Pair Of Optimal Strategies Research Articles

The Convergence of a Cooperation Markov Decision Process System.

In a general Markov decision progress system, only one agent’s learning evolution is considered. However, considering the learning evolution of a single agent in many problems has some limitations, more and more applications involve multi-agent. There are two types of cooperation, game environment among multi-agent. Therefore, this paper introduces a Cooperation Markov Decision Process system with two agents, which is suitable for the learning evolution of cooperative decision between two agents. It is further found that the value function in the system also converges in the end, and the convergence value is independent of the choice of the value of the initial value function. This paper presents an algorithm for finding the optimal strategy pair in the system, whose fundamental task is to find an optimal strategy pair and form an evolutionary system . Finally, an example is given to support the theoretical results.

A Policy Improvement Algorithm for Solving a Mixture Class of Perfect Information and AR-AT Semi-Markov Games

Zero-sum two-person discounted semi-Markov games with finite state and action spaces are studied where a collection of states having Perfect Information (PI) property is mixed with another collection of states having Additive Reward–Additive Transition and Action Independent Transition Time (AR-AT-AITT) property. For such a PI/AR-AT-AITT mixture class of games, we prove the existence of an optimal pure stationary strategy for each player. We develop a policy improvement algorithm for solving discounted semi-Markov decision processes (one player version of semi-Markov games) and using it we obtain a policy-improvement type algorithm for computing an optimal strategy pair of a PI/AR-AT-AITT mixture semi-Markov game. Finally, we extend our results when the states having PI property are replaced by a subclass of Switching Control (SC) states.

Differential game model with discretized solution for the use of limited water resources

It is indispensable for the sustainable water supply of the human society to distribute the available water resources with maximal efficiency among the different consumers while assuring the needed time for recharge. The theoretically established, effective tool for the solution of this allocation problem is mathematical modelling with special regard to the methods of game theory.In the present paper, a differential game is proposed to describe the consumption and the (natural) recharge of a limited water resource. Then the general course of the solution of the allocation problem is given aiming at maximizing the players’ yields through the discretization of the strategy sets. In case of two consumers, the solution is given in more details along with several practical examples. The Pareto optimality of the non-cooperative solution (Nash equilibrium) is checked and the Pareto optimal strategy pair, which provides maximal sum payoff for the players, is suggested for cooperation (after a possible re-distribution of the payoffs). This cooperative solution generally provides higher payoff for each player than if they consumed according to conflicting non-cooperative behaviour. In addition, examples with three consumers are provided similarly.

Zero-Sum Markov Games with Random State-Actions-Dependent Discount Factors: Existence of Optimal Strategies

This work deals with a class of discrete-time zero-sum Markov games under a discounted optimality criterion with random state-actions-dependent discount factors of the form $$\tilde{\alpha }(x_{n},a_{n},b_{n},\xi _{n+1})$$ , where $$x_{n}, a_{n}, b_{n}$$ , and $$\xi _{n+1}$$ are the state, the actions of players, and a random disturbance at time n, respectively, taking values in Borel spaces. Assuming possibly unbounded payoff, we prove the existence of a value of the game as well as a stationary pair of optimal strategies.

Zero-Sum Stochastic Games with Partial Information and Average Payoff

We consider a discrete time partially observable zero-sum stochastic game with average payoff criterion. We study the game using an equivalent completely observable game. We show that the game has a value and also we present a pair of optimal strategies for both the players.

Stackelberg strategy solution for optimal software release policies

We present a software release policy which is based on the Stackelberg strategy solution concept. The model formulated assumes the existence of two type of producers in the market, the leader and follower. The resulting release policy combines both cost factors and a loss of opportunity factor which is the result of competition between the rival producers. We define a Stackelberg strategy pair in the context of our model and, through a series of preliminary results, show that an optimal strategy pair exists. We also present a numerical example which utilizes a software reliability growth model based on the nonhomogeneous Poisson process. Finally, we explore the relative leadership property of the optimal strategies.

Correlated Equilibrium in Two-Person Zero-Sum Games

but any convex combination of pairs of optimal strategies such that p(2, 2) = 0 satisfies p(1, 1) > 2 (with the obvious notation p(i, j) for the induced probability of row i and column j). However, the following is easily checked. Let I and J be the sets of pure strategies of player 1 and player 2 respectively in a zero-sum game G with value v. Then p = [p(i, i)I(, J) IXJ is a correlated equilibrium distribution for G if and only if for every E J such that p(jo) > 0, the conditional probability of player 2 over player l's actions given jo' [p (iIjo)]I , is an optimal strategy for player 1, yielding exactly v against jo and similarly for [p(jlio)]jEj, io E , p(io) > 0. Hence as conjectured by R. Aumann, if a pure strategy pair occurs with positive probability in a correlated equilibrium, then it occurs with positive probability in a pair of optimal strategies. Also, if one of the players has a unique optimal strategy, then every correlated equilibrium distribution concentrates on a pair of optimal strategies.

A Simplified Two-Person Poker with a Noisy Communication Channel Between Players

Abstract In this note, we consider a version of two-person zero-sum poker game, la Relance in which player I’s decision is transmitted to player II through a channel with noise. We obtain a pair of optimal strategies and the game value, and then consider the relation between a channel matrix and our solution.

A model for real poker with an upper bound of assets

This paper considers a continuous model of two-person poker, where the maximal amount of betB is assumed and the player who acts first chooses the amount of bet in the game. We analyze a model, in which the range of the amount of bet β is a finite interval [0,B], 0≤B<+∞, to obtain a saddle point of the payoff function as a pair of optimal strategies among mixed strategies. We compare our results with those of Karlin and Restrepo and those of Newman.

Qualification of optimal strategies and singular surfaces of a class of differential games

At each point of a regular region, there exist two tangent cones which are complementary to each other. The two vectograms defined by the pair of optimal strategies must be contained separately in the two tangent cones. The velocity vector resulting from the selection of the optimal strategies consequently must represent a semipermeable direction. These conditions, which reveal a fundamental separating property of the optimal velocity vector, are weaker than that of Isaacs' main equation. However, they hold even on singular surfaces. A singular surface arises only when the resulting velocity vector of the previous optimal strategies cannot continuously represent a semipermeable direction after the optimal path crosses over the surface. This observation yields a necessary condition for a singular surface to occur. The localization of singular surfaces is then possible. Finally, a smooth condition with respect to optimal paths is given.

Existence of optimal open-loop strategies for a class of differential games

In this paper, the class of differential games with linear system equations and a quadratic performance index is investigated for saddlepoint solutions when one or both of the players use open-loop control. For each formulation of the game, a necessary and sufficient condition is obtained for the existence of an optimal strategy pair that generates a regular optimal path. For those cases where a solution exists, the unique saddle-point solution is presented. Also, relationships are established between the time intervals of existence of solutions for the various formulations of the game.

Finite state stochastic games: Existence theorems and computational procedures

Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{X_{n}}</tex> be a Markov process with finite state space and transition probabilities <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p_{ij}(u_{i}, v_{i})</tex> depending on u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v_{i}.</tex> State 0 is the capture state (where the game ends; <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p_{oi} \equiv \delta_{oi})</tex> ; <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u = {u_{i}}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v = {v_{i}}</tex> are the pursuer and evader strategies, respectively, and are to be chosen so that capture is advanced or delayed and the cost <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C_{i^{u,v}} = E[\Sum_{0}^{\infty} k (u(X_{n}), v(X_{n}), X_{n}) | X_{0} = i]</tex> is minimaxed (or maximined), where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k(\alpha, \beta, 0) \equiv 0</tex> . The existence of a saddle point and optimal strategy pair or e-optimal strategy pair is considered under several conditions. Recursive schemes for computing the optimal or ε-optimal pairs are given.

A differential game of pursuit and evasion

This note deals with a two player, zero-sum differential game. The pursuer endeavors to minimize while the evader seeks to maximize the distance between them at a given time. The controls at the disposal of the players are bounded forces per unit mass. An optimal strategy pair—that is, a saddle point—is deduced by invoking necessary conditions. Optimality is ascertained by employing sufficiency conditions.