Round-robin tournaments with limited resources

We propose a theoretical model of a round-robin tournament with limited resources motivated by the fact that in real-world sport round-robin tournaments, participating teams are sometimes forced to distribute their effort over multiple matches. We assume that participating teams have a limited amount of effort that must be distributed between all matches of the tournament. We model the outcome of each match as a first-price all-pay auction. The prizes are awarded according to the ranking at the end of the tournament, based on the number of wins. The tournament with three teams is solved. In the case of identical budgets and low first prize, we find two types of equilibria—‘effort-saving’ and ‘burning-out’—both leading to unequal payoffs; when the first prize is large, we find multiple families of equilibria with both equal and non-equal payoffs. In the case of non-identical budgets, we show that there exist equilibria where a team with a lower budget gets a higher payoff than a team with a larger budget.