So the last will be the first Ruud H. Koning Manon Grevinga april - - PowerPoint PPT Presentation

Rules Theory: a contest model Data Results

So the last will be the first

Ruud H. Koning Manon Grevinga april 2018

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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.

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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.

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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.

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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.

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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

• rder 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.

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Rules Theory: a contest model Data Results

Plan

1 Rules 2 Theory: a contest model 3 Data 4 Results

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Rules Theory: a contest model Data Results

Skating 101

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Rules Theory: a contest model Data Results

World Cup events

Regular throughout the season at different locations, Monetary and reputation incentives,

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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.

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Rules Theory: a contest model Data Results

Tournaments

World Allround Championships, European Championships, World Sprint Championships, Four distances, Reputation incentives, sponsoring

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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.

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Rules Theory: a contest model Data Results

Single distances

World Single Distance Championships, Olympic Games, One distance, Reputation incentives, sponsoring, most prestigeous

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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.

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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
• rder/information!

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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.

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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. yi = xi + ǫi, i = 1, 2. (1)

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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. yi = xi + ǫi, i = 1, 2. (1) Effort is costly for each skater, and following Brown and Minor (2014) we model costs by the concave function 1

2cix2 i , where the

parameter ci is interpreted as measuring talent: a more talented athlete has a lower value for ci, so it is easier for him to exert more effort.

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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. yi = xi + ǫi, i = 1, 2. (1) Effort is costly for each skater, and following Brown and Minor (2014) we model costs by the concave function 1

2cix2 i , where the

parameter ci is interpreted as measuring talent: a more talented athlete has a lower value for ci, so it is easier for him to exert more

• effort. Expected utility for skater 1 is

EU(x1) = Pr(y1 > y2)V − 1 2c1x2

1,

(2)

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Rules Theory: a contest model Data Results

Contest model-simultaneous game

The first skater maximizes his expected utility: EU(x1) = Pr(y2 < y1)V − 1 2c1x2

1

= Pr(ǫ2 − ǫ1 < x1 − x2)V − 1 2c1x2

1

= Φ x1 − x2 √ 2

• V − 1

2c1x2

1.

(3)

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Rules Theory: a contest model Data Results

Contest model-simultaneous game

The first skater maximizes his expected utility: EU(x1) = Pr(y2 < y1)V − 1 2c1x2

1

= Pr(ǫ2 − ǫ1 < x1 − x2)V − 1 2c1x2

1

= Φ x1 − x2 √ 2

• V − 1

2c1x2

1.

(3) Only the distribution of the differences of the random terms is relevant.

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Rules Theory: a contest model Data Results

Contest model-simultaneous game

1 √ 2 φ x1 − x2 √ 2

• V − c1x1

= 0, (4) 1 √ 2 φ x2 − x1 √ 2

• V − c2x2

= 0. (5)

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Rules Theory: a contest model Data Results

Contest model-simultaneous game

1 √ 2 φ x1 − x2 √ 2

• V − c1x1

= 0, (4) 1 √ 2 φ x2 − x1 √ 2

• V − c2x2

= 0. (5) Using symmetry of the normal density, these conditions can be solved to x2 x1 = c1 c2

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Rules Theory: a contest model Data Results

Contest model-simultaneous game

1 √ 2 φ x1 − x2 √ 2

• V − c1x1

= 0, (4) 1 √ 2 φ x2 − x1 √ 2

• V − c2x2

= 0. (5) Using symmetry of the normal density, these conditions can be solved to x2 x1 = c1 c2 so the skater with lowest cost will exert more effort. Normalizing total effort to x1 + x2 = 1, we get equilibrium effort for both skaters x∗

1 =

c2 c1 + c2 , x∗

2 =

c1 c1 + c2 . (6)

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Rules Theory: a contest model Data Results

Contest model-sequential game

The second skater observes output of the first skater (y1), he maximizes EU(x2) = Pr(x2 + ǫ2 > y1)V − 1 2c2x2

2

= (1 − Pr(ǫ2 < y1 − x2))V − 1 2c2x2

2

= (1 − F2(y1 − x2))V − 1 2c2x2

2,

(7)

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Rules Theory: a contest model Data Results

Contest model-sequential game

The second skater observes output of the first skater (y1), he maximizes EU(x2) = Pr(x2 + ǫ2 > y1)V − 1 2c2x2

2

= (1 − Pr(ǫ2 < y1 − x2))V − 1 2c2x2

2

= (1 − F2(y1 − x2))V − 1 2c2x2

2,

(7) Assuming interior optimum, the optimal choice is f2(y1 − x2)V − c2x2 = 0. (8)

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Rules Theory: a contest model Data Results

Contest model-sequential game

The second skater observes output of the first skater (y1), he maximizes EU(x2) = Pr(x2 + ǫ2 > y1)V − 1 2c2x2

2

= (1 − Pr(ǫ2 < y1 − x2))V − 1 2c2x2

2

= (1 − F2(y1 − x2))V − 1 2c2x2

2,

(7) Assuming interior optimum, the optimal choice is f2(y1 − x2)V − c2x2 = 0. (8) So solution will be x∗

2 = x∗ 2(y1, V , c2), which solves FOC. Note:

marginal distribution of ǫ2 has become relevant.

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Rules Theory: a contest model Data Results

Contest model-sequential game

Differentiating FOC with respect to y1, we find

• 1 − dx∗

2

dy1

• f ′

2(y1 − x∗ 2(y1, V , c2))V − c2

dx∗

2

dy1 = 0, so that dx∗

2

dy1 = f ′

2(y1 − x∗ 2(y1, V , c2))

c2/V + f ′

2(y1 − x∗ 2(y1, V , c2)).

(9) This expression shows how the optimal response of the second skater varies with a small increase of the output (ie., time) of the first skater. The sign of this expression depends on the sign of f ′

2(y1 − x∗ 2(y1, V , c2)) and the magnitude of c2/V relative to this

derivative.

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Rules Theory: a contest model Data Results

Contest model-simultaneous game

Suppose we follow Brown and Minor (2014): ǫ2 is uniformly distributed on [− 1

2a, 1 2a]. The density is f2(ǫ2) = 1 a and its

derivative with respect to its argument is 0. Hence, for this case dx∗

2

dy1 = 0, in other words, the distributional assumption imposes that the second skater does not respond to the output of the first skater.

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Rules Theory: a contest model Data Results

Contest model-sequential game

In the second case, assume that ǫ2 follows a standard normal distribution, as above. The standardnormal density is increasing when its argument is negative, so y1 < x∗

2(y1, V , c2) ⇒ dx∗ 2

dy1 > 0.

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Rules Theory: a contest model Data Results

Contest model-sequential game

In the second case, assume that ǫ2 follows a standard normal distribution, as above. The standardnormal density is increasing when its argument is negative, so y1 < x∗

2(y1, V , c2) ⇒ dx∗ 2

dy1 > 0. In case y1 > x∗

2(y1, V , c2), we have that the optimal response is

increasing in y1 if c2 < (y1 − x∗

2(y1, V , c2))φ(y1 − x∗ 2(y1, V , c2)V

The second skater will increase his effort if output of the first skater is increased if cost of effort of the second skater is low enough.

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Rules Theory: a contest model Data Results

Contest model-sequential game

It may seem counterintuitive if the second skater would not respond with at least as much effort as the first skater, since his marginal cost of effort c2 is assumed to be lower than the marginal cost of the first skater. However, we consider the derivative of

• ptimal effort of the second skater with respect to the output of

the first skater. The first skater may have been very lucky (a large value of ǫ1). In that case, it may be unattractive for the second skater to try to beat the luck of the first skater, as the marginal cost of effort is too high.

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Rules Theory: a contest model Data Results

1 2 3 4 1 2 3 4

• utput skater 1
• ptimal effort skater 2

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Rules Theory: a contest model Data Results

Data

From www.isuresults.eu; we use the results of ISU speed skating tournaments from seasons 2008/2009 to 2017/2018 (partly). The following speed skating tournaments are used: European Championships Allround World Cup Final European Speed Skating World Cup Championships World Single Distances Olympic Winter Games Championships World Allround Championships World Sprint Championships

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Rules Theory: a contest model Data Results

Data

The ISU speed skating tournaments are organized in: Astana Changchun Hamar Kolomna Richmond Berlin Chelyabinsk Harbin Moscow Salt Lake City Budapest Collalbo Heerenveen Nagano Sochi Calgary Erfurt Inzell Obihiro Gangneung Minsk Seoul Stavanger Faster times are skated on covered high altitude rings, such as the

• nes in Calgary (1105m above sea level) and Salt Lake City

(1425m). Slower times are skated on rings at sea level, such as Heerenveen (1m) and Moscow (167m). All world records for men and women have been realized in either Calgary or Salt Lake City.

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Rules Theory: a contest model Data Results

Data

The dataset contains 561 different events (where we distinguish between a Division A event and a Division B event, even though they are may be skated on the same day at the same venue), and in total 11881 individual skating times are recorded. On average, 21 performances are recorded in each event. The dataset has 1000m and 1500m times of 716 skaters, 209 appear in one season

• nly, while 24 appear in all ten seasons. How do we take individual

development into account?

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Rules Theory: a contest model Data Results

Data

• F

M 8 7 6 5 4 3 2 1 8 7 6 5 4 3 2 1 35.0 37.5 40.0 42.5

Pair (1 is last pair) time per 500m (sec)

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Rules Theory: a contest model Data Results

Simple test

If later skaters have no informational advantage, the pair with the winning time should be uniformly distributed in events where skaters are randomly allocated to pairs. Following the discussion of the rules, skaters are randomly allocated in the World Sprint Championship (the 1000m on the first day), Olympic Games, and in the World Single Distances Championship. Out of 24 suitable events in our dataset, the winning time was generated in the last pair in ten cases. This test may have low power if there is significant individual heterogeneity.

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Rules Theory: a contest model Data Results

Simple test, p = 0.15 (up to 2013/14

2 4 6 8 10 4 3 2 1

pair (1 is last) frequency of winning time

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Rules Theory: a contest model Data Results

Best time so far, Vancouver Olympic Games 2010, 1000m men

• 34.50

34.75 35.00 35.25 5 10 15 20

pair (1 is last) best so far

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Rules Theory: a contest model Data Results

Model

Important components: individual heterogeneity, location/date specific effects. Model 1 is standard in the literature as serves as

• ur starting point. Skater i skates on date t in event e in season s

in location l. log yietsl = α0 + αi +

• l

γlRtl +

• s

φsSts +ζDSDSiet + ζBBiet + ζOOiet + ψSBOSBOiet +ψppairiet + ψBSFBSF iet + +ψb.peersB.PEERSet + ǫietsl.

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Rules Theory: a contest model Data Results

Model

Important components: individual heterogeneity, location/date specific effects. Model 1 is standard in the literature as serves as

• ur starting point. Skater i skates on date t in event e in season s

in location l. log yietsl = α0 + αi +

• l

γlRtl +

• s

φsSts +ζDSDSiet + ζBBiet + ζOOiet + ψSBOSBOiet +ψppairiet + ψBSFBSF iet + +ψb.peersB.PEERSet + ǫietsl. Main hypothesis: H0 : ψp = ψBSF = 0. Both peer- and information effects.

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Rules Theory: a contest model Data Results

Model

Important components: individual heterogeneity, location/date specific effects. Model 1 is standard in the literature as serves as

• ur starting point. Skater i skates on date t in event e in season s

in location l. log yietsl = α0 + αi +

• l

γlRtl +

• s

φsSts +ζDSDSiet + ζBBiet + ζOOiet + ψSBOSBOiet +ψppairiet + ψBSFBSF iet + +ψb.peersB.PEERSet + ǫietsl. Main hypothesis: H0 : ψp = ψBSF = 0. Both peer- and information effects. Model 2: αis, model 3: αis and βtl.

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Rules Theory: a contest model Data Results

Results, α = 0.01

×100% model 1 model 2 model 3 distance1500m 2.078 2.199 2.357 divisionB 0.906 0.458 0.517 eventsingle.race −0.097 0.054 laneO 0.074 0.069 0.073 sbo 0.058 0.021 0.027 pair.r 0.033 −0.012 0.000 bsf 0.427 0.321 0.176 best.sb.peers 0.000 0.000 0.002 model 1: individual FE, season and location effects estimated model 2: season-individual FE, location effects estimated model 3: season-individual FE, date-location effects estimated.

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Rules Theory: a contest model Data Results

Results, incidental parameters?

10 20 30 50 100 150

number of observarions per location−date combination frequency

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Rules Theory: a contest model Data Results

Model 4, identification

Consider skaters i and j, and two dates t and t′. The model is Tietsl = αis + βlt + γ′xietsl + ǫietsl. Taking differences over time ∆tTietsl = βlt − βlt′ + γ′∆txietsl + ∆tǫietsl ∆tTjetsl = βlt − βlt′ + γ′∆txjetsl + ∆tǫjetsl

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Rules Theory: a contest model Data Results

Model 4, identification

Consider skaters i and j, and two dates t and t′. The model is Tietsl = αis + βlt + γ′xietsl + ǫietsl. Taking differences over time ∆tTietsl = βlt − βlt′ + γ′∆txietsl + ∆tǫietsl ∆tTjetsl = βlt − βlt′ + γ′∆txjetsl + ∆tǫjetsl So ∆tTietsl −∆tTjetsl = γ′(∆txietsl −∆txjetsl)+∆tǫietsl −∆tǫjetsl. All fixed effects are gone!

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Rules Theory: a contest model Data Results

Model 4, identification

We need to identify skaters who have participated in the same tournaments, and relate difference in improvements over time to differences of variation in covariates over time. Within the same season.

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Rules Theory: a contest model Data Results

Model 4, identification

We need to identify skaters who have participated in the same tournaments, and relate difference in improvements over time to differences of variation in covariates over time. Within the same season. Only the effect of within-season time-varying covariates is identified, provided the variation between individual skaters

• varies. Examples: bsf and sbo.

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Rules Theory: a contest model Data Results

Model 4, identification

We need to identify skaters who have participated in the same tournaments, and relate difference in improvements over time to differences of variation in covariates over time. Within the same season. Only the effect of within-season time-varying covariates is identified, provided the variation between individual skaters

• varies. Examples: bsf and sbo.

No longer identified: best.sb.peers (no variation between i and j), distance, tournament vs single race etc.

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Rules Theory: a contest model Data Results

Conclusion

Theory: information disclosure matters. Tested hypothesis of informational advantage on truly randomized data: suggestive but not conclusive.

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Rules Theory: a contest model Data Results

Conclusion

Theory: information disclosure matters. Tested hypothesis of informational advantage on truly randomized data: suggestive but not conclusive. After separation of individual ability, location specific effects, and disclosure of information in model, bsf has significant and meaningful effect. Order and disclosure of information matters, despite absolute measure of performance.

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