The paper considers a model of competition among firms that produce a homogeneous good in a networked environment. A bipartite graph determines which subset of markets a firm can supply to. Firms compete a la Cournot and decide how to allocate their production output to the markets they are directly connected to. We assume that markets have inverse linear demand and firms have quadratic production costs. First, we show that the resulting Cournot game has a unique equilibrium for any given network and provide a characterization of the production quantities at equilibrium. Our results identify a close connection between the equilibrium outcome and supply paths in the underlying network structure. In particular, we show that whether two firms see their output in different markets as strategic substitutes or complements depends critically on the paths between those markets in the line graph induced by the original bipartite network. Armed with a characterization of the equilibrium supply decisions, we explore the effect of changes in the network structure on firms' profits and consumer welfare. First, we study the question of a firm entering a new market. We show that entry may not be beneficial for either the firm or the consumers as such a move affects the entire vector of production quantities. The firm might face a more aggressive competition in its original markets due to its entry to a new market. Moreover, the effect on other firms and consumers also depends on their location in the network. This is in stark contrast with standard results in Cournot oligopoly where entry implies more competition in the market and thus higher consumer welfare. Similarly, the effect of a merger between two firms on profits and overall welfare largely depends on the structure of competition in the original Cournot market. In particular, we show that insights from analyzing mergers in a single market do not carry over in a networked environment. Market concentration indices are insufficient to correctly account for the network effect of a merger and one should not restrict attention to the set of markets that the firms participating in the merger supply to. Finally, we study the operations of a cartel including the entire set of firms. We show that the cartel maximizes its profits by appropriately segmenting the markets among its members so that a firm supplies solely to the ones allocated to it, and we provide an algorithm that computes the optimal production quantities for each firm in the cartel. © 2014 Authors

The paper considers a model of competition among firms that produce a homogeneous good in a networked environment. A bipartite graph determines which subset of markets a firm can supply to. Firms competeà la Cournot and decide how to allocate their production output to the markets they are directly connected to. We assume that markets have inverse linear demand and firms have quadratic production costs. First, we show that the resulting Cournot game has a unique equilibrium for any given network and provide a characterization of the production quantities at equilibrium. Our results identify a close connection between the equilibrium outcome and supply paths in the underlying network structure. In particular, we show that whether two firms see their output in different markets as strategic substitutes or complements depends critically on the paths between those markets in the line graph induced by the original bipartite network.Armed with a characterization of the equilibrium supply decisions, we explore the effect of changes in the network structure on firms' profits and consumer welfare. First, we study the question of a firm entering a new market. We show that entry may not be beneficial for either the firm or the consumers as such a move affects the entire vector of production quantities. The firm might face a more aggressive competition in its original markets due to its entry to a new market. Moreover, the effect on other firms and consumers also depends on their location in the network. This is in stark contrast with standard results in Cournot oligopoly where entry implies more competition in the market and thus higher consumer welfare.Similarly, the effect of a merger between two firms on profits and overall welfare largely depends on the structure of competition in the original Cournot market. In particular, we show that insights from analyzing mergers in a single market do not carry over in a networked environment. Market concentration indices are insufficient to correctly account for the network effect of a merger and one should not restrict attention to the set of markets that the firms participating in the merger supply to.Finally, we study the operations of a cartel including the entire set of firms. We show that the cartel maximizes its profits by appropriately segmenting the markets among its members so that a firm supplies solely to the ones allocated to it, and we provide an algorithm that computes the optimal production quantities for each firm in the cartel.

Innovation contests have emerged as a viable alternative to the standard research and development process. They are particularly suited for settings that feature a high degree of uncertainty regarding the actual feasibility of the end goal. The objective of the contest designer is to maximize the probability of reaching the innovation goal while minimizing the time it takes to complete the project. Obviously here the important question is how to best design these contests. This paper departs from prior literature through three key modeling features. First, in our model, an agent's progress towards the goal is not a deterministic function of effort. As is typically the case in real-world settings, progress is positively correlated with effort but the mapping involves a stochastic component. Secondly and quite importantly, it is possible that the innovation in question is not attainable, either because the goal is actually infeasible or because it requires too much effort and resources that it makes little economic sense to pursue. We model such a scenario by having an underlying state of the world (whether the innovation is attainable or not) over which participants have some prior belief. Taken together, these two features imply that an agent's lack of progress may be attributed to either an undesirable underlying state (the innovation is not attainable) or simply to the fact that the agent was unlucky in how her effort was stochastically mapped to progress. Thirdly, we consider a dynamic framework that captures how competition between agents evolves over time and incorporates the fact that agents learn from each other's partial progress to discern the underlying reason for their own lack of progress. In particular, our modeling setup includes well-defined intermediate milestones that constitute partial progress towards the end goal.This setup enables us to study the complex role that information plays in an innovation tournament. Information about the progress of one of the participants has the following interesting dual role: it makes agents more optimistic about the state of the world, as the goal is more likely to be attainable and thus agents have a higher incentive to exert costly effort. We call this the encouragement effect. At the same time, such information implies that one of the participants is leading the contest, which might negatively affect effort provision from the remaining agents, as the likelihood of them beating the leader and winning the prize becomes slimmer. We refer to this as the competition effect. These two effects interact with each other in subtle ways over the duration of the contest, and understanding this interaction is of paramount importance for successful contest design.

We consider a network creation game in which each player (vertex) has a fixed budget to establish links to other players. In our model, each link has unit price and each agent tries to minimize its cost, which is either its local diameter or its total distance to other players in the (undirected) underlying graph of the created network. Two versions of the game are studied: in the MAX * A preliminary version of this paper appeared in Proceedings of the 23rd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2011), pp. 207-214 version, the cost incurred to a vertex is the maximum distance between the vertex and other vertices, and in the SUM version, the cost incurred to a vertex is the sum of distances between the vertex and other vertices. We prove that in both versions pure Nash equilibria exist, but the problem of finding the best response of a vertex is NP-hard. We take the social cost of the created network to be its diameter, and next we study the maximum possible diameter of an equilibrium graph with n vertices in various cases. When the sum of players' budgets is n − 1, the equilibrium graphs are always trees, and we prove that their maximum diameter is Θ(n) and Θ(log n) in MAX and SUM versions, respectively. When each vertex has unit budget (i.e. can establish link to just one vertex), the diameter of any equilibrium graph in either version is Θ(1). We give examples of equilibrium graphs in the MAX version, such that all vertices have positive budgets and yet the diameter is Ω( √ log n). This interesting (and perhaps counter-intuitive) result shows that increasing the budgets may increase the diameter of equilibrium graphs and hence deteriorate the network structure. Then we prove that every equilibrium graph in the SUM version has diameter 2 O( √ log n) . Finally, we show that if the budget of each player is at least k, then every equilibrium graph in the SUM version is k-connected or has diameter smaller than 4.

This paper studies multitier supply chain networks in the presence of disruption risk. Firms decide how much to source from their upstream suppliers so as to maximize their expected profits, and prices of intermediate goods are set so that markets clear. We provide an explicit characterization of (expected) equilibrium profits, which allows us to derive insights into how the network structure—that is, the number of firms in each tier, production costs, and disruption risk—affect firms’ profits. Furthermore, we establish that networks that maximize profits for firms that operate in different stages of the production process—that is, for upstream suppliers and downstream retailers—are structurally different. In particular, the latter have relatively less diversified downstream tiers and generate more variable output than the former. Finally, we consider supply chains that are formed endogenously. Specifically, we study a setting where firms decide whether to engage in production by considering their (expected) postentry profits. We argue that endogenous entry may lead to chains that are inefficient in terms of the number of firms that engage in production. This paper was accepted by Vishal Gaur, operations management.