsic preferences over their choices. The set of all feasible choices may be continuous or discrete, whereas the strategy sets of different players can only differ in their upper and lower bounds. The resulting strategic game with discontinuous utilities does not satisfy the assumptions of any general theorem known as of today. Nonetheless, the existence of a (pure strategy) Nash equilibrium, as well as the "finite best response improvement property," are established

Decentralized Convergence to Nash Equilibria in Constrained Deterministic Mean Field Control

This paper considers decentralized control and optimization methodologies for large populations of systems, consisting of several agents with different individual behaviors, constraints and interests, and affected by the aggregate behavior of the overall population. For such large-scale systems, the theory of aggregative and mean field games has been established and successfully applied in various scientific disciplines. While the existing literature addresses the case of unconstrained agents, we formulate deterministic mean field control problems in the presence of heterogeneous convex constraints for the individual agents, for instance arising from agents with linear dynamics subject to convex state and control constraints. We propose several model-free feedback iterations to compute in a decentralized fashion a mean field Nash equilibrium in the limit of infinite population size. We apply our methods to the constrained linear quadratic deterministic mean field control problem and to the constrained mean field charging control problem for large populations of plug-in electric vehicles.Comment: IEEE Trans. on Automatic Control (cond. accepted

Cournot tatonnement and Nash equilibrium in binary status games

We study a rather simplified game model of competition for status. Each player chooses a scalar variable (say, the level of conspicuous consumption), and then those who chose the highest level obtain the "high" status, while everybody else remains with the "low" status. Each player strictly prefers the high status, but they also have intrinsic preferences over their choices. The set of all feasible choices may be continuous or discrete, whereas the strategy sets of different players can only differ in their upper and lower bounds. The resulting strategic game with discontinuous utilities does not satisfy the assumptions of any general theorem known as of today. Nonetheless, the existence of a (pure strategy) Nash equilibrium, as well as the "finite best response improvement property," are established

Enabling Privacy in a Distributed Game-Theoretical Scheduling System for Domestic Appliances

Demand side management (DSM) makes it possible to adjust the load experienced by the power grid while reducing the consumers' bill. Game-theoretic DSM is an appealing decentralized approach for collaboratively scheduling the usage of domestic electrical appliances within a set of households while meeting the users' preferences about the usage time. The drawback of distributed DSM protocols is that they require each user to communicate his/her own energy consumption patterns, which may leak sensitive information regarding private habits. This paper proposes a distributed privacy-friendly DSM system that preserves users' privacy by integrating data aggregation and perturbation techniques: users decide their schedule according to aggregated consumption measurements perturbed by means of additive white Gaussian noise. We evaluate the noise power and the number of users required to achieve a given privacy level, quantified by means of the increase of the information entropy of the aggregated energy consumption pattern. The performance of our proposed DSM system is compared to the one of a benchmark system that does not support privacy preservation in terms of total bill, peak demand, and convergence time. Results show that privacy can be improved at the cost of increasing the peak demand and the number of game iterations, whereas the total bill is only marginally incremented

POTENTIAL GAMES WITH AGGREGATION IN NON-COOPERATIVE GENERAL INSURANCE MARKETS

AbstractIn the global insurance market, the number of product-specific policies from different companies has increased significantly, and strong market competition has boosted the demand for a competitive premium. Thus, in the present paper, by considering the competition between each pair of insurers, an N-player game is formulated to investigate the optimal pricing strategy by calculating the Nash equilibrium in an insurance market. Under that framework, each insurer is assumed to maximise its utility of wealth over the unit time interval. With the purpose of solving a game of N-players, the best-response potential game with non-linear aggregation is implemented. The existence of a Nash equilibrium is proved by finding a potential function of all insurers' payoff functions. A 12-player insurance game illustrates the theoretical findings under the framework in which the best-response selection premium strategies always provide the global maximum value of the corresponding payoff function.</jats:p