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 League – which are based on the results of the national championships (round-robin tournaments) and the national cups (knockout 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 (4 October, 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 the top places (1-4, depending on the country's ranking) in the national championship, a round-robin tournament, qualify for the Champions League, the most important and profitable club tournament, while the next tier qualifies for the Europa League, the second tournament. The winner of the national cup, a knockout tournament, qualifies for the Europa League. If the winner of the national cup qualifies for the Champions League, then the runner-up in the cup enters the Europa League. We argue that the described rule creates a possibility that, in certain circumstances, a team might benefit from deliberately losing a game in the championship. Furthermore, we show that a whole class of such redistribution rules is inherently flawed.

The intuition behind the misalignment of incentives is straightforward. A strategic loss by one team might help another team, that otherwise enters to the Europa League as the cup winner, to advance to the Champions League, giving the cup runner-up a place in the Europa League. The simplest of such situations is when the cup runner-up might prefer to lose to the cup winner in the national championship to help the latter to advance to the Champions League and free a place in the Europa League for itself.

In economics, the problem of the aggregation of results in sports tournaments is connected to the classic problem of the aggregation of voter preferences. Kenneth Arrow formulated in his seminal work several highly desired properties of aggregation rules of voter preferences (Arrow, 1963). Ariel Rubinstein then used a similar approach for the problem of ranking participants in a round-robin tournament (Rubinstein 1980; see also Gibbard 1973; Satherwhite 1975; Duggan and Schwartz 2000).

A similar question arises in connection with aggregation of tournaments results: under the given ranking rule, is there a team that has a positive incentive to lose a game deliberately due to strategic issues? Of course, if only one tournament is being played, then under every reasonable ranking rule a team can't be better off by losing instead of winning.

# The logic of misaligned incentives

Our theoretical analysis demonstrates that incentive incompatibility necessarily arises under any reasonable ranking method if it involves the possible redistribution of slots from a round-robin tournament. The following simple example illustrates the basic logic of the argument.

Let there be two round-robin tournaments, ‘Tournament 1’ and ‘Tournament 2’, and four teams, A, B, C and D, participating in both of them. Two teams are going to advance (e.g. qualify for the Champions League). These two slots are granted to the top teams of Tournament 1 and Tournament 2. However, it might happen that both tournaments are won by one team. In this case, the following rule is in place: the vacant slot goes to the 2nd team in Tournament 1.

Now let us construct the situation when team B is better off losing the game versus team A under any circumstances.

**Table 1**. Team B is better off by losing a game to team A in Tournament 2

Under any reasonable ranking methods, team A will be ranked first and team B second in Tournament 1. In Tournament 2, teams A and C compete for the first position. If team B loses to A in the last match of Tournament 2, then team A wins both tournaments, and B advances as the second team of Tournament 1. If, instead, team B wins over A, team C wins Tournament 2 instead of A and advances. Thus, team B is better off losing the game against A.

This idea can be easily expanded to the general case with more than three teams, more international competitions to advance to and arbitrary ‘reasonable’ redistribution rules.

If vacant slots are always redistributed in favour of a championship, there are no incentives to lose a game in the championship due to the monotonicity of the ranking rule in the championship and the impossibility of awarding any extra places to the cup. Thus, there is an important practical implication: if one wants to avoid deliberate losses, define the redistribution rule in such a way that all vacant places are awarded to the teams from the round-robin tournament.

# An unlikely, yet possible situation: European qualification for the 2014 World Cup

The most recent example deals with the qualification tournament in the UEFA zone for the 2014 FIFA World Cup. There are 53 teams competing for 12 European places at the World Cup. These teams are split into eight groups consisting of six teams, and one group consisting of five teams. Teams from one group play each other twice on a home-away basis. Points are awarded as always: three points for a win, one point for a draw, 0 points for a loss. Each team finishing first in its group automatically qualifies for the World Cup. The worst of nine second-placed teams is out. The other eight second-placed teams are split into pairs and the winner from each pair also qualifies for the final tournament. The problem might arise because of the rule that determines the best eight second-placed teams. According to the rules, for each second-placed team the number of points gained versus the 1st, 3rd, 4th and 5th teams is calculated, and all second-placed teams are ranked with respect to this number.

The current situation (4 October, 2013) can be found here: http://en.wikipedia.org/wiki/2014_FIFA_World_Cup_qualification. Let us focus on Austria, playing in Group C, where the position is as follows:

Germany is first with 22 points, Sweden second with 17, Austria has 14, Ireland 11, Kazakhstan 4, and Faroe Islands 0 points. Given that Austria has already beaten Faroe Islands 6-0 and has lost two points playing against Kazakhstan (4-0, 0-0), Austria will benefit if their result against Faroe Islands is taken into account, while the one with the Kazakh team is not. A draw, instead of an expected easy win, against Faroe Islands might accomplish just this.

While the circumstances that might give Austria such a chance are highly improbable, they are not impossible. Indeed, suppose that Faroe Islands defeat Kazakhstan 10-0, Austria defeats Sweden, Germany defeats Ireland and Sweden. Now, Sweden will be placed behind Austria because of the goal difference. Finally, Ireland soundly defeats Kazakhstan so that the former is placed the last in the group.

The remaining game is Faroe Islands vs. Austria. If Austria wins, they will compete with other second-placed teams with 14 points and a goal difference that includes +4 with Kazakhstan. If, instead, the match is a draw, then the Faroe Islands took the place of Kazakhstan among the teams with which the games are accounted for in the 2nd teams competition. Thus, Austria will have the same 14 points, but the goal difference will include +6 with Faroe Islands.

Still, it remains to show that this goal difference might play a role in determining who goes to Brazil. This will be the case if there is at least one another team that ends up 2nd in another group with 14 points with all other 2nd placed teams grabbing at least 15 points. Such a situation is possible with Hungary finishing with 14 points in Group D, though to satisfy the second part of the condition, Bulgaria must score at least 14 points against the 1st, 3rd, 4th and 5th teams in group B. The latter is only possible in the case of Malta winning over Denmark and the Czech Republic, which is only a theoretical possibility, and a further loss of the Czech Republic to Bulgaria.

# References

Arrow, K (1963) *Social Choice and Individual Values*, 2nd edition (New York: Wiley, 1963).

Dagaev, D and K Sonin (2013). “Winning by Losing: Incentive Incompatibility in Multiple Qualifiers,” CEPR Discussion Paper 9373, 10 March 2013

Duggan, J and T Schwartz (2000) ‘Strategic Manipulability without Resoluteness or Shared Beliefs: Gibbard-Satterthwaite Generalized’, *Social Choice and Welfare* 17 (2000): 85–93.

Gibbard, A (1973) `Manipulation of Voting Schemes', *Econometrica* 41 (1973): 587–601.

Kovash, K and S Levitt (2009). “Professionals Do Not Play Minimax: Evidence from Major League Baseball and the National Football League,” NBER Working Paper No. 15347, September 2009

Rubinstein, A (1980) ‘Ranking the Participants in a Tournament’, *SIAM Journal on Applied Mathematics *38(1): 108–11.

Satterthwaite, M (1975) ‘Strategy-proofness and Arrow's Conditions’,* Journal of Economic Theory* 10 (1975): 187–217.