Game Theory at the Universities of Milano (III) May Politecnico di Milano, Italy

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1 Game Theory at the Universities of Milano (III) May Politecnico di Milano, Italy

2 Game Theory at the Universities of Milano (III) follows the first two meetings held at Politecnico di Milano, the first one in 2009 and the second one in Each past edition hosted a group of wellknown international scholars, presenting their research activities. The meeting represents also the conclusion of a cross-sectorial course of the Ph.D. School of Politecnico, entitled Game Theoretical Models in Engineering. Thus, even if the purpose of the meeting is to present results at the frontier of the research in this area, the main speakers will be invited to provide a broad overview of the subject of their talk. Shorter communications of young researches will be also scheduled. Registration, program, and schedule are available at Organizers Nicola Gatti Roberto Lucchetti Sponsor PRIN 2009 Variational and Topological methods in the study of Nonlinear Phenomena Dipartimento di Matematica del Politecnico di Milano FARB Politecnico di Milano List of invited speakers Steve Alpern (University of Warwick) Danilo Ardagna (Politecnico di Milano) Konstantin Avratchenkov (INRIA) Nicola Basilico (Politecnico di Milano) Pierpaolo Battigalli (Università Bocconi) Giuditta Caffarra (Politecnico di Milano) Emiliano Catonini (Università Bocconi) Michela Chessa (Università degli Studi di Milano) Laura Cottatellucci (Eurecom) Francisco Facchinei (Università La Sapienza) Josep Freixas (Universitat Politècnica de Catalunya) Astrid Gamba (Università degli Studi di Milano-Bicocca) Mario Gilli (Università degli Studi di Milano-Bicocca) Fabio Maccheroni (Università Bocconi) Elena Manzoni (Università degli Studi di Milano-Bicocca) Stefano Moretti (Université Paris-Dauphine) Marco Rocco (Politecnico di Milano)

3 SCHEDULE Thursday May 23 9:45- Registration and opening Francisco Facchinei (Università di Roma 1) Complex variational inequalities, generalized Nash equilibrium problems and applications 11:00-11:30 Coffee break Steve Alpern (University of Warwick) Patrolling games Nicola Basilico (Politecnico di Milano) Computing patrolling strategies in leader-follower adversarial settings 13:00-15:00 Lunch 15: Stefano Moretti (Paris Dauphine) On the problem of coordinating self-interested agents placed on a network to realize a socially optimal system Michela Chessa (Università degli Studi di Milano) A generating functions approach for computing the Public Good index efficiently 16:15-16:45 Coffee break Emiliano Catonini (Università Bocconi) Non-binding agreements and forward induction reasoning Giuditta Caffarra (Politecnico di Milano) From orders on a set to orders on its power set using semivalues 18:00-18:30 Danilo Ardagna (Politecnico di Milano) Service provisioning problem in cloud and multi-cloud systems: a Generalized Nash equilibrium model Friday May 24 9:45-10:30 Konstantin Avratchenkov (INRIA-Sophia Antipolis) Non-cooperative stochastic games and applications Laura Cottatellucci (EURECOM) Cooperative stochastic games and applications 11:15-11:30 Coffee break Astrid Gamba (Università degli Studi di Milano-Bicocca) Preference-dependent learning in the Centipede Game: a theoretical and experimental study Elena Manzoni (Università degli Studi di Milano-Bicocca) Incomplete information models of guilt aversion in the trust game 12:30-13:00 Marco Rocco (Politecnico di Milano) Which mechanism in sponsored search auctions with externalities? 13:00-14:30 Lunch 14: Pierpaolo Battigalli (Università Bocconi) Selfconfirming long run biases Josep Freixas (Universitat Politècnica de Catalunya) On the qualitative inverse problem for voting rules with abstention 16:00-16:30 Coffee break Mario Gilli (Università degli Studi di Milano-Bicocca) Accountability in autocracies: the role of revolution threat Fabio Maccheroni (Università Bocconi) Mixed extensions of decision problems under uncertainty

4 Book of Abstracts Patrolling games Steve Alpern, University of Warwick Joint work with: Alec Morton and Katarina Papadaki A key operational problem for those charged with the security of vulnerable facilities (such as airports or art galleries) is the scheduling and deployment of patrols. Motivated by the problem of optimizing randomized, and thus unpredictable, patrols, we present a class of patrolling games. The facility to be patrolled can be thought of as a network or graph Q of interconnected nodes (e.g., rooms, terminals), and the Attacker can choose to attack any node of Q within a given time T. He requires m consecutive periods there, uninterrupted by the Patroller, to commit his nefarious act (and win). The Patroller can follow any path on the graph. Thus, the patrolling game is a winlose game, where the Value is the probability that the Patroller successfully intercepts an attack, given best play on both sides. We determine analytically either the Value of the game, or bounds on the Value, for various classes of graphs, and we discuss possible extensions and generalizations. Service provisioning problem in cloud and multi-cloud systems: a Generalized Nash uquilibrium model Danilo Ardagna, Politecnico di Milano In recent years the evolution and the widespread adoption of virtualization, service-oriented architectures, autonomic and utility computing have converged letting a new paradigm to emerge: The Cloud Computing. Cloud Computing aims at streamlining the on-demand provisioning of software, hardware, and data as services, providing end-user with flexible and scalable services accessible through the Internet. Since the Cloud offer is currently becoming wider and more attractive to business owners, the development of efficient resource provisioning policies for Cloud-based services becomes increasingly challenging. Indeed, modern Cloud services operate in an open and dynamic world characterized by continuous changes where strategic interaction among different economic agents takes place. This seminar aims to discuss the hourly basis service provisioning and capacity allocation problem through the formulation of a mathematical model based on noncooperative-game-theoretic approach. We take the perspective of Software as a Service (SaaS) providers which want to minimize the costs associated with the virtual machine instances allocated in a multi-iaass (Infrastructure as a Service) scenario, while avoiding incurring in penalties for requests execution failures and providing quality of service guarantees. SaaS providers compete and bid for the use of infrastructural resources, while the IaaSs want to maximize their revenues obtained providing virtualized resources. The problem has been modeled as a Generalized Nash Equilibrium Problem (GNEP). Thanks to a deep analysis of the game under study, we demonstrate the social equilibria existence for the corresponding generalized potential game. The best-reply solution is pursued heuristically with the implementation of two different algorithms suitable for a distributed implementation. We demonstrate the effectiveness of our approach by performing numerical analyses, considering multiple workloads and system configurations. Results show that our algorithms are scalable. Equilibria efficiency is quantitatively analysed in terms of Price of Anarchy (PoA). Furthermore, a multiple IaaS analysis points out the SaaS benefits in exploiting multiple IaaS deployment. Non-cooperative stochastic games and applications Konstantin Avratchenkov, INRIA-Sophia Antipolis

5 We define stochastic games with discounted, average and Blackwell type criteria. We discuss existence and uniqueness results. Then, we describe special classes of stochastic games for which efficient numerical procedures exist. For most described special classes of stochastic games we provide illustrations by telecommunication applications. Computing patrolling strategies in leader-follower adversarial settings Nicola Basilico, Politecnico di Milano Protecting environments from intentional threats is a task that characterizes many security applications and that nowadays constitutes a topic of crucial practical relevance. One of the main problems in this domain is given by the limited resources (such as robots, cameras, check points) that can be employed to monitor an environment in order to prevent attacks or intrusions. This limitation poses the problem of finding effective resource allocation strategies that could deal with such limits providing a good protection level. In the very last years, this problem has been addressed in a game theoretical fashion leading to the definition of a new class of Security Games. In this approach, the presence of an rational attacker is explicitly considered in formulating the problem of finding an effective strategy. In this talk, I will present a general introduction on Security Games and I will focus on a particular kind of games that embrace robotic patrolling as application of reference: Patrolling Security Games (PSGs). The general idea behind a PSG is that of modeling a patrolling situation as a leader-follower competitive game, played by a defender and an attacker, and of studying the equilibria of this game to derive effective patrolling strategies. In this talk, I will present an overview of our recent works on PSGs, with a particular attention to algorithms for computing equilibrium strategies and to the definition of different techniques to enable the employment of these models in realistic settings. Selfconfirming long run biases Pierpaolo Battigalli, Università Bocconi We consider an uncertainty averse and myopic decision maker facing a recurrent decision problem. We study selfconfirming strategies. We show that a long run bias emerges that favors "tested" actions, that is, actions on which information has been collected over time. In so doing we provide, inter alia, a learning foundation for the selfconfirming equilibrium with model uncertainty of Battigalli et al. (2011, IGIER w.p. 428). From orders on a set to orders on its power set using semivalues Giuditta Caffarra, Politecnico di Milano The problem of extending a preference relation over a set of objects to its power set has been carried out in literature with the purpose of characterizing rankings over subsets with a system of axioms formulated according to the context. This approach prevents any kind of interaction among the objects and cooperative games have been used to handle this issue. A cooperative game can be naturally interpreted as the numerical representation of the preferences over the power set elements, in particular, we consider those extensions such that the ranking induced by a semivalue over the set of objects preserves the original preferences. We will compare the axiomatic approach with the properties which characterize the semivalue extensions and we will study how certain semivalues characteristics impact on the extension problem. Non-binding agreements and forward induction reasoning Emiliano Catonini, Università Bocconi

6 The paper studies the credibility of pre-play, non-binding, possibly incomplete agreements among players in dynamic non-cooperative games. In dynamic games, players may observe a deviation from the agreement before the game is over. The attempt to rationalize the deviation may lead players to revise their beliefs about co-players' behavior in the continuation of the game. This instance of forward induction reasoning is based not just on beliefs about rationality, but also on interactive beliefs about the compliance with the agreement itself. Here I study the effects of such rationalization on the self-enforceability of an agreement, that is on the possibility that the agreement is commonly believed and that players comply with it. A generating functions approach for computing the Public Good index efficiently Michela Chessa, Università degli Studi di Milano In the past years, a combinatorial method based on generating functions was introduced to compute Shapley-Shubik, Banzhaf and other indices for weighted majority games exactly and efficiently. In this work, taking inspiration from what it has already been done, in view of the efficiency of the generating functions method, we define a generating function for computing the Public Good index, maintaining the property of exactness of the resulting algorithm. The main difference with the existing algorithms derives from the fact that the Public Good index takes into account only minimal winning coalitions and counts how many swings of a player involve them. Moreover, we study the computational complexity of the algorithm. Cooperative stochastic games and applications Laura Cottatellucci, EURECOM In this tutorial we extend key concepts of cooperative games such as Core and Shapley values to dynamic systems described by Markov Decision Chains (MDC) where the payoffs are distributed among players at each stage of the game. Fundamental properties for dynamic systems such as time consistency, greedy player satisfaction, and coalition maintenance are illustrated. For cooperative stochastic games (CST) with transferable utilities, we propose a stationary payoff procedure satisfying the above mentioned properties. For the case of CST with non-transferable utilities, we focus on an endogenous Markov process, i.e. players strategies do not influence the transition probabilities, with the feasible points of each stage belonging to a polymatroid. This case models a system of primary relevance in information and communication theory: the achievable rate region of multiple access channels. For this system, we discuss the payoff distribution according to CSG and optimal fairness criteria. Complex variational inequalities, generalized Nash equilibrium problems and applications Francisco Facchinei, Università di Roma 1 We introduce and study variational inequalities in the complex domain, along with some technical tools useful in their study. We then extend to the complex domain some recent developments in the field of the distributed solution of (generalized) Nash equilibrium problems. In order to illustrate our techniques we consider some new MIMO games over vector Gaussian Interference Channels, modeling some distributed resource allocation problems in MIMO cognitive radio systems and femtocells. These games are examples of Nash equilibrium problems that can not be handled by current methodologies.

7 On the qualitative inverse problem for voting rules with abstention Josep Freixas, Universitat Politècnica de Catalunya The first part of the presentation introduces voting games with abstention and more generally voting games with several levels of approval in input and output. Weighted notions and desirability relations are introduced to this context and the precise relationships between weightedness and completeness are established. Several notions of power for these games are discussed and a final study on allowable hierarchies for these games is provided. We conclude by listing which rankings of power are achievable for games of this type. Thus, we ascertain which voting systems of this type can de designed with a given ranking of power. Preference-dependent learning in the Centipede Game: a theoretical and experimental study Astrid Gamba, Università degli studi di Milano-Bicocca We study the interaction of individuals with heterogeneous preference types in the strategic context of a six-stage Centipede game. We adopt a version of Self-confirming equilibrium, whereby individuals maximize their subjective expected utilities, given their preference types (e.g., selfish or altruistic) and their type-specific and possibly wrong conjectures on the opponent's strategies. Such equilibrium with off-path prediction errors has a plausible learning interpretation (i.e., fictitious play) under the assumption that individuals learn their opponent's strategies via their own observations of past interactions. We test our learning interpretation in an experiment, where we study the learning dynamics of a heterogeneous population of individuals, who are classified according to their social preferences and interact anonymously along 40 repetitions of the six-stage Centipede game. We study how the long-run type-specific conjectures and behavior change under different ex post information structures. We find that if individuals rely only on their own observations of co-players' past moves, in the long run heterogeneous behavior resembles a Self-confirming equilibrium. Off-path prediction errors turn out to be type-specific: selfish types have coarser conjectures about the opponent's behavior with respect to more altruistic types. When feedbacks on past play are more informative (aggregate frequencies of moves actually taken by individuals in the co-player s role or aggregate frequencies of plans of actions adopted by individuals in the co-player s role) the long run play is closer to a Bayesian Nash equilibrium with a common and correct belief about the play. Accountability in autocracies: the role of revolution threat Mario Gilli, Università degli studi di Milano-Bicocca The purpose of this paper is to explore the joint work of two mechanisms that might constrain autocratic rulers: the threat of a coup d état by the political elite and of a revolution by the citizens. Our results will help explain a well established and crucial fact, i.e. that autocracies are far more likely than democracies to be either the best or the worst performers in terms of growth and that they are far more heterogenous than democracies in public goods policies. To this aim, we focus on accountability within dictatorships where by accountability we mean a mechanism that involves an agency relationship between politicians and citizens aimed at screening the rulers different types and providing an effective incentivazion system. We use a common agency model where the political elite and the citizens are the principals and the autocrat is the agent; the tools the principals can use are the leader s removal through a coup and a revolt, respectively. Our results highlight that the role of checks and balances in disciplining autocrats turns out to be subtle: congruent policies in autocracies require either an intermediate strength of the state, so that the revolution threat by the citizens is credible only when there are poor outcomes, or a

8 strong enough political elite, so that a coup threat is credible only when economic policies are non-congruent. Otherwise, we will have inefficient economic policies. Hence, both excessively strong and excessively weak rulers lead to poor economic performances: in autocracies there need to be a balanced distribution of de facto political power to incentivize the ruler to choose congruent economic policies. This requirement explains the huge variability in economic performances in autocracies: they depend on the above details of political institutions and how they evolve over time and across countries. Mixed extensions of decision problems under uncertainty Fabio Maccheroni, Università Bocconi Incomplete information models of guilt aversion in the trust game Elena Manzoni, Università degli Studi di Milano-Bicocca Joint work with: Giuseppe Attanasi and Pierpaolo Battigalli In the theory of psychological games it is assumed that players preferences on material consequences depend on endogenous beliefs. Most of the applications of this theoretical framework assume that the psychological utility functions representing such preferences are common knowledge. But this is often unrealistic. In particular, it cannot be true in experimental games where players are subjects drawn at random from a population. Therefore an incompleteinformation methodology is called for. We take a first step in this direction, focusing on models of guilt aversion in the Trust Game. We consider two alternative modeling assumptions: (i) guilt aversion depends on the role played in the game, because only the trustee can feel guilt for letting the co-player down, (ii) guilt aversion is independent of the role played in the game. We show how the set of Bayesian equilibria changes as the upper bound on guilt sensitivity varies, and we compare with complete-information equilibria. Our analysis illustrates the incompleteinformation approach to psychological games and can help organize experimental results in the Trust Game. On the problem of coordinating self-interested agents placed on a network to realize a socially optimal system Stefano Moretti, Paris Dauphine In this talk, we introduce some strategic games based on connection situations with the objective to coordinate self-interested agents placed on the nodes of a graph to realize a more efficient communication network. We address the problem of the design of cost allocation protocols that may guarantee the convergence of the best response dynamic and we analyze the effects of cost monotonicity and other state-dependent properties on the optimality of a protocol. Which mechanism is sponsored search auctions with externalities? Marco Rocco, Politecnico di Milano Sponsored search is one of the most successful applications of economic mechanisms in real life. A crucial issue is the modeling of the user behavior to provide the best targeting of ads to each user. Experimental studies show that the click through rate of an ad is dramatically affected by both its position and the other displayed ads. However, these externalities rise severe currently open computational issues in the determination of the best allocation and of the payments, preventing their adoption in practice so far. In the present paper, we provide a number of results when the most famous externality model, the cascade model, is adopted: we design the first exact algorithm

9 for computing the efficient allocation, we show that the previously presented constantapproximation algorithm does not lead to any incentive compatible mechanism, we design a monotonic constant-approximation algorithm for finding the allocation and two different polynomial-time algorithms for the payments, each with different properties, leading to incentive compatible mechanisms. Finally, we provide a thorough experimental evaluation of the presented algorithms with Yahoo! Webscope A3 dataset to identify which mechanism should be adopted in concrete applications.