When is the value of public information positive in a game?

The value of public information is studied by considering the equilibrium selections that maximize the weighted sum of players' payoffs. We show that the value of information can be deduced from the deterministic games where the uncertain parameters have given values. If the maximal weighted sum of equilibrium payoffs in deterministic games is convex then the value of information in any Bayesian game derived from the deterministic games is positive with respect to the selection. We also show the converse result that positive value of information implies convexity. Hence, the convexity of maximal weighted sum of payoffs in deterministic games fully characterizes the value of information with respect to considered selections. We also discuss the implications of our results when positive value of information means that for any equilibrium in a game with less information there is a Pareto dominant equilibrium in any game with more information.     

Piecewise suboptimal control laws for differential games

This paper is concerned with optimal parameter selection in differential games. Necessary and sufficient conditions are derived for the existence of a saddle point for a general two-person zero-sum differential game when one or both the players use suboptimal control laws of specified form. The specified forms for the controls consist of weighted sums of the state variables, the weighting factors being products of known time-varying functions and of piecewise-constant functions to be determined in an optimal manner. The controls which are formed in this way are referred to as piecewise control laws. The time intervals associated with the piecewise control laws can be different for each player. The general results are applied to linear-quadratic games, and for this class of differential games, an additional development is given to obtain piecewise control law parameters that are independent of initial conditions, so that a saddle point with respect to the expected value of the performance index is obtained. Consideration has also been given to the problem of optimizing the gain change points. The results are applied to scalar and vector dynamic systems, and numerical solutions are presented.     

Pursuit–Evasion Games with incomplete information in discrete time

Pursuit–Evasion Games (in discrete time) are stochastic games with nonnegative daily payoffs, with the final payoff being the cumulative sum of payoffs during the game. We show that such games admit a value even in the presence of incomplete information and that this value is uniform, i.e. there are e{\epsilon}-optimal strategies for both players that are e{\epsilon}-optimal in any long enough prefix of the game. We give an example to demonstrate that nonnegativity is essential and expand the results to Leavable Games.     

Approximation and representation of the value for some differential games with asymmetric information

We consider differential games with incomplete information. For special games with dynamics independent of the state of the system and linear payoffs, we give a representation formula for the value similar to the value of repeated games with lack of information on both sides. For general games, this representation formula does not hold and we introduce an approximation of the value: we build a sequence of functions converging to the value function.     

Adaptive allocation rules for hypergraph games

Cooperative games with hypergraph structure, or hypergraph games, assume that all players in a hyperlink or conference have to be present before communication. Contrary to this situation, assuming that whenever players leave a conference the remaining players can still communicate with each other, adaptive allocation rules for hypergraph games, being alternative extensions of the Myerson value and the position value respectively, are introduced in this paper. Axiomatic characterizations are also provided by considering players' absence.     

On the role of chance moves and information in two-person games

The value of information has been the subject of many studies in a strategic context. The central question in these studies is how valuable the information hidden in the chance moves of a game is for one or more of the players. Generally speaking, only the extra possibilities that are beneficial for the players have been considered so far. In this note we study the value of information for a special class of two-person games. For these games we also investigate how “badly” the players can do, both with and without knowing the result of the chance move. In this way one can determine to what extent the players are restricted in their possibilities by the fact that some information is hidden in the chance moves of the games. This allows for a comparison of the influence of the chance move to the control that the players have over the game result.     

Information market games

In this paper information markets with perfect patent protection and only one initial owner of the information are studied by means of cooperative game theory. To each information market of this type a cooperative game with sidepayments is constructed. These cooperative games are called information (market) games. The set of all information games with fixed player set is a cone in the set of all cooperative games with the same player set. Necessary and sufficient conditions are given in order that a cooperative game is an information game. The core of this kind of games is not empty and is also the minimal subsolution of the game. The core is the image of an (n-1)-dimensional hypercube under an affine transformation, (= hyperparallellopiped), the nucleolus and -value coincide with the center of the core. The Shapley value is computed and may lie inside or outside the core. The Shapley value coincides with the nucleolus and the -value if and only if the information game is convex. In this case the core is also a stable set.     

On stochastic programmed design of strategies in a differential game

A differential game /1–17/ is analyzed, in which the strategies form controls on the basis of information on the motion's history. The computation of this game's value is discussed, as also is the construction of optimal strategies on the basis of auxiliary programmed constructions which contain an artificially introduced random element. Thus, a method of stochastic programmed design, proposed in /18,19/ for differential games, is examined here from a certain general viewpoint.     

New results in the theory of differential games with information time lag

The results contained herein provide a rigorous formulation of a broad class of differential games with information time lag and present a theoretical analysis for treating such games. This analysis extends the so-called Hamilton-Jacobi theory of optimal control and the main equation analysis developed by Isaacs to treat differential games with information time lag. Necessary and sufficient conditions satisfied by thepotential value function are developed to indicate the strategy-synthesis procedure for differential games with information time lag.     

Existence of a value in semidynamical games

The games (generalizing differential games) in which the dynamics of players is described by k-semidynamical systems are called semidynamical games. For such games two theorems on the existence of a value in the class of piecewise program strategies are proved. Examples are given to show that the conditions of these theorems impose very weak restrictions on the set of admissible controls of the players and, in the games with a fixed duration, on the set of trajectories of semidynamical systems. Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 401–410, September, 1977.     

Alfa-cut based linear programming methodology for constrained matrix games with payoffs of trapezoidal fuzzy numbers

The purpose of this paper is to develop an effective methodology for solving constrained matrix games with payoffs of trapezoidal fuzzy numbers (TrFNs), which are a type of two-person non-cooperative games with payoffs expressed by TrFNs and players’ strategies being constrained. In this methodology, it is proven that any Alfa-constrained matrix game has an interval-type value and hereby any constrained matrix game with payoffs of TrFNs has a TrFN-type value. The auxiliary linear programming models are derived to compute the interval-type value of any Alfa-constrained matrix game and players’ optimal strategies. Thereby the TrFN-type value of any constrained matrix game with payoffs of TrFNs can be directly obtained through solving the derived four linear programming models with data taken from only 1-cut and 0-cut of TrFN-type payoffs. Validity and applicability of the models and method proposed in this paper are demonstrated with a numerical example of the market share game problem.     

On the saddle-point solution of a class of stochastic differential games

This paper deals with the saddle-point solution of a class of stochastic differential games described by linear state dynamics and quadratic objective functionals. The information structure of the problem is such that both players have access to a common noisy linear measurement of the state and they are permitted to utilize only this information in constructing their controls. The saddle-point solution of such differential game problems has been discussed earlier in Ref. 1, but the conclusions arrived there are incorrect, as is explicitly shown in this paper. We extensively discuss the role of information structure on the saddle-point solution of such stochastic games (specifically within the context of an illustrative discrete-time example) and then obtain the saddle-point solution of the problem originally formulated by employing an indirect approach.This work was done while the author was on sabbatical leave at Twente University of Technology, Department of Applied Mathematics, Enschede, Holland, from Applied Mathematics Division, Marmara Scientific and Industrial Research Institute, Gebze, Kocaeli, Turkey.     

An operator solution of stochastic games

A class of zero-sum, two-person stochastic games is shown to have a value which can be calculated by transfinite iteration of an operator. The games considered have a countable state space, finite action spaces for each player, and a payoff sufficiently general to include classical stochastic games as well as Blackwell’s infiniteG _δ games of imperfect information. Research supported by National Science Foundation Grants DMS-8801085 and DMS-8911548.     

The Shapley value for bicooperative games

The aim of the present paper is to study a one-point solution concept for bicooperative games. For these games introduced by Bilbao (Cooperative Games on Combinatorial Structures, 2000) , we define a one-point solution called the Shapley value, since this value can be interpreted in a similar way to the classical Shapley value for cooperative games. The main result of the paper is an axiomatic characterization of this value.     

On the Time Consistency of Equilibria in a Class of Additively separable Differential Games

A class of state-redundant differential games is detected, where players can be partitioned into two groups, so that the state dynamics and the payoff functions of all players are additively separable w.r.t. controls and states of any two players belonging to different groups. We prove that, in this class of games, open-loop Nash and feedback Stackelberg equilibria coincide, both being strongly time consistent. This allows us to bypass the issue of the time inconsistency that typically affects the open-loop Stackelberg solution.     

Additional aspects of the Stackelberg strategy in nonzero-sum games

The Stackelberg strategy in nonzero-sum games is a reasonable solution concept for games where, either due to lack of information on the part of one player about the performance function of the other, or due to different speeds in computing the strategies, or due to differences in size or strength, one player dominates the entire game by imposing a solution which is favorable to himself. This paper discusses some properties of this solution concept when the players use controls that are functions of the state variables of the game in addition to time. The difficulties in determining such controls are also pointed out. A simple two-stage finite state discrete game is used to illustrate these properties.This work was supported in part by the U.S. Air Force under Grant No. AFOSR-68-1579D, in part by NSF under Grant No. GK-36276, and in part by the Joint Services Electronics Program under Contract No. DAAB-07-72-C-0259 with the Coordinated Science Laboratory, University of Illinois, Urbana, Illinois.     

Performance versus informativeness in linear-quadratic Gaussian noncooperative games

In this paper, we study the impact of informativeness on the performance of linear quadratic Gaussian Nash and Stackelberg games. We first show that, in two-person static Nash games, if one of the players acquires more information, then this extra information is beneficial to him, provided that it is orthogonal to both players' information. A special case is that when one of the players is informationally stronger than the other, then any new information is beneficial to him. We then show that a similar result holds for dynamic Nash games. In the dynamic games, the players use strategies that are linear functions of the current estimates of the state, generated by two Kalman filters. The same properties are proved to hold in static and feedback Stackelberg games as well.This work was partially supported by the US Air Force Office of Scientific Research under Grant No. AFOSR-82-0174.     

A gradient projection method for solving continuous games

This paper develops a procedure for numerically solving continuous games (and also matrix games) using a gradient projection method in a general Hilbert space setting. First, we analyze the symmetric case. Our approach is to introduce a functional which measures how far a strategy deviates from giving zero value (i.e., how near the strategy is to being optimal). We then incorporate this functional into a nonlinear optimization problem with constraints and solve this problem using the gradient projection algorithm. The convergence is studied via the corresponding steepest-descent differential equation. The differential equation is a nonlinear initial-value problem in a Hilbert space; thus, we include a proof of existence and uniqueness of its solution. Finally, nonsymmetric games are handled using the symmetrization techniques of Ref. 1.     

An approach via state estimation to differential games with information time lag

This paper deals with zero-sum two-person differential games in which one player has a deferred information on the state vector. This player mends this lack of information by using an adaptative deterministic extrapolation to estimate the plant state, and then, makes his decisions by means of the datas so obtained. An analysis of the phenomenon yields a criterion for optimizing the estimation which is based upon the Hamiltonian estimation of the perfect information game. A class of extrapolators is given by its dynamical equation. Then, the initial game is reduced to a new game containing pure time delay in the state and the controls.