Cooperation in partly observable networked markets☆
Introduction
In many markets, successful execution of mutually beneficial economic transactions relies on informal contracts that are enforced by social pressure and reputation.1 Informal enforcement mechanisms include personal and community enforcement mechanisms.2 It is now widely recognized that personal enforcement is highly effective when two parties interact frequently and need each other to interact at all, that is, when bilateral interaction opportunities are exclusive and frequent. Third-party observability can facilitate community enforcement to overcome the limitations of personal enforcement.3 However, despite the abundance of research on repeated games and community enforcement, the frequency and especially the exclusivity of interaction opportunities are treated mostly as “black boxes.” Similarly, the level of third-party observability is modeled for highly specialized cases.
To be specific, in much of the repeated-games and community-enforcement literature, either any two parties interact in every period or random matching is assumed.4 The common theme is, therefore, that interactions are non-competing. That is, two individuals have an opportunity to interact with each other is independent of whether and when each of them interacts with additional individuals. By contrast, we note that in many market environments, interactions are competing—a buyer may want to purchase a limited quantity of a good, and a borrower may need a loan of up to some given size, independent of how many sellers or lenders are in the market. As a result, the frequency and exclusivity of interactions in two-sided markets with buyers and sellers, or investors and entrepreneurs, are inseparable from the structure of the market—the subset of “agents” to whom each “client” has access, and with whom he chooses to interact.5
In addition, much of the emerging literature on repeated games in networks considers either public monitoring or a monitoring structure that is identical to the interaction structure (i.e., an individual observes interactions in which she participates).6 By contrast, in many markets, each client learns about the outcomes of a different subset of the interactions in the market and that the patterns of third-party observability may not be subject to the same restrictions as the interaction structure.
This paper develops a new model and new techniques for the study of repeated games in a market with clients and agents, in which interactions are competing and the monitoring structure is not tied to the interaction patterns. To this end, we model the market structure as a two-sided interaction network, and allow interactions to be competing. The observability structure is then modeled as a separate observability network.
In our model, each client can initially interact with (e.g., purchase a good from or make a loan to) only a subset of agents to whom he is connected. The initial connections between clients and agents define a two-sided interaction network . Clients can also decide to eliminate their connections with agents they do not trust. As a result, the interaction network may evolve over time. In every period, agents meet sequentially with clients who are connected to them and decide whether to cooperate or defect. Each agent (client) is able to interact with, at most, one client (agent) in a given period, and the interaction outcome between an agent a and a client c is observable to a subset of the clients (including c); such clients are said to be connected to client c. The connections between clients define an observability network R that captures the third-party observability in the market (R can be thought of as a reduced-form object that captures the diffusion of information that results from word-of-mouth (WOM), reputation systems, or any other mechanism that facilitates third-party observability). The combined network, , captures the market structure.
Analyzing the model requires overcoming several challenges. First, the competing nature of interactions implies an agent's incentives to cooperate depend on the entire network structure. If a network is large, calculating the expected payoff of any individual v is computationally challenging. Second, even if the network is not large, the challenge remains to derive comparative statics relating the network structure and the ability to sustain cooperation, because there is no obvious way to incorporate the full dimensionality of a network.7
Our main methodological contribution, which allows us to overcome the aforementioned challenges, is the introduction of two new graph-theoretic techniques to the study of repeated games in networks:
- 1.
A new method for reducing questions about the global properties of a network (e.g., characterizing payoffs that depend on the entire network) to questions about the local properties of the network. This approach allows us to provide conditions under which the incentives of an agent a to cooperate with a client c depend only on her beliefs with respect to her local neighborhood—a subnetwork that includes agent a and is of a size that is independent of the size of the entire network. Thus, we are able to analyze large networks as if they were small. To derive these “local conditions,” we make use of recent results in the graph theory literature by Gamarnik and Goldberg (2010), hereafter GG.
- 2.
A local approximation technique, which, when combined with (1) and applied to a model in which individuals have significant uncertainty regarding the network structure, allows us to prove the following result: if the network N is large and all other agents always cooperate, the incentives of agent a to cooperate in N can be approximated by the incentives of a to cooperate in a simpler network—a random tree with known distributions over the numbers of connections of clients and agents. This result is based on a key graph-theoretic lemma that we prove: consider a large bipartite graph G that is chosen uniformly at random (u.a.r.), conditional on the (finite support) distributions of the number of links attached to nodes in the graph; then, G is asymptotically locally like a random tree.8
We note that our random-tree characterization is quite general and can be extended to many settings. When applied to the study of the effect of the interaction and observability structures on cooperation, it allows us to ask (and answer) the following questions for any fixed level of agents' patience: What structures of the network N can be sustained indefinitely in equilibria in which all interactions end in cooperation? For what structures of the interaction network G does an observability network R exists such that can be sustained indefinitely and allow for an equilibrium with full cooperation? The answers define a set of networks in which a connection between client c and agent a has the interpretation that c is able and willing to interact with a and that a always cooperates with c. Our focus is therefore different from much of the existing literature, in which the equivalents of G and R are held fixed (and are often identical to each other), and in which the goal is to construct strategies that sustain cooperation when individuals are sufficiently patient.9
Section 2 presents a model of a networked market, and the notion of a Totally Cooperative strict -Nash Equilibrium with Ostracizing strategies (TCEO) is defined in Section 3. In Section 4, we derive our first main result and provide conditions under which the incentives of an agent to cooperate depend only on her local network structure. In Section 5, we propose a specific model of beliefs with respect to the network structure and derive our second main result, reducing the characterization of the structure of cooperation networks to an equivalent question in a simpler family of networks. Section 6 reviews related literature and offers a discussion of the main methodological contributions of the paper.
Section snippets
Model
Consider a market with a set of clients and a set of agents . Time is discrete, and clients and agents live forever and have a common discount factor δ. In any given period, clients and agents meet in a random order that is governed by the structure of an interaction network that connects clients and agents. When a client and an agent meet, they engage in a trust-based interaction, and then both exit the market for the remainder of that period. If the agent defects
Equilibrium
An equilibrium requires strategies that describe clients' link-elimination actions and agents' cooperation actions, for any history of play, given and , such that every individual v's strategy maximizes her expected payoff given the strategies of all individuals in . It is easy to verify an equilibrium exists. To see that, consider the following strategies: at any time , every client c deletes all of his links, and every agent a defects in all of her interactions.
Cooperation based on local beliefs
To state our results, we develop a general framework of beliefs about the network structure. At the heart of the framework is the distinction between global beliefs, which generally refer to an individual's beliefs about the potential interactions between all clients and agents in the network, which may be arbitrarily large, and local beliefs, which generally refer to her beliefs about the potential interactions between clients and agents separated from her in the network by a distance that is
The Global Fractions (GF) model
Theorem 1 paves the path to a tractable solution to the following question: For what structures of the interaction network G does an observability network R exists such that a TCEO with the network exists?
However, the residual problem still requires computational analysis. For example, if and the network is commonly known, we are still required to calculate for networks with some radius d, which, though much smaller than the entire network, may not be
The (un)importance of global beliefs
Much of the previous work on games in networks analyzes static network games (e.g., Galeotti et al., 2010, Ballester et al., 2006; and Bramoullé et al., 2014). In static network games, a player's payoff depends only on the actions taken by her immediate neighbors. As a result, a player uses beliefs on the network structure only to establish a prior over the actions her neighbors will take, and, as Galeotti et al. (2010) show, assuming a player has incomplete knowledge of the network structure
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Cited by (0)
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We are grateful to Susan Athey, Drew Fudenberg, and Al Roth, for many discussions and invaluable comments. We also benefited from the help of Ilan Lobel and Theo Weber, and from the useful comments of Nageeb Ali, James Burns, Ben Golub, Roberto Serrano, Adrien Vigier, and seminar participants at Harvard University. This paper was significantly revised during a summer that Fainmesser spent in Microsoft Research New England. The paper was previously circulated under the title: “Bilateral and Community Enforcement in a Networked Market with Simple Strategies.”