Extensive form games

The aim of this work is to propose a logical framework for representing interacting agents in the context of extensive form games. Because of the importance of the temporal dimension provided by such games, we create a modal epistemic logic that allows to quantify over both strategies and vertices within the game tree. The first part of the article is devoted to the logic itself with the definition of its language and its semantics. In order to illustrate the use of this logic, we define, in the following part, the concept of rationality in the case of extensive form games and the backward induction concept, as they are defined by Robert Aumann. Based on these definitions, we then provide a syntactic proof of Aumann's theorem that states the following: "for any non degenerate game of perfect information, common knowledge of rationality implies the backward induction solution". We finally propose an in-depth formal analysis of the hypotheses that are needed to prove such a theorem.

Many experimental studies indicate that people are motivated by reciprocity. Rabin (1993) develops techniques for incorporating such concerns into game theory and economics. His model, however, does not fare well when applied to situations with an interesting dynamic structure (like many experimental games), because it is developed for normal form games in which information about the sequential structure of a strategic situation is suppressed.

When studying extensive-form games it is commonly assumed that players make their decisions individually. One usually does not allow the possibility for the players to negotiate their respective strategies and formally commit themselves to future moves. As a consequence, many non-zero-sum games have been shown to have equilibrium outcomes that are suboptimal and arguably counter-intuitive. For this reason we feel there is a need to explore a new line of research in which game-playing agents are allowed to negotiate binding agreements before they make their moves. We analyze what happens under such assumptions and define a new equilibrium solution concept to capture this. We show that this new solution concept indeed yields solutions that are more efficient and, in a sense, closer to what one would expect in the real world. Furthermore, we demonstrate that our ideas are not only theoretical in nature , but can also be implemented on bounded rational agents, with a number of experiments conducted with a new algorithm that combines techniques from Automated Negotiations, (Algorithmic) Game Theory, and General Game Playing. Our algorithm, which we call Monte Carlo Negotiation Search, is an adaptation of Monte Carlo Tree Search that equips the agent with the ability to negotiate. It is completely domain-independent in the sense that it is not tailored to any specific game. It can be applied to any non-zero-sum game, provided that its rules are described in Game Description Language. We show with several experiments that it strongly outperforms non-negotiating players, and that it closely approximates the theoretically optimal outcomes, as defined by our new solution concept.

Typical models of strategic interactions in computer science use simultaneous move games. However, in applications simultaneity is often hard or impossible to achieve. In this paper, we study the robustness of the Nash Equilibrium when the assumption of simultaneity is dropped. In particular we propose studying the sequential price of anarchy: the quality of outcomes of sequential versions of games whose simultaneous counterparts are prototypical in algorithmic game theory. We study different classes of games with high price of anarchy, and show that the subgame perfect equilibrium of their sequential version is a much more natural prediction, ruling out unreasonable equilibria, and leading to much better quality solutions.

By a player splitting we mean a mechanism that distributes the information sets of a player among so-called agents. A player splitting is called independent if each path in the game tree contains at most one agent of every player. Following Mertens (1989), a solution is said to have the player splitting property if, roughly speaking, the solution of an extensive form game does not change by applying independent player splittings. We show that Nash equilibria, perfect equilibria, Kohlberg-Mertens stable sets and Mertens stable sets have the player splitting property. An example is given to show that the proper equilibrium concepts does not satisfy the player splitting property. Next, we give a definition of invariance under (general) player splittings which is an extension of the player splitting property to the situation where we also allow for dependent player splittings. We come to the conclusion that none of the solutions above are invariant under any dependent player splitting. The results are used to give several characterizations of the class of independent player splittings and the class of single appearance structures by means of invariance of solution concepts under player splittings.

By a player splitting we mean a mechanism that distributes the information sets of a player among so-called agents. A player splitting is called independent if each path in the game tree contains at most one agent of every player. Following Mertens (1989), a solution is said to have the player splitting property if, roughly speaking, the solution of an extensive form game does not change by applying independent player splittings. We show that Nash equilibria, perfect equilibria, Kohlberg-Mertens stable sets and Mertens stable sets have the player splitting property. An example is given to show that the proper equilibrium concepts does not satisfy the player splitting property. Next, we give a definition of invariance under (general) player splittings which is an extension of the player splitting property to the situation where we also allow for dependent player splittings. We come to the conclusion that none of the solutions above are invariant under any dependent player splitting. The results are used to give several characterizations of the class of independent player splittings and the class of single appearance structures by means of invariance of solution concepts under player splittings.

We present the logic G.STRAT for reasoning about group strategies. To enable a modal semantics for reasoning about group strategies, strategy profiles will be viewed as structured possible worlds. Although the general logic is undecidable and not finitely axiomatizable for systems with 3 agents or more, we identify more well-behaved logics like ATL, and temporal stitlogics as fragments. We explain our aim to apply the logics to reasoning about multi-agent programs, such as the programs of 2APL. Finally, we discuss several properties of (multi-)agency expressible in the logic. Among these properties is an axiom characterizing a basic form of autonomy.