The Position Value for Partition Function Form Network Games

• Author: ANNE NOUWELAND,MARCO SLIKKER
• Date: 01 April 2016
• Published date: 01 April 2016
• DOI: http://doi.org/10.1111/jpet.12138

THE POSITION VALUE FOR PARTITION FUNCTION FORM

NETWORK GAMES

ANNE VAN DEN NOUWELAND

University of Oregon

MARCO SLIKKER

Eindhoven University of Technology

Abstract

We use the axiomatization of the position value for network situations

in van den Nouweland and Slikker (2012) to define a position value

for partition function form network situations. We do this by generaliz-

ing the axioms to the partition function form value function setting as

studied in Navarro (2007) and then showing that there exists a unique

allocation rule satisfying these axioms. We call this allocation rule the

position value for partition function form network situations.

1. Introduction

The public goods literature contains many examples that feature networks with exter-

nalities across components, including the private provision of public goods in diverse

societies (Allouch 2013), market sharing agreements (Belleflamme and Bloch 2004),

preferential trade agreements (Lake 2012), and the use of word of mouth to dissemi-

nate information (Bouchard St-Amant 2013). In such situations, the network induces a

partition function form game, as in Navarro (2007). Understanding partition function

form network situations thus provides important insights into a plethora of network

situations with externalities.

Partition function form network situations have, to our knowledge, been studied

only in Navarro (2007), who introduces the two properties component efficiency and

fairness and identifies a unique allocation rule satisfying these properties. The resulting

extension of the Myerson value (cf. Myerson 1977) is an allocation rule that is player-

based and considers the importance of each player with all his or her connections in

the network. The current paper contributes to this literature by defining an extension

of the position value (cf. Borm, Owen, and Tijs 1992), which is an allocation rule that is

based on the importance of the connections (links) between the players and distributes

the worth of each link among the two players who maintain it. In situations where

maintaining and using a link between two players takes effort by both of these players,

the position value thus provides valuable insight into the contributions of the links to

Anne van den Nouweland, Department of Economics, University of Oregon, Eugene, OR 97403-1285

(annev@uoregon.edu). Marco Slikker,School of Industrial Engineering, Eindhoven University of Tech-

nology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands (M.Slikker@tue.nl).

Received August 22, 2014; Accepted September 21, 2014.

C2014 Wiley Periodicals, Inc.

Journal of Public Economic Theory, 18 (2), 2016, pp. 226–247.

226

Partition Function Form Network Games 227

players’ payoffs. We identify an appropriate extension of the position value as follows.

First we take the axioms used in the axiomatization of the position value for network sit-

uations in van den Nouweland and Slikker (2012) and we consider extensions of these

to partition function form network situations. Then, we show that there exists a unique

allocation rule satisfying these extensions of the axioms. We also address potential alter-

native extensions of the axioms, and show that these do not lead to a possible alternative

definition of a position value for partition function form network situations.

The setup of this paper is as follows. In the next section, we explain the terminol-

ogy and notations that we use for networks, partition function form value functions,

and partition function form network situations and we also identify a basis of the space

of partition function form value functions. In Section 3, we define allocation rules for

partition function form network situations and we extend the axioms of van den Nouwe-

land and Slikker (2012) to the setting of partition function form network situations. In

Section 4, we then use these axioms and identify a unique allocation rule—the partition

function form position value—that satisfies them. In Section 5, we address alternative

extensions of the axioms and investigate the feasibility of using these other extensions

to define an allocation rule. We conclude in Section 6 with an explanation of how the

results in the paper demonstrate that the partition function form position value that we

define in Section 4 is, from an axiomatic perspective, the only candidate for an exten-

sion of the position value for network situations to the setting of partition function form

network situations.

2. Preliminaries

In this section, we explain the terminology and notations that we use for networks,

partition function form value functions, and partition function form network situations.

We also identify a basis of the space of partition function form value functions.

2.1. Networks

Throughout this paper, we consider a fixed set of players N={1,...,n}. A link is a

subset {i,j}of two different players i,j∈N,i= j. As is customary in the literature, we

often denote a link l={i,j}by ij and we refer to las “the link between players iand

j.” For any coalition of players S⊆N, we denote the set of all possible links between

players in Sby gS={ij|i,j∈S,i= j}.

A network consists of a set of nodes and a set of links between these nodes. In

this paper, we will only consider networks on the set of nodes Nand therefore we can

identify a network with its links. Thus, a network is a set of links g⊆gN.Wedenotethe

set of all possible networks by G={g|g⊆gN}and the set of nonempty networks, that

is, networks that include at least one link, by Gne ={g⊆gN|g=∅}.

A coalition of players S⊆Nis said to be connected in a network g∈Gif for any

two players i,j∈S, there is a sequence of links in gthat form a path from ito j,that

is, there exist an m∈{1,2,...,n}and i1,i2,...,im∈Nsuch that i1=i,ikik+1∈gfor

each k∈{1,2,...,m−1},andim=j. A network g∈Ginduces a partition C(g)ofthe

player set Ninto connected coalitions that are set-inclusion maximal with respect to

this property, namely, C(g)={S⊆N|Sis connected in gand for any T⊆Nwith S⊆

Tit holds that either T=Sor Tis not connected in g}. A network g∈Gis said to be

connected if the set of all players Nis connected in g. Thus, a network g∈Gis con-

nected if and only if C(g)={N}.