A NOTE ON REPEATED GAMES WITH VANISHING ACTIONS

2005 ◽  
Vol 07 (01) ◽  
pp. 107-115 ◽  
Author(s):  
REINOUD JOOSTEN
Keyword(s):  
Repeated Games ◽  
Relative Frequency ◽  
Time Horizon ◽  
Convex Combination ◽  
Pure Strategy ◽  
Repeated Game ◽  
Entire Time ◽  
Long Run ◽  
Average Rewards ◽  
Threat Point

A two-person general-sum repeated game with vanishing actions is an infinitely repeated game where the players face the following restrictions. Each action must be used by player k ∈ {1,2} at least once in every rk ∈ ℕ consecutive stages, otherwise the action vanishes for the remaining play. We assume that the players wish to maximize their limiting average rewards over the entire time-horizon. A strategy-pair is jointly convergent if for each action pair a number exists to which the relative frequency with which this action pair is chosen, converges with probability one. A pair of feasible rewards is called individually rational if each player receives at least the threat-point reward, i.e., the amount which he can guarantee himself regardless of what the opponent does given r1, r2 and the actions available in the long run. In a repeated game with vanishing actions, there may exist multiple threat points which are endogenous to the play. We prove that all individually-rational jointly-convergent pure-strategy rewards can be supported by an equilibrium. Furthermore, each convex combination of individually-rational jointly-convergent pure-strategy rewards, can be supported by an equilibrium for m × n-games provided r1 > m ≥ 2, r2 > n ≥ 2.

Reciprocal Social Strategy in Social Repeated Games and Emergence of Social Norms

2017 ◽  
Vol 26 (01) ◽  
pp. 1760007
Author(s):  
Chi-Kong Chan ◽  
Jianye Hao ◽  
Ho-Fung Leung
Keyword(s):  
Social Norms ◽  
Repeated Games ◽  
Repeated Game ◽  
Social Utility ◽  
Learning Mechanism ◽  
Artificial Society ◽  
Social Strategy ◽  
Long Run ◽  
The Individual ◽  
High Level

In an artificial society where agents repeatedly interact with one another, effective coordination among agents is generally a challenge. This is especially true when the participating agents are self-interested, and that there is no central authority to coordinate, and direct communication or negotiation are not possible. Recently, the problem was studied in a paper by Hao and Leung, where a new repeated game mechanism for modeling multi-agent interactions as well as a new reinforcement learning based agent learning method were proposed. In particular, the game mechanism differs from traditional repeated games in that the agents are anonymous, and the agents interact with randomly chosen opponents during each iteration. Their learning mechanism allows agents to coordinate without negotiations. The initial results had been promising. However, extended simulation also reveals that the outcomes are not stable in the long run in some cases, as the high level of cooperation is eventually not sustainable. In this work, we revisit he problem and propose a new learning mechanism as follows. First, we propose an enhanced Q-learning-based framework that allows the agents to better capture both the individual and social utilities that they have learned through observations. Second, we propose a new concept of \social attitude" for determining the action of the agents throughout the game. Simulation results reveal that this approach can achieve higher social utility, including close-to-optimal results in some scenarios, and more importantly, the results are sustainable with social norms emerging.

scholarly journals A complete folk theorem for finitely repeated games

2020 ◽  
Vol 49 (4) ◽  
pp. 1129-1142
Author(s):  
Ghislain-Herman Demeze-Jouatsa
Keyword(s):  
Repeated Games ◽  
Pure Strategy ◽  
Complete Information ◽  
Repeated Game ◽  
Complete Characterization ◽  
Folk Theorem ◽  
Equilibrium Payoff ◽  
Perfect Monitoring ◽  
Special Case

AbstractThis paper analyzes the set of pure strategy subgame perfect Nash equilibria of any finitely repeated game with complete information and perfect monitoring. The main result is a complete characterization of the limit set, as the time horizon increases, of the set of pure strategy subgame perfect Nash equilibrium payoff vectors of the finitely repeated game. This model includes the special case of observable mixed strategies.

Two Non-senses of “Probability”

2021 ◽  
pp. 47-76
Author(s):  
Wayne C. Myrvold
Keyword(s):  
Relative Frequency ◽  
The Other ◽  
Objective Probability ◽  
Sequence Of Events ◽  
Principle Of Indifference ◽  
Long Run ◽  
Eighteenth And Nineteenth Centuries

This chapter engages in some ground-clearing. Two concepts have been proposed to play the role of objective probability. One is associated with the idea that probability involves mere counting of possibilities (often wrongly attributed to Laplace). The other is frequentism, the idea that probability can be defined as long-run relative frequency in some actual or hypothetical sequence of events. Associated with the idea that probability is merely a matter of counting of possibilities is a temptation to believe that there is a principle, called the Principle of Indifference, which can generate probabilities out of ignorance. In this chapter the reasons that neither of these approaches can achieve its goal are rehearsed, with reference to historical discussions in the eighteenth and nineteenth centuries. It includes some of the prehistory of discussions of what has come to be known, misleadingly, as Bertrand’s paradox.

A Game Theoretic Approach to Multi-Period Newsvendor Problems with Censored Markovian Demand

2019 ◽  
Vol 01 (01) ◽  
Author(s):  
Farzaneh Mansourifard ◽  
Parisa Mansourifard ◽  
Bhaskar Krishnamachari
Keyword(s):  
Lower Bound ◽  
Convex Combination ◽  
Repeated Game ◽  
Theoretic Approach ◽  
Worst Case ◽  
Point Distribution ◽  
Demand Distribution ◽  
Deterministic Solution ◽  
Uncertainty Set ◽  
Game Theoretic

This paper studies the Newsvendor problem for a setting in which (i) the demand is temporally correlated, (ii) the demand is censored, (iii) the distribution of the demand is unknown. The correlation is modeled as a Markovian process. The censoring means that if the demand is larger than the action (selected inventory), only a lower bound on the demand can be revealed. The uncertainty set on the demand distribution is given by only the upper and lower bound on the amount of the change from a time to the next time. We propose a robust approach to minimize the worst-case total cost and model it as a min-max zero-sum repeated game. We prove that the worst-case distribution of the adversary at each time is a two-point distribution with non-zero probabilities at the extrema of the uncertainty set of the demand. And the optimal action of the decision-maker can have any of the following structures: (i) a randomized solution with a two-point distribution at the extrema, (ii) a deterministic solution at a convex combination of the extrema. Both above solutions balance over-utilization and under-utilization costs. Finally, we extend our results to uni-model cost functions and present numerical results to study the solution.

On the Strategic Equivalence of Linear Dynamic and Repeated Games

2015 ◽  
Vol 17 (03) ◽  
pp. 1550006
Author(s):  
Joachim Hubmer
Keyword(s):  
Repeated Games ◽  
Dynamic Game ◽  
Repeated Game ◽  
Transition Function ◽  
The State ◽  
State Variables ◽  
Equivalent Representation ◽  
Deterministic Dynamic ◽  
Subgame Perfect Equilibria ◽  
Perfect Equilibria

Dynamic (or stochastic) games are, in general, considerably more complicated to analyze than repeated games. This paper shows that for every deterministic dynamic game that is linear in the state, there exists a strategically equivalent representation as a repeated game. A dynamic game is said to be linear in the state if it holds for both the state transition function as well as for the one-period payoff function that (i) they are additively separable in action profiles and states and (ii) the state variables enter linearly. Strategic equivalence refers to the observation that the two sets of subgame perfect equilibria coincide, up to a natural projection of dynamic game strategy profiles on the much smaller set of repeated game histories. Furthermore, it is shown that the strategic equivalence result still holds for certain stochastic elements in the transition function if one allows for additional signals in the repeated game or in the presence of a public correlating device.

Simulation of a Random Variable and its Application to Game Theory

2021 ◽  
Author(s):  
Mehrdad Valizadeh ◽  
Amin Gohari
Keyword(s):  
Repeated Games ◽  
Side Information ◽  
Random Variable ◽  
Long Run ◽  
Variation Distance ◽  
N Stage ◽  
Target Source ◽  
Stage Game ◽  
The Difference ◽  
Zero Sum

We provide a new tool for simulation of a random variable (target source) from a randomness source with side information. Considering the total variation distance as the measure of precision, this tool offers an upper bound for the precision of simulation, which is vanishing exponentially in the difference of Rényi entropies of the randomness and target sources. This tool finds application in games in which the players wish to generate their actions (target source) as a function of a randomness source such that they are almost independent of the observations of the opponent (side information). In particular, we study zero-sum repeated games in which the players are restricted to strategies that require only a limited amount of randomness. Let be the max-min value of the n stage game. Previous works have characterized [Formula: see text], that is, the long-run max-min value, but they have not provided any result on the value of vn for a given finite n-stage game. Here, we utilize our new tool to study how vn converges to the long-run max-min value.

The Bayesian Approach to Statistics

1966 ◽  
Vol 92 (3) ◽  
pp. 326-339 ◽  
Author(s):  
P. G. Moore
Keyword(s):  
Bayesian Approach ◽  
Relative Frequency ◽  
Frequency Theory ◽  
Long Run ◽  
Alternative Approach ◽  
Frequency Concept Of Probability ◽  
The Bayesian Approach ◽  
Statistical Definition

For the last thirty years the teaching of statistics in universities in this country has been dominated by the relative frequency theory of probability, exemplified in Richard von Mises's book, Probability, Statistics and Truth. This statistical definition along the lines of the long run frequency concept of probability can be illustrated by asking what meaning is to be given to the statement ‘the probability of getting a head on a single toss of a penny is one half.’ The relative frequency adherent would answer something like ‘in a long sequence of repeated tosses, the proportion of outcomes that are heads is one half’. In the last ten years, however, an alternative approach has come to the fore, under the general title of the Bayesian Approach to Statistics. A Bayesian adherent would answer something along the lines that he would be ‘prepared to offer even money on getting a head on a single toss’.

scholarly journals Repeated Games with Long-Run and Short-Run Players

1990 ◽  
Vol 57 (4) ◽  
pp. 555 ◽  
Author(s):  
Drew Fudenberg ◽  
David M. Kreps ◽  
Eric S. Maskin
Keyword(s):  
Repeated Games ◽  
Long Run ◽  
Short Run

scholarly journals On the Determinants of Cooperation in Infinitely Repeated Games: A Survey

2018 ◽  
Vol 56 (1) ◽  
pp. 60-114 ◽  
Author(s):  
Pedro Dal Bó ◽  
Guillaume R. Fréchette
Keyword(s):  
Experimental Data ◽  
Repeated Games ◽  
Multiple Equilibria ◽  
Prisoner’S Dilemma ◽  
Personal Characteristics ◽  
Repeated Game ◽  
Experimental Literature ◽  
Imperfect Monitoring ◽  
Strategic Uncertainty ◽  
Infinitely Repeated Games

A growing experimental literature studies the determinants of cooperation in infinitely repeated games, tests different predictions of the theory, and suggests an empirical solution to the problem of multiple equilibria. To provide a robust description of the literature's findings, we gather and analyze a metadata set of experiments on infinitely repeated prisoner's dilemma games. The experimental data show that cooperation is affected by infinite repetition and is more likely to arise when it can be supported in equilibrium. However, the fact that cooperation can be supported in equilibrium does not imply that most subjects will cooperate. High cooperation rates will emerge only when the parameters of the repeated game are such that cooperation is very robust to strategic uncertainty. We also review the results regarding the effect of imperfect monitoring, changing partners, and personal characteristics on cooperation and the strategies used to support it. (JEL C71, C73, D81, D83)

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