Dependence in games and dependence games

Davide Grossi 0 Paolo Turrini 0 0 D. Grossi Institute for Logic , Language and Computation, University of Amsterdam , Amsterdam, The Netherlands In the multi-agent systems community, dependence theory and game theory are often presented as two alternative perspectives on the analysis of agent interaction. The paper presents a formal analysis of a notion of dependence between players, given in terms of standard game-theoretic notions of rationality such as dominant strategy and best response. This brings the notion of dependence within the realm of game theory providing it with the sort of mathematical foundations which still lacks. Concretely, the paper presents two results: first, it shows how the proposed notion of dependence allows for an elegant characterization of a property of reciprocity for outcomes in strategic games; and second, it shows how the notion can be used to define new classes of coalitional games, where coalitions can force outcomes only in the presence of reciprocal dependencies. Dependence theory was born within the social sciences [7] and brought into multi-agent systems (henceforth, MAS) by [5,6] where, in the last decade, its ideas have made their way into several lines of research (e.g., [1,3,17,18]). Dependence theory has, at the moment, several versions and no unified theory. Yet, the aim of the theory is clear: One of the fundamental notions of social interaction is the dependence relation among agents. In our opinion, the terminology for describing interaction in a multi-agent world is necessarily based on an analytic description of this relation. Starting from such a - terminology, it is possible to devise a calculus to obtain predictions and make choices that simulate human behavior [6, p. 2]. Therefore, dependence theory addresses two issues: (i) the representation of dependence relations among the agents in a system (to the nature of this relation we will come back in Sect. 3); (ii) the use of such information as a means to obtain predictions about the behavior of the system. While all contributions to dependence theory have focused on the first point, the second challenge, devise a calculus to obtain predictions, has been mainly left to computer simulation [17]. The present paper takes up these two challenges outlining a theory of dependence in strategic interaction which makes use of standard game-theoretic notions and techniques. The theory we propose builds up on a notion of dependence relations in strategic games, inspired from informal literature on dependence theory in MAS (e.g., [5,6]): Player i depends on player j for reaching outcome s, within a given game, if and only if j plays a strategy, in the profile determining s, which is a best response (or a dominant strategy) for i . The aim of the paper is to provide a thorough analysis of the above informal definition, which will be made formal in Definition 9. Concretely, the paper presents two results. First, it shows that this notion of dependence allows for the characterization of an original notion of reciprocity for strategic games (Theorem 1), i.e. how players in a strategic interaction can profitably make use of their dependence relation. Second, it shows that the notion of dependence can be fruitfully applied to ground coalition formation. The class of coalitional games where coalitions can force outcomes only in the presence of reciprocity here called dependence gamescan be directly linked with standard solution concepts used in cooperative game theory (Theorems 2, 3). The paper generalizes results presented in [9], discusses further examples, and puts the results in perspective with game theory and with related work in MAS. Definitions and results will be thoroughly illustrated by examples and, in particular, by discussing their application to two- and three-person games. As to the motivation behind the research presented here, it can be said that the paper moves from the authors impression that, within the MAS community, dependence theory and game theory are erroneously considered to be alternative, when not incompatible, paradigms for the analysis of social interaction. It is our conviction that the theory of games and that of dependence are highly compatible endeavors which can highly benefit from one another in the field of MAS. On the one hand dependence theory can be incorporated into the highly developed mathematical framework of game theory, obtaining the sort of foundations that still lacks. On the other hand, game theory can incorporate a dependence-theoretic perspective on the analysis of strategic interaction. Section 2 briefly introduces the basic notions of game theory with which we will work in the rest of the paper. Section 3 introduces our formal analysis of the concept of dependence, relating it to informal definitions available in the literature on dependence theory. It then moves on to the characterization of reciprocal outcomes in games. Starting from the notion of reciprocity, Section 4 introduces a notion of agreement among players in a game, and on this ground it defines and studies a specific class of coalitional games. Section 5 describes in detail some further examples of our analysis proposing also an application of the framework Fig. 1 Examples of two player strategic games in game matrices. Ordinal preferences are represented, as usual, by means of numerical payoffs. U and D denote the strategies up and, respectively, down for the row player. L and R denote the strategies left and right for the column player 2 Preliminaries: game theory The present section introduces the basic game-theoretic notions used in the paper. All definitions will be based on an ordinal notion of preference. Main sources for this preliminary section are [12,13]. 2.1 Strategic games, sub-games and solution concepts Let us start with the definition of a game in strategic form. Definition 1 (Game) A (strategic form) game is a tuple G = (N , S, i , i , o)1 where: N is a non-empty set of players; S is a non-empty set of outcomes; i is a non-empty set of strategies for each player i N ; i is a total preorder on S; o : iN i S is a function from the set iN i of strategy profiles to the set of outcomes S.2 Examples of games represented as payoff matrices are given in Fig. 1. The terms agent and player will be used interchangeably. Before we continue with the next definition, let us first introduce some further notation. Strategy profiles will be denoted , . . . . Given a strategy profile and a player i, i denotes the strategy chosen by i in , i.e., the i th projection of , and i denotes the profile consisting of all the strategies of the players except i . So, a profile can be seen as the juxtaposition of i and i , in symbols, = (i , i ). More generally, C , for a coalition C N , denotes the tuple of strategies performed by the agents in C in profile , i.e. an element of iC i . Given a coalition C , in order to denote the set of agents not be (...truncated)