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 2164-6066

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Volume 4, 2017

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Journal of Dynamics & Games

2017 , Volume 4 , Issue 4

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Game theoretical modelling of a dynamically evolving network I: General target sequences
Mark Broom and Chris Cannings
2017, 4(4): 285-318 doi: 10.3934/jdg.2017016 +[Abstract](154) +[HTML](126) +[PDF](1062.6KB)
Abstract:

Animal (and human) populations contain a finite number of individuals with social and geographical relationships which evolve over time, at least in part dependent upon the actions of members of the population. These actions are often not random, but chosen strategically. In this paper we introduce a game-theoretical model of a population where the individuals have an optimal level of social engagement, and form or break social relationships strategically to obtain the correct level. This builds on previous work where individuals tried to optimise their number of connections by forming or breaking random links; the difference being that here we introduce a truly game-theoretic version where they can choose which specific links to form/break. This is more realistic and makes a significant difference to the model, one consequence of which is that the analysis is much more complicated. We prove some general results and then consider a single example in depth.

Stability of the replicator dynamics for games in metric spaces
Saul Mendoza-Palacios and Onésimo Hernández-Lerma
2017, 4(4): 319-333 doi: 10.3934/jdg.2017017 +[Abstract](385) +[HTML](108) +[PDF](417.9KB)
Abstract:

In this paper we study the stability of the replicator dynamics for symmetric games when the strategy set is a separable metric space. In this case the replicator dynamics evolves in a space of measures. We study stability criteria with respect to different topologies and metrics on the space of probability measures. This allows us to establish relations among Nash equilibria (of a certain normal form game) and the stability of the replicator dynamics in different metrics. Some examples illustrate our results.

Nonlinear dynamics from discrete time two-player status-seeking games
Borun Shi and Robert A. Van Gorder
2017, 4(4): 335-359 doi: 10.3934/jdg.2017018 +[Abstract](330) +[HTML](306) +[PDF](802.6KB)
Abstract:

We study the dynamics of two-player status-seeking games where moves are made simultaneously in discrete time. For such games, each player's utility function will depend on both non-positional goods and positional goods (the latter entering into "status"). In order to understand the dynamics of such games over time, we sample a variety of different general utility functions, such as CES, composite log-Cobb-Douglas, and King-Plosser-Rebelo utility functions (and their various simplifications). For the various cases considered, we determine asymptotic dynamics of the two-player game, demonstrating the existence of stable equilibria, periodic orbits, or chaos, and we show that the emergent dynamics will depend strongly on the utility functions employed. For periodic orbits, we provide bifurcation diagrams to show the existence or non-existence of period doubling or chaos resulting from bifurcations due to parameter shifts. In cases where multiple feasible solution branches exist at each iteration, we consider both cases where deterministic or random selection criteria are employed to select the branch used, the latter resulting in a type of stochastic game.

A new perspective on the classical Cournot duopoly
Rabah Amir and Igor V. Evstigneev
2017, 4(4): 361-367 doi: 10.3934/jdg.2017019 +[Abstract](166) +[HTML](72) +[PDF](272.8KB)
Abstract:

The paper provides new conditions for the existence, uniqueness, and symmetry of pure-strategy Nash equilibrium in the classical Cournot duopoly.

A probability criterion for zero-sum stochastic games
Xiangxiang Huang, Xianping Guo and Jianping Peng
2017, 4(4): 369-383 doi: 10.3934/jdg.2017020 +[Abstract](145) +[HTML](75) +[PDF](405.5KB)
Abstract:

This paper introduces a probability criterion for two-person zero-sum stochastic games, and focuses on the probability that the payoff before the first passage time to some target state set exceeds a level formulated by both players, which shows the security for player 1 and the risk for player 2. For the game model based on discrete-time Markov chains, under a suitable condition on the game's primitive data, we establish the Shapley equation, from which the existences of the value of the game and a pair of optimal policies are ensured. We also provide a recursive way of computing (or at least approximating) the value of the game. At last, the application of our main result is exhibited via an inventory system.

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