The Shapley Value of Tuples in Query Answering

Abstract

We investigate the application of the Shapley value to quantifying the contribution of a tuple to a query answer. The Shapley value is a widely known numerical measure in cooperative game theory and in many applications of game theory for assessing the contribution of a player to a coalition game. It has been established already in the 1950s, and is theoretically justified by being the very single wealth-distribution measure that satisfies some natural axioms. While this value has been investigated in several areas, it received little attention in data management. We study this measure in the context of conjunctive and aggregate queries by defining corresponding coalition games. We provide algorithmic and complexity-theoretic results on the computation of Shapley-based contributions to query answers; and for the hard cases we present approximation algorithms.

The Impact of Negation on the Complexity of the Shapley Value in Conjunctive Queries

“…Following earlier notions of formal causality by Halpern and Pearl [15], Salimi et al [27] proposed the causal effect: assuming endogenous facts are randomly removed independently and uniformly, what is the difference in the expected query answer between assuming the presence and the absence of f ? A recent framework has proposed to adopt the Shapley value to the task [20].…”

“…In the framework of Livshits et al [20], query answering is viewed as a cooperative game where the players are the database facts and the utility function is the query answer, in the case of aggregate queries, or 0/1 in the case of Boolean queries. They showed that the (precise or approximate) evaluation of the Shapley value on common aggregate queries amounts to the evaluation of the Shapley value for a Boolean query.…”

“…However, while the Shapley value of F facts can be either positive or zero (since they can only help in satisfying the query), the Shapley value of G facts can be either negative or zero (since they can only help in violating the query). As explained by Livshits et al [20], understanding the complexity of the Shapley value for Boolean queries such as (1) is also necessary and sufficient for understating the complexity of the Shapley value for aggregate queries such as Count{c | F (m), E (m,p, c), ¬G (c, p)} that counts the countries that import one or more products that they do not grow.…”

“…As in previous work on quantification of contribution of facts [20,23,27], we view the database as consisting of two types of facts: endogenous facts and exogenous facts. Exogenous facts are taken as given (e.g., inherited from external sources) without questioning, and are beyond experimentation with hypothetical or counterfactual scenarios.…”

“…Livshits et al [20] have studied the complexity of computing the Shapley value, and their work is restricted to positive CQs and UCQs (and aggregates thereof). In this paper, we study the impact of negation on this complexity.…”