Nash Equilibrium Seeking in Quadratic Noncooperative Games Under Two Delayed Information-Sharing Schemes

In this paper, we propose non-model-based strategies for locally stable convergence to Nash equilibrium in quadratic noncooperative games where acquisition of information (of two different types) incurs delays. Two sets of results are introduced: (a) one, which we call cooperative scenario, where each player employs the knowledge of the functional form of his payoff and knowledge of other players’ actions, but with delays; and (b) the second one, which we term the noncooperative scenario, where the players have access only to their own payoff values, again with delay. Both approaches are based on the extremum seeking perspective, which has previously been reported for real-time optimization problems by exploring sinusoidal excitation signals to estimate the Gradient (first derivative) and Hessian (second derivative) of unknown quadratic functions. In order to compensate distinct delays in the inputs of the players, we have employed predictor feedback. We apply a small-gain analysis as well as averaging theory in infinite dimensions, due to the infinite-dimensional state of the time delays, in order to obtain local convergence results for the unknown quadratic payoffs to a small neighborhood of the Nash equilibrium. We quantify the size of these residual sets and corroborate the theoretical results numerically on an example of a two-player game with delays.

     

Distributed consensus in noncooperative inventory games

This paper deals with repeated nonsymmetric congestion games in which the players cannot observe their payoffs at each stage. Examples of applications come from sharing facilities by multiple users. We show that these games present a unique Pareto optimal Nash equilibrium that dominates all other Nash equilibria and consequently it is also the social optimum among all equilibria, as it minimizes the sum of all the players’ costs. We assume that the players adopt a best response strategy. At each stage, they construct their belief concerning others probable behavior, and then, simultaneously make a decision by optimizing their payoff based on their beliefs. Within this context, we provide a consensus protocol that allows the convergence of the players’ strategies to the Pareto optimal Nash equilibrium. The protocol allows each player to construct its belief by exchanging only some aggregate but sufficient information with a restricted number of neighbor players. Such a networked information structure has the advantages of being scalable to systems with a large number of players and of reducing each player’s data exposure to the competitors.     

Unilateral vs. bilateral link-formation: A transition without decay

We provide a model that bridges the gap between the simplest variation of two benchmark models of strategic network formation: Bala and Goyal’s two-way flow model without decay, where links can be unilaterally formed, and a variation of Jackson and Wolinsky’s model based on bilateral formation of links assuming no decay. In the model introduced and studied here a link can be created unilaterally. When it is only supported by one of the two players the flow through the link suffers some degree of decay, but when it is supported by both the flow runs without friction. When the decay in links supported by only one player is maximal (i.e. there is no flow) we have a simple variation of Jackson and Wolinsky’s connections model assuming no decay, while when flow in those links is perfect we have Bala and Goyal’s two-way flow model without decay. We study Nash, strict Nash and pairwise Nash stability for the intermediate models. Efficiency and dynamics are also examined.     

Internal correlation in repeated games

This paper characterizes the set of all the Nash equilibrium payoffs in two player repeated games where the signal that the players get after each stage is either trivial (does not reveal any information) or standard (the signal is the pair of actions played). It turns out that if the information is not always trivial then the set of all the Nash equilibrium payoffs coincides with the set of the correlated equilibrium payoffs. In particular, any correlated equilibrium payoff of the one shot game is also a Nash equilibrium payoff of the repeated game.For the proof we develop a scheme by which two players can generate any correlation device, using the signaling structure of the game. We present strategies with which the players internally correlate their actions without the need of an exogenous mediator.     

Variational inequality formulation for the games with random payoffs

We consider an n-player non-cooperative game with random payoffs and continuous strategy set for each player. The random payoffs of each player are defined using a finite dimensional random vector. We formulate this problem as a chance-constrained game by defining the payoff function of each player using a chance constraint. We first consider the case where the continuous strategy set of each player does not depend on the strategies of other players. If a random vector defining the payoffs of each player follows a multivariate elliptically symmetric distribution, we show that there exists a Nash equilibrium. We characterize the set of Nash equilibria using the solution set of a variational inequality (VI) problem. Next, we consider the case where the continuous strategy set of each player is defined by a shared constraint set. In this case, we show that there exists a generalized Nash equilibrium for elliptically symmetric distributed payoffs. Under certain conditions, we characterize the set of a generalized Nash equilibria using the solution set of a VI problem. As an application, the random payoff games arising from electricity market are studied under chance-constrained game framework.     

Nondominated Nash points: application of biobjective mixed integer programming

We study the connection between biobjective mixed integer linear programming and normal form games with two players. We first investigate computing Nash equilibria of normal form games with two players using single-objective mixed integer linear programming. Then, we define the concept of efficient (Pareto optimal) Nash equilibria. This concept is precisely equivalent to the concept of efficient solutions in multi-objective optimization, where the solutions are Nash equilibria. We prove that the set of all points in the payoff (or objective) space of a normal form game with two players corresponding to the utilities of players in an efficient Nash equilibrium, the so-called nondominated Nash points, is finite. We demonstrate that biobjective mixed integer linear programming, where the utility of each player is an objective function, can be used to compute the set of nondominated Nash points. Finally, we illustrate how the nondominated Nash points can be used to determine the disagreement point of a bargaining problem.     

Identity and fragmentation in networks

This paper examines the role of identity in the fragmentation of networks by incorporating the choice of commitment to identity characteristics into a non-cooperative network formation game. The Nash network features divisions based on identity, with layers within these divisions. Using the refinement of strictness I find structures where stars of highly committed players are linked together by less committed players.     

Probabilities of Pure Nash Equilibria in Matrix Games when the Payoff Entries of One Player Are Randomly Selected

The Nash equilibrium in pure strategies represents an important solution concept in nonzero sum matrix games. Existence of Nash equilibria in games with known and with randomly selected payoff entries have been studied extensively. In many real games, however, a player may know his own payoff entries but not the payoff entries of the other player. In this paper, we consider nonzero sum matrix games where the payoff entries of one player are known, but the payoff entries of the other player are assumed to be randomly selected. We are interested in determining the probabilities of existence of pure Nash equilibria in such games. We characterize these probabilities by first determining the finite space of ordinal matrix games that corresponds to the infinite space of matrix games with random entries for only one player. We then partition this space into mutually exclusive spaces that correspond to games with no Nash equilibria and with r Nash equilibria. In order to effectively compute the sizes of these spaces, we introduce the concept of top-rated preferences minimal ordinal games. We then present a theorem which provides a mechanism for computing the number of games in each of these mutually exclusive spaces, which then can be used to determine the probabilities. Finally, we summarize the results by deriving the probabilities of existence of unique, nonunique, and no Nash equilibria, and we present an illustrative example.     

A solution for noncooperative games

In this paper, we study solutions of strict noncooperative games that are played just once. The players are not allowed to communicate with each other. The main ingredient of our theory is the concept of rationalizing a set of strategies for each player of a game. We state an axiom based on this concept that every solution of a noncooperative game is required to satisfy. Strong Nash solvability is shown to be a sufficient condition for the rationalizing set to exist, but it is not necessary. Also, Nash solvability is neither necessary nor sufficient for the existence of the rationalizing set of a game. For a game with no solution (in our sense), a player is assumed to recourse to a standard of behavior. Some standards of behavior are examined and discussed.This work was sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and by the National Science Foundation under Grant No. MCS-75-17385-A01. The author is grateful to J. C. Harsanyi for his comments and to S. M. Robinson for suggesting the problem.     

On Nash equilibrium points and games of imperfect information

This paper considers a class of two-player, nonzero-sum games in which the players have only local, as opposed to global, information about the payoff functions. We study various modes of behavior and their relationship to different stability properties of the Nash equilibrium points.     

One-way flow network formation under constraints

We study the effects of institutional constraints on stability and efficiency in the “one-way flow” model of network formation. In this model the information that flows through a link between two players runs only towards the player that initiates and supports the link, so in order for it to flow in both directions, both players must pay whatever the unit cost of a directional link is. We assume that an exogenous “societal cover” consisting of a collection of possibly overlapping subsets covering the set of players specifies the social organization in different groups or “societies,” so that a player may initiate links only with players that belong to at least one society that he/she also belongs to, thus restricting the feasible strategies and networks. In this setting, we examine the impact of such societal constraints on stable/efficient architectures and on dynamics.     

Stochastic Discrete-Time Nash Games with Constrained State Estimators

In this paper, we consider stochastic linear-quadratic discrete-time Nash games in which two players have access only to noise-corrupted output measurements. We assume that each player is constrained to use a linear Kalman filter-like state estimator to implement his optimal strategies. Two information structures available to the players in their state estimators are investigated. The first has access to one-step delayed output and a one-step delayed control input of the player. The second has access to the current output and a one-step delayed control input of the player. In both cases, statistics of the process and statistics of the measurements of each player are known to both players. A simple example of a two-zone energy trading system is considered to illustrate the developed Nash strategies. In this example, the Nash strategies are calculated for the two cases of unlimited and limited transmission capacity constraints.     

Bounded rationality, strategy simplification, and equilibrium

It is frequently suggested that predictions made by game theory could be improved by considering computational restrictions when modeling agents. Under the supposition that players in a game may desire to balance maximization of payoff with minimization of strategy complexity, Rubinstein and co-authors studied forms of Nash equilibrium where strategies are maximally simplified in that no strategy can be further simplified without sacrificing payoff. Inspired by this line of work, we introduce a notion of equilibrium whereby strategies are also maximally simplified, but with respect to a simplification procedure that is more careful in that a player will not simplify if the simplification incents other players to deviate. We study such equilibria in two-player machine games in which players choose finite automata that succinctly represent strategies for repeated games; in this context, we present techniques for establishing that an outcome is at equilibrium and present results on the structure of equilibria.     

How long to Pareto efficiency?

We consider uncoupled dynamics (each player knows only his own payoff function) that reach outcomes that are Pareto efficient and individually rational. We show that in the worst case the number of periods it takes to reach these outcomes must be exponential in the number of players and hence the same number of periods it takes to reach Nash equilibria. For social welfare maximizing outcomes we provide a tight bound on the minimal number of steps required for reaching such an outcome by uncoupled dynamics.     

Adaptive expectations and dynamic adjustment in noncooperative games with incomplete information

This paper is concerned with the analysis of a Nash equilibrium of a noncooperative game. It can be shown that, without complete information about the other players' objectives or interests, the group of players, as a whole, can reach a Nash equilibrium by adopting a class of adaptive expectation and dynamic adjustment processes. It is shown that, if the expectation and adjustment processes are made continuously, the stability of the overall dynamic process is independent of the specific mechanisms of the expectation and the adjustment, but depends on the properties of each player's objective or payoff function. If, however, expectation and adjustment processes are made at discrete time intervals, the stability of the discrete process depends on the speed of adjustment chosen by each player.This research was supported by ONR Contract No. N00014-75-C-0738. The authors are indebted to the referee for several valuable comments and suggestions for improvement.     

Analysis of decision-making in economic chaos control

In some economic chaotic systems, players are concerned about whether their performance is improved besides taking some methods to control chaos. In the face of chaos occurring in competition, whether one player takes controlling measures or not affects not only their own earning but also other opponents’ income. An output duopoly competing evolution model with bounded rationality is introduced in this paper. Using modern game theory, decision-making analyses about chaos control of the model are taken by taking aggregate profits as players’ payoff. It is found that the speed of players’ response to the market and whether the decisive parameters are in the stable region of the Nash equilibrium or not have a distinct influence on the results of the game. The impact of cost function’ type on results of the game is also found. The mechanism of influences is discovered by using numerical simulation.     

A Characterization of Nash Equilibrium for the Games with Random Payoffs

We consider a two-player random bimatrix game where each player is interested in the payoffs which can be obtained with certain confidence. The payoff function of each player is defined using a chance constraint. We consider the case where the entries of the random payoff matrix of each player jointly follow a multivariate elliptically symmetric distribution. We show an equivalence between the Nash equilibrium problem and the global maximization of a certain mathematical program. The case where the entries of the payoff matrices are independent normal/Cauchy random variables is also considered. The case of independent normally distributed random payoffs can be viewed as a special case of a multivariate elliptically symmetric distributed random payoffs. As for Cauchy distribution, we show that the Nash equilibrium problem is equivalent to the global maximization of a certain quadratic program. Our theoretical results are illustrated by considering randomly generated instances of the game.     

Single-payoff farsighted stable sets in strategic games with dominant punishment strategies

We investigate farsighted stable sets in a class of strategic games with dominant punishment strategies. In this class of games, each player has a strategy that uniformly minimizes the other players’ payoffs for any given strategies chosen by these other players. We particularly investigate a special class of farsighted stable sets, each of which consists of strategy profiles yielding a single payoff vector. We call such a farsighted stable set as a single-payoff farsighted stable set. We propose a concept called an inclusive set that completely characterizes single-payoff farsighted stable sets in strategic games with dominant punishment strategies. We also show that the set of payoff vectors yielded by single-payoff farsighted stable sets is closely related to the strict \(\alpha \)-core in a strategic game. Furthermore, we apply the results to strategic games where each player has two strategies and strategic games associated with some market models.     

Equilibrium selection and altruistic behavior in noncooperative social networks

Given their importance in determining the outcome of many economic interactions, different models have been proposed to determine how social networks form and which structures are stable. In Bala and Goyal (Econometrica 68, 1181–1229, 2000), the one-sided link formation model has been considered, which is based on a noncooperative game of network formation. They found that the empty networks, the wheel in the one-way flow of benefits case and the center-sponsored star in the two-way flow case play a fundamental role since they are strict Nash equilibria of the corresponding games for a certain class of payoff functions. In this paper, we first prove that all these network structures are in weakly dominated strategies whenever there are no strict Nash equilibria. Then, we exhibit a more accurate selection device between these network architectures by considering “altruistic behavior” refinements. Such refinements that we investigate here in the framework of finite strategy sets games have been introduced by the authors in previous papers.