Dynamic exchange networks

Abstract

Considering the theoretical and empirical untenability of static exchange networks, researchers have asked how exchange outcomes change when links are added or deleted. The present paper assesses the validity of seemingly sensible propositions concerning the effects of adding and deleting a link on (i) the payoffs of the actors in the link, (ii) the payoffs of actors in neighboring links and (iii) the variance of payoffs in the exchange network. The propositions were examined by applying expected value theory (EVT) to all 13,597 networks up to size 8. All propositions were falsified. Some falsifications of propositions could be attributed to EVTs prediction that actors use sub-optimal exchange relations. Since other well-known theories of exchange, like power-dependence theory and network exchange theory, also predict that actors use sub-optimal relations, these results are robust to selection of the theory of exchange.

Introduction

Research on exchange networks has mainly focused on the effects of network structure on outcomes in bilateral exchange (e.g., Cook and Emerson, 1978, Molm, 1997, Willer, 1999). Network structure determines which actors can exchange with whom, where a link signifies a bilateral exchange relation. An exchange relation usually constitutes, as in the present paper, the opportunity to split a common resource pool of 24 points. Successful exchange occurs if two connected actors can agree on a division. There is an upper limit to the number of exchanges that an actor can engage in. Usually this limit is one (the so-called “one-exchange rule”), as also in this paper. It is through this limit that network structure becomes relevant for how resource pools are split. It gives certain actors a credible threat in the bargaining process, namely to turn to an alternative exchange partner, thereby possibly excluding the current exchange partner and leaving him empty-handed. The main result of both theoretical and empirical research on exchange networks is that network structure has a huge impact on what actors earn. Examples of exchange networks are presented in Fig. 1.

In some bilateral exchange relations, actors obtain equally profitable exchange outcomes (Bonacich and Bienenstock, 1995, Bonacich and Bienenstock, 1997), for example, all actors in the Box (Fig. 1). In other bilateral exchange relations, one actor has a slight advantage over the other actor in the relation, e.g. B (C) in the Line4 (Fig. 1) obtains a larger exchange outcome than A (D) in their exchange relation (e.g., Willer, 1999). Finally, in some networks, some actors do have a clear advantage and obtain most of the profit as opposed to other actors who gain almost nothing, e.g., B in the Line3 (Fig. 1) obtains most of the profit in his exchange with either A or C (e.g., Willer, 1999), and D and E obtain most of the profit in their exchanges in the 2S3B (Fig. 1) network (Willer and Willer, 2000, Corominas-Bosch, 2004).

What the examples make clear is that a single link can be the difference between earning almost nothing, half the resource pool, and almost everything. Even if changing one's links is rather costly, an actor will attempt to do so, e.g., by adding a beneficial exchange relation to his network. For example, if A and C in the Line3 are both interested in the car B has to sell, it is in both As and Cs advantage to search for an additional seller of a car in order to avoid being exploited by B. And, if the Line4 represents a friendship network, A (D) might be tempted to look for another friend because if his friend B (C) goes out together or plays tennis with C (D), A (B) is excluded.

Benefits of additional links make network dynamics likely, especially when the benefits are large. Not only such theoretical considerations imply network change, also exchange networks (buyer–seller networks, friendship networks, or other social structures that can be considered exchange networks) outside the laboratory are observed to be dynamic. Traders do search for alternative exchange partners to improve their bargaining position in their current exchange relation. People do look for new friends if they feel they put a lot into the friendship and receive little in return. Considering the theoretical and empirical untenability of stasis, a key question becomes how profits change when links are added or deleted.

Yet the question of marginal benefits of links has received little attention in the literature (Kollock, 1994). Only a handful of theoretical studies (Leik, 1991, Leik, 1992, Willer and Willer, 2000, Bonacich, 2001, Bonacich, 2004) and not a single empirical study on dynamic exchange networks has been carried out. The neglect of dynamic exchange networks is particularly striking in light of the very extensive experimental investigations of static exchange networks (Willer and Willer, 2000, p. 252). In these experiments, static networks are exogenously determined by the experimenters. By fixing the network, one of the most powerful forms of strategy to enhance outcomes of exchange is ignored: negotiating changes in the network itself (Leik, 1992, p. 309). Therefore, a desirable step in expanding theory on network exchange is to consider an actor's potential to manipulate (i.e., to delete and add) the links themselves, thereby enhancing his bargaining power in subsequent exchanges, and thus, indirectly increasing his expected payoff.

A few studies have sought answers to the question how profits change when links are added or deleted. Leik, 1991, Leik, 1992 considers exchange networks in which a manipulator is randomly appointed who can propose link changes. Leik derives various propositions on what types of link changes will occur and how these changes will affect payoffs or power differentials in the network, i.e., exchange outcomes at the macro-level. Willer and Willer (2000) assume that actors themselves add and delete links. They evaluate Leik's propositions and add some to them that concern the effects of a change in one actor's link(s) on his own and others’ payoffs. Many of their propositions will be referred to in Section 3.

The propositions in Leik (1992) and Willer and Willer (2000) and other propositions are in the present paper systematically investigated using one framework; the effects will be assessed of adding and deleting a link on the payoff (i) of actors in the link (micro), (ii) of the neighbors of these actors (meso), and (iii) differences of all actors in the network (macro).

Our study differs in two important respects from the work of Leik (1992) and Willer and Willer (2000). Firstly, the two aforementioned studies make use of a specific theory of network exchange only to a limited extent. The propositions in those papers are too general to prove without choosing a specific theory of how actors exchange, that is, whether a proposition is true or not might depend on the theory of network exchange that is used to evaluate these propositions. Leik did not use any theory of exchange. Although Willer and Willer (2000) employed network exchange theory (NET), they did not prove all of their propositions and left some conjectures for future research (e.g., their Conjectures 1 (p. 261) and 3 (p. 268)). In the present study, we chose to use Friedkin's expected value theory as the theory for network exchange so that their propositions can be unambiguously evaluated. We motivate our selection of EVT in the next section, but will also argue later that at least some of our findings are robust across exchange theories. Secondly, in the evaluation of their propositions, Willer and Willer as well as Leik only refer to a limited number of exchange networks. In the present study all 13,597 connected and unconnected exchange networks with up to 8 actors are investigated. Examining all small networks has the advantage that one obtains an overview of how often a proposition is violated, if ever, and one might discover principles of dynamic exchange by inspecting networks in which propositions are violated.

In Section 2 the scope conditions underlying our analysis are discussed, our selection of Friedkin, 1992, Friedkin, 1993, Friedkin, 1995 expected value theory (EVT) is motivated, and EVT is discussed in detail. In Section 3 the propositions to be tested are explicated and related to propositions in Willer and Willer (2000) and Leik (1992). In Section 4, the results of our analyses are presented. We conclude with a discussion in Section 5.