Bi‐cooperative network games: A link‐based allocation rule

• Author: Surajit Borkotokey,Diganta Mukherjee,Loyimee Gogoi
• Date: 01 June 2018
• DOI: http://doi.org/10.1111/ijet.12148
• Published date: 01 June 2018

doi: 10.1111/ijet.12148

Bi-cooperative network games: A link-based allocation rule

Surajit Borkotokey,∗Loyimee Gogoi∗and Diganta Mukherjee†

We study the notion of a bi-cooperative network game and obtain a link-based allocation rule

for the corresponding class. Unlike classical network games where players have single motive

of forming (or not forming) a network, a player under a bi-cooperative set up has two more

variations: all her links attribute exclusively to her positiveor negative contributions, or few links

are positive while few others are negative.We extend the position value, an important link-based

allocation rule in classical network games, and show that it is the only rule for all the three

variations of a bi-cooperative network game.

Key wor ds bi-cooperative game, network game, bi-cooperative network game, position value,

LG value

JEL classification C71, C72

Accepted 25 July2016

1 Introduction

A bi-cooperative game (Bilbao 2000) is an extension of a classical cooperative game where players

can support an issue, oppose it, or remain indifferent. It assumes that the value accrued by a group

of players (coalition) depends on the action of the remaining players, with possible absentees. Thus

the player set is divided into a partition of three groups: the positive contributors who support an

issue, the negative contributors who oppose the issue, and the absentees who remain indifferent

to (silent on) the issue. Consider, for example, situations of international trade negotiation, public

policy-making in a direct democracy setting, crowdsourcing of knowledge, etc. (details follow).

The characteristic function (equivalently bi-characteristic function; see Fujimoto 2014) repre-

senting the game assigns a value to the positive players that depends on their opponents, the negative

players. Network games `alaJackson and Wolinsky (1996) are graph-restricted cooperative games,

where players can generate a value only when they are linked through networks. Borgotokey and

Gogoi (2014) recently introduced a network-restricted bi-cooperative game (bi-cooperative net-

work game, for short) in which interactions among players with bipolar motives are considered

only through some exogenously given network. Such games are extensions of classical network

games. There are many real-life situations that borrow features of a network game (Jackson2005)

but the players exhibit bipolarity in cooperation. Such situations can be modeled neither by the

notion of a network game nor by a bi-cooperative game alone, rather by a hybrid of these two

notions.

∗Department of Mathematics, Dibrugarh University,Dibrugarh, India. Email: surajitbor@yahoo.com

†Sampling and Official Statistics Unit, ISI-Kolkata,India,

The work done in this paper is under the UGC Major Research Project UGC-India 42-26/2013(SR). The authors ac-

knowledge Krishnendu Ghosh Dastidar and Michel Grabisch for their insightful suggestions.

International Journal of Economic Theory 14 (2018) 103–128 © IAET 103

International Journal of Economic Theory

Bi-cooperative network games: Link-based rule Surajit Borkotokey et al.

In this paper we propose a more general model of bi-cooperative network games that involves

positive and negativeplayers and links in the same network, and we assume that a player can even have

both positive and negative links simultaneously.We provide some motivating examples to highlight

the relevance of bi-cooperative network games under three different settings.

1.1 Motivation

As a first instance, consider a geopolitical issue on which countries with different motives agree to

negotiate (see, for example, Park 2000, where countries are of different sizes). The negotiations are

facilitated/influenced by all existing pairwise trade relationships among these countries. A positive

link represents a complementary trade relation, whereas a negative link implies a competitive sit-

uation. The motivational differences will influence which treaties are suitable and which will call

for amendments. Thus, this is a situation where the countries are connected by the trade network

and the negotiation game is played out subject to the given network. In contrast, we might think of

another situation where the players with opposing stance do not have any information about each

other and bargaining between randomly matched players leads to possible link formation. Manea

(2000) shows that such a game may create components in a network called mutually estranged sets.

Note that there is a thirdpossibilit y: that playersmay simultaneously have positive and negative links.

Consider the curious example of double agents working for various spying agencies (Isby 2004) or

the duality of involvement of players in different networks according to their affinity to different

groups of people. A strict physical instructor may be a very jovial person among friends. In Figure 1

we depict these three models. It is interesting to note that an ordinary bi-cooperative game cannot

account for these three distinctly posed situations as it does not consider the networks involving the

players. On the other hand, the notion of bipolarity prevents classical network games from modeling

such situations.

Let us take a more specific example.We consider a situation where a companyF1asks for feedback

on one of its products by posting on a social media site, Facebook for example. Some, but not all,

Figure 1 Three types of bi-cooperative network games (Solid lines: positive links; Dash lines: negative links;

Dotted lines: absentee links).

104 International Journal of Economic Theory 14 (2018) 103–128 © IAET