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.

We introduce a network creation game in which each player (vertex) has a fixed budget to establish links to other players. In this model, each link has a unit price, and each agent tries to minimize its cost, which is either its eccentricity or its total distance to other players in the underlying (undirected) graph of the created network. Two versions of the game are studied: In the MAX 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 a unit budget (i.e., can establish a 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 counterintuitive) 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 a diameter smaller than 4.

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.

We consider a network creation game in which, each player (vertex) has a limited budget to establish links to other players. In our model, each link has a unit cost and each agent tries to minimize its cost which is its local diameter or its total distance to other players in the (undirected) underlying graph of the created network. Two variants of the game are studied: in the MAX version, the cost incurred to a vertex is the maximum distance between that vertex and other vertices, and in the SUM version, the cost incurred to a vertex is the sum of distances between that 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.