Sequential contests with synergy and budget constraints

We study a sequential Tullock contest with two stages and two identical prizes. The players compete for one prize in each stage and each player may win either one or two prizes. The players have either decreasing or increasing marginal values for the prizes, which are commonly known, and there is a constraint on the total effort that each player can exert in both stages. We analyze the players' allocations of efforts along both stages when the budget constraints (effort constraints) are either restrictive, nonrestrictive or partially restrictive. In particular, we show that when the players are either symmetric or asymmetric and the budget constraints are restrictive, independent of the players' values for the prizes, each player allocates his effort equally along both stages of the contest.