Dmitry Dagaev & Konstantin Sonin
The rules of sports are complex and involve the interaction of many self-interested agents. Sports are thus ripe for game theoretical analysis (Kovash and Levitt 2009). Indeed, this can sometimes make a real change.
In March 2013, we wrote that the rules that are commonly used in European countries for the allocation of slots in the UEFA Champions League and the Europa Leaguewhich are based on the results of the national championships (round-robin tournaments) and the national cups (knock-out tournaments)are inherently manipulatable (Dagaev and Sonin 2013). That is, there is always the possibility of a situation where a team will be strictly better off by losing a game. This is a general result. Any redistribution rule that allows the cup's runner-up to advance in the case that the cup's winner advances based on its place in a championship has the same drawback. In September 2013, UEFA announced a change in the rules, effective starting in 2015, so that the runner-up will not get a slot under any circumstances, thus removing the drawback.
Yet the problem of misaligned incentives is not restricted to national tournaments. Competition rules of the European qualification tournament for the 2014 FIFA World Cup in Brazil, which will be completed this autumn, suffer from the same problem. And the perverse incentives situation is not merely a theoretical possibility. In fact, as of now (October 4, 2013), when four-fifths of the World Cup qualification games are complete, there is still a scenario under which a team might need to achieve a draw instead of winning to go to Brazil. Admittedly, this is highly unlikely: among other things, it requires Malta, currently ranked 142th in the world rankings, to beat Denmark and the Czech Republic.
Sport tournaments as a strategic game
In any sport tournament, the rules define a strategic interaction between participants. Ideally, these rules should be structured so that a team cannot advance by losing instead of winning a game. In practice, those who design the rules might overlook adverse consequences for incentives that the rules create, as in most real-world situations the corresponding (game-theoretical) game is not easy to solve.
Consider the following set of rules that is common in European football (52 out of 53 UEFA countries have used a variation of this system up until now). Most countries hold more than one tournament that allows teams to qualify for international leagues. Typically, teams that win