The Value for Action-set Games

Uncertainty in Artificial Intelligence, 2004

Action-graph games (AGGs) are a fully expres- sive game representation which can compactly express both strict and context-specific indepen- dence between players' utility functions. Ac- tions are represented as nodes in a graph G, and the payoff to an agent who chose the action s depends only on the numbers of other agents who chose actions connected to s. We

Mathematics of Operations Research

Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero-sum case, they may have different values at different equilibriums. Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value [Formula: see text]). Similar to the standard value function in control literature, it enjoys many nice properties, such as regularity, stability, and more importantly, the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closed-loop controls (instead of open-loop controls); and (ii) we must allow for path dependent controls, even if the problem is in a state-dependent (Markovian) setting. We shall consider both discrete and continuous time models with finite time horizon. For the latter, we will also provide a du...

International Journal of Game Theory, 1997

The reactive bargaining set (Granot [1994]) is the set of outcomes for which no justified objection exists. Here, in a justified objection the objector first watches how the target tries to act (if he has such an option), and then reacts by making a profit and ruining the target's attempt to maintain his share. In this paper we explore properties of the reactive bargaining set, set up the system of inequalities that defines it, and construct a dynamic system in the sense of Stearns' transfer scheme that leads the players to this set. We also extend the definition of the reactive bargaining set to NTU games in a way that keeps it nonempty. To shed light on its nature and its relative ease of computation, we compute the reactive bargaining set for games that played important role in the game theory literature.

International Journal of Game Theory, 1997

We prove that the reactive bargaining set coincides with the core of simple flow games, and it essentially coincides with the kernel of simple superadditive games.

Mathematical Methods of Operations Research, 2001

We extend the consistency principle for strategic games (Peleg and Tijs (1996)) to apply to solutions which assign to each game a collection of product sets of strategies. Such solutions turn out to satisfy desirable properties that solutions assigning to each game a collection of strategy profiles lack. Our findings lead us to propose a new direction for normative game theory.

1996

We investigate the equivalence between several notions of bargaining sets which occur in the literature and the core of simple games.

Games and Economic Behavior, 2011

Representing and reasoning with games becomes difficult once they involve large numbers of actions and players, because the space requirement for utility functions can grow unmanageably. Action-Graph Games (AGGs) are a fully-expressive game representation that can compactly express utility functions with structure such as context-specific independence, anonymity, and additivity. We show that AGGs can be used to compactly represent all games that are compact when represented as graphical games, symmetric games, anonymous games, congestion games, and polymatrix games, as well as games that require exponential space under all of these existing representations. We give a polynomial-time algorithm for computing a player's expected utility under an arbitrary mixed-strategy profile, and show how to use this algorithm to achieve exponential speedups of existing methods for computing sample Nash equilibria. We present results of experiments showing that using AGGs leads to a dramatic increase in the size of games accessible to computational analysis. 1 (Current affiliation: BMO Capital Markets). nbhat@physics.utoronto.ca 1 We gratefully acknowledge Moshe Tennenholtz for his co-authorship of a paper on Local Effect Games , an action-centric graphical model for games that inspired our work on AGGs.

2006

Action-Graph Games (AGGs) are a fully expressive game representation which can compactly express strict and context-specific independence and anonymity structure in players' utility functions. We present an efficient algorithm for computing expected payoffs under mixed strategy profiles. This algorithm runs in time polynomial in the size of the AGG representation (which is itself polynomial in the number of players when the in-degree of the action graph is bounded). We also present an extension to the AGG representation which allows us to compactly represent a wider variety of structured utility functions. 1 An action-graph game (AGG) is a tuple N, S, ν, u . Let N = {1, . . . , n} denote the set of agents. Denote by S = i∈N S i the set of action profiles, where is the Cartesian product and S i is agent i's set of actions. We denote by s i ∈ S i one of agent i's actions, and s ∈ S an action profile.

Games and Economic Behavior, 2013

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Review of Economic Design, 1999

This paper provides a non-cooperative interpretation for bargaining sets concepts in economic environments. We investigate the implementability of the Aumann-Maschler and Mas-Colell bargaining sets, and provide mechanisms whose subgame perfect equilibrium outcomes realize these sets. These mechanisms, in contrast to general mechanisms suggested in the implementation literature, have a natural structure closely related to that of the rationale underlying the bargaining sets. Furthermore, the strategy sets consist mainly of allocations and coalitions (thus avoiding any reference to preference parameters) and are finite dimensional.