On a class of differential games without saddle-point solutions

The minimax solution of a linear regulator problem is considered. A model representing a game situation in which the first player controls the dynamic system and selects a suitable, minimax control strategy, while the second player selects the aim of the game, is formulated. In general, the resulting differential game does not possess a saddle-point solution. Hence, the minimax solution for the player controlling the dynamic system is sought and obtained by modifying the performance criterion in such a way that (a) the minimax strategy remains unchanged and (b) the modified game possesses a saddle-point solution. The modification is achieved by introducing a regularization procedure which is a generalization of the method used in an earlier paper on the quadratic minimax problem. A numerical algorithm for determining the nonlinear minimax strategy in feedback form, in which Pagurek's result on open-loop and closed-loop sensitivity is used to nontrivially simplify the computational aspects of the problem, is presented and applied on a simple example.     

Bifurcation analysis of a non-cooperative differential game with one weak player

We study a bifurcation problem for a system of two differential equations in implicit form. For each value of the parameter θ, the solution yields a pair of Nash equilibrium strategies in feedback form, for a non-cooperative differential game. When θ=0, the second player has no power to influence the dynamics of the system, and his optimal strategy is myopic. The game thus reduces to an optimal control problem for the first player. By studying the bifurcation in the solutions to the corresponding system of Hamilton-Jacobi equations, one can establish existence and multiplicity of solutions to the differential game, as θ becomes strictly positive.     

The construction of stable bridges in differential games with phase constraints

A differential approach-and-evasion game in a finite time interval is considered [1]. It is assumed that the positions of the game are constricted by certain constraints which represent a closed set in the space of the positions. In the case of the first player, it is necessary to ensure that the phase point falls into the terminal set at a finite instant of time and, in the case of the second player, that this terminal set is evaded at this instant [1]. A method is proposed for the approximate construction of the positional absorption set, that is, the set of all positions belonging to a constraint from which the problem of approach facing the first player is solvable. Relations are written out which determine the system of sets which approximates the positional absorption set. The main result is a proof of the convergence of the approximate system of sets to the positional absorption set and the construction of a computational procedure for constructing the approximate system of sets.     

Existence theorems for games of survival

A two-person, zero-sum differential game of survival with general type phase constraints is investigated. The dynamics of both players is governed by a system of differential inclusions. Player II can choose any strategy in the Varaiya-Lin sense, while player I can select any lower -strategy (Ref. 1, p. 400). The existence of a value and an optimal strategy for player II is proved under the assumptions that the set of all player II's trajectories is compact in the Banach space of all continuous mappings and that some capturability condition is fulfilled.     

The solution of evolutionary games using the theory of Hamilton-Jacobi equations

A dynamical model of a non-antagonistic evolutionary game for two coalitions is considered. The model features an infinite time span and discounted payoff functionals. A solution is presented using differential game theory. The solution is based on the construction of a value function for auxiliary antagonistic differential games and uses an approximate grid scheme from the theory of generalized solutions of the Hamilton-Jacobi equations. Together with the value functions the optimal guaranteeing procedures for control on the grid are computed and the Nash dynamic equilibrium is constructed. The behaviour of trajectories generated by the guaranteeing controls is investigated. Examples are given.     

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.     

连续对策的最大熵策略密度对策解

对于正方形[0,2]×[0,2]上的连续对策,将局中人的非纯策略(概率分布函数)的导数称为这个局中人的策略密度(概率密度函数).建立了这种连续对策的最大熵理论.主要证明了当每个局中人都没有最优纯策略时,具有最大熵的最优策略密度集合的非空紧凸性,研究了最优策略密度的最大熵,给出一类带有最大熵的连续对策.     

A linear differential game on a plane with a minimum-type functional

A differential game on a plane with a functional in the form of the minimum, with respect to time, of a certain prescribed phase vector function (quality function) is considered. It is proved that the game value is constant outside a certain bounded region, consisting of two parts. In the first subregion, the value is equal to the quality function, and in the second it satisfies Bellman's equation. For the constant-value region, where the players’ optimum strategies are not unique, single-valued guaranteeing players’ strategies are proposed. The results of a numerical investigation of the problem are presented.     

Dominance elimination procedures on finite alternative games

A general definition of dominance elimination procedures for finite games is given which includes the multistage game representation of sophisticated voting for binary procedures. There then follows a proof that the set of outcomes resulting from the successive application of any dominance elimination procedure to any game in which each player has strict preferences over the alternatives, and in which the alternative and strategy spaces are finite, contains the set of outcomes attained by applying a procedure previously defined byFarquharson [1969].     

Single-type differential games with convex integral payoff

Single-type differential games are considered. The problem of taking a phase point to a disk of a fixed radius at a given time is studied. The payoff is the integral of a convex function depending on the norm of the first player’s control.     

Differential games with fixed terminal time and estimation of the instability degree of sets in these games

This paper contributes to the theory of differential games. A game problem of bringing a conflict-controlled system to a compact target set is analyzed. Sets in the position space that terminate on the target set and are not stable bridges are considered. The notion of stability defect of these sets is examined. It is demonstrated how the notion of stability defect can be used to construct sets with relatively good geometry that are at the same time convenient for the first player to play the game successfully.     

Nonzero-sum stochastic games with unbounded costs: Discounted and average cost cases

We treat non-cooperative stochastic games with countable state space and with finitely many players each having finitely many moves available in a given state. As a function of the current state and move vector, each player incurs a nonnegative cost. Assumptions are given for the expected discounted cost game to have a Nash equilibrium randomized stationary strategy. These conditions hold for bounded costs, thereby generalizing Parthasarathy (1973) and Federgruen (1978). Assumptions are given for the long-run average expected cost game to have a Nash equilibrium randomized stationary strategy, under which each player has constant average cost. A flow control example illustrates the results. This paper complements the treatment of the zero-sum case in Sennott (1993a).     

Absolutely cooperative solution for a linear,multiplayer differential game

If the players of ann-player differential game agree to cooperate, then the solutions to the game should be confined to undominated ones. A property of an undominated or Pareto-optimal solution is that, when compared locally with any other solution, at least one player must do worse or all do the same if they use a solution other than the Pareto-optimal one.Closely related to the concept of a Pareto-optimal solution is the concept of an absolutely cooperative solution. The absolutely cooperative solution is given the property that, when compared locally with any other solution, every player will do no better if a solution other than the absolutely cooperative one is used.A set of necessary and sufficient conditions for an absolutely cooperative solution is presented in this paper. The circumstance under which the control variables may take on interior values is also included.This research was supported in part by NASA Grant No. NGR-03-002-011.     

Control of an aircraft landing in windshear

The problem of the feedback control of an aircraft landing in the presence of windshear is considered. The landing process is investigated up to the time when the runway threshold is reached. It is assumed that the bounds on the wind velocity deviations from some nominal values are known, while information about the windshear location and wind velocity distribution in the windshear zone is absent. The methods of differential game theory are employed for the control synthesis.The complete system of aircraft dynamic equations is linearized with respect to the nominal motion. The resulting linear system is decomposed into subsystems describing the vertical (longitudinal) motion and lateral motion. For each subsystem, an, auxiliary antagonistic differential game with fixed terminal time and convex payoff function depending on two components of the state vector is formulated. For the longitudinal motion, these components are the vertical deviation of the aircraft from the glide path and its time derivative; for the lateral motion, these components are the lateral deviation and its time derivative. The first player (pilot) chooses the control variables so as to minimize the payoff function; the interest of the second player (nature) in choosing the wind disturbance is just opposite.The linear differential games are solved on a digital computer with the help of corresponding numerical methods. In particular, the optimal (minimax) strategy is obtained for the first player. The optimal control is specified by means of switch surfaces having a simple structure. The minimax control designed via the auxiliary differential game problems is employed in connection with the complete nonlinear system of dynamical equations.The aircraft flight through the wind downburst zone is simulated, and three different downburst models are used. The aircraft trajectories obtained via the minimax control are essentially better than those obtained by traditional autopilot methods.     

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.     

三步矩阵对策上的无中生有计及其实例

基于一个历史实例及假定:①三步矩阵对策中赢得矩阵都不变,②每步都是局中人1先行动,③对于每步对策,局中人2观测不到对手究竟使用了何策略;但局中人1可以观测到对手所用的策略,建立了三步矩阵对策上的无中生有计(《三十六计》中的第七计)的对策模型.研究了当局中人2中计,半识破和完全识破对手的无中生有计时的赢得和所用的策略的情况.并用上述实例对模型作了说明.     

On Pontryagin’s first direct method in pursuit problems

We describe an example of a three-dimensional linear differential game with convex compact sets of control. In this example, the integrand in Pontryagin’s first direct method is discontinuous on a set of positive measure.     

Proper belief revision and rationalizability in dynamic games

In this paper we develop an epistemic model for dynamic games in which players may revise their beliefs about the opponents’ utility functions as the game proceeds. Within this framework, we propose a rationalizability concept that is based upon the following three principles: (1) at every instance of the game, a player should believe that his opponents are carrying out optimal strategies, (2) a player, at information set h, should not change his belief about an opponent’s relative ranking of two strategies s and s′ if both s and s′ could have led to h, and (3) the players’ initial beliefs about the opponents’ utility functions should agree on a given profile u of utility functions. Common belief in these events leads to the concept of persistent rationalizability for the profile u of utility functions. It is shown that for a given game tree with observable deviators and a given profile u of utility functions, every properly point-rationalizable strategy is a persistently rationalizable strategy for u. This result implies that persistently rationalizable strategies always exist for all game trees with observable deviators and all profiles of utility functions. We provide an algorithm that can be used to compute the set of persistently rationalizable strategies for a given profile u of utility functions. For generic games with perfect information, persistent rationalizability uniquely selects the backward induction strategy for every player.     

On finite tetraprimary groups

A three-player game is considered in which the first and second players have dynamic superiority over the third player. Two fixed time points are specified. The game ends if either the first player captures the third player at the first time point, or the second player captures the third player at the second time point. We analyze a situation when the initial positions in the game are such that neither the first nor the second player alone can capture the third player at the specified points of time. We propose sufficient conditions on the parameters of the game under which, for given initial states of the players, the first and second players by applying some controls can guarantee that one of them will meet the third player at the prescribed moment. Simulation results for a model example are also presented.