Manipulation of moves in sequential contests

In two-player Tullock contests with endogenous timing of moves, the weak player moves early and the strong player follows. This order of moves is the third-best outcome for a contest organiser as it leads to a contest with lower aggregate effort compared to a contest where the players move in reverse order (first best) or simultaneously (second best). We propose that if the contest organiser-who does not know ex ante which player is the strong and which is the weak one-offers a lower price (effort cost) to the player(s) who choose(s) to exert effort early, she can achieve a greater payoff by affecting the contestants' sequence of moves. We show that there exists no price that generates the first-best outcome. However, there is a price (or a range of prices) that induces both players to move early and receive it, leading to the second-best outcome. We also discuss the case where both players move early but only one receives the low price, e.g., lobbying or Instagram "giveaway" contests.