Winning by Losing. Incentive Incompatibility in Multiple Qualiers. Dmitry Dagaev, Konstantin Sonin. Warsaw November 30, 2012

Transcription

1 ¾ : Incentive Incompatibility in Multiple Qualiers Dmitry Dagaev 1 Konstantin Sonin 2 1 Higher School of Economics 2 New Economic School Warsaw November 30, 2012

2 Plan The problem of manipulability of football matches outcomes The real-world example Formal theorem Possible xes

3 Problem Do the competing teams have perverse incentives to lose a game deliberately? ¾Yes, if team is bribed ranking rule or distribution of prizes is not monotonous the game is coalitional What if tournament is fair? No problems in a single tournament. But in multiple tournaments...

4 Real-world example 1 : Russia-2011/2012 Standings as of May 8, 2012: Place Team Pts 1 Zenit St.Petersburg 85 2 CSKA Moscow 73 3 Spartak Moscow 72 4 Dynamo Moscow 71 5 Anzhi Makhachkala 70 6 Lokomotiv Moscow 66 7 Rubin Kazan' 65 8 Kuban' Krasnodar 60 Remaining matches (May 13): Kuban' Dynamo, Rubin CSKA, Lokomotiv Spartak, Anzhi Zenit. Cup nal (May 9): Rubin Dynamo. 1 Credits to Dr. Andrei Brichkin who initially noticed this drawback

5 Real-world example: Russia-2011/2012 Two major European international tournaments: UEFA Champions League and UEFA Europa League Berths are distributed according to the following rules: 1. Teams that are ranked 1st and 2nd in Russian national championship qualify for the Champions League. 2. Teams that are ranked 3rd to 5th in the national championship, qualify for the Europa League. 3. Russian Cup winner qualies for the Europa League. 4. If Cup winner is ranked 1st or 2nd in the national championship, then this team qualies for the Champions League, and the Cup runner-up team qualies for Europa League. 5. If Cup winner is ranked 3rd to 5th in the national championship, then the team ranked 6th qualies for Europa League.

6 Real-world example: Russia-2011/2012 Suppose that Dynamo wins the Russian Cup and beats Kuban', and Rubin CSKA is a draw. Anzhi Zenit game is irrelevant. Lokomotiv's win Team Pts 1 Zenit 85 2 Dynamo 74 3 CSKA 74 4 Spartak 72 5 Anzhi 70 6 Lokomotiv 69 7 Rubin 66 8 Kuban 60 Draw Team Pts 1 Zenit 85 2 Dynamo 75 3 CSKA 74 4 Spartak 72 5 Anzhi 70 6 Lokomotiv 67 7 Rubin 66 8 Kuban 60 Lokomotiv's loss Team Pts 1 Zenit 85 2 Spartak 75 3 Dynamo 74 4 CSKA 74 5 Anzhi 70 6 Lokomotiv 66 7 Rubin 66 8 Kuban 60 Lokomotiv has all the incentives to lose!

7 Literature review (Harary, Moser, 1966) Ranking teams = Aggregation of voter preferences (Arrow, 1963) Several highly desired properties of aggregation of voter preferences rules. Arrow's impossibility theorem. (Rubinstein, 1985) Axiomatical approach for the problem of ranking participants in the round-robin tournament (Gibbard, 1973; Satterthwaite, 1975; Duggan, Schwartz, 2000) Under ¾good enough aggregation rules there always exists a voter who can protably deviate from his truly preferences (Russell, Walsh, 2009) Coalitional manipulating in cup and round robin competitions

8 Theory denitions Tournament is a pair (X, v(x, y)), where X is a nonempty nite set of the teams and v(x, y) is a function which satises the following three conditions: 1) v(x, y) is dened on the set (X X ) \ {(x, y) x = y}; 2) image of v(x, y) is a subset of the set { 1, 0, 1}; 3) for each x 0, y 0 X, x 0 y 0, the equality v(x 0, y 0 ) = v(y 0, x 0 ) holds. Ranking method S is a rule that orders the participating teams in accordance with the results of all matches. If X = n, then S(v) = (s 1 (v),..., s n (v)), where s i (v) is the place of i-th team.

9 Example Let X = {A, B, C, D} and S = S(v) is the following ranking method: 1) Victory = 3 points, Draw = 1 point, Defeat = 0 points; 2) More points ranked higher; 3) Same number of points matches between these teams are considered; 4) Initial seeding: A B C D. For any v ranking S denes a totally ordered set S(v) of the teams from X. Consider characteristic function v 0 : A B C D A B C D Here S(v 0 ) = D B A C.

10 Theory denitions Nv 1 (i), Nv 0 (i) and Nv 1 (i) = numbers of wins, draws and losses of team i respectively. Ranking method S satises monotonicity property, i for any characteristic function v and for any two teams i, j X such that N 1 v (x) N 1 v (y), N 1 v (x) + N 0 v (x) N 1 v (y) + N 0 v (y), (1) where at least one of the inequalities in (1) is strict, holds. s x (v) < s y (v) (2)

11 Theory denitions One international tournament and N domestic tournaments take place, N 2. Tickets to international tournament are the only prizes in domestic tournaments. X = {1, 2,..., K}, K 1. b i is the number of tickets into international tournament laying on the line in tournament i, i = 1,..., K.

12 Theory denitions What if a certain team gets the ticket into international tournament more than once? In the extreme case there will be only max b i contested tickets instead of i b i. Redistribution rule must be dened i

13 Theorem Theorem. Let the following ve conditions hold simultaneously: 1) N 2; 2) b i 1 for( each i = ) 1,..., N; 3) K > max b i, 3 ; i 4) for each i = 1,..., N and for each characteristic function v the set S i (v) is strictly totally ordered; 5) for each i = 1,..., N ranking method S i satises monotonicity property. Then for any ranking methods S 1 (v),..., S N (v) and for any redistributing rule R there exist such characteristic functions v 1,..., v N, w and i, 1 i N, that the following four conditions hold simultaneously:

14 Theorem i) there exists the collection (x 0, y 0 ), such that v i (x 0, y 0 ) = 1 and w(x 0, y 0 ) = 1; ii) for any collection (x, y), dierent from (x 0, y 0 ), holds the equality w(x, y) = v i (x, y); iii) according to the standings S 1 (v 1 ),..., S i 1 (v i 1 ), S i (v i ), S i+1 (v + 1),..., S N (v N ) team x gets a ticket to international tournament; iv) according to the standings S 1 (v 1 ),..., S i 1 (v i 1 ), S i (w), S i+1 (v + 1),..., S N (v N ) team x doesn't get a ticket to international tournament. In other words, for each ¾good ranking methods and redistribution rule it is possible to construct a situation, when a team prefers to lose

15 Generalizations 1. Most of European football national championships: two rounds on the home-away basis 2. Sometimes teams compete for tickets to several tournaments. Example: Champions League and Europa League. 3. Play-o tournaments No incentives to lose in a play-o tournament If there is one play-o tournament (Cup) and one round-robin tournament (Championship) it could be protable to lose in round-robin tournament in order to push a certain team that is successful in both tournaments higher and to gain from redistribution Remark: however, it is possible only if redistribution favours round-robin tournament

16 Advice to UEFA -How to prevent deliberate losses? -Redistribution rule favouring Championship, not Cup.

17 Thank you