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Article Excerpt
1. Introduction
One of the most active domains of research in management today concerns the ways in which ongoing relations influence the patterns of exchange and the distributions of resources in and among organizations. (1) Although any attempt to summarize this literature would require its own paper-length treatment, recent studies have, for example, found that prior relations influence the recruitment, retention, and utilization of employees (Fernandez et al. 2000), the terms of exchange between buyers and suppliers (Kollock 1994), and the patterns of trade between nations (Ingram et al. 2005). Indeed, this large and growing literature has usefully documented a plethora of cases in which the patterns of existing relations appear to favor some firms at the expense of others.
Despite the substantial progress that has been made, many have criticized this literature as being overly static (e.g., Salancik 1995): Although certain positions appear valuable, research has remained relatively silent on how actors might come to occupy them. As a consequence, many questions remain open: Do patterns of relations themselves confer benefits, or do they merely reflect underlying actor-level heterogeneity in endowments? Perhaps only those with valuable resources come to occupy positions of intermediation; the apparent benefit of being a broker might then stem from those resources rather than the position. To the extent that actors do profit from their positions, should these returns prove long-lasting or fleeting? To answer these and other questions, research must adopt a more dynamic perspective on how relational networks affect exchange. (2)
Although any progress on this front will require both empirical investigations and theoretical elaboration, here, we focus on the latter by building an analytical model of how firms form relations and then use those relations to create and capture value. Our analysis is premised on the notion that firms attempt to maximize their expected profits both when forming connections to other parties and when cooperating and competing with them. (3) Although we recognize that this purely instrumental view of how firms form and manage their relationships almost certainly overstates the actual extent of profit-motivated behavior--at least some interfirm relations undoubtedly emerge from, or come to embody, more social interactions--it nonetheless provides a useful baseline for understanding the assumptions required for a theory of competitive advantage based on a particular position in a relational network.
Our analytical approach could address almost any type of "positional" advantage, but space constraints prevent us from considering them all here. We therefore focus on understanding one of the positional advantages most commonly considered in the literature: the "broker"--one that profits by intermediating between two or more parties (Simmel 1950, Burt 1992; see Burt 2000, for a review of the empirical literature). Individuals, for example, in positions of intermediation have been found to have greater influence (Padgett and Ansell 1993, Fernandez and Gould 1993, Burt 2004) and to receive larger bonuses and faster promotions (Burt 1992, Podolny and Baron 1997) than the average employee. Similarly, firms in such positions earn larger margins (Burt 1983, Talmud 1994) and appear able to protect those margins (Fernandez-Mateo 2006). In addition to being one of the most widely studied types of positions, it is also one particularly well suited to analysis by game theory because competitive scarcity--an issue that coalitional game theory addresses quite precisely--sits at the heart of both the control and information benefits posited as the sources of the broker's advantage.
We use the "biform game" (a two-stage model) as methodological scaffolding. (4) Our paper begins by describing the model. We then turn to the analysis of the second stage of the model, considering the conditions necessary for a broker to enjoy a competitive advantage. Our model allows us to distinguish the effects of position from those of actor-level endowments. Finally, we build in the first stage, allowing actors to form relationships endogenously (on the basis of the incentives implied by the second stage), to determine whether the broker's competitive advantage remains stable in a world of rational relational investments.
Although our formal model examines stylized versions of the exchange relations one typically observes, we nonetheless believe that the insights it generates point to the value of such an approach. In particular, we find that several conditions must hold for brokers to profit from their positions: (1) The other actors involved must not have equally attractive alternatives for creating value that do not include the broker. (2) The broker's position must allow it to mediate between at least three parties. (3) The broker must enjoy sufficiently attractive alternatives so that its threat not to engage in any particular coalition remains credible. If these conditions hold, brokers can exploit their ability to intermediate in agreements among rational actors. If, however, actors build their networks with full understanding of how these relations could affect the creation and distribution of value, brokers cannot expect to profit from their positions under most conditions. Some actor always has an incentive to "close" any "structural hole." After we introduce our modeling framework, we discuss the intuitions behind and the implications of our propositions in [section]4 through [section]6.
2. Analytical Framework
Our model builds on the intellectual infrastructure of the biform game, introduced by Brandenburger and Stuart (2007). Biform games synthesize the two branches of game theory--noncooperative and coalitional--in a manner that retains the strengths of each. (5)
Although a recent development, this approach has enormous potential as a formal methodology for understanding the foundational issues of business policy and strategy because firm performance typically depends on both noncooperative (strategic) and coalitional (competitive) interactions. On the one hand, performance depends on the firm's ability to capture value. Firms, suppliers, and customers compete with one another in markets to both produce and appropriate value. Coalitional game theory maps well onto these situations and provides tools for determining which transactions should occur, how much value the system should create, and who should receive what share as a result. On the other hand, a firm anticipating its ability to compete can initiate a variety of moves--such as entering a new market, introducing a new product, or changing pricing policy--designed to alter the competitive landscape to its advantage. Strategic behavior of this sort fits better with the assumptions of noncooperative game theory. By splitting the analysis into two stages, one corresponding to the strategic phase and the other to the competitive phase, the biform game incorporates both types of interactions. In the first stage, actors engage in strategic actions (modeled using noncooperative game theory) with the express purpose of establishing themselves as effective competitors in the second, competitive stage (modeled using coalitional game theory).
In our case, the initial stage involves the creation of relations as a result of strategic machinations on the part of the actors, while the later stage allows those actors to compete over participation in value-creating projects--via the relationships formed in the first stage--in return for shares of the value created. Because our actors' first-stage, relationship-building activities depend on the anticipated value of those relationships, we begin with an analysis of the second stage. In other words, treating the network as a given, we first identify the conditions necessary for brokers to capture value in the market. In [section]5, we then introduce the first stage and analyze the stability of these positions in the presence of strategic relationship formation.
3. Competitive Stage
In the second stage, coalitional game, actors liaise and bargain with one another to create and capture value. As with any coalitional game, the "inputs" consist of the set of actors and an enumeration of the various ways in which they might interact to produce value. In our case, the latter stem from the combination of the network--formed in the first stage but considered fixed in the second stage--with a set of value-generating "projects" that require the participation of sets of connected actors. For those less familiar with coalitional games, we begin by reviewing some of their general features in [section]3.1 and then, in [section]3.2, explain how these features emerge from our setup.
3.1. Coalitional Game Preliminaries
A set of actors, N, combined with, for each group G [??] N, a number, [v.sub.G], define a coalitional game. (6) Each [v.sub.G], or value available to the group, represents the total economic value the actors in G can produce by transacting only among themselves--in other words, by ignoring any and all opportunities outside the group. In most applications, analysis assumes these values. In some cases, such as ours, however, where the details of value creation play a central role, the [v.sub.G] arise from other, more basic, assumptions. Game theorists refer to the list of all the values available, v [equivalent to] ([v.sub.G])[.sub.G[??]N], as a game's characteristic function; together with the set of agents, N, this characteristic function completely describes a coalitional game.
In these games, one can quantify any actor's overall contribution to the production of value. Specifically, an actor's added value is the difference between the value available to the group that includes all actors and the value produced when that particular actor does not participate (i.e., a[v.sub.i] [equivalent to] [v.sub.N] - [v.sub.N.sub.-i], where [N.sub.-i] denotes the set of all agents except for i).
The output of a coalitional game includes the aggregate value produced ([v.sub.N]) and the various ways that the actors involved might split that value. A distribution of value, [pi] = ([[pi].sub.1],..., [[pi].sub.n]), is a vector of real numbers in which [[pi].sub.i] indicates the amount of value received by actor i in exchange for its participation in value production. Which distributions one considers possible outcomes depends on the solution concept applied--the core, the nucleolus, Shapley value, kernal, stable set, Myerson value, etc. (7)
We adopt the preferred approach in strategy applications and focus on the core. We consider a distribution of value a competitive outcome if it satisfies: (i) [[summation].sub.i[member of]N][[pi].sub.i] [less than or equal to] [v.sub.N], and (ii) for all G [??] N, [[summation].sub.i[member of]G][[pi].sub.i] [greater than or equal to] [v.sub.G]. The first condition imposes a budget constraint; actors cannot split among themselves more value than they produce. The second condition, meanwhile, ensures that every actor, and every potential group of actors, receives sufficient rewards to prevent them from defecting from the generation of [v.sub.N] to produce value on their own. The core (C), then, is the set of all competitive outcomes. Note that, in some cases, the aggregate value, [v.sub.N], cannot support any competitive distributions (in which case, C = [empty set]). (8)
When a nonempty core exists, each actor faces a range of competitive outcomes [[[pi].sub.i.sup.min], [[pi].sub.i.sup.max]], where [[pi].sub.i.sup.min] = [[pi].sub.i.sup.max] and even [[pi].sub.i.sup.min] = [[pi].sub.i.sup.max] = are possibilities. At least one competitive distribution results in actor i receiving [[pi].sub.i.sup.min], at least one results in actor i receiving [[pi].sub.i.sup.max], and other distributions deliver all values between these bounds. The core offers an attractive solution concept for strategy applications because it allows one to distinguish the effects of competition from those due to extra-competitive forces (e.g., norms of fairness and reciprocity, institutional arrangements, and bargaining acumen); competition determines the bounds of the range, while extra-competitive factors establish the specific payoff received within this range.
3.2. Our Setup
We build our second stage, coalitional game, from two elements: the set of relationships formed in the first stage and a set of value-producing projects (an exogenous feature of the second stage). The value that a group of actors can create depends on the projects available to it, and whether or not the relationships linking them permit the completion of those projects. Our description begins by elaborating the details of value creation. We then turn to an explication of our assumptions regarding value appropriation. Our description uses standard graph-theoretic terminology; for those unfamiliar with this vocabulary, please see Table 1.
We define the network, inherited from the first stage, as an undirected graph (N, R) composed of a set of actors, indexed by N [equivalent to] {1,..., n}, and a set of dyadic relationships, R. Typical element (i, j) signals that actors i and j have a relationship. (9) The set of actors can include a wide range of economic agents whose interaction generates economic value: producers, customers, employees, etc. On the relationship side, meanwhile, our imagery is more of an acquaintance than a friend--an actor with whom one might interact regularly and therefore have private information on or access to, but not one to whom one would attach emotional feelings.
Let P represent a set of available projects, with typical project p. For each project, we assume that its completion requires the involvement of a set of participant contributors, [X.sub.p] [??] N, and that upon completion, the project delivers net economic value in the amount of [u.sub.p], a scalar (so, p = ([X.sub.p], [u.sub.p])). To allow for the possibility that actors do nothing, we also include a single "null" project, [p.sub.[empty set]] [equivalent to] ([empty set], 0).
Although a simple formulation, our notion of a "project" accommodates a wide variety of value-creating activities. For example, it might represent a project in the common sense of the word, such as bringing a film to market. It that case, [X.sub.p] would reflect the specific combinations of inputs required to do so, not only capabilities and resources across the value chain--production, distribution, and exhibition--but also potentially a variety of competencies at each level--production, for instance, requires producers, writers, directors, actors, cameramen, grips, editors, etc. Alternatively, projects might also represent the economic opportunities available to partnerships, buyer-supplier relationships, or other forms of bilateral or multilateral exchange. (10)
Note that the net values, [u.sub.p], implicitly...
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