EARLY ROUND UPSETS AND CHAMPIONSHIP BLOWOUTS

I. INTRODUCTION Popular discussions of tournament play often celebrate the success of underdogs in early rounds and decry the prevalence of unexciting finals. Such a pattern might be just a statistical illusion. There are many early round matches, so some upsets are inevitable, and there is only one final, so it is unlikely to be the most exciting match of the tournament. In this article we consider instead whether the pattern might have a real foundation in the strategic allocation of effort by tournament participants. We analyze a two-round tournament in which a favorite and underdog play in each of the two semifinals, and then one player from each match advances to the final. Victory in a match is probabilistic in that the chance of winning is increasing in each player's relative quality and in his relative effort expenditure. (1) Each player has a fixed amount of effort to exert over the two rounds, unused effort is of no value, and the only payoff is from winning the tournament. To maximize the chance of winning, each player must balance out the benefits of expending more effort in the semifinal against the opportunity cost of having less effort available for the final. Favorites and underdogs both have an incentive to conserve resources for the final, but the trade-offs they face are different. A favorite plays a weak opponent in the semifinal, but is likely to face a tough opponent in the final, so it has a strong incentive to hold back. Conversely, an underdog already plays a tough opponent in the semifinal, so it has less incentive to conserve effort for the final. (2) Because of these different incentives, we find that in any symmetric Nash equilibrium underdogs exert more effort than favorites do in the semifinal. The extra effort does not fully compensate for lower ability, and underdogs are still likely to lose, but the chance of an upset rises. In the final round each player expends all its remaining resources, so differences in abilities are no longer compressed by strategic considerations. Instead, an underdog who makes it to the final has fewer resources left to spend than the favorite, so differences in abilities are amplified and the chance of a blowout by the favorite rises. If a favorite loses to an underdog in the early rounds it is often accused of looking past the underdog to its next match. Even if a player's strategy is optimal ex ante, it might turn out to be unsuccessful ex post, so such criticism is often unfair. Our results imply that it is rational for the favorite to hold back on resources in the semifinal even if it correctly anticipates that in equilibrium the underdog will be playing harder. Sometimes the strategy will backfire, but on average the favorite benefits from being in a better position for the final. These results are derived using a standard contest model in which each player's probability of victory in a match is a strictly increasing function of his ability and effort. (3) Such models were first developed to analyze rent-seeking (Tullock 1967, 1980), and similar models appear in a wide variety of areas including patent races (Loury 1979), election campaigns (Snyder 1989), compensation schemes (Nalebuff and Stiglitz 1983), career ladders (Lazear and Rosen 1981), lobbying (Baye et al. 1993), and sports contests (Szymanski 2003). (4) More specifically, we model the contest as a sequential elimination ladder tournament in which players first compete in separate groups and the winners then compete against each other (Rosen 1986). Our result that underdogs do indeed try harder than favorites in the semifinals contrasts with that of Rosen (1986, P. 707-8). Because the favorite has a better chance of victory in the final, Rosen finds that winning a semifinal match is more valuable to the favorite, so the favorite tries harder than the underdog. We reach the opposite conclusion because we assume that there is a fixed supply of effort to be used across rounds and unused effort in the tournament has no value. …