Markov games : an annotated bibliography

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Solving generic nonarchimedean semidefinite programs using stochastic game algorithms

A general issue in computational optimization is to develop combinatorial algorithms for semidefinite programming. We address this issue when the base field is nonarchimedean. We provide a solution for a class of semidefinite feasibility problems given by generic matrices. Our approach is based on tropical geometry. It relies on tropical spectrahedra, which are defined as the images by the valuation of nonarchimedean spectrahedra. We establish a correspondence between generic tropical spectrahedra and zero-sum stochastic games with perfect information. The latter have been well studied in algorithmic game theory. This allows us to solve nonarchimedean semidefinite feasibility problems using algorithms for stochastic games. These algorithms are of a combinatorial nature and work for large instances.Comment: v1: 25 pages, 4 figures; v2: 27 pages, 4 figures, minor revisions + benchmarks added; v3: 30 pages, 6 figures, generalization to non-Metzler sign patterns + some results have been replaced by references to the companion work arXiv:1610.0674

Dynkin games with incomplete and asymmetric information

We study the value and the optimal strategies for a two-player zero-sum optimal stopping game with incomplete and asymmetric information. In our Bayesian set-up, the drift of the underlying diffusion process is unknown to one player (incomplete information feature), but known to the other one (asymmetric information feature). We formulate the problem and reduce it to a fully Markovian setup where the uninformed player optimises over stopping times and the informed one uses randomised stopping times in order to hide their informational advantage. Then we provide a general verification result which allows us to find the value of the game and players' optimal strategies by solving suitable quasi-variational inequalities with some non-standard constraints. Finally, we study an example with linear payoffs, in which an explicit solution of the corresponding quasi-variational inequalities can be obtained.Comment: 31 pages, 5 figures, small changes in the terminology from game theor

Player aggregation in the traveling inspector model

We consider a model of dynamic inspection/surveillance of a number of facilities in different geographical locations. The inspector in this process travels from one facility to another and performs an inspection at each facility he visits. His aim is to devise an inspection/ travel schedule which minimizes the losses to society (or to his employer) resulting both from undetected violations of the regulations and from the costs of the policing operation. This model is formulated as a non-cooperative, single-controller, stochastic game. The existence of stationary Nash equilibria is established as a consequence of aggregating all the inspectees into a single “aggregated inspectee”. It is shown that such player aggregation causes no loss of generality under very mild assumptions. A notion of an “optimal Nash equilibrium” for the inspector is introduced and proven to be well-defined in this context. The issue of the inspector’s power to “enforce” such an equilibrium is also discussed

Advances in Zero-Sum Dynamic Games

International audienceThe survey presents recent results in the theory of two-person zero-sum repeated games and their connections with differential and continuous-time games. The emphasis is made on the following(1) A general model allows to deal simultaneously with stochastic and informational aspects.(2) All evaluations of the stage payoffs can be covered in the same framework (and not only the usual Cesàro and Abel means).(3) The model in discrete time can be seen and analyzed as a discretization of a continuous time game. Moreover, tools and ideas from repeated games are very fruitful for continuous time games and vice versa.(4) Numerous important conjectures have been answered (some in the negative).(5) New tools and original models have been proposed. As a consequence, the field (discrete versus continuous time, stochastic versus incomplete information models) has a much more unified structure, and research is extremely active

Making Sense of Unexpected Preferences

This dissertation includes three papers using quantitative models to sensibly describe what kinds of preferences political actors will or actually do hold when existing theory offers no insight. The first two papers use evolutionary game theory to predict ways in which politicians, artificially selected on the basis of good performance to remain in office, will in the long run diverge from instrumental rationality as ordinarily assumed in game theory. The first sets out a general principle for producing models of preference evolution in games as political models, namely, that the information about opponent preferences necessary for evolution of non-rational preferences comes from opponents\u27 previous plays, and applies it to two simple games. The second uses the same principles in more detail on a bargaining game that models the plea negotiations between a prosecutor and a defense attorney, leading to a conclusion that failure to learn from setbacks during a trial is an evolutionarily favored trait among prosecutors. The third paper addresses the ideological preferences of Supreme Court justices, which existing statistical models do not effectively compare to those of elected officials since the two groups never vote on the same items, by identifying a set of political actors with whom both groups commonly interact: organized interest groups who vote on Supreme Court cases with amicus curiae briefs and on electoral candidates using campaign donations