The Shapley and Banzhaf values in microarray games

doi:10.1016/j.cor.2009.02.020

Computers & Operations Research

Volume 37, Issue 8, August 2010, Pages 1406-1412

https://doi.org/10.1016/j.cor.2009.02.020 Get rights and content

Abstract

We provide a characterization of the Banzhaf value on the class of the microarray games, and another characterization of the Shapley value, on the same class, in the spirit of [Moretti S, Patrone F, Bonassi S. The class of microarray games and the relevance index for genes. TOP 2007;15:256–80]. Then we compare the results given by the two indices when applied to a colon tumor data-set published in literature.

Introduction

It is well known that several diseases have a genetic origin. Recently, a new tool, called microarray technology, has been implemented to generate a lot of information on gene expression of human beings. These data can in principle be used to identify genes which are, for instance, responsible of a particular disease. Many models for data analysis have been presented in the literature for inferring, from a matrix of gene expression data, the role of genes, their interactions and their behaviors when changes in condition of the biological system occur (see for instance the book [12]). An alternative method for gene expression analysis based on coalitional games has been proposed in [8]. The main advantage of this approach is the possibility to compute a numerical index, called a relevance index, which represents the relevance of each gene under a certain condition (e.g., a tumor) taking into account the expression behaviors of the other genes. In [8], the authors use the Shapley value with the aim of ranking the genes according to their relevance and provide an axiomatic characterization of the Shapley value on the class of the so-called microarray games.

In literature, other approaches using game theory for gene expression analysis have been introduced. In [7], a method based on the framework of minimum cost spanning trees (MCST) has been introduced to represent the interactions between all possible pairs of genes and extended to implement the notion of association for coalitions of genes. In [4], classification games with genes in the role of players have been studied to analyze the power of groups of genes to classify samples into the right classes (for instance, the class of normal tissues or the class of tumor tissues). A biological validation for the use of the Shapley value ([13]; see also [9]) of microarray games as a relevance index for genes is presented in [1], where a set of genes involved in the pathogenesis of neuroblastic tumors has been selected according to the Shapley value of a microarray game. The problem to compare the relevance of genes under two different conditions has been recently studied in [10] where a statistical method called comparative analysis of Shapley (shortly, CASh) value is applied to real gene expression data concerning the gene expression in children differentially exposed to air pollution.

In this paper we consider another important one-point solution in game theory, namely the Banzhaf value [3]. In particular, taking also inspiration from [6], we offer a pool of properties characterizing it, and we give an alternative characterization for the Shapley value, always on the class of the microarray games. Finally, we compare the results given by the two indices on a set of data relative to patients affected by colon tumor.

Section snippets

Preliminary notations

Let us introduce some basic game theoretical notations. A coalitional game or characteristic-form game is a pair $(N, v)$ , where $N$ denotes the finite set of players and $v : 2^{N} \to R$ the characteristic function, with $v (\emptyset) = 0$ . If the set $N$ of players is fixed, we identify a coalitional game $(N, v)$ with the corresponding characteristic function $v$ . A group of players $T \subseteq N$ is called a coalition and $v (T)$ is called the value of this coalition. A coalitional game $(N, w)$ such that $w : 2^{N} \to {0, 1}$ is called a ${0, 1}$ -game.

Axiomatic characterization for microarray games

In [8] it was proved that the Shapley value is the only one-point solution, on the class of microarray games, fulfilling a pool of reasonable properties that we shall describe later. In this paper we are interested in giving another pool of properties characterizing the Banzhaf value. In doing this, we also find another alternative characterization of the Shapley value. To start with, we introduce a new definition, motivated by an analogous one given in [5], for general cooperative games.

Definition 1

Let $v \in$

Some thoughts on Banzhaf versus Shapley

The two relevance indices we deal with in this paper are suitable to rank genes potentially responsible of a genetic disease. In general, they will give different ranking. How can we interpret this fact? In this section we briefly comment on this.

It is clear that the differences in the two indices arise from the differences of their behavior with respect to the unanimity games. So, what is the basic difference among them, when dealing with this type of games? Of course, they do assign zero to

Colon data analysis

Moretti et al. [8] introduced a preliminary application of the Shapley value for a microarray game defined on a tumor/normal data-set published in [2]⁵ containing expression levels of a set $N$ of 2000 genes measured using Affymatrix oligonucleotide microarrays for a set of $40$ tumor samples and a set of 22 normal samples, in total 62 samples from colon tissues. In that application, after the preprocessing stage performed by the

Acknowledgments

The authors are grateful to two anonymous referees for their helpful comments. Stefano Moretti gratefully acknowledges the financial support of the EU project NewGeneris, European Union 6th FP (FOOD-CT-2005-016320).

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¹: The research of this author was partially supported by Ministero dell’Istruzione, dell’Università e della Ricerca Scientifica (COFIN 2005).

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The Shapley and Banzhaf values in microarray games

Abstract

Introduction

Section snippets

Preliminary notations

Axiomatic characterization for microarray games

Some thoughts on Banzhaf versus Shapley

Colon data analysis

Acknowledgments

Computers & Mathematics with Applications

Identification of low intratumoral gene expression heterogeneity in neuroblastic tumors by wide-genome expression analysis and game theory

Cancer

Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissue probed by oligonucleotide arrays

Proceedings of the National Academy of Sciences of the United States of America

Weighted voting doesn’t work: a game theoretic approach

Rutgers Law Review

Shapley–Shubik and Banzhaf indices revisited

Mathematics of Operations Research