Robust Exponential Worst Cases for Divide-et-Impera Algorithms for Parity Games

The McNaughton-Zielonka divide et impera algorithm is the simplest and most flexible approach available in the literature for determining the winner in a parity game. Despite its theoretical worst-case complexity and the negative reputation as a poorly effective algorithm in practice, it has been shown to rank among the best techniques for the solution of such games. Also, it proved to be resistant to a lower bound attack, even more than the strategy improvements approaches, and only recently a family of games on which the algorithm requires exponential time has been provided by Friedmann. An easy analysis of this family shows that a simple memoization technique can help the algorithm solve the family in polynomial time. The same result can also be achieved by exploiting an approach based on the dominion-decomposition techniques proposed in the literature. These observations raise the question whether a suitable combination of dynamic programming and game-decomposition techniques can improve on the exponential worst case of the original algorithm. In this paper we answer this question negatively, by providing a robustly exponential worst case, showing that no intertwining of the above mentioned techniques can help mitigating the exponential nature of the divide et impera approaches.Comment: In Proceedings GandALF 2017, arXiv:1709.0176

Game Theory

The Special Issue “Game Theory” of the journal Mathematics provides a collection of papers that represent modern trends in mathematical game theory and its applications. The works address the problem of constructing and implementation of solution concepts based on classical optimality principles in different classes of games. In the case of non-cooperative behavior of players, the Nash equilibrium as a basic optimality principle is considered in both static and dynamic game settings. In the case of cooperative behavior of players, the situation is more complicated. As is seen from presented papers, the direct use of cooperative optimality principles in dynamic and differential games may bring time or subgame inconsistency of a solution which makes the cooperative schemes unsustainable. The notion of time or subgame consistency is crucial to the success of cooperation in a dynamic framework. In the works devoted to dynamic or differential games, this problem is analyzed and the special regularization procedures proposed to achieve time or subgame consistency of cooperative solutions. Among others, special attention in the presented book is paid to the construction of characteristic functions which determine the power of coalitions in games. The book contains many multi-disciplinary works applied to economic and environmental applications in a coherent manner

Strategic interaction between futures and spot markets.

There is a literature (e.g., Allaz and Vila, 1992 and Hughes and Kao, 1997) showing that in an oligopolistic context, the presence of a futures market induces firms to use it in order to increase its market share. The consequence of this behavior is that the total quantity supplied by the industry increases, thus making the oligopolistic outcome closer to the competitive equilibrium. In the present work, we propose a model to study the interaction of spot and futures markets that does not imply this pro-competitive effect. The model is the same as in Allaz and Vila in the sense that firms have infinitely many moments to trade in the futures market before the spot market takes place. We analyze the equilibria in the infinite case directly and show that many equilibria emerge in a kind of folk-theorem result (but ours is not a repeated game). The equilibrium in which firms do not use the forward market is particularly robust as it satisfies the most demanding definition of renegotiation-proofuess. Furthermore, if firms are allowed to buy in the futures market, they can sustain the monopolistic outcome in a renegotiation-proof equilibrium (notice that there is only one period in the spot market). We also study the role of information in the model and argue that our results fit better stylized facts of some industries like the power market in the U.K.Futures markets; Cournot competition; Collusion;

The Dynamic Reform of Political Institutions

This paper introduces a class of games designed to study dynamic, endogenous reform of political institutions. Dynamic political games (DPGs) are dynamic games in which institutional choice is both recursive and instrumental. Future political aggregation rules are decided under current ones, and institutional choices do not affect payoffs or technology directly. We examine properties of the Markovian equilibria of DPGs. In any equilibrium, institutional reform occurs if the subsequent political rule is chosen to be different than the present one. Which environments exhibit institutional reform and which tend toward institutional stability? Private (public) sector decisions are said to be inessential if, roughly, they can always be replaced by decisions in the public (private) sector in a social planner's payoff. We show that if the private sector is inessential, then institutional reform never occurs. However, if public sector decisions are inessential, then institutional reform must occur. The result suggests that an ineffective private sector is conducive to institutional stability, while an ineffective public sector is conducive to change. We also address the ``political fixed point problem" that arises in a model of recursive institutional choice. Namely, the current political rule (e.g., majority voting) admits a solution only if all feasible political rules admit solutions in all future dates. If the class of political rules is dynamically consistent then DPGs are shown to admit political fixed points. This result is used to prove two equilibrium existence theorems, one of which implies that all decision rules are smooth functions of the economic stateRecursive, dynamic political games, institutional reform, political fixed points, inessential.

Achievable Outcomes of Dynamic Contribution Games, Second Version

This paper concerns multistage games, with and without discounting, in which each player can increase the level of an action over time so as to increase the other players’ future payoffs. An action profile is achievable if it is the limit point of a subgame perfect equilibrium path. Necessary conditions are derived for achievability under relatively general conditions. They imply that any efficient profile that is approximately achievable must be in the core of the underlying coalitional game. In some but not all games with discounting, the necessary conditions for achievability are also sufficient for a profile to be the limit of achievable profiles as the period length shrinks to zero. Consequently, in these games when the period length is very short, (i) the set of achievable profiles does not depend on the move structure; (ii) an efficient profile can be approximately achieved if and only if it is in the core; and (iii) any achievable profile can be achieved almost instantly.dynamic games, monotone games, core, public goods, voluntary contribution, gradualism

Time-consistency of cooperative solutions

Management in complex social and economics networks includes elements of cooperative behavior or full cooperation between agents involved in decision making process. The most appropriate mathematical tool for modeling in this case is the mathematical theory of cooperative games. Unfortunately the classical cooperative game theory considers cooperation as one-shot interaction between the decision makers and for this reason can not be used for modeling of dynamic interactions arising in long term strategic management. The theory of cooperative differential games is the most adequate tool for modeling strategic management development on a given time interval. The use of this theory from the beginning poses the problems connected with dynamic stability (time-consistency) of optimal cooperative solutions. The consideration of optimality principles taken from the classical cooperative game theory shows the time-inconsistency and thus non applicability of these principles in strategic management. In this paper the methods of construction of time consistent solutions is proposed for the problems of strategic management in social and economic networks. The authors tried to present a rather complicated material on acceptable level. Theory is illustrated with a number of examples and a more comprehensive analysis of joint venture.

Collective Bargaining and Walrasian Equilibrium

This paper contributes to the research agenda on non-cooperative foundations ofWalrasian Equilibrium. A class of barganing games in which agents bargain over prices and maximum trading con- straints is considered: It is proved that all the Stationary Sub- game Perfect Equilibria of these games implement Walrasian al- locations as the bargaining frictions vanish. The main novelty of the result is twofold: (1) it holds for any number of agents; (2) it is robust to di¤erent speci�cations of the bargaining process.strategic bargaining; Walrasian Equilibrium

A theory of regular Markov perfect equilibria in dynamic stochastic games: genericity, stability, and purification

This paper studies generic properties of Markov perfect equilibria in dynamic stochastic games. We show that almost all dynamic stochastic games have a finite number of locally isolated Markov perfect equilibria. These equilibria are essential and strongly stable. Moreover, they all admit purification. To establish these results, we introduce a notion of regularity for dynamic stochastic games and exploit a simple connection between normal form and dynamic stochastic games.Dynamic stochastic games, Markov perfect equilibrium, regularity, genericity, finiteness, strong stability, essentiality, purifiability, estimation, computation, repeated games