A Characterization of Stationary Nash Equilibria of Single Controller Constrained Stochastic Games

2015 ◽  
Vol 17 (02) ◽  
pp. 1540018
Author(s):  
Vikas Vikram Singh ◽  
N. Hemachandra
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Mathematical Program ◽  
Transition Probabilities ◽  
Stochastic Game ◽  
Quadratic Program ◽  
Linear Programs ◽  
Average Case ◽  
Single Controller ◽  
Stationary Nash Equilibrium

We consider a two player finite state-action general sum single controller constrained stochastic game with both discounted and average cost criteria. We consider the situation where player 1 has subscription-based constraints and player 2, who controls the transition probabilities, has realization-based constraints which can also depend on the strategies of player 1. It is known that a stationary Nash equilibrium for discounted case exists under strong Slater condition, while, for the average case, stationary Nash equilibrium exists if additionally the Markov chain is unichain. For each case we show that the set of stationary Nash equilibria of this game has one to one correspondence with the set of global minimizers of a certain nonconvex mathematical program. If the constraints of player 2 do not depend on the strategies of player 1, then the mathematical program reduces to a quadratic program. The known linear programs for zero sum games of this class can be obtained as a special case of above quadratic programs.

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.

BLACKWELL OPTIMALITY IN STOCHASTIC GAMES

2013 ◽  
Vol 15 (04) ◽  
pp. 1340025
Author(s):  
VIKAS VIKRAM SINGH ◽  
N. HEMACHANDRA ◽  
K. S. MALLIKARJUNA RAO
Keyword(s):  
Nash Equilibrium ◽  
Stochastic Games ◽  
Stochastic Game ◽  
Discount Factor ◽  
State Action ◽  
Blackwell Optimality ◽  
Finite State ◽  
Markov Decision ◽  
Stationary Nash Equilibrium ◽  
General Stochastic

Blackwell optimality in a finite state-action discounted Markov decision process (MDP) gives an optimal strategy which is optimal for every discount factor close enough to one. In this article we explore this property, which we call as Blackwell–Nash equilibrium, in two player finite state-action discounted stochastic games. A strategy pair is said to be a Blackwell–Nash equilibrium if it is a Nash equilibrium for every discount factor close enough to one. A stationary Blackwell–Nash equilibrium in a stochastic game may not always exist as can be seen from "Big Match" example where a stationary Nash equilibrium does not exist in undiscounted case. For a Single Controller Additive Reward (SC-AR) stochastic game, we show that there exists a stationary deterministic Blackwell–Nash equilibrium which is also a Nash equilibrium for undiscounted case. For general stochastic games, we give some conditions which together are sufficient for any stationary Nash equilibrium of a discounted stochastic game to be a Blackwell–Nash equilibrium and it is also a Nash equilibrium of an undiscounted stochastic game. We illustrate our results on general stochastic games through a variant of the pollution tax model.

A Theory of Patronized Goods: Inefficient and Efficient Equilibria

2011 ◽  
pp. 65-87 ◽  
Author(s):  
A. Rubinstein
Keyword(s):  
Nash Equilibrium ◽  
Social Policy ◽  
Nash Equilibria ◽  
Institutional Environment ◽  
Social Motivation ◽  
Pareto Optimal ◽  
Market Failures ◽  
Equilibrium Conditions ◽  
Policy Issues

The article considers some aspects of the patronized goods theory with respect to efficient and inefficient equilibria. The author analyzes specific features of patronized goods as well as their connection with market failures, and conjectures that they are related to the emergence of Pareto-inefficient Nash equilibria. The key problem is the analysis of the opportunities for transforming inefficient Nash equilibrium into Pareto-optimal Nash equilibrium for patronized goods by modifying the institutional environment. The paper analyzes social motivation for institutional modernization and equilibrium conditions in the generalized Wicksell-Lindahl model for patronized goods. The author also considers some applications of patronized goods theory to social policy issues.

scholarly journals Expressiveness and Nash Equilibrium in Iterated Boolean Games

2021 ◽  
Vol 22 (2) ◽  
pp. 1-38
Author(s):  
Julian Gutierrez ◽  
Paul Harrenstein ◽  
Giuseppe Perelli ◽  
Michael Wooldridge
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Infinite Sequence ◽  
Multi Agent Systems ◽  
Temporal Logics ◽  
Agent Systems ◽  
Temporal Properties ◽  
Multi Agent ◽  
Game Theoretic ◽  
Boolean Games

We define and investigate a novel notion of expressiveness for temporal logics that is based on game theoretic equilibria of multi-agent systems. We use iterated Boolean games as our abstract model of multi-agent systems [Gutierrez et al. 2013, 2015a]. In such a game, each agent  has a goal  , represented using (a fragment of) Linear Temporal Logic ( ) . The goal  captures agent  ’s preferences, in the sense that the models of  represent system behaviours that would satisfy  . Each player controls a subset of Boolean variables , and at each round in the game, player is at liberty to choose values for variables in any way that she sees fit. Play continues for an infinite sequence of rounds, and so as players act they collectively trace out a model for , which for every player will either satisfy or fail to satisfy their goal. Players are assumed to act strategically, taking into account the goals of other players, in an attempt to bring about computations satisfying their goal. In this setting, we apply the standard game-theoretic concept of (pure) Nash equilibria. The (possibly empty) set of Nash equilibria of an iterated Boolean game can be understood as inducing a set of computations, each computation representing one way the system could evolve if players chose strategies that together constitute a Nash equilibrium. Such a set of equilibrium computations expresses a temporal property—which may or may not be expressible within a particular fragment. The new notion of expressiveness that we formally define and investigate is then as follows: What temporal properties are characterised by the Nash equilibria of games in which agent goals are expressed in specific fragments of  ? We formally define and investigate this notion of expressiveness for a range of fragments. For example, a very natural question is the following: Suppose we have an iterated Boolean game in which every goal is represented using a particular fragment of : is it then always the case that the equilibria of the game can be characterised within ? We show that this is not true in general.

scholarly journals A Kinetic Theory Model of the Dynamics of Liquidity Profiles on Interbank Networks

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 363
Author(s):  
Marina Dolfin ◽  
Leone Leonida ◽  
Eleonora Muzzupappa
Keyword(s):  
Kinetic Theory ◽  
Network Formation ◽  
Transition Probabilities ◽  
Stochastic Game ◽  
Theory Model ◽  
Point Of View ◽  
Network Efficiency ◽  
Modelling Framework ◽  
Wide Range ◽  
Complex Adaptive

This paper adopts the Kinetic Theory for Active Particles (KTAP) approach to model the dynamics of liquidity profiles on a complex adaptive network system that mimic a stylized financial market. Individual incentives of investors to form or delete a link is driven, in our modelling framework, by stochastic game-type interactions modelling the phenomenology related to policy rules implemented under Basel III, and it is exogeneously and dynamically influenced by a measure of overnight interest rate. The strategic network formation dynamics that emerges from the introduced transition probabilities modelling individual incentives of investors to form or delete links, provides a wide range of measures using which networks might be considered “best” from the point of view of the overall welfare of the system. We use the time evolution of the aggregate degree of connectivity to measure the time evolving network efficiency in two different scenarios, suggesting a first analysis of the stability of the arising and evolving network structures.

scholarly journals Semidefinite Programming and Nash Equilibria in Bimatrix Games

2020 ◽  
Author(s):  
Amir Ali Ahmadi ◽  
Jeffrey Zhang
Keyword(s):  
Nash Equilibrium ◽  
Semidefinite Programming ◽  
Nash Equilibria ◽  
Valid Inequalities ◽  
Bimatrix Games ◽  
Competitive Game ◽  
Approximate Nash Equilibria ◽  
Hard Problems ◽  
Rank 2

We explore the power of semidefinite programming (SDP) for finding additive ɛ-approximate Nash equilibria in bimatrix games. We introduce an SDP relaxation for a quadratic programming formulation of the Nash equilibrium problem and provide a number of valid inequalities to improve the quality of the relaxation. If a rank-1 solution to this SDP is found, then an exact Nash equilibrium can be recovered. We show that, for a strictly competitive game, our SDP is guaranteed to return a rank-1 solution. We propose two algorithms based on the iterative linearization of smooth nonconvex objective functions whose global minima by design coincide with rank-1 solutions. Empirically, we demonstrate that these algorithms often recover solutions of rank at most 2 and ɛ close to zero. Furthermore, we prove that if a rank-2 solution to our SDP is found, then a [Formula: see text]-Nash equilibrium can be recovered for any game, or a [Formula: see text]-Nash equilibrium for a symmetric game. We then show how our SDP approach can address two (NP-hard) problems of economic interest: finding the maximum welfare achievable under any Nash equilibrium, and testing whether there exists a Nash equilibrium where a particular set of strategies is not played. Finally, we show the connection between our SDP and the first level of the Lasserre/sum of squares hierarchy.

scholarly journals Limit Points of Endogenous Misspecified Learning

Econometrica ◽  
2021 ◽  
Vol 89 (3) ◽  
pp. 1065-1098
Author(s):  
Drew Fudenberg ◽  
Giacomo Lanzani ◽  
Philipp Strack
Keyword(s):  
Nash Equilibrium ◽  
High Probability ◽  
Nash Equilibria ◽  
Prior Belief ◽  
Positive Probability ◽  
Initial Belief ◽  
Long Run ◽  
Limit Points

We study how an agent learns from endogenous data when their prior belief is misspecified. We show that only uniform Berk–Nash equilibria can be long‐run outcomes, and that all uniformly strict Berk–Nash equilibria have an arbitrarily high probability of being the long‐run outcome for some initial beliefs. When the agent believes the outcome distribution is exogenous, every uniformly strict Berk–Nash equilibrium has positive probability of being the long‐run outcome for any initial belief. We generalize these results to settings where the agent observes a signal before acting.

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