# American Institute of Mathematical Sciences

ISSN:
2164-6066

eISSN:
2164-6074

All Issues

## Journal of Dynamics & Games

April 2014 , Volume 1 , Issue 2

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2014, 1(2): 181-254 doi: 10.3934/jdg.2014.1.181 +[Abstract](2504) +[PDF](897.5KB)
Abstract:
Blackwell approachability, regret minimization and calibration are three criteria used to evaluate a strategy (or an algorithm) in sequential decision problems, described as repeated games between a player and Nature. Although they have at first sight not much in common, links between them have been discovered: for instance, both consistent and calibrated strategies can be constructed by following, in some auxiliary game, an approachability strategy.
We gather seminal and recent results, develop and generalize Blackwell's elegant theory in several directions. The final objectives is to show how approachability can be used as a basic powerful tool to exhibit a new class of intuitive algorithms, based on simple geometric properties. In order to be complete, we also prove that approachability can be seen as a byproduct of the very existence of consistent or calibrated strategies.
2014, 1(2): 255-281 doi: 10.3934/jdg.2014.1.255 +[Abstract](1633) +[PDF](488.4KB)
Abstract:
We study the deterministic dynamics of networks ${\mathcal N}$ composed by $m$ non identical, mutually pulse-coupled cells. We assume weighted, asymmetric and positive (cooperative) interactions among the cells, and arbitrarily large values of $m$. We consider two cases of the network's graph: the complete graph, and the existence of a large core (i.e. a large complete subgraph). First, we prove that the system periodically eventually synchronizes with a natural "spiking period" $p \geq 1$, and that if the cells are mutually structurally identical or similar, then the synchronization is complete ($p= 1$) . Second, we prove that the amount of information $H$ that ${\mathcal N}$ generates or processes, equals $\log p$. Therefore, if ${\mathcal N}$ completely synchronizes, the information is null. Finally, we prove that ${\mathcal N}$ protects the cells from their risk of death.
2014, 1(2): 283-298 doi: 10.3934/jdg.2014.1.283 +[Abstract](1795) +[PDF](430.1KB)
Abstract:
Bergstrom, Blume and Varian [4] provides an elegant game-theoretic model of an economy with one private good and one public good. Strategies of players consist of voluntary contributions of the private good to public good production. Without relying on first order conditions, as in prior literature, the authors demonstrate existence of Nash equilibrium. The assumption of one-private good greatly facilities the results. We provide an analogue of the Bergstrom, Blume and Varian result in a model allowing multiple private and public goods. In addition, we relate the strategic market game equilibrium to the private-provision equilibrium of Villanacci and Zenginobuz [17], which provides a counter-part to the Walrasian equilibrium for a public goods economy. To obtain our results we introduce a model of a strategic market game with public goods. Our approach also incorporates, into the strategic market game literature, economies with production.
2014, 1(2): 299-330 doi: 10.3934/jdg.2014.1.299 +[Abstract](1700) +[PDF](396.9KB)
Abstract:
We study existence and turnpike properties of approximate solutions for a class of dynamic discrete-time two-player zero-sum games without using convexity-concavity assumptions. We describe the structure of approximate solutions which is independent of the length of the interval, for all sufficiently large intervals and show that approximate solutions are determined mainly by the objective function, and are essentially independent of the choice of interval and endpoint conditions.