Doping in Sport
Due to the economic stakes in modern competitive sports, problem of doping in sport becomes serious. Thus doping becomes an economic issue. Doping makes the competition unfair and do harm to the health of the athletes who have doped. We would analyze the problem of doping under game theory. What is needed to deter from doping includes not only improving the test system, but also measures on the number and prizes of events and prevention. It is necessary that either tests are strong and prizes can then be large, or tests are not so efficient and prizes must be low.
Inhaltsverzeichnis 
Introduction
Motivation
We develop a gametheoretic model of doping which focuses on the economic aspects of competitive sports. According to the model, incentives for athletes to use doping increase when
 the efficiency of the drug test system is low,
 the number of competitions during one season is high,
 the range of prizes from sports events is large,
 the perceived health cost is low.
Assumptions
The use of doping is analyzed under the game theory so as to capture the economic factors of competitive sports. We develop a twoplayer game based on the following assumptions:
 a health cost is incurred when athletes use doping,
 doping makes any athlete to improve his results during the season,
 if both athletes dope, the order of positions remains unchanged,
 a doped athlete has a positive probability to be caught by a drug test and, then to be punished.
Basic factors
In order to analyze the athletes’ motivation to use doping, we are to deal with four basic factors.:
 the efficiency of the test system,
 the number of events,
 the range of prizes from sports events,
 prevention measures.
Model of the game
definition
Let us consider two athletes, A and B competing in a season with n sports events. They are assumed to be identical in terms of physical abilities and preferences.
The extensive form game is run as follows: firstly, each athlete chooses the action doping (D) or not doping (ND). Then the season is played. After that each athlete has a drug test. Any doped athlete has a positive probability p to be caught. Any undoped athlete has no risk to be falsely recognized as doped. Finally, the prizes are delivered.
Let w_{1} and w_{2} be the prizes respectively for the first and the second place in any event with nw_{1}>nw_{2}. Doping involves a cost c which is the health cost. If an athlete is caught as doped, he is banned and not paid for any events. We suppose that both athletes have a utility function u increasing in revenue. If the athletes are both doped, neither of them gain any events. If the athletes are both undoped, they are assumed to win each half of the events. However, if one athlete dopes while his competitor does not, he is sure to win all the events. The strategies of the athletes are to choose between D and ND, in the first stage. Then four cases have to be considered (where U_{A} and U_{B} denote the payoffs for athlete A and B):
 (D, D): Both athletes dope. Each athlete wins n/2 events and incurs cost c. With probability p^{2}, both are caught by the doping tests and thus have no reward. With probability (1p) ^{2}, neither is caught and each thus gets n(w_{1}+w_{2})/2. With probability p(1p), only one is caught: the athlete recognized as doped gets nothing while the other one (falsely recognized as undoped) gets all the prizes, namely nw1. The payoffs of any athlete are:
 U_{A}(D,D) = U_{B}(D,D)

 (ND, ND): Both athletes do not dope. Each wins half of the competitions, thus the payoffs are:
 (D, ND): Athlete A dopes while athlete B does not. Athlete A wins all the events. However, with probability p, he is caught by the doping test, thus getting nothing while athlete B then recovers the prizes for the winner, namely nw_{1}. With probability 1p, athlete A is not caught by the test and thus gains nw_{1}, while athlete B then gains nw_{2}. The payoffs are
 U_{A}(D,ND) = (1 − p)u(nw_{1} − c) + pu( − c),
 U_{B}(D,ND) = (1 − p)u(nw_{2}) + pu(nw_{1}).
 (ND, D): This case is symmetric to the previous one:
 U_{A}(D,ND) = (1 − p)u(nw_{2}) + pu(nw_{1}),
 U_{B}(D,ND) = (1 − p)u(nw_{1} − c) + pu( − c).
The Nash equilibrium
We conclude by solving the game. Pair (ND, ND) is a Nash equilibrium of the game iff
 U_{A}(D,ND) = U_{B}(ND,D) < = U_{A}(ND,ND) = U_{B}(ND,ND),
 U_{A}(ND,D) = U_{B}(D,ND) > U_{A}(D,D) = U_{B}(D,D),
Proposition 1
If the athletes are riskaverse, pair (ND, ND) is the (unique) Nash equilibrium of the game iff .
Proof
We have to prove that (1) implies (2). (1) implies



 ,
 ,
In the case that the athletes are riskaverse, function u is concave, so . For the same reason, we have

 which is strictly smaller than (1p)u(nw_2)+pu(nw_1)</math> since

 (because function u is increasing and w_{1}>w_{2})
The possible equilibrium of the game are represented in terms of {p, w1} in Figure 1. Ordinate of point A is w2+2c/n (when condition (1) is equality with p=0), which is independent from utility function u. Curve Aa, on which relation (1) is an equality, departs the situations where (ND, ND) is an equilibrium; curve Ab, on which relation (2) is an equality, departs the situations where (D, D) is an equilibrium. Prop. 1 states that zones D and ND do not overlap when the athletes are riskaverse.
 In zone D, (D, D) is the only Nash equilibrium; the payoffs from using doping outweigh the health costs. It is clearly a case of Prisoner's Dilemma. Although it is better for each athlete not to dope, each one finds it optimal to use doping while his competitor does not. Doping is here a dominating strategy, which leads to a preferred outcome regardless of the strategy used by the competitor.
 In zone ND, (ND, ND) is the only Nash equilibrium; the health costs outweigh the gain from doping. Here not doping is a dominating strategy.
 In zone D/ND, both (ND, D) and (D, ND) are Nash equilibrium. In this case both doped and undoped athletes coexist. If one athlete uses doping, his competitor may have no interest to also dope since the later will only win half of the events incurring the health cost.
Proposition 2
Let’s define
 If the athletes are riskaverse, .
 If the athletes are riskneutral, .
 If the athletes are risklike, .
If the athletes are riskaverse, function u is concave, . , hence . Therefor the other two cases are also correct.
As a result, when the athletes are riskaverse, it is sufficient to have a doping test of efficiency k<1/2 to prevent them from doping. It is more difficult to deter from doping when athletes are risklike than when they are riskaverse: the required test system efficiency should be higher.
Antidoping policy
From Figure 1, we can derive some implications for antidoping policy. Graphically, the problem is how to reach the zone ND, starting from zone D. The model identifies 4 factors which have to be controlled in order to deter from doping.
 The drug test efficiency, p:
 The number of sports events, n:
 The range of prize, w_{1}w_{2}
 The health cost, c
References
[1] Bird, E.J., Wagner, G.G. 1997. Sports as a common property resource: A solution to the dilemmas of doping. Journal of Conflict Resolution, 41, 749766.
[2]Berivik, G., 1987, The doping dilemma – Some game theoretical and philosophical considerations, Sportwissenshaft, 17, 8394.