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Volume 1, 2014

Journal of Dynamics & Games

October 2014 , Volume 1 , Issue 4

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Nonzero-sum stochastic differential games with additive structure and average payoffs
Beatris Adriana Escobedo-Trujillo and José Daniel López-Barrientos
2014, 1(4): 555-578 doi: 10.3934/jdg.2014.1.555 +[Abstract](1039) +[PDF](527.0KB)
Abstract:
This paper deals with nonzero-sum stochastic differential games with an additive structure and long-run average payoffs. Our main objective is to give conditions for the existence of Nash equilibria in the set of relaxed stationary strategies. Such conditions also ensure the existence of a Nash equilibrium within the set of stationary Markov (deterministic) strategies, and that the values of the average payoffs for these equilibria coincide almost everywhere with respect to Lebesgue's measure. This fact generalizes the results in the controlled (single player game) case found by Raghavan [47] and Rosenblueth [48]. We use relaxation theory and standard dynamic programming techniques to achieve our goals. We illustrate our results with an example motivated by a manufacturing system.
Investment under uncertainty, competition and regulation
Adrien Nguyen Huu
2014, 1(4): 579-598 doi: 10.3934/jdg.2014.1.579 +[Abstract](881) +[PDF](531.1KB)
Abstract:
We investigate a randomization procedure undertaken in real option games which can serve as a basic model of regulation in a duopoly model of preemptive investment. We recall the rigorous framework of M. Grasselli, V. Leclère and M. Ludkovsky (Priority Option: the value of being a leader, International Journal of Theoretical and Applied Finance, 16, 2013), and extend it to a random regulator. This model generalizes and unifies the different competitive frameworks proposed in the literature, and creates a new one similar to a Stackelberg leadership. We fully characterize strategic interactions in the several situations following from the parametrization of the regulator. Finally, we study the effect of the coordination game and uncertainty of outcome when agents are risk-averse, providing new intuitions for the standard case.
A game theoretic analysis of the cops and robber game
Georgios Konstantinidis
2014, 1(4): 599-619 doi: 10.3934/jdg.2014.1.599 +[Abstract](635) +[PDF](454.5KB)
Abstract:
We provide a game theoretic framework for the game of cops and robbers. Within this framework we study certain assumptions which underlie the concepts of optimal strategies and capture time. We show rigorously that these assumptions are justified. Our results are based on the theory developed by C. Alos-Ferrer and K. Ritzberger on the existence of subgame perfect equilibria in extensive form games which are infinite and / or have infinite horizon.
Payoff performance of fictitious play
Georg Ostrovski and Sebastian van Strien
2014, 1(4): 621-638 doi: 10.3934/jdg.2014.1.621 +[Abstract](605) +[PDF](444.1KB)
Abstract:
We investigate how well continuous-time fictitious play in two-player games performs in terms of average payoff, particularly compared to Nash equilibrium payoff. We show that in many games, fictitious play outperforms Nash equilibrium on average or even at all times, and moreover that any game is linearly equivalent to one in which this is the case. Conversely, we provide conditions under which Nash equilibrium payoff dominates fictitious play payoff. A key step in our analysis is to show that fictitious play dynamics asymptotically converges to the set of coarse correlated equilibria (a fact which is implicit in the literature).
A limit theorem for Markov decision processes
Mathias Staudigl
2014, 1(4): 639-659 doi: 10.3934/jdg.2014.1.639 +[Abstract](819) +[PDF](506.5KB)
Abstract:
In this paper I prove a deterministic approximation theorem for a sequence of Markov decision processes with finitely many actions and general state spaces as they appear frequently in economics, game theory and operations research. Using viscosity solution methods no a-priori differentiabililty assumptions are imposed on the value function.

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