Shipwrecks and Treasure Hunters

Abstract

We examine dynamic search as a game in which two rivals explore (an island) for a hidden prize of known value. In every period until its discovery, the players decide how much of the unsearched area to comb. If a player finds the prize alone he wins it and the game ends. Players have a per-period discount factor and costs proportional to the area they search. First, as a benchmark for efficiency, we solve the monopoly problem. Second, in the duopoly setting we show that if players are sufficiently impatient they can inefficiently over-search in the unique symmetric Markov perfect equilibrium (SMPE) – a result akin to the tragedy of the commons. On the other hand, if players are patient: the SMPE is unique except for possibly one point; and either over- or under-search can result. Finally, with patient players, several counter-intuitive results can arise: for example, players might be better off searching a larger island or looking for a less valuable prize.