No abstract

The Shapley value is a key solution concept for coalitional games in general and voting games in particular. Its main advantage is that it provides a unique and fair solution, but its main drawback is the complexity of computing it (e.g for voting games this complexity is #P-complete). However, given the importance of the Shapley value and voting games, a number of approximation methods have been developed to overcome this complexity. Among these, Owen's multi-linear extension method is the most time efficient, being linear in the number of players. Now, in addition to speed, the other key criterion for an approximation algorithm is its approximation error. On this dimension, the multi-linear extension method is less impressive. Against this background, this paper presents a new approximation algorithm, based on randomization, for computing the Shapley value of voting games. This method has time complexity linear in the number of players, but has an approximation error that is, on average, lower than Owen's. In addition to this comparative study, we empirically evaluate the error for our method and show how the different parameters of the voting game affect it. Specifically, we show the following effects. First, as the number of players in a voting game increases, the average percentage error decreases. Second, as the quota increases, the average percentage error decreases. Third, the error is different for players with different weights; players with weight closer to the mean weight have a lower error than those with weight further away. We then extend our approximation to the more general k-majority voting games and show that, for n players, the method has time complexity O(k 2 n) and the upper bound on its approximation error is O(k 2 / √ n).

This paper studies bilateral multi-issue negotiation between self-interested autonomous agents. Now, there are a number of different procedures that can be used for this process; the three main ones being the package deal procedure in which all the issues are bundled and discussed together, the simultaneous procedure in which the issues are discussed simultaneously but independently of each other, and the sequential procedure in which the issues are discussed one after another. Since each of them yields a different outcome, a key problem is to decide which one to use in which circumstances. Specifically, we consider this question for a model in which the agents have time constraints (in the form of both deadlines and discount factors) and information uncertainty (in that the agents do not know the opponent's utility function). For this model, we consider issues that are both independent and those that are interdependent and determine equilibria for each case for each procedure. In so doing, we show that the package deal is in fact the optimal procedure for each party. We then go on to show that, although the package deal may be computationally more complex than the other two procedures, it generates Pareto optimal outcomes (unlike the other two), it has similar earliest and latest possible times of agreement to the simultaneous procedure (which is better than the sequential procedure), and that it (like the other two procedures) generates a unique outcome only under certain conditions (which we define).

This paper analyzes the process of automated negotiation between two competitive agents that have firm deadlines and incomplete information about their opponent. Generally speaking, the outcome of a negotiation depends on many parameters-including the agents' preferences, their reservation limits, their attitude toward time and the strategies they use. Although in most realistic situations it is not possible for agents to have complete information about each of these parameters for its opponent, it is not uncommon for agents to have partial information about some of them. Under such uncertainty, our aim is to determine how an agent can exploit its available information to select an optimal strategy. Here, in particular, the optimal strategies are determined considering all possible ways in which time can effect negotiation. Moreover, we list the conditions for convergence when both agents use their respective optimal strategies and study the effect of time on negotiation outcome.

This paper presents a new model for multi-issue negotiation under time constraints in an incomplete information setting. In this model the order in which issues are bargained over and agreements are reached is determined endogenously as part of the bargaining equilibrium. We show that the sequential implementation of the equilibrium agreement gives a better outcome than a simultaneous implementation when agents have like, as well as conflicting, time preferences. We also show that the equilibrium solution possesses the properties of uniqueness and symmetry, although it is not always Pareto-optimal.