The Shapley value provides a unique solution to coalition games and is used to evaluate a player's prospects of playing a game. Although it provides a unique solution, there is an element of uncertainty associated with this value. This uncertainty in the solution of a game provides an additional dimension for evaluating a player's prospects of playing the game. Thus, players want to know not only their Shapley value for a game, but also the associated uncertainty. Given this, our objective is to determine the Shapley value and its uncertainty and study the relationship between them for the voting game. But since the problem of determining the Shapley value for this game is #P-complete, we first present a new polynomial time randomized method for determining the approximate Shapley value. Using this method, we compute the Shapley value and correlate it with its uncertainty so as to allow agents to compare games on the basis of both their Shapley values and the associated uncertainties. Our study shows that, a player's uncertainty first increases with its Shapley value and then decreases. This implies that the uncertainty is at its minimum when the value is at its maximum, and that agents do not always have to compromise value in order to reduce uncertainty.
[1]
Alvin E. Roth,et al.
The Shapley Value as a von Neumann-Morgenstern Utility
,
1977
.
[2]
Vladislav Kargin.
Uncertainty of the Shapley Value
,
2003
.
[3]
Andre Francis.
Advanced Level Statistics
,
1988
.
[4]
Xiaotie Deng,et al.
On the Complexity of Cooperative Solution Concepts
,
1994,
Math. Oper. Res..
[5]
L. Shapley.
A Value for n-person Games
,
1988
.
[6]
Alvin E. Roth,et al.
The Shapley value: The expected utility of playing a game
,
1988
.
[7]
Ariel Rubinstein,et al.
A Course in Game Theory
,
1995
.
[8]
Jeffrey S. Rosenschein and Gilad Zlotkin.
Rules of Encounter
,
1994
.
[9]
Victor R. Lesser,et al.
Coalitions Among Computationally Bounded Agents
,
1997,
Artif. Intell..
[10]
J. Nash.
THE BARGAINING PROBLEM
,
1950,
Classics in Game Theory.
[11]
J. Harsanyi.
Rational behavior and bargaining equilibrium in games and social situations: Frontmatter
,
1977
.
[12]
Sarit Kraus,et al.
Methods for Task Allocation via Agent Coalition Formation
,
1998,
Artif. Intell..