Two-person red-and-black stochastic games

1997 ◽  
Vol 34 (1) ◽  
pp. 107-126 ◽  
Author(s):  
Piercesare Secchi
Keyword(s):  
Stochastic Games ◽  
Stochastic Game ◽  
Gambling Problem ◽  
Stationary Strategy ◽  
Gambling Theory ◽  
Suitable Generalization

We define a leavable stochastic game which is a possible two-person generalization of the classical red-and-black gambling problem. We show that there are three basic possibilities for a two-person red-and-black game which, by analogy with gambling theory, we call the subfair, the fair and the superfair cases. A suitable generalization of what in gambling theory is called bold play is proved to be a uniformly ε-optimal stationary strategy for player I in the fair and the subfair cases whereas a generalization of timid play is shown to be ε-optimal for player I in the superfair possibility.

Two-person red-and-black stochastic games

1997 ◽  
Vol 34 (01) ◽  
pp. 107-126 ◽  
Author(s):  
Piercesare Secchi
Keyword(s):  
Stochastic Games ◽  
Stochastic Game ◽  
Gambling Problem ◽  
Stationary Strategy ◽  
Gambling Theory ◽  
Suitable Generalization

We define a leavable stochastic game which is a possible two-person generalization of the classical red-and-black gambling problem. We show that there are three basic possibilities for a two-person red-and-black game which, by analogy with gambling theory, we call the subfair, the fair and the superfair cases. A suitable generalization of what in gambling theory is called bold play is proved to be a uniformly ε-optimal stationary strategy for player I in the fair and the subfair cases whereas a generalization of timid play is shown to be ε-optimal for player I in the superfair possibility.

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.

scholarly journals A formula for the value of a stochastic game

2019 ◽  
Vol 116 (52) ◽  
pp. 26435-26443 ◽  
Author(s):  
Luc Attia ◽  
Miquel Oliu-Barton
Keyword(s):  
Dynamic Model ◽  
Discount Rate ◽  
Stochastic Games ◽  
Stochastic Game ◽  
Solution Concept ◽  
Robust Solution ◽  
General Dynamic

In 1953, Lloyd Shapley defined the model of stochastic games, which were the first general dynamic model of a game to be defined, and proved that competitive stochastic games have a discounted value. In 1982, Jean-François Mertens and Abraham Neyman proved that competitive stochastic games admit a robust solution concept, the value, which is equal to the limit of the discounted values as the discount rate goes to 0. Both contributions were published in PNAS. In the present paper, we provide a tractable formula for the value of competitive stochastic games.

ON THE DIFFICULT INTERPLAY BETWEEN LIFE, "COMPLEXITY", AND MATHEMATICAL SCIENCES

2013 ◽  
Vol 23 (10) ◽  
pp. 1861-1913 ◽  
Author(s):  
N. BELLOMO ◽  
D. KNOPOFF ◽  
J. SOLER
Keyword(s):  
Kinetic Theory ◽  
Complex Systems ◽  
Case Studies ◽  
Mathematical Theory ◽  
Stochastic Games ◽  
Mathematical Structure ◽  
Living Systems ◽  
Active Particles ◽  
Suitable Generalization ◽  
Mathematical Sciences

This paper presents a revisiting, with developments, of the so-called kinetic theory for active particles, with the main focus on the modeling of nonlinearly additive interactions. The approach is based on a suitable generalization of methods of kinetic theory, where interactions are depicted by stochastic games. The basic idea consists in looking for a general mathematical structure suitable to capture the main features of living, hence complex, systems. Hopefully, this structure is a candidate towards the challenging objective of designing a mathematical theory of living systems. These topics are treated in the first part of the paper, while the second one applies it to specific case studies, namely to the modeling of crowd dynamics and of the immune competition.

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.

scholarly journals Two-person red-and-black with bet-dependent win probabilities

2005 ◽  
Vol 37 (1) ◽  
pp. 75-89 ◽  
Author(s):  
Laura Pontiggia
Keyword(s):  
Nash Equilibrium ◽  
Stochastic Game ◽  
The Other ◽  
Gambling Problem ◽  
Weight Parameter ◽  
Laws Of Motion

We present two variations of a two-person, noncooperative stochastic game, inspired by the famous red-and-black gambling problem presented by Dubins and Savage. Two players each hold an integer amount of money and they each aim to win the other player's fortune. At every stage of the game they simultaneously bid an integer portion of their current fortune, and their probabilities of winning depend on these bids. We describe two different laws of motion specifying this dependency. In one version of the game, the players' probabilities of winning are proportional to their bets. In the other version, the probabilities of winning depend on the size of their bets and a weight parameter w. For each version we give a Nash equilibrium, in which the player for which the game is subfair (w ≤ ½) plays boldly and the player for which the game is superfair (w ≥ ½) plays timidly.

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 On the Existence and Determining Stationary Nash Equilibria for Switching Controller Stochastic Games

2021 ◽  
Vol 14 ◽  
pp. 290-301
Author(s):  
Dmitrii Lozovanu ◽  
◽  
Stefan Pickl ◽  
Keyword(s):  
Nash Equilibrium ◽  
Normal Form ◽  
Nash Equilibria ◽  
Stochastic Games ◽  
Stochastic Game ◽  
Static Game ◽  
Stationary Strategies ◽  
Switching Controller ◽  
Optimal Stationary Strategies ◽  
Stationary Nash Equilibrium

In this paper we consider the problem of the existence and determining stationary Nash equilibria for switching controller stochastic games with discounted and average payoffs. The set of states and the set of actions in the considered games are assumed to be finite. For a switching controller stochastic game with discounted payoffs we show that all stationary equilibria can be found by using an auxiliary continuous noncooperative static game in normal form in which the payoffs are quasi-monotonic (quasi-convex and quasi-concave) with respect to the corresponding strategies of the players. Based on this we propose an approach for determining the optimal stationary strategies of the players. In the case of average payoffs for a switching controller stochastic game we also formulate an auxiliary noncooperative static game in normal form with quasi-monotonic payoffs and show that such a game possesses a Nash equilibrium if the corresponding switching controller stochastic game has a stationary Nash equilibrium.

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.

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