### Abstract

We propose the study of computing the Shapley value for a new class of cooperative games that we call budgeted games, and investigate in particular knapsack budgeted games, a version modeled after the classical knapsack problem. In these games, the “value” of a set S of agents is determined only by a critical subset T ⊆ S of the agents and not the entirety of S due to a budget constraint that limits how large T can be. We show that the Shapley value can be computed in time faster than by the naｨıve exponential time algorithm when there are sufficiently many agents, and also provide an algorithm that approximates the Shapley value within an additive error. For a related budgeted game associated with a greedy heuristic, we show that the Shapley value can be computed in pseudo-polynomial time. Furthermore, we generalize our proof techniques and propose what we term algorithmic representation framework that captures a broad class of cooperative games with the property of efficient computation of the Shapley value. The main idea is that the problem of determining the efficient computation can be reduced to that of finding an alternative representation of the games and an associated algorithm for computing the underlying value function with small time and space complexities in the representation size.

Original language | English (US) |
---|---|

Pages (from-to) | 106-119 |

Number of pages | 14 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 8877 |

State | Published - Jan 1 2014 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*,

*8877*, 106-119.

**The shapley value in knapsack budgeted games.** / Bhagat, Smriti; Weinsberg, Udi; Kim, Anthony; Muthukrishnan, Shanmugavelayutham.

Research output: Contribution to journal › Article

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*, vol. 8877, pp. 106-119.

}

TY - JOUR

T1 - The shapley value in knapsack budgeted games

AU - Bhagat, Smriti

AU - Weinsberg, Udi

AU - Kim, Anthony

AU - Muthukrishnan, Shanmugavelayutham

PY - 2014/1/1

Y1 - 2014/1/1

N2 - We propose the study of computing the Shapley value for a new class of cooperative games that we call budgeted games, and investigate in particular knapsack budgeted games, a version modeled after the classical knapsack problem. In these games, the “value” of a set S of agents is determined only by a critical subset T ⊆ S of the agents and not the entirety of S due to a budget constraint that limits how large T can be. We show that the Shapley value can be computed in time faster than by the naｨıve exponential time algorithm when there are sufficiently many agents, and also provide an algorithm that approximates the Shapley value within an additive error. For a related budgeted game associated with a greedy heuristic, we show that the Shapley value can be computed in pseudo-polynomial time. Furthermore, we generalize our proof techniques and propose what we term algorithmic representation framework that captures a broad class of cooperative games with the property of efficient computation of the Shapley value. The main idea is that the problem of determining the efficient computation can be reduced to that of finding an alternative representation of the games and an associated algorithm for computing the underlying value function with small time and space complexities in the representation size.

AB - We propose the study of computing the Shapley value for a new class of cooperative games that we call budgeted games, and investigate in particular knapsack budgeted games, a version modeled after the classical knapsack problem. In these games, the “value” of a set S of agents is determined only by a critical subset T ⊆ S of the agents and not the entirety of S due to a budget constraint that limits how large T can be. We show that the Shapley value can be computed in time faster than by the naｨıve exponential time algorithm when there are sufficiently many agents, and also provide an algorithm that approximates the Shapley value within an additive error. For a related budgeted game associated with a greedy heuristic, we show that the Shapley value can be computed in pseudo-polynomial time. Furthermore, we generalize our proof techniques and propose what we term algorithmic representation framework that captures a broad class of cooperative games with the property of efficient computation of the Shapley value. The main idea is that the problem of determining the efficient computation can be reduced to that of finding an alternative representation of the games and an associated algorithm for computing the underlying value function with small time and space complexities in the representation size.

UR - http://www.scopus.com/inward/record.url?scp=84914165798&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84914165798&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84914165798

VL - 8877

SP - 106

EP - 119

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -