A Game-Theoretic Model Motivated by the DARPA Network Challenge

In this paper we propose a game-theoretic model to analyze events similar to the 2009 \emph{DARPA Network Challenge}, which was organized by the Defense Advanced Research Projects Agency (DARPA) for exploring the roles that the Internet and social networks play in incentivizing wide-area collaborations. The challenge was to form a group that would be the first to find the locations of ten moored weather balloons across the United States. We consider a model in which $N$ people are located in the space with a fixed coverage volume around each person's geographical location, and these people can join together to form groups. A balloon is placed in the space and a group wins if it is the first one to report the location of the balloon. A larger team has a higher probability of finding the balloon, but the prize money is divided equally among the team members and hence there is a competing tension to keep teams as small as possible. We analyze this model under a natural set of utilities, and under the assumption that the players are \emph{risk averse}. Risk aversion is the reluctance of a person to accept a bargain with an uncertain payoff rather than another bargain with a more certain, but possibly lower, expected payoff. We are interested in analyzing the structures of the groups in Nash equilibria for our model. We show if the game is played in the one-dimensional space (line), or more generally in the $d$-dimensional Euclidean space for any $d\geq 1$, then in any Nash equilibrium there always exists a group covering a constant fraction of the total volume. In the discrete version, the players are located at the vertices of a graph and each vertex can cover itself and all its neighbors. For the class of bounded-degree regular graphs, we show in any Nash equilibrium there always exists a group covering a constant fraction of the total number of vertices.