scholarly journals On a Simplified Method of Defining Characteristic Function in Stochastic Games

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1135
Author(s):  
Elena Parilina ◽  
Leon Petrosyan
Keyword(s):  
Characteristic Function ◽  
Stochastic Games ◽  
Stochastic Game ◽  
New Method ◽  
Characteristic Functions ◽  
Zero Sum Games ◽  
Subgame Consistency ◽  
The Core ◽  
Simplified Method ◽  
Zero Sum

In the paper, we propose a new method of constructing cooperative stochastic game in the form of characteristic function when initially non-cooperative stochastic game is given. The set of states and the set of actions for any player is finite. The construction of the characteristic function is based on a calculation of the maximin values of zero-sum games between a coalition and its anti-coalition for each state of the game. The proposed characteristic function has some advantages in comparison with previously defined characteristic functions for stochastic games. In particular, the advantages include computation simplicity and strong subgame consistency of the core calculated with the values of the new characteristic function.

scholarly journals Cooperative Stochastic Games with Mean-Variance Preferences

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 230
Author(s):  
Elena Parilina ◽  
Stepan Akimochkin
Keyword(s):  
Standard Deviation ◽  
Stochastic Games ◽  
Stochastic Game ◽  
Stochastic Variable ◽  
Expected Payoff ◽  
The Core ◽  
Risk Sensitive ◽  
Payoff Functions ◽  
Cooperative Solutions ◽  
Mean Variance

In stochastic games, the player’s payoff is a stochastic variable. In most papers, expected payoff is considered as a payoff, which means the risk neutrality of the players. However, there may exist risk-sensitive players who would take into account “risk” measuring their stochastic payoffs. In the paper, we propose a model of stochastic games with mean-variance payoff functions, which is the sum of expectation and standard deviation multiplied by a coefficient characterizing a player’s attention to risk. We construct a cooperative version of a stochastic game with mean-variance preferences by defining characteristic function using a maxmin approach. The imputation in a cooperative stochastic game with mean-variance preferences is supposed to be a random vector. We construct the core of a cooperative stochastic game with mean-variance preferences. The paper extends existing models of discrete-time stochastic games and approaches to find cooperative solutions in these games.

Irrational-Behavior-Proof Conditions Based on Limit Characteristic Functions

2019 ◽  
Vol 7 (1) ◽  
pp. 1-16
Author(s):  
Cui Liu ◽  
Hongwei Gao ◽  
Ovanes Petrosian ◽  
Juan Xue ◽  
Lei Wang
Keyword(s):  
Characteristic Function ◽  
Cooperative Game ◽  
Shapley Value ◽  
Cooperative Games ◽  
Characteristic Functions ◽  
The Core ◽  
The Shapley Value ◽  
Irrational Behavior ◽  
The Individual

Abstract Irrational-behavior-proof (IBP) conditions are important aspects to keep stable cooperation in dynamic cooperative games. In this paper, we focus on the establishment of IBP conditions. Firstly, the relations of three kinds of IBP conditions are described. An example is given to show that they may not hold, which could lead to the fail of cooperation. Then, based on a kind of limit characteristic function, all these conditions are proved to be true along the cooperative trajectory in a transformed cooperative game. It is surprising that these facts depend only upon the individual rationalities of players for the Shapley value and the group rationalities of players for the core. Finally, an illustrative example is given.

Continuous stochastic games

1973 ◽  
Vol 10 (3) ◽  
pp. 597-604 ◽  
Author(s):  
Matthew J. Sobel
Keyword(s):  
Equilibrium Point ◽  
Stochastic Games ◽  
Metric Spaces ◽  
Stochastic Game ◽  
Zero Sum

Nonzero-sum N-person stochastic games are a generalization of Shapley's two-person zero-sum terminating stochastic game. Rogers and Sobel showed that an equilibrium point exists when the sets of states, actions, and players are finite. The present paper treats discounted stochastic games when the sets of states and actions are given by metric spaces and the set of players is arbitrary. The existence of an equilibrium point is proven under assumptions of continuity and compactness.

Continuous stochastic games

1973 ◽  
Vol 10 (03) ◽  
pp. 597-604 ◽  
Author(s):  
Matthew J. Sobel
Keyword(s):  
Equilibrium Point ◽  
Stochastic Games ◽  
Metric Spaces ◽  
Stochastic Game ◽  
Zero Sum

Nonzero-sum N-person stochastic games are a generalization of Shapley's two-person zero-sum terminating stochastic game. Rogers and Sobel showed that an equilibrium point exists when the sets of states, actions, and players are finite. The present paper treats discounted stochastic games when the sets of states and actions are given by metric spaces and the set of players is arbitrary. The existence of an equilibrium point is proven under assumptions of continuity and compactness.

scholarly journals Acyclic Gambling Games

2020 ◽  
Vol 45 (4) ◽  
pp. 1237-1257
Author(s):  
Rida Laraki ◽  
Jérôme Renault
Keyword(s):  
Repeated Games ◽  
Stochastic Games ◽  
Stochastic Game ◽  
Compact Metric Space ◽  
System Of Functional Equations ◽  
State Variable ◽  
Limit Value ◽  
Lack Of Information ◽  
Zero Sum ◽  
Continuous Running

We consider two-player, zero-sum stochastic games in which each player controls the player’s own state variable living in a compact metric space. The terminology comes from gambling problems in which the state of a player represents its wealth in a casino. Under standard assumptions (e.g., continuous running payoff and nonexpansive transitions), we consider for each discount factor the value vλ of the λ-discounted stochastic game and investigate its limit when λ goes to zero. We show that, under a new acyclicity condition, the limit exists and is characterized as the unique solution of a system of functional equations: the limit is the unique continuous excessive and depressive function such that each player, if the player’s opponent does not move, can reach the zone when the current payoff is at least as good as the limit value without degrading the limit value. The approach generalizes and provides a new viewpoint on the Mertens–Zamir system coming from the study of zero-sum repeated games with lack of information on both sides. A counterexample shows that under a slightly weaker notion of acyclicity, convergence of (vλ) may fail.

scholarly journals Zero-sum games for pure jump processes with risk-sensitive discounted cost criteria

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chandan Pal ◽  
Somnath Pradhan
Keyword(s):  
Stochastic Games ◽  
Jump Processes ◽  
General State ◽  
Discounted Cost ◽  
Zero Sum Games ◽  
Risk Sensitive ◽  
General State Space ◽  
Zero Sum ◽  
Saddle Point Equilibrium ◽  
Markov Strategies

<p style='text-indent:20px;'>In this paper we study zero-sum stochastic games for pure jump processes on a general state space with risk sensitive discounted criteria. We establish a saddle point equilibrium in Markov strategies for bounded cost function. We achieve our results by studying relevant Hamilton-Jacobi-Isaacs equations.</p>

scholarly journals China-CEEC Cooperation: China’s Building of a New Type of International Relations

2017 ◽  
Vol 23 (78) ◽  
pp. 19-34 ◽  
Author(s):  
Liu Zuokui
Keyword(s):  
International Relations ◽  
International Cooperation ◽  
Third Parties ◽  
Zero Sum Games ◽  
The Core ◽  
New Type ◽  
Zero Sum

Abstract The article analyzes how the 16+1 Cooperation promotes the Chinese new type of international relations from four perspectives: firstly, the “16+1 Cooperation” insists on not rejecting third parties and promotes the idea of open and inclusive international cooperation; Secondly, the cooperation framework adheres to the principle of mutually-beneficial and win-win cooperation, and proposes to wisely handle differences and divergences; Thirdly, this framework never engages in zero-sum games, instead, it fully respects and closely watches the core interests and major concerns of the relevant parties; Fourthly, it is committed to creating a cooperative platform through consultation, to meet the interests of all. The article also makes an analysis of the challenges facing 16+1 Cooperation and gives some suggestions.

scholarly journals Balancing Two-Player Stochastic Games with Soft Q-Learning

2018 ◽  
Author(s):  
Jordi Grau-Moya ◽  
Felix Leibfried ◽  
Haitham Bou-Ammar
Keyword(s):  
Network Architecture ◽  
Stochastic Games ◽  
Decision Makers ◽  
High Dimensional ◽  
Neural Network Architecture ◽  
Rational Agents ◽  
Q Learning ◽  
Zero Sum Games ◽  
Team Games ◽  
Zero Sum

Within the context of video games the notion of perfectly rational agents can be undesirable as it leads to uninteresting situations, where humans face tough adversarial decision makers. Current frameworks for stochastic games and reinforcement learning prohibit tuneable strategies as they seek optimal performance. In this paper, we enable such tuneable behaviour by generalising soft Q-learning to stochastic games, where more than one agent interact strategically. We contribute both theoretically and empirically. On the theory side, we show that games with soft Q-learning exhibit a unique value and generalise team games and zero-sum games far beyond these two extremes to cover a continuous spectrum of gaming behaviour. Experimentally, we show how tuning agents' constraints affect performance and demonstrate, through a neural network architecture, how to reliably balance games with high-dimensional representations.

Pareto Optimality in Electoral Competition

1971 ◽  
Vol 65 (4) ◽  
pp. 1141-1145 ◽  
Author(s):  
Peter C. Ordeshook
Keyword(s):  
Pareto Optimality ◽  
Pareto Optimal ◽  
Mixed Strategies ◽  
Zero Sum Games ◽  
The Core ◽  
Equilibrium Strategies ◽  
Optimal Policies ◽  
Optimal Allocations ◽  
Zero Sum ◽  
Pareto Optimal Allocations

The core of welfare economics consists of the proof that, for certain classes of goods, perfectly competitive markets are efficient in that they provide Pareto optimal allocations of these goods. In this paper, the efficiency of competitive elections is examined. Elections are modeled as two-candidate zero-sum games, and three kinds of equilibria for such games are identified: pure, risky, and mixed strategies. It is shown, however, that regardless of which kind of equilibrium prevails, if candidates adopt equilibrium strategies, an election is efficient in the sense that the candidates advocate Pareto optimal policies. But one caveat to this analysis is that while an election is Pareto optimal, citizens can unanimously prefer markets to elections as a mechanism for selecting future policies.

MARKOV STRATEGIES ARE BETTER THAN STATIONARY STRATEGIES

1999 ◽  
Vol 01 (01) ◽  
pp. 9-31 ◽  
Author(s):  
J. FLESCH ◽  
F. THUIJSMAN ◽  
O. J. VRIEZE
Keyword(s):  
Stochastic Games ◽  
Stochastic Game ◽  
Initial State ◽  
Stationary Strategies ◽  
Finite State ◽  
Zero Sum ◽  
Action Spaces ◽  
Markov Strategies ◽  
Better Than

We examine the use of stationary and Markov strategies in zero-sum stochastic games with finite state and action spaces. It is natural to evaluate a strategy for the maximising player, player 1, by the highest reward guaranteed to him against any strategy of the opponent. The highest rewards guaranteed by stationary strategies or by Markov strategies are called the stationary utility or the Markov utility, respectively. Since all stationary strategies are Markov strategies, the Markov utility is always larger or equal to the stationary utility. However, in all presently known subclasses of stochastic games, these utilities turn out to be equal. In this paper, we provide a colourful example in which the Markov utility is strictly larger than the stationary utility and we present several conditions under which the utilities are equal. We also show that each stochastic game has at least one initial state for which the two utilities are equal. Several examples clarify these issues.

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