Rules Theory: a contest model Data Results So the last will be the first Ruud H. Koning Manon Grevinga april 2018 1 / 32
Rules Theory: a contest model Data Results Introduction Sport fans value the uncertainty of the outcome of a sporting contest, in fact, one may argue that this uncertainty is one of the defining characteristics of sports Fort (2006), Szymanski (2003). Uncertainty of outcome can be regulated by choosing a certain competition format. 2 / 32
Rules Theory: a contest model Data Results Introduction Sport fans value the uncertainty of the outcome of a sporting contest, in fact, one may argue that this uncertainty is one of the defining characteristics of sports Fort (2006), Szymanski (2003). Uncertainty of outcome can be regulated by choosing a certain competition format. Different tournament types have different incentives for the participants. It is well known that average effort provided by equally skilled athletes decreases with the number of participants in a winner take all contest. 2 / 32
Rules Theory: a contest model Data Results Introduction Sport fans value the uncertainty of the outcome of a sporting contest, in fact, one may argue that this uncertainty is one of the defining characteristics of sports Fort (2006), Szymanski (2003). Uncertainty of outcome can be regulated by choosing a certain competition format. Different tournament types have different incentives for the participants. It is well known that average effort provided by equally skilled athletes decreases with the number of participants in a winner take all contest. Some tournament types base ranking on some absolute measure of performance, so essentially all athletes compete against each other, even though they may not compete simultaneously. In such a case, incentives are similar to the ones in a single rank order tournament, and so is effort provided. 2 / 32
Rules Theory: a contest model Data Results Introduction In this paper we focus on speed skating. A speed skating event (say, 1500m men) usually has approximately 20 participants, who skate against each other in pairs. Finishing time is recorded and after all participants have skated, the one with the fastest time wins. Even though each skater has only one direct opponent, he effectively competes against all participants. However, skaters in later pairs have an advantage over earlier skaters: they know the time to beat in order to lead the ranking at that moment. 3 / 32
Rules Theory: a contest model Data Results Introduction To assess the informational advantage of later skaters, we estimate a flexible model where their performance depends-among other things-on the best time skated so far. In most tournaments the order of skaters is determined by performance in earlier events, so in our model we allow for this by incorporating individual effects. We find a small and significant effect of the best time skated so far: if the best time skated so far decreases by 1 second, performance of the skater improves by approximately 0.17%-0.42% (depending on the specification). Even though this effect appears to be very small, it may be significant as the time difference between top places of important tournaments may be tiny. 4 / 32
Rules Theory: a contest model Data Results Plan 1 Rules 2 Theory: a contest model 3 Data 4 Results 5 / 32
Rules Theory: a contest model Data Results Skating 101 6 / 32
Rules Theory: a contest model Data Results World Cup events Regular throughout the season at different locations, Monetary and reputation incentives, 7 / 32
Rules Theory: a contest model Data Results World Cup events Regular throughout the season at different locations, Monetary and reputation incentives, End of season: World Cup final, Order: first meeting based on perception by coaches, Later meetings: inverse to current ranking, Final ranking: cumulative points, Division A and Division B events, promotion relegation. 7 / 32
Rules Theory: a contest model Data Results Tournaments World Allround Championships, European Championships, World Sprint Championships, Four distances, Reputation incentives, sponsoring 8 / 32
Rules Theory: a contest model Data Results Tournaments World Allround Championships, European Championships, World Sprint Championships, Four distances, Reputation incentives, sponsoring Once a year, Order (1500m): second day, related to current ranking, order 1000m (first day): draw within groups, second day related to ranking, Final ranking: weighted average of times. 8 / 32
Rules Theory: a contest model Data Results Single distances World Single Distance Championships, Olympic Games, One distance, Reputation incentives, sponsoring, most prestigeous 9 / 32
Rules Theory: a contest model Data Results Single distances World Single Distance Championships, Olympic Games, One distance, Reputation incentives, sponsoring, most prestigeous Once a year/once every four years, Order: draw within groups, Final ranking: time. 9 / 32
Rules Theory: a contest model Data Results Rules The order of skaters in a tournament is not random. Instead, it is partly based on seeding, and partly based on a draw. In general, better skaters tend to skate in later pairs in tournaments, so the fact that later skaters tend to be faster cannot be attributed to informational advantage only: they may be just better under any circumstance. So we need to separate individual ability and order/information! 10 / 32
Rules Theory: a contest model Data Results Contest model setup To fix ideas how information affects behaviour of utility maximizing athletes, we develop a simple contest model. First, we consider two skaters who engage in a simultaneous contest. 11 / 32
Rules Theory: a contest model Data Results Contest model setup To fix ideas how information affects behaviour of utility maximizing athletes, we develop a simple contest model. First, we consider two skaters who engage in a simultaneous contest. y i = x i + ǫ i , i = 1 , 2 . (1) 11 / 32
Rules Theory: a contest model Data Results Contest model setup To fix ideas how information affects behaviour of utility maximizing athletes, we develop a simple contest model. First, we consider two skaters who engage in a simultaneous contest. y i = x i + ǫ i , i = 1 , 2 . (1) Effort is costly for each skater, and following Brown and Minor (2014) we model costs by the concave function 1 2 c i x 2 i , where the parameter c i is interpreted as measuring talent: a more talented athlete has a lower value for c i , so it is easier for him to exert more effort. 11 / 32
Rules Theory: a contest model Data Results Contest model setup To fix ideas how information affects behaviour of utility maximizing athletes, we develop a simple contest model. First, we consider two skaters who engage in a simultaneous contest. y i = x i + ǫ i , i = 1 , 2 . (1) Effort is costly for each skater, and following Brown and Minor (2014) we model costs by the concave function 1 2 c i x 2 i , where the parameter c i is interpreted as measuring talent: a more talented athlete has a lower value for c i , so it is easier for him to exert more effort. Expected utility for skater 1 is E U ( x 1 ) = Pr( y 1 > y 2 ) V − 1 2 c 1 x 2 1 , (2) 11 / 32
Rules Theory: a contest model Data Results Contest model-simultaneous game The first skater maximizes his expected utility: E U ( x 1 ) = Pr( y 2 < y 1 ) V − 1 2 c 1 x 2 1 = Pr( ǫ 2 − ǫ 1 < x 1 − x 2 ) V − 1 2 c 1 x 2 1 � x 1 − x 2 � V − 1 2 c 1 x 2 = Φ 1 . (3) √ 2 12 / 32
Rules Theory: a contest model Data Results Contest model-simultaneous game The first skater maximizes his expected utility: E U ( x 1 ) = Pr( y 2 < y 1 ) V − 1 2 c 1 x 2 1 = Pr( ǫ 2 − ǫ 1 < x 1 − x 2 ) V − 1 2 c 1 x 2 1 � x 1 − x 2 � V − 1 2 c 1 x 2 = Φ 1 . (3) √ 2 Only the distribution of the differences of the random terms is relevant. 12 / 32
Rules Theory: a contest model Data Results Contest model-simultaneous game � x 1 − x 2 � 1 √ φ √ V − c 1 x 1 = 0 , (4) 2 2 1 � x 2 − x 1 � √ φ √ = 0 . (5) V − c 2 x 2 2 2 13 / 32
Rules Theory: a contest model Data Results Contest model-simultaneous game � x 1 − x 2 � 1 √ φ √ V − c 1 x 1 = 0 , (4) 2 2 1 � x 2 − x 1 � √ φ √ = 0 . (5) V − c 2 x 2 2 2 Using symmetry of the normal density, these conditions can be solved to x 2 = c 1 x 1 c 2 13 / 32
Rules Theory: a contest model Data Results Contest model-simultaneous game � x 1 − x 2 � 1 √ φ √ V − c 1 x 1 = 0 , (4) 2 2 1 � x 2 − x 1 � √ φ √ = 0 . (5) V − c 2 x 2 2 2 Using symmetry of the normal density, these conditions can be solved to x 2 = c 1 x 1 c 2 so the skater with lowest cost will exert more effort. Normalizing total effort to x 1 + x 2 = 1, we get equilibrium effort for both skaters c 2 c 1 x ∗ x ∗ 1 = , 2 = . (6) c 1 + c 2 c 1 + c 2 13 / 32
Rules Theory: a contest model Data Results Contest model-sequential game The second skater observes output of the first skater ( y 1 ), he maximizes E U ( x 2 ) = Pr( x 2 + ǫ 2 > y 1 ) V − 1 2 c 2 x 2 2 = (1 − Pr( ǫ 2 < y 1 − x 2 )) V − 1 2 c 2 x 2 2 = (1 − F 2 ( y 1 − x 2 )) V − 1 2 c 2 x 2 2 , (7) 14 / 32
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