Competition and Networks of Collaboration

Transcription

1 Competition and Networks of Collaboration Nikita Roketskiy February 2015 Abstract I develop a model of collaboration between tournament participants. In this model, agents collaborate in pairs, and an endogenous structure of collaboration is represented by a network. Agents value both their collaboration with others and higher tournament rankings. In contrast with most of the networkformation literature, I assume that the agents are forward-looking and capable of coordination. I use von Neumann-Morgenstern stable sets as a solution. First, I find stable sets of outcomes in which networks of collaboration consist of multiple groups. Members of the same group collaborate with each other at the maximum level. To gain an advantage in the tournament, the larger groups refuse to collaborate with the smaller groups; as a result, there is no collaboration between the groups. In these outcomes, absence of collaboration between groups leads to efficiency loss. Second, I provide a necessary and sufficient condition for the stability of an efficient network of collaboration in winner-takes-all tournaments. Finally, I introduce transfers into the model to allow the agents to pay for restoring missing collaboration. I show that the inefficient outcomes that feature multiple groups remain stable in the model with transfers. Keywords: networks, collaboration, farsighted agents, stable sets, tournaments JEL Codes: D85, C71 This paper is a revised version of a chapter of my Ph.D. Dissertation at NYU. I am very grateful to my advisors, Alessandro Lizzeri and Debraj Ray, for their guidance, support and encouragement. I thank (in alphabetical order) Heski Bar-Isaac, Joyee Deb, Guillaume Fréchette, Sergei Izmalkov, Philippe Jehiel, Melis Kartal, Sergei Kovbasyuk, Guy Laroque, John Leahy, Stefania Minardi, Leonardo Pejsachowicz, Michael Richter, Ariel Rubinstein, Burkhard Schipper, Karl Schlag, Ennio Stacchetti, Kevin Thom, Matan Tsur and all seminar participants at NYU, UCL, HEC, UPF, CEU, UC3M, Cambridge-INET Workshop and Vienna Joint Economics seminar for their insightful comments and suggestions. All errors are my own. Department of Economics, University College London, Gower Street, London, UK, WC1E 6BT, n.roketskiy@ucl.ac.uk, web: uctpnr1/. 1

2 1 Introduction We often observe collaboration between direct competitors. For instance, firms that compete in the market for a final product often collaborate at the R&D stage. Similarly, co-workers who compete for a promotion, usually collaborate with their rivals. Agents in these environments face a dilemma: If they collaborate, they become stronger competitors, but they also strengthen their rivals positions. Under what conditions do competitors collaborate at the efficient level? If those conditions do not hold, what are stable patterns of collaboration? Does competition suppress collaboration, and if it does, will agents use transfers to buy collaboration and restore efficiency? I address these questions with a model in which an endogenous structure of collaboration is represented by a network, i.e. I assume that a quantum of collaboration is a bilateral contract. To make my model tractable, I restrict my attention to situations in which competition can be modeled as a tournament. I make a further simplification by assuming that higher level of collaboration (i.e., having more collaboration partners) results in a better performance and, therefore, into a higher tournament ranking. In my model, a finite population of agents participates in a tournament. Each agent may enter collaboration agreements with any subset of his opponents. Each agreement is bilateral which means that it requires the consent from both partners. Once all agreements are in place, all agents are ranked according to their performance, which is increasing in the number of their collaboration partners. Agents value collaboration agreements directly and indirectly through their preferences for higher tournament ranks. If a network of collaboration is fixed, adding a missing link between any two agents preserves their relative ranking and weakly moves them both up in the aggregate ranking. I focus on a protocol-free negotiation of collaboration agreements. To model this process, I take a cooperative route: I look at all possible suggestions that agents can collectively make and test them against possible objections by other agents. This process results in stable sets of outcomes (networks of collaboration) that are immune to objections. Requiring a stable outcome to be immune to all objections is too strong, so I require only that a stable outcome is immune to objections that lead to other stable outcomes. Formally, I study von Neumann-Morgenstern stable sets of outcomes defined for a farsighted blocking relation. My findings are threefold. First, I find stable networks of collaboration that have a group structure. When tournament prizes are large enough, agents are endogenously 2

3 divided into several groups. Agents collaborate at the maximum possible level within each group, but collaboration across groups is absent. Put differently, these groups form complete components. Any complete component is strictly larger in size than a union of all complete components that are smaller in size. In particular, the largest complete component always contains a strict majority of all agents. When tournament prizes are small, the complete network of collaboration is stable. The intuition behind this result builds upon the observation that a large enough group can guarantee top tournament rankings for its members irrespective of what the rest of the agents do. Roughly speaking, large enough group has a collective maxmin strategy that yields a high payoff for its members. Indeed, members of this group can refuse to collaborate with outsiders. If a group constitutes a majority, there are more collaboration opportunities within the group than outside of it, hence the group members have a competitive advantage in the tournament. The size of each group is endogenous. It can be found by maximizing an agent s payoffs across complete components of various sizes (assuming that the agent is part of these complete components). For example, the size of the largest group maximizes a participant s payoff across all possible groups that can be formed by a strict majority. One interesting interpretation of this criterion is the following: Imagine a by-invitation-only union where all participants collaborate with each other. Start with union that is formed by the smallest strict majority (this is a minimal requirement for the union members to dominate the tournament). Since all agents are identical, the preferences of the insiders are perfectly aligned and they will invite new members only if their own payoffs increase. Such a union will stop growing as soon as it reaches the size of the largest group in my model. My second and most important finding is a necessary and sufficient condition for stability of the efficient outcome in winner-takes-all tournaments. I show that there exists a stable set that contains an efficient outcome if and only if a payoff of an agent in the complete network is weakly larger than a payoff of an agent in any complete component that constitute a strict majority; Moreover, if such a stable set exists, it is a singleton. For winner-takes-all tournaments, this condition is equivalent to the prize in the tournament being sufficiently small. In my framework, efficiency implies maximum collaboration. Intuitively, if the tournament prize is large enough, a strict subset of agents would want to trade links value for larger share of the prize. Full collaboration opens doors for this trade because agents have many ways to alter each other s payoffs. To the best of my knowledge, this result does not appear in the literature (with the notable exception of Dutta et al. (1998); however, similar observation in their paper is derived only for a three-agent example and it does not generalize). To 3

4 stress this finding further, consider a network formation game in strategic form in which players have the same payoff functions as in my model. In this game, there is always an equilibrium in weakly dominant strategies that admits the efficient and complete network of collaboration as a unique equilibrium outcome. In my model, the inefficiency arises because agents engage in collective, coordinated actions. The important driving force behind these two results is an externality caused by competition. Consider a complete network of collaboration in which all agents tie for all rankings in the tournament. Severing a link between two agents moves both of them all the way down to the bottom two positions of the ranking or, equivalently, moves the rest of the agents away from the bottom two positions. In this case, two agents who lose the link bear the opportunity cost that equals to a value of the link and the value of the top ranking positions. At the same time these agents impose a positive externality on the rest of the agents since the rankings of the latter improve. Clearly, agents cannot exploit this positive externality to their benefit unilaterally, but collectively such an exploitation may be possible. For instance, consider all agents severing a link with agent i. These agents internalize the effect of a positive externality they impose on each other. Consider the same intuition from a slightly different angle. Every missing link comes with an opportunity cost that is divided across two agents. A majority that follows the collective tactics described above manages to concentrate a large portion of opportunity costs of many missing links on a small set of outsiders. At the same time, the majority enjoys the benefit of missing links that comes in the form of unequal distribution of tournament rankings. There are many collectively profitable tactics that involve severing links in this model. These tactics resemble sabotage, a phenomenon widely studied in the literature on tournaments (see Lazear (1989), Chen (2003) and Konrad (2000)). A sabotage can be defined as a destructive activity that inhibits rivals performance. Usually a saboteur incurs a cost. Since sabotage decreases the tournament ranking of targeted agents, it may be individually profitable for a saboteur if the cost of sabotage is relatively low. In my model, sabotage is never individually profitable because the effect of collaboration is symmetric across the participants. One of the innovations of this paper is that it identifies a self-enforcing collective analog of sabotage that requires a group of agents to coordinate their sabotage activity on a common target. This allows agents to split the costs of sabotage and hence can make sabotage profitable. When I look at the efficiency of stable outcomes, I find a negative result. A natural question then is whether one can allow agents to buy missing collaboration from each other and restore efficiency. In particular, there are large gains from such a trade in 4

5 stable outcomes in which networks of collaboration feature group structure. In my most general version of the model, I allow agents to use monetary transfers to pay each other for collaboration. Transfers are modeled as part of bilateral agreements. A pair of agents jointly decide on the amount of money one agent pays to the other. I show that inefficient stable networks that I find are robust to the introduction of transfers. In these stable networks agents in larger complete components refuse to collaborate with agents in smaller complete components, which results in efficiency losses. It turns out that even if we allow to split the gains of restored links endogenously, without any restrictions, missing links are not restored. Intuitively, agents are (perfect) substitutes for each other. When negotiating a price of a missing link, they undercut each other until this price is zero. An important assumption in this part of the model is that the transfers are part of self-enforcing bilateral agreements 1 and therefore are set in a decentralized manner. If one were to introduce a benevolent central planner who can tax the agents, efficiency would be immediately restored by always enforcing an egalitarian outcome. My findings are consistent with properties of networks of collaboration that are observed in practice despite the fact that the model is highly stylized. One salient illustration that supports my theoretical results is a study of early GSM market by Bekkers et al. (2002). In their paper, Bekkers et al. (2002) examine the emergence of GSM technology in 1990s. They document that large portfolios of standard-essential patents for GSM technology were owned by several companies: Nokia, Motorola, Alcatel, Phillips, Bull, Telia and others. Five of these companies Ericsson, Nokia, Siemens, Motorola, and Alcatel signed numerous cross-licensing agreements that allowed them to use each other s patents without paying royalties. This network of cross-licensing agreements provided its participants with a market advantage over firms that were not included in it. Not surprisingly, the same five companies later dominated the market for GSM infrastructure and terminals, having a total market share of 85 percent in At the same time, three other companies, Phillips, Bull, and Telia, held roughly as many patents as Alcatel and were not able to convert them into a significant market share. Moreover, they performed worse than Ericsson and Siemens, which had considerably smaller patent portfolios and, yet, were ranked the largest and the third-largest GSM companies in My model suggests that if the stakes in the winner-takes-all competition are high enough, the efficient network of collaboration, in which agents sign all available 1 This assumption implies that the agents cannot commit to conditioning the transfers on the level of collaboration. Nevertheless, the agents are allowed to have transfers that are endogenously conditional on the level of collaboration. In particular, they can react to a termination of collaboration by reneging on transfers. 5

6 collaboration agreements, is not stable. Moreover, there are stable networks in which a group of firms that dominates the market (let us call them insiders ) do not collaborate with other, outsider firms. Despite the fact that this tactics destroys value of collaboration between insiders and outsiders, it is profitable for the insiders because it allows them to maintain their dominant position in the market. Indeed, Bekkers et al. (2002) claims that the structure of cross-licensing agreements in GSM industry in the 1990s, directed by Motorola, was instrumental in crowding out potential rivals such as Phillips. This story is not unique: For instance, smartphone patent war has similar features. More generally, my model provides several important insights on such phenomena as patent wars and other types of market competition outside of the price domain. First, bilateral agreements such as cross-licensing are a powerful instrument in shaping a landscape for future market competition. For instance, they can be used to create persistent asymmetric market outcomes in symmetric environments. Second, if stakes in the competition are high, asymmetric inefficient outcomes (e.g. an inefficient level of cross-licensing) are inevitable. Finally, the prospect of these outcomes forces firms to join exclusive alliances in which bilateral agreements play the role of a skeleton that holds alliances together. Another application of my model is related to the theory proposed by McAdams (1995). This theory suggests that racial discrimination in the U.S. is fueled by the desire to maintain the gap in social status between white majority and ethnic and racial minorities. According to McAdams (1995), if people value high social status, they may sacrifice mutually beneficial interracial interactions in order to gain higher status. Note, that in this theory, race is a marker that is irrelevant for fundamental economic characteristics of agents. However, since it is easily observable, it is convenient to use it for specifying social norms that support the difference in social status. In other countries, in which the population is racially homogeneous, other markers, such as nationality, ethnicity or religion are used for discrimination. Sometimes, the markers are almost artificial and are not derived from observable characteristics of an individual. Example of such markers are castes in India, Pakistan, Nepal and Sri Lanka. McAdams (1995) provides evidence that discrimination is often sustained through threats of exile. If a member of a discriminating majority interacts with members of a discriminated minority, he or she risks being ostracized. My paper provides a plausible mechanism for sustaining such social norms when agents are allowed to undertake collective deviations from the social norm. The rest of the paper is structured as follows: The related literature is discussed in detail in Section 2. Section 3 contains a simple three-agent example outlining the 6

7 main findings of the paper. The setup of the general model in Section 4 is followed by the results in Sections 5 and 6. Concluding remarks are presented in Section 7. Selected proofs and lemmata can be found in the Appendix. 2 Related Literature This paper contributes to the literature on network formation and its applications to R&D collaboration, discrimination and tournaments. This paper is closely related to Goyal and Joshi (2003), Goyal and Moraga- Gonzalez (2001) and Marinucci and Vergote (2011). These papers develop various models of R&D collaboration between market competitors. In these models, firms can resort to joint research to save on R&D costs. The common findings in this literature is that networks of collaboration consisting of several components are possible in equilibrium. My model produces several important results that do not appear in the literature on R&D collaboration. First, I argue that under certain conditions, efficient outcomes maybe unstable. More precisely, I provide a necessary and sufficient condition for the existence of a farsighted stable set that contains the unique efficient outcome (which is a complete network of collaboration in my model). If this condition is not satisfied, the efficient outcome cannot be stable. The previous results in the literature cannot rule out the efficient network as an equilibrium outcome. Second, in my model, the sizes of the complete components in stable networks are uniquely determined by the shape of payoff functions, whereas in Goyal and Joshi (2003) and Marinucci and Vergote (2011), local incentives of individual agents put bounds on sizes of the components. In addition, it is worth pointing out that Goyal and Joshi (2003), Goyal and Moraga-Gonzalez (2001) and Marinucci and Vergote (2011) model competition differently from the current paper (the closest being Marinucci and Vergote (2011) who model competition as a winner-takes-all tournament with stochastic outcomes). Finally, my model settles the difference in results between R&D models that use coalition- and network-formation approach. There exists an extensive literature on collaboration between firms that looks at the coalitions of firms rather than bilateral agreements between them (e.g. Bloch (1995), Bloch (1996), Yi (1998), Yi (1997), Yi and Shin (2000) and Joshi (2008)). Surveys of the literature can be found in Bloch (2002) or Ray (2007). The predictions obtained in this literature are different from the findings obtained in network-formation models discussed above. In particular the grand coalition (which is the analog of the complete network) is usually not stable because there exists a smaller coalition that prefers to reduce the amount 7

8 of collaboration in exchange for greater market power. These results are obtained under assumption that participation in a coalition is exclusive. The results I obtain are similar, but I do not use this assumption: In my paper, groups are endogenously exclusive. Therefore my model is useful in understanding the relationship between coalition-formation and network-formation models. There are several ways to model formation of networks. The vast majority of the literature employs pairwise stability notion developed by Jackson and Wolinsky (1996). A network is considered pairwise stable whenever there is no pair of agents who want to create their joint missing link and there is no agent who wants to delete his existing link. According to this definition, the desire to create or delete links is dictated by the immediate benefit from this network modification. In many applications, this definition produces reasonable predictions. However, there are models (mine being an example) in which it makes sense to wander beyond pairwise deviations and myopic agents. In my model I relax the restriction to pairwise deviations. I also assume that agents are concerned with long-run consequences of their actions. This means they understand that their opponents are strategic. This assumption, to the best of my knowledge, was first examined in Harsanyi (1974). Harsaniy s approach was developed further by Chwe (1994), Ray and Vohra (1997), Diamantoudi and Xue (2007) for coalition formation problems and by Greenberg (1990) for theory of social situations. These papers use various versions of farsighted stable sets among other solution concepts (such as the largest consistent set). In the network-formation setting, farsighted stable sets are studied in Herings et al. (2009), Grandjean et al. (2010), Grandjean et al. (2011) and Mauleon et al. (2011). The solution used in my paper is a farsighted stable set. The definition of a farsighted stable set that I use is very similar to some definitions that appear in the aforementioned literature. It differs from existing solutions in several details that are specific to the problem that I study. Formally, my framework does not fit into a definition of a pure network-formation model because the agreement between two agents in my framework covers both collaboration and transfers. Pairwise stability and farsighted stable sets give different predictions in my model. In the most interesting cases, the set of pairwise stable networks and farsighted stable sets are disjoint. The relationship between farsighted stable sets and pairwise stability is discussed further in Section 5.1. There are other cooperative solutions that utilize farsighted agents. Most of them, such as the largest consistent set in Chwe (1994) or pairwise farsightedly stable set in Herings et al. (2009) are obtained from a von Neumann-Morgenstern stable set by substituting either internal stability or external stability with properties of set-inclusion minimality or maximality to ensure existence. I use a constructive 8

9 argument to show that a farsighted stable set always exists in my settings, so I do not need to relax the stability notion to use one of the existence results from this literature. An alternative approach to the network-formation problem is to model the process as a noncooperative game in which agents sequentially create and terminate links. The seminal paper on noncooperative network formation is Aumann and Myerson (1988). In their model, there is a deterministic rule that defines active players. Agents can only create, but not terminate links, and the game ends when no pair agents wants to create a missing link. In contrast to my model, in Aumann and Myerson (1988) and other papers on dynamic network-formation games such as Dutta et al. (2005) and Konishi and Ray (2003), the equilibrium outcome naturally depends on the sequences of active agents. This paper is also related to the program proposed in Salop and Scheffman (1983). In their paper, Salop and Scheffman (1983) state that firms can capture the market by increasing the costs of production for their rivals. In another paper, Salop and Scheffman (1987) describe various strategies that firms can use to raise their competitors costs. They find that some of those strategies can be more effective than predatory pricing. The mechanism described in my model, can be used by a coalition of firms to gain control over the market. A coalition of firms does not need to engage in predatory pricing to raise the joint share of the market. Instead, it can limit an access to its intellectual property and, hence, create a competitive advantage for its members. The findins in my paper are complementary to the results in the literature on sabotage in tournaments. Lazear (1989), Chen (2003) and Konrad (2000) suggest that agents may sabotage their rivals if the cost of sabotage is low. I argue that if costs are large, agents still can sabotage their rivals, but they have to coordinate their actions to save on costs. This gives rise to a collective sabotage. I show that, when competition is over a large prize, collective sabotage is self-enforcing and often unavoidable, i.e. it takes place in every stable outcome. 3 Simple example In this section, I present a simple three-agent example that illustrates main findings. Consider three engineers, Antony, Brutus and Caesar, participating in a winnertakes-all tournament with a prize R. The objective of the tournament is to select the best design for a smartphone. Each engineer is an expert on a particular smartphone 9

10 module: Antony s specialty is touchscreens, Brutus one is batteries and Caesar s one is mobile processors and memory modules. The engineers can ask each other to design high-quality proprietary modules for their smartphones or they can source low-quality generic modules from the market. When two engineers, say Antony and Brutus, agree to collaborate, Antony can use a battery design developed by Brutus in exchange for his own touchscreen design. 2 In this case, their products will have identical proprietary touchscreens and batteries. It is convenient to represent a structure of bilateral collaboration by a network (see Figure 1) in which nodes correspond to agents and links correspond to collaborations. For simplicity, assume that the quality of a final product is strictly increasing in the number of proprietary modules and does not depend on any other characteristics. Therefore an engineer whose smartphone has the largest number of proprietary modules wins the tournament. Moreover, assume that even if an engineer does not win the tournament, he can use his prototype in the future, therefore developing a high-quality prototype is valuable: Let f(k 1) be a value of a prototype with k proprietary components. In this very stylized tournament, there is only one decision that each engineer has to make: whom to collaborate with. Consider Antony and Brutus. Collaboration between them does not change their relative positions in the tournament. Suppose Antony has a better prototype than Brutus. I assume that if they collaborate with each other, Antony s prototype will still be better than Brutus one. Moreover, collaboration contributes towards the value of both prototypes and makes them more competitive, when compared to Caesar s prototype. The same can be said about any pair of engineers. If the competitors are myopic, i.e., if they evaluate their actions based on immediate consequences they will fully collaborate and all three prototypes will be built with proprietary components. More formally, the unique pairwise stable network of collaboration is a complete one (see Figure 1a). Note that this outcome is also the unique efficient outcome since the tournament is a constant-sum game and the value of prototypes is increasing with collaboration. However, if engineers can coordinate their actions, the complete network of collaboration is not always a plausible prediction. For instance, suppose R is large and all three engineers are collaborating with each other. Any two engineers (e.g. Antony and Brutus) have a jointly profitable deviation. If they simultaneously refuse to share their modules with Caesar, the value of their prototypes drops from f(2) to f(1), but their individual chance of winning the tournament increases from 1/3 to 2 Such a collaboration is essentially a cross-licensing agreement when engineers have patent protection for their proprietary components. 10

11 1/2, since Caesar s prototype becomes strictly worse than the other two prototypes (see Figure 1b). If R/6 > f(2) f(1), such a deviation is mutually beneficial for Antony and Brutus. Antony Antony Brutus Caesar (a) Complete network Antony Brutus Caesar (b) Stable network Antony Brutus (c) A deviation Caesar Brutus Caesar (d) Another stable network Figure 1: Various networks of collaboration Naturally, one may cast doubt on the credibility of this deviation. For instance, both Brutus and Caesar prefer to restore their missing link in order to proceed from the outcome depicted in Figure 1b to the outcome depicted in Figure 1c. Note that the credibility of the latter deviation is also not obvious as both Antony and Caesar would like to seize their collaboration with Brutus and restore their missing link in order to proceed from the outcome depicted in Figure 1c to the one depicted in Figure 1d. It is easy to see that there are no outcomes in this example that are immune to all coalitional deviations. To resolve this problem, I relax a stability requirement. Suppose that stable outcomes are those that are immune only to credible coalitional deviations (i.e., to deviations towards other stable outcomes). 3 If R/6 > f(2) f(1), a set of all collaboration networks with exactly one link is stable. To show this, consider the following two arguments. First, there is no coalition of engineers who can and want to proceed from the outcome depicted in Figure 1b to the one depicted in Figure 1d. Indeed, the only engineer who wants to follow this path is Caesar and he cannot do 3 This definition implies that a set of stable outcomes must be self-enforcing. 11

12 anything to make this transition happen (he needs Antony s active participation, but Antony does not gain anything from this transition). Second, for any outcome with zero, two or three links, there is a coalition of two engineers who want to proceed to an outcome in which they collaborate only with each other. Moreover, these two engineers can always implement this transition without relying on the third one. On top of immunity to credible deviations, the three networks that have a single link possess a property that any seemingly profitable action is followed by a punishment. Consider an immediately profitable deviation from the stable network in Figure 1b to the outcome in Figure 1c. Brutus payoff increases from f(1) + R/2 to f(2) + R as a result. However, this deviation is followed by another one that leads to the stable network in Figure 1d. It turns out that Brutus original deviation triggered a chain of events and in the final outcome of this chain his payoff is f(0) < f(1) + R/2. The relaxed requirements for stability imply that any profitable deviation from a stable outcome is followed by counter-deviation back into the set of stable outcomes and at least one of the original deviators is worse off in the new stable outcome compared to the starting point. When is the efficient level of collaboration stable in this example? All three engineers share their design when competition is not too fierce compared to direct benefits from collaboration i.e. when R 6(f(2) f(1)). This condition can be rewritten as N arg max{v k }, k>n/2 where V k = f(k 1) + R/k is a payoff of an agent participating in a large fully collaborating group of size k and n is the total number of players (n = 3 for this example). Intuitively, when group is formed, its size is determined by the utility of its representative member. Additional members are added only when the current members benefit from it and existing members are excluded if the remaining ones benefit from it. If any member of an existing group deviates from this norm others threaten the deviant with exclusion from the group. Note that when someone deviates from this norm, other members of the collaborating majority suffer from the decrease in their payoff. This provides an incentive for commencing a punishment. There is a connection between this criterion for efficiency and union mentality. This connection appears in the literature on clubs and coalition formation. In this setup, however, the notion of a group is not defined exogenously and the set of feasible outcomes is much richer than a collection of partitions of N. It is interesting that competitive forces of the tournament give rise to endogenous structures of collaboration that resemble unions. Most of the findings presented in this section do not depend on the simplifying 12

13 assumptions about three players and winner-takes-all tournament. In the next section, I present a much richer model followed by formal results that generalize the observations discussed here. 4 Model Let N = {1,..., n} be a set of identical agents competing in a tournament. The agents may engage in nonexclusive bilateral collaboration. The outcome of the tournament depends on the number of collaboration partners for each agent. I assume that bilateral collaboration requires the consent of both partners. For instance, if collaboration between two agents exists and one of them decides to quit, the collaboration is terminated immediately. I follow a cooperative approach i.e. I define a set of outcomes, agents preferences and a binary blocking relation on this set. Using these components, I study stable outcomes in the sense of von Neumann and Morgenstern (see von Neumann and Morgenstern (1944)). An outcome in this game is a pair (F, T ) where F {0, 1} N N is an adjacency matrix that describes a network of collaboration and T R N N + is a matrix that describes a system of transfers between agents. For any outcome (F, T ), matrix F is assumed to be symmetric i.e. it is assumed that collaboration is mutual. 4 F i,j = 1 means that agents i and j are collaborating with each other. The following notation will be useful: for M N, I(M) {0, 1} N N is an adjacency matrix such that for all i j : [I(M)] i,j = 1 if {i, j} M and [I(M)] i,j = 0 otherwise. In particular, matrix I( ) describes the empty network and I(N) describes the complete one. For two matrices X and Y, by X Y denote their Hadamard product: i, j : [X Y ] i,j = X i,j Y i,j. Matrix T describes the transfers between agents. I assume that T i,j 0 is the amount agent i pays to agent j in the outcome (F, T ). By (F, 0) I denote an outcome with zero transfers. Finally, by U denote a set of all feasible outcomes. The result of the tournament depends on the amount of collaboration between agents. In particular, given an outcome (F, T ), players are ranked according to their degree (number of links) in the network F in descending order. Ties are resolved randomly using uniform distribution. Let R : N R be a tournament prize schedule, i.e. R(k) is the prize for an agent ranked k-th in the tournament. I assume that R 4 Alternatively, I could assume that F is not restricted and interpret F i,j = 1 as help offered by agent i to agent j. The findings of the paper remain the same and, in stable outcomes that I find, symmetry of F arises endogenously. 13

14 is decreasing and convex (the latter means that R(k) R(k + 1) is decreasing in k). For any i, j : 1 i j n let r(i, j) = 1 j i + 1 j R(k) be an expected prize for an agent who is randomly placed between rankings i and j in the tournament (by construction, this agent ties with j i other agents). Agents payoffs are additive in a tournament prize, a value of collaboration and transfers. Let f : N R + be a strictly increasing function. Then, the payoff of agent i in outcome (F, T ) is ( n ) n n U i (F, T ) = r (p i (F ), q i (F )) + f F i,j + T j,i T i,j, where p i and q i denote the lower and the upper bound on possible rankings for agent i in the tournament. These bounds are defined as follows: { } n n p i (F ) = k N : F i,j < F k,j + 1 and j=1 { q i (F ) = n k N : j=1 k=i j=1 n F i,j > j=1 j=1 } n F k,j. By U M (F, T ), I denote a vector of utilities for the set of agents M in outcome (F, T ). Also, for two vectors U M, V M I say that U M V M if i M : U i > V i. In this specification, f stands for the value of collaboration net its effect on a tournament outcome. In the vast majority of the literature (see Goyal and Joshi (2003), Goyal and Moraga-Gonzalez (2001),Marinucci and Vergote (2011) and others), collaboration (links) is assumed to be costly. I assume the opposite (since f is increasing, each link in F comes with a benefit) for two reasons: first, this assumption relates better to applications that I discuss in the Introduction, and second, it makes forces I study more salient. If I assume that collaboration is costly, the results change very little. Networks that consist of several components remain stable with only difference that these components are not complete. In addition to that the sizes of the components may change. In the current specification of the model, I assume that the agents derive a value only from their own links. The results are similar if I assume that indirect connections 14 j=1 j=1

15 are also valuable for the agents. One way to do it is to assume that the value of a link is increasing in the degree of an agent on the opposite side of this link. Since agents utilities are linear in transfers and f is strictly increasing, the set of efficient outcomes consists of all outcomes in which all agents collaborate at the maximum extent. Proposition 1. An outcome (F, T ) is efficient if and only if F = I(N) i.e. if a corresponding network of collaboration is complete. Proof. Start with the observation that the Pareto frontier is a straight line with a slope of forty five degrees. Therefore, one can use utilitarian welfare criterion. Consider an outcome (F, T ). Observe that n n T i,j = 0. The social welfare in this outcome is W = n U i (F, T ) = i=1 i=1 j=1 ( n n ) f F i,j + i=1 j=1 n R(i). The social welfare is strictly increasing in F and does not depend on T. The main result of this paper studies the stability of efficient outcomes in this model. As shown in Proposition 1, besides efficiency, these outcomes have another potentially desirable property completeness of the network of collaboration. For instance, if this model is used to study cross-licensing agreements between firms, the efficiency criterion used in Proposition 1 does not include consumer welfare. If prices are fixed, it is natural to assume that consumer welfare is increasing in the quality of the product, which, in turn, increases in the pool of available patents. Therefore, modulo the effect on prices, the consumers most preferable outcome is the complete network of collaboration. In other applications, for example in promotion tournaments, the complete network of collaboration is also desirable. 4.1 Network formation and stability When modelling network formation, I follow the the usual practice in cooperative games. I define a notion of stability using a binary blocking relation on the set of feasible outcomes. To understand the idea behind the blocking relation consider a groups (or a coalition) of players carrying out a transition from one outcome to another. Once, the transition takes place, farsighted agents expect further transitions. Eventually, as the result of a sequence of such transitions, the agents arrive at the terminal outcome from which no further transitions are attempted. A necessary condition for 15 i=1

16 rational agents to engage in such a sequence of transitions is that ultimately, in the terminal outcome, they are better off. I implicitly assume that the agents do not derive the utility from transitory outcomes along a transition. More precisely, if there are two different transitions between outcomes (F, T ) and (F, T ), agents do not distinguish between these two transitions because the final destination is the same. One way to justify this assumption is to interpret the transitions as proposals and counterproposals (or objections) that agents make to each other without engaging in actual modification of physical outcomes. These proposals are meant to convince everyone to proceed to a stable outcome right away. The following definition formalizes the idea of a feasible transition i.e. what each coalition can do in terms of shaping outcomes. Note that the feasibility of a transition does not depend on agents preferences. Definition 1. A coalition M can enforce a transition from outcome (F, T ) to outcome (F, T ), i.e. (F, T ) M (F, T ) if for all i, j N: (i) F i,j > F i,j or T i,j > T i,j implies i, j M, (ii) F i,j < F i,j or T i,j < T i,j implies i M or j M. In this definition, it is postulated that any collaboration or transfer between two players requires the consents of both participants. Notice that a reduction in the amount of money transferred can be done unilaterally, either by refusing to pay (on the side of the sender) or by refusing to accept (on the side of the receiver). The next definition introduces a blocking relation that formalizes, among other things, the assumption that agents are rational and farsighted. Definition 2. An outcome (F, T ) U setwise farsightedly blocks (F, T ) U or (F, T ) (F, T ) if there exists a finite sequence {(S k, F k, T k )} K k=1, k = 1,..., K : S k (F k, T k ) U such, that N and (i) (F, T ) = (F 1, T 1 ) S 1 (F 2, T 2 ) S 2... S K (F, T ) (ii) U Sk (F, T ) U Sk ((F k, T k )) for all k K. The blocking relation makes little sense on its own, and its adequacy should not be judged in the absence of stability concept. It is implicitly assumed that all agents who participate in a sequence of transitions from (F, T ) to (F, T ) believe 16

17 that the latter outcome is final or, in other words, stable. When this definition is used to check for stability, a blocking outcome is always stable and a blocked one is arbitrary. A stability notion that I use in conjunction with this blocking relation is von Neumann-Morgenstern stable set defined for an abstract problem (U, ). 5 Definition 3. A set of outcomes R U is farsighted stable 6, if it satisfies internal and external stability conditions: (IS) for any (F, T ), (F, T ) R : (F, T ) (F, T ); (ES) for any (F, T ) R there exist (F, T ) R : (F, T ) (F, T ). A stable set of outcomes is a collection of all outcomes that are unblocked by elements of this stable set. Let Y : 2 U 2 U be a function that for a set of outcomes X returns a set Y (X ) of all outcomes that are unblocked by any outcome in X : Y (X ) = {(F, T ) U : (F, T ) (F, T ), (F, T ) X }. Then, R is farsighted stable if and only if R = Y (R). Farsighted stability is a set-valued solution. An element of a stable set is not considered stable in isolation (unless, the stable set is a singleton). A stability of a single element hinges upon the stability of all other elements in the stable set. This means, for instance, that there can be more than one stable set. Moreover, the same outcome can belong to one stable set and lie outside of some other one. This logical construction is further reinforced by the consistency property of a stable set. Consistency of a set of outcomes means that any profitable deviation from an outcome in this set is followed by a path back into a set; Moreover, a path back is such that one of the original deviators is punished. Chwe (1994) shows that a farsighted stable set possesses the consistency property. In the original proposition, Chwe (1994) formulates this property for one-step deviations, but it can be easily extended to sequential deviations. Proposition 2 (Chwe, 1994). Let R be a farsighted stable set. Take (F, T ) R and any (F, T ) (F, T ). For any sequence underlying this blocking {(S k, F k, T k )} K k=1 and for any outcome (F, T ) R such that (F, T ) (F, T ), there exists an agent i K S k such that k=1 U i (F, T ) U i (F, T ). 5 One can also define an abstract core for (U, ). However, in my model, for the most interesting values of parameters, this abstract core is empty. 6 There is little agreement in naming various stability concepts in the recent literature on cooperative games. This name is chosen following Ray and Vohra (2014). 17

18 Proof. By contradiction, suppose that for all i K S k k=1 U i (F, T ) < U i (F, T ). Note that (F, T ) = (F 1, T 1 ) S 1 (F 2, T 2 ) S 2... S K (F, T ) implies that (F, T ) S (F, T ) where S = K S k. Then, (F, T ) (F, T ), which contradicts the internal stability for R. k=1 Intuitively, this property of farsighted stable sets means that there is a punishment for any profitable deviation from a stable outcome. Any deviation from a stable outcome ultimately results in a transition back to another outcome in a stable set and there is always at least one player among the original deviators, who is worse off. 5 Stable sets in the absence of transfers In this section, I restrict my attention to a model without transfers. I assume that U = { (F, T ) {0, 1} N N R N N + : T i,j = 0, i, j } There are two main results for the model without transfers. The first result (Proposition 3) presents a stable set of networks that have group structure and feature structural holes. The second result (Theorem 1) provides necessary and sufficient conditions for stability of an efficient outcome in winner-takes-all tournaments. The findings for the general model with transfers are presented in the next section. In contrast with a common intuition about transfers restoring efficiency, the ability to use transfers has no such effect when agents are farsighted. In some cases, transfers can even be harmful. I find outcomes that feature minimal welfare loss subject to unequal ex ante distribution of tournament prizes across agents. Some of these outcomes are stable in the model without transfers and not stable in the general model. The following definition proves to be very useful in solving the model. Definition 4. Consider a sequence {m k } K k=1. Let M 0 = 0, and for k 1, let M k = k m i. The sequence {m k } K k=1 is group-optimal if M K = n and for all k 1 i=1 m k arg max n M k 1 2 <m n M k 1 {r(1 + M k 1, m + M k 1 ) + f(m 1)}. 18

19 For clarity of exposition, I assume that group-optimal sequence is unique. All the results easily generalize to multiple group optimal sequences by taking a union across these sequences. Given a group-optimal sequence {m k } K k=1, let V k = r (1 + M k 1, M k ) + f (m k 1). The definition of m 1 and V 1 considers a set of all collaboration networks in which a majority group of size m forms a complete component. The size of the majority group m that maximizes the payoff of a single member across this set is m 1 and the maximum payoff is V 1. The criterion for m k is identical to the criterion for m 1 formulated with respect to a residual problem in which the sizes and a structure of all larger groups are fixed. Definition 5. A collaboration network F has a group structure induced by a sequence {m k } K k=1 if there exists a partition N = {N 1,..., N K } of the set N such that (i) k : N k = m k, (ii) F = K I(N i ). i=1 An example of a network that satisfies Definition 5 is given in Figure 2. This network is induced by a sequence {5, 3, 1}. There are three complete components of size 5, 3 and 1 in this example Figure 2: A group-structured network induced by a sequence {5, 3, 1}. The following proposition finds farsighted stable sets that consist of networks that have group structure. Proposition 3. A set of all outcomes (F, 0) U in which F has a group structure induced by a group optimal sequence is farsighted stable. 19

20 Proof. Let R be a set of all outcomes (F, 0) U in which F has a group structure induced by a group-optimal sequence. I show that the set R satisfies internal and external stability. I show that for any (F, 0), (G, 0) R, (F, 0) (G, 0). Let H = {H 1,...} be a partition that induces a network of collaboration F and G = {G 1,...} be a partition that induces G. Also, let K = {i N : U i (F, 0) > U i (G, 0)}. Denote an index of a largest set infiltrated by agents from K in F by k, i.e. for all j < k : K H j = and K H k. Let M = G j, and note that M > n/2. For any S N \ M and for any j k (F, 0) : (G, 0) S (F, 0), we have U M (G, 0) = U M (F, 0). Hence, if (F, 0) (G, 0), it must be that U M (G, 0) = U M (F, 0), which contradicts K M. I show external stability by construction. For every state (F, 0) R I find (G, 0) R : (G, 0) (F, 0). Consider (F, 0) R, such that F does not contain a complete component of size m 1. By Lemma 1, a set A(F, 0) = {i : U i (F, 0) < V 1 } is nonempty. Moreover, if A(F, 0) < n/2, one can apply Lemma 2 repeatedly to obtain a sequence of outcomes {(F i, 0)} such that the last element of the sequence, (F L, 0), satisfies m 1 A(F L, 0) n/2. Consider an outcome (F L+1, 0) such that F L+1 = I(A(F L, 0)) + F L I(N \ A(F L, 0)). Clearly N = A(F L+1, 0). Take a set N 1 : A(F L, 0) N 1 and N 1 = m 1 and consider an outcome (F L+2, 0) such that F L+2 = I(N 1 ) + F L+1 I(N \ N 1 ). This procedure obtains a sequence of outcomes (F i, 0) and a sequence of coalitions S i that satisfy the following properties for any i {1,..., L + 2}: 1. S i N 1 2. (F i, 0) S i (F i+1, 0) 3. j S i : U j (F i, 0) < U j (F L+2, 0) = V 1. In this part of the sequence, the largest group, i.e. the group of size m 1 formed a complete component. The rest of the sequence is constructed by induction: suppose there exists a sequnce along which the largest k groups form complete components. I use the argument above to construct part of the sequence along which k +1th largest group forms a component. There exists N k+1 N \ N j : N k+1 = m k+1 such that this part of the sequence, enumerated by i {I k + 1,..., I k+1 }, satisfies the following three conditions for any i {I k + 1,..., I k+1 }: 20 j k