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

Competitive multi-player stochastic games with applications to multi-person financial contracts

Competitive Multi-Player Stochastic Games with Applications to Multi-Person Financial Contracts Ivan Guo Abstract In the financial market, almost all traded derivatives only involve two parties. The aim of this thesis is to design and evaluate financial contracts involving multiple parties. This is done by utilising and extending concepts from game theory, financial mathematics and backward stochastic differential equations. The thesis is divided into two parts: multi-player stochastic competitive games and multi-person financial contracts. The first part of the thesis proposes two novel classes of multi-period multi-player stopping games: the multi-player redistribution game and the multi-player affine game. Both formulations are generalisations of the classic two-player Dynkin game, with a focus on designing the dependence between the payoffs of all players and their stopping decisions. These games are shown to be weakly unilaterally competitive, and sufficient conditions are given for the existence of optimal equilibria (a new solution concept motivated by financial applications), individual values and coalition values. The second part of the thesis introduces the notion of multi-person financial contracts by extending the two-person game option. These contracts may involve an arbitrary number of parties and each party is allowed to make a wide array of decisions, which then determines the settlement date as well as the payoffs. The generalised Snell envelope is introduced for the valuation of multi-person contracts and sufficient conditions for the existence of unique and additive arbitrage prices are provided. Finally, a new class of multi-dimensional reflected backward stochastic differential equations are proposed to model multi-person affine game options under market friction

Optimal search in discrete locations:extensions and new findings

A hidden target needs to be found by a searcher in many real-life situations, some of which involve large costs and significant consequences with failure. Therefore, efficient search methods are paramount. In our search model, the target lies in one of several discrete locations according to some hiding distribution, and the searcher's goal is to discover the target in minimum expected time by making successive searches of individual locations. In Part I of the thesis, the searcher knows the hiding distribution. Here, if there is only one way to search each location, the solution to the search problem, discovered in the 1960s, is simple; search next any location with a maximal probability per unit time of detecting the target. An equivalent solution is derived by viewing the search problem as a multi-armed bandit and following a Gittins index policy. Motivated by modern search technology, we introduce two modes---fast and slow---to search each location. The fast mode takes less time, but the slow mode is more likely to find the target. An optimal policy is difficult to obtain in general, because it requires an optimal sequence of search modes for each location, in addition to a set of sequence-dependent Gittins indices for choosing between locations. For each mode, we identify a sufficient condition for a location to use only that search mode in an optimal policy. For locations meeting neither sufficient condition, an optimal choice of search mode is extremely complicated, depending both on the hiding distribution and the search parameters of the other locations. We propose several heuristic policies motivated by our analysis, and demonstrate their near-optimal performance in an extensive numerical study. In Part II of the thesis, the searcher has only one search mode per location, but does not know the hiding distribution, which is chosen by an intelligent hider who aims to maximise the expected time until the target is discovered. Such a search game, modelled via two-person, zero-sum game theory, is relevant if the target is a bomb, intruder, or, of increasing importance due to advances in technology, a computer hacker. By Part I, if the hiding distribution is known, an optimal counter strategy for the searcher is any corresponding Gittins index policy. To develop an optimal search strategy in the search game, the searcher must account for the hider’s motivation to choose an optimal hiding distribution, and consider the set of corresponding Gittins index policies. %It follows that an optimal search strategy in the search game must be some Gittins index policy if the hiding distribution is assumed to be chosen optimally by the hider. However, the searcher must choose carefully from this set of Gittins index policies to ensure the same expected time to discover the target regardless of where it is hidden by the hider. %It follows that an optimal search strategy in the search game must be a Gittins index policy applied to a hiding distribution which is optimal from the hider's perspective. However, to avoid giving the hider any advantage, the searcher must carefully choose such a Gittins index policy among the many available. As a result, finding an optimal search strategy, or even proving one exists, is difficult. We extend several results for special cases from the literature to the fully-general search game; in particular, we show an optimal search strategy exists and may take a simple form. Using a novel test, we investigate the frequency of the optimality of a particular hiding strategy that gives the searcher no preference over any location at the beginning of the search