Pde brno
1 like291 views
This document summarizes a game theory model of doping in competitive sports. It makes the following key points: 1) The model frames doping as a prisoner's dilemma game, where the dominant strategy for both athletes is to dope, even though not doping would be better for both. 2) For the non-doping equilibrium to occur, the probability of getting caught r must be greater than half the benefit of winning divided by the cost of getting caught c. 3) If one athlete is believed to be inherently better, doping may not have a pure strategy equilibrium, only a mixed strategy one, making doping even harder to prevent. 4) Uncertainty over drug effects
Pde brno
- 1. The Performance-Enhancing Drug Game by Kjetil K. Haugen Molde University College Servicebox 8, N-6405 Molde, Norway E-mail: Kjetil.Haugen@hiMolde.no Journal of Sports Economics (2004) 5:67-86 1
- 2. Athens Olympics 2004 Shawn Crawford: winning the gold on 200 meters 2
- 3. Outline 1) A Brief discussion on Doping history and extent 2) The Performance-Enhancing Drug Game 3) Nash Equilibrium Characteristics 4) Relaxing model assumptions 5) Policy implications and conclusions 3
- 4. Doping history Old phenomenon: Philostratus and Galerius reports on doping from ancient Olympics in the third century B.C. More Recent doping creativity: Oxygen Strychnine Brandy and cocaine mixed Today: Shift from proving existence of unnatural substances towards proving too high levels of natural substances Testosterone Growth hormones EPO 4
- 5. Doping extent Sports o鍖cials: Signi鍖cant improvement less doping now than before. Atlanta 96: 2 cases (Andrews, 1998) Performance decrease (swimming, athletics typically force events and female events) Improved testing, WADA etc. Still, recall the situation in Athens, 24 caught, and a lot either caught before the games, or simply not daring to enter 5
- 6. Doping extent - alternative angle Chicago doctor, Bob Goldman asked 198 US athletes the following questions: (Andrews 1998) 1) Given the choice of taking a drug with certain e鍖ect (certain win) and no probability of being caught, what would you do? (99% YES) 2) If the same situation could be repeated over 5 consecutive years, but with a certain death caused by side e鍖ects after 5 years, would you still take the drug? (more than 50% YES) 6
- 7. The Performance-Enhancing Drug Game Some relevant previous research: Breivik, G. (1987) The doping dilemma: Some game theoretical and philosophical considerations. Sportswissenschaft, 17, 8394 Eber, N. & Th卒pot, J. (1999) Doping e in sport and competition. Louvain Economic Review, 435446 Berentsen, A. (2002) The economics of doping. European Journal of Political Economy, 18, 109127 7
- 8. A simple Game model assumptions (1) 2 athletes (of equal strength) compete against each other in some sports event The athletes possible strategic choices are to use dope or not. Hence, we assume only one available drug. The drug is assumed e鍖ective, that is if one agent takes the drug and the other does not, the drug-taker wins the competition with certainty. The drug is also assumed to have equal e鍖ect on both athletes. That is, if both athletes takes the drug, they are again equal in strength. Both agents must decide (simultaneously) before the competition on whether to take the drug or not, and this decision is made only once (one-shot). 8
- 9. A simple Game model assumptions (2) The agents pay-o鍖s are de鍖ned according to three interesting outcomes for each agent: W: Agent i wins the competition L: Agent i looses the competition E: Agent i is exposed as a drug abuser To simplify calculations, without loss of generality, we de鍖ne the following utilities for each of the above outcomes: ｧ ｪ ui(W) = a, a > 0 ｨ u() = ui(L) = 0 ｪ ｩ u (E) = c, c > 0 i (1) Note that the above de鍖nition implicitly makes the assumption that both agents have a symmetrical utility structure. That is, the value of winning, loosing or being exposed is the same for both agents. 9
- 10. A simple Game model assumptions (3) The probability of being exposed as a drug abuser, r, is assumed to be a nature call in this game, that is both players know it and can not in any way in鍖uence it, neither the value nor the actual test which takes place after the competition. The probability of being exposed as a drug abuser if drugs are not used, is assumed zero. (Unlike real doping tests, we hence assume perfectness.) Furthermore, we assume for simplistic reasons that the pay-o鍖 received by any agent, is kept even if this agent is caught in doping. Finally, both agents know all there is to know (every assumption de鍖ned above complete information). Even though the latter events in Salt Lake City may prove this to be a fairly realistic assumption. 10
- 11. A Two-Player Simultaneous Game AGENT 2 D ND 1 2 a-rc 0 D AGENT 1 1 2 a-rc a-rc 1 2 ND a-rc 0 1 2 a a Expected utility (D,D) case: 1 1 1 揃 a + 揃 0 r 揃 c = a rc 2 2 2 (2) Expected utility (D,ND) case: (AGENT 1) 1 揃 a + 0 揃 0 r 揃 c = a rc (3) 11
- 12. Nash Equilibria (1) Crucial assumption: Sign of 1 a rc. Reason2 1 a rc > 0, no doping ably to assume that 2 would take place if not. Then, 1 a rc > 0 2 (4) 1 1 1 1 a rc + a > a a rc > a 2 2 2 2 (5) 12
- 13. Nash equilibria (2) AGENT 2 D ND 1 2 a-rc 0 D AGENT 1 1 2 a-rc a-rc a-rc ND 1 2 0 1 2 a a 13
- 14. Some simple conclusions: Everybody use drugs (D,D) is a unique Nash equilibrium Under reasonable assumptions of r, c > 0, 1 arc < 1 a, or the Nash equilibrium (D,D) 2 2 is of Prisoners Dilemma type. Hence, regulation or anti-doping work is necessary. A necessary condition for E鍖cient antia doping work is: 1 arc < 0 or r > 2c . That 2 is, r must be increased su鍖ciently, unless, a pareto worsening is the e鍖ect. (regulators use more money on anti-doping work, same number of dopers) Anti-doping work should be strongly differentiated between sports activities. If asoccer >> acurling and csoccer ccurling then, rsoccer >> rcurling . 14
- 15. Relaxing equal strength assumption Assume now: AGENT1 is better than AGENT2 with common knowledge probability p. AGENT 2 D ND (1-p)a-rc 0 D AGENT 1 pa-rc ND a-rc 0 a-rc (1-p)a pa In the (D,D) and (ND,ND) cases, AGENT1 beats AGENT2 with probability p the drug has still equal e鍖ect. In the (ND,D) and (D,ND) cases, we still assume (to make things simple) that the drugtaker wins with certainty. That is, the drug is magical. 15
- 16. Nash equilibria: unequal strength (1) AGENT1 better Consequently: if than AGENT2 p > 1 . 2 1 a rc > 0 then, 2 pa rc > 0 (6) Adding (1 p)a to each side of (6) yields: a rc > (1 p)a or (7) AGENT 2 D ND (1-p)a-rc 0 D AGENT 1 pa-rc ND a-rc 0 a-rc (1-p)a pa 16
- 17. Nash equilibria: unequal strength (2) The rest of the Best Reply functions are determined by the sign of the expression: (1 p)a rc (8) If (1 p)a rc > 0. its back to the initial case. However, if (1 p)a rc < 0, No Nash equilibrium in pure strategies exist, hence, a unique Nash equilibrium in mixed strategies is the Game Theoretic prediction. Consequently, regulation within the boundaries of (1 p)a rc < 0 In practical terms: if an athlete believes strongly enough in the magical e鍖ectiveness of a drug, doping can not be fought. It is possible to prove (see paper) that if the assumption of a magic drug is relaxed, an unique Nash equilibrium of (ND, ND)-type may at least exist. 17
- 18. Policy implications and conclusions Athletes belief in magical drug problem in anti-doping work. Hence, increased uncertainty of outcome or increased competition is an interesting and not much discussed anti-doping strategy. May also serve as an explanation on why certain uncompetitive sports like athletics has more problems with doping than f.i. soccer. Common knowledge on actual doping effects may have similar e鍖ects. However, incentive problems here. Increasing c, the disutility of being caught, obvious! Could it be that the sports industry has incentives not to 鍖ght doping? 18