EQUILIBRIUM STRATEGIES IN STOCHASTIC GAMES WITH ADDITIVE COST AND TRANSITION STRUCTURE

1999 ◽  
Vol 01 (02) ◽  
pp. 131-147 ◽  
Author(s):  
HEINZ-UWE KÜENLE
Keyword(s):  
Weight Function ◽  
Stochastic Games ◽  
Optimal Strategies ◽  
Cost Structure ◽  
Equilibrium Strategy ◽  
Transition Structure ◽  
Equilibrium Strategies ◽  
Zero Sum ◽  
Action Spaces ◽  
Treated State

Two-person stochastic games with additive transition and cost structure and the criterion of expected total costs are treated. State space and action spaces are standard Borel, and unbounded costs are allowed. For the zero-sum case, it is shown that there are stationary deterministic εη-optimal strategies for every ε>0 and a certain weight function η if some semi-continuity and compactness conditions are fulfilled. Using these results, the existence of so-called quasi-stationary deterministic εη-equilibrium strategy pairs under corresponding conditions is proven.

scholarly journals Characterization and simplification of optimal strategies in positive stochastic games

2018 ◽  
Vol 55 (3) ◽  
pp. 728-741 ◽  
Author(s):  
János Flesch ◽  
Arkadi Predtetchinski ◽  
William Sudderth
Keyword(s):  
State Space ◽  
Optimal Strategy ◽  
Stochastic Games ◽  
Optimal Strategies ◽  
Action Space ◽  
Positive Zero ◽  
Countable State ◽  
Zero Sum ◽  
Action Spaces

Abstract We consider positive zero-sum stochastic games with countable state and action spaces. For each player, we provide a characterization of those strategies that are optimal in every subgame. These characterizations are used to prove two simplification results. We show that if player 2 has an optimal strategy then he/she also has a stationary optimal strategy, and prove the same for player 1 under the assumption that the state space and player 2's action space are finite.

PIVOTING ALGORITHMS FOR SOME CLASSES OF STOCHASTIC GAMES: A SURVEY

2001 ◽  
Vol 03 (02n03) ◽  
pp. 253-281 ◽  
Author(s):  
S. R. MOHAN ◽  
S. K. NEOGY ◽  
T. PARTHASARATHY
Keyword(s):  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Stochastic Games ◽  
Optimal Strategies ◽  
Linear Complementarity ◽  
Dual Simplex Method ◽  
Special Cases ◽  
Dual Simplex ◽  
Finite Step ◽  
Zero Sum

In this paper, we survey the recent literature on computing the value vector and the associated optimal strategies of the players for special cases of zero-sum stochastic games, or in computing a Nash equilibrium point and the corresponding stationary strategies of the players for special cases of nonzero-sum stochastic games, using finite-step algorithms based on pivoting. Examples of finite-step pivoting algorithms are the various simplex-type algorithms, such as the primal simplex or dual simplex method for solving the linear programming problem or Lemke's or Lemke-Howson's algorithm for solving the linear complementarity problem. Also included are Lemke-type algorithms for solving various generalisations of the linear complementarity problem. The survey also includes a few new results and observations.

scholarly journals On the existence of stationary Nash equilibria in average stochastic games with finite state and action spaces

2020 ◽  
Vol 13 ◽  
pp. 304-323
Author(s):  
Dmitrii Lozovanu ◽  
◽  
Stefan Pickl ◽  
Keyword(s):  
Nash Equilibria ◽  
Stochastic Games ◽  
Optimal Strategies ◽  
Stationary Strategies ◽  
Finite State ◽  
Optimal Stationary Strategies ◽  
Action Spaces

We consider infinite n-person stochastic games with limiting average payoffs criteria for the players. The main results of the paper are concerned with the existence of stationary Nash equilibria and determining the optimal strategies of the players in the games with finite state and action spaces. We present conditions for the existence of stationary Nash equilibria in the considered games and propose an approach for determining the optimal stationary strategies of the players if such strategies exist.

scholarly journals Equilibrium and Optimal Strategies in M/M/1 Queues with Working Vacations and Vacation Interruptions

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Ruiling Tian ◽  
Linmin Hu ◽  
Xijun Wu
Keyword(s):  
Optimal Strategies ◽  
Cost Structure ◽  
Single Server ◽  
Equilibrium Strategies ◽  
Markovian Queues ◽  
Multiple Working Vacations ◽  
Working Vacations ◽  
System States ◽  
Partially Observable ◽  
Stationary Behavior

We consider the customers equilibrium and socially optimal joining-balking behavior in single-server Markovian queues with multiple working vacations and vacation interruptions. Arriving customers decide whether to join the system or balk, based on a linear reward-cost structure that incorporates their desire for service, as well as their unwillingness for waiting. We consider that the system states are observable, partially observable, and unobservable, respectively. For these cases, we first analyze the stationary behavior of the system and get the equilibrium strategies of the customers and compare them to socially optimal balking strategies numerically.

scholarly journals Algorithms for uniform optimal strategies in two-player zero-sum stochastic games with perfect information

2012 ◽  
Vol 40 (1) ◽  
pp. 56-60 ◽  
Author(s):  
Konstantin Avrachenkov ◽  
Laura Cottatellucci ◽  
Lorenzo Maggi
Keyword(s):  
Stochastic Games ◽  
Optimal Strategies ◽  
Perfect Information ◽  
Zero Sum

scholarly journals Subgame Maxmin Strategies in Zero-Sum Stochastic Games with Tolerance Levels

2021 ◽  
Author(s):  
János Flesch ◽  
P. Jean-Jacques Herings ◽  
Jasmine Maes ◽  
Arkadi Predtetchinski
Keyword(s):  
Stochastic Games ◽  
Sufficient Conditions ◽  
Payoff Function ◽  
Surprising Result ◽  
Countable State ◽  
Finite Action ◽  
Zero Sum ◽  
Necessary And Sufficient ◽  
Action Spaces ◽  
Special Case

AbstractWe study subgame $$\phi $$ ϕ -maxmin strategies in two-player zero-sum stochastic games with a countable state space, finite action spaces, and a bounded and universally measurable payoff function. Here, $$\phi $$ ϕ denotes the tolerance function that assigns a nonnegative tolerated error level to every subgame. Subgame $$\phi $$ ϕ -maxmin strategies are strategies of the maximizing player that guarantee the lower value in every subgame within the subgame-dependent tolerance level as given by $$\phi $$ ϕ . First, we provide necessary and sufficient conditions for a strategy to be a subgame $$\phi $$ ϕ -maxmin strategy. As a special case, we obtain a characterization for subgame maxmin strategies, i.e., strategies that exactly guarantee the lower value at every subgame. Secondly, we present sufficient conditions for the existence of a subgame $$\phi $$ ϕ -maxmin strategy. Finally, we show the possibly surprising result that each game admits a strictly positive tolerance function $$\phi ^*$$ ϕ ∗ with the following property: if a player has a subgame $$\phi ^*$$ ϕ ∗ -maxmin strategy, then he has a subgame maxmin strategy too. As a consequence, the existence of a subgame $$\phi $$ ϕ -maxmin strategy for every positive tolerance function $$\phi $$ ϕ is equivalent to the existence of a subgame maxmin strategy.

scholarly journals Algorithm for Computing Approximate Nash Equilibrium in Continuous Games with Application to Continuous Blotto

Games ◽  
2021 ◽  
Vol 12 (2) ◽  
pp. 47
Author(s):  
Sam Ganzfried
Keyword(s):  
Nash Equilibrium ◽  
Imperfect Information ◽  
Pure Strategy ◽  
Strategy Space ◽  
Equilibrium Strategies ◽  
Approximate Nash Equilibrium ◽  
Blotto Game ◽  
Nash Equilibrium Strategies ◽  
Zero Sum ◽  
Action Spaces

Successful algorithms have been developed for computing Nash equilibrium in a variety of finite game classes. However, solving continuous games—in which the pure strategy space is (potentially uncountably) infinite—is far more challenging. Nonetheless, many real-world domains have continuous action spaces, e.g., where actions refer to an amount of time, money, or other resource that is naturally modeled as being real-valued as opposed to integral. We present a new algorithm for approximating Nash equilibrium strategies in continuous games. In addition to two-player zero-sum games, our algorithm also applies to multiplayer games and games with imperfect information. We experiment with our algorithm on a continuous imperfect-information Blotto game, in which two players distribute resources over multiple battlefields. Blotto games have frequently been used to model national security scenarios and have also been applied to electoral competition and auction theory. Experiments show that our algorithm is able to quickly compute close approximations of Nash equilibrium strategies for this game.

Zero-sum two-person semi-Markov games

1992 ◽  
Vol 29 (01) ◽  
pp. 56-72 ◽  
Author(s):  
Arbind K. Lal ◽  
Sagnik Sinha
Keyword(s):  
Stochastic Games ◽  
Optimal Strategies ◽  
Existence Of A Solution ◽  
Average Case ◽  
Optimality Equation ◽  
Markov Games ◽  
Special Cases ◽  
Ergodic Condition ◽  
Markov Decision ◽  
Zero Sum

Semi-Markov games are investigated under discounted and limiting average payoff criteria. The issue of the existence of the value and a pair of stationary optimal strategies are settled; the optimality equation is studied and under a natural ergodic condition the existence of a solution to the optimality equation is proved for the limiting average case. Semi-Markov games provide useful flexibility in constructing recursive game models. All the work on Markov/semi-Markov decision processes and Markov (stochastic) games can be viewed as special cases of the developments in this paper.

Customer Strategic Joining Behavior in Markovian Queues with Working Vacations and Vacation Interruptions Under Bernoulli Schedule

2019 ◽  
Vol 36 (01) ◽  
pp. 1950003
Author(s):  
Qingqing Ma ◽  
Yiqiang Q. Zhao ◽  
Weiqi Liu ◽  
Jihong Li
Keyword(s):  
Stationary Distribution ◽  
Optimal Strategies ◽  
Queuing System ◽  
Cost Structure ◽  
Single Server ◽  
Working Vacation ◽  
Equilibrium Strategies ◽  
Working Vacations ◽  
Bernoulli Schedule ◽  
Bernoulli Vacation

This study considers a single-server Markovian working vacation queuing system with Bernoulli vacation interruptions. Based on a linear reward-cost structure, the customer strategic joining behavior is analyzed under different information levels available to the arriving customers, namely fully observable, almost unobservable, and fully unobservable. For these cases, we first obtain the system stationary distribution. Thereafter, we determine the customer equilibrium strategies and compare them numerically with socially optimal strategies.

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