# 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.

# Introduction

## Motivation

We develop a game-theoretic 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 two-player 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 un-doped athlete has no risk to be falsely recognized as doped. Finally, the prizes are delivered.

Let w1 and w2 be the prizes respectively for the first and the second place in any event with nw1>nw2. 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 un-doped, 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 UA and UB 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 p2, both are caught by the doping tests and thus have no reward. With probability (1-p) 2, neither is caught and each thus gets n(w1+w2)/2. With probability p(1-p), only one is caught: the athlete recognized as doped gets nothing while the other one (falsely recognized as un-doped) gets all the prizes, namely nw1. The payoffs of any athlete are:
UA(D,D) = UB(D,D)
$=(1-p)^2u(\frac{n(w_1+w_2)}{2}-c)+p(1-p)(u(nw_1-c)+u(-c))+p^2u(-c)$
$=(1-p)^2u(\frac{n(w_1+w_2)}{2}-c)+p(1-p)u(nw_1-c)+pu(-c)$

• (ND, ND): Both athletes do not dope. Each wins half of the competitions, thus the payoffs are:
$U_A(ND, ND)=U_B(ND, ND)=u(\frac{n(w_1+w_2)}{2})$
• (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 nw1. With probability 1-p, athlete A is not caught by the test and thus gains nw1, while athlete B then gains nw2. The payoffs are
UA(D,ND) = (1 − p)u(nw1c) + pu( − c),
UB(D,ND) = (1 − p)u(nw2) + pu(nw1).
• (ND, D): This case is symmetric to the previous one:
UA(D,ND) = (1 − p)u(nw2) + pu(nw1),
UB(D,ND) = (1 − p)u(nw1c) + pu( − c).

## The Nash equilibrium

We conclude by solving the game. Pair (ND, ND) is a Nash equilibrium of the game iff

UA(D,ND) = UB(ND,D) < = UA(ND,ND) = UB(ND,ND),
i.e. iff
$\frac{u(n(w_1+w_2))}{2}\ge(1-p)u(nw_1-c)+pu(-c) -------------------- (1)$
Let $f(w_1)=\frac{u(nw_1-c)-u(\frac{n(w_1+w_2)}{2})}{u(nw_1-c)+u(-c)}$. Condition (1) is equivalent to $p\ge f(w_1)$. Pair (D, D) is not a Nash equilibrium iff
UA(ND,D) = UB(D,ND) > UA(D,D) = UB(D,D),
i.e. iff
$(1-p)^2u(\frac{n(w_1+w_2)}{2}-c)+p(1-p)u(nw_1-c)+pu(-c)<(1-p)u(nw_2)+pu(nw_1) -- (2)$

Proposition 1
If the athletes are risk-averse, pair (ND, ND) is the (unique) Nash equilibrium of the game iff $p\ge f(w_1)$.
Proof
We have to prove that (1) implies (2). (1) implies

$(1-p)^2u(\frac{n(w_1+w_2)}{2}-c)+(1-p)u(nw_1-c)+pu(-c)\le$
$(1-p)^2u(\frac{n(w_1+w_2)}{2}-c)+(1-p)u(nw_1-c)+u(\frac{n(w_1+w_2)}{2})-(1-p)u(nw_1-c)$
$=(1-p)^2u(\frac{n(w_1+w_2)}{2}-c)+ u(\frac{n(w_1+w_2)}{2})-(1-p)^2u(nw_1-c)$
$=(1-p)^2(u(\frac{n(w_1+w_2)}{2})-k_1)+u(\frac{n(w_1+w_2)}{2})-(1-p)^2(u(nw_1)-k_2)$,
$=((1-p)^2+1)u(\frac{n(w_1+w_2)}{2})-(1-p)^2(u(nw_1)+k_1-k_2)$,
with $k_1=u(\frac{n(w_1+w_2)}{2})-u(\frac{n(w_1+w_2)}{2}-c)$ and k2 = u(nw1) − u(nw1c).

In the case that the athletes are risk-averse, function u is concave, so $k_1\ge k_2$. For the same reason, we have

$((1-p)^2+1)u(\frac{n(w_1+w_2)}{2})-(1-p)^2(u(nw_1)+k_1-k_2)\le$
$((1-p)^2+1)\frac{u(nw_1+u(nw_2)}{2}-(1-p)^2(u(nw_1)+k_1-k_2)$ which is strictly smaller than (1-p)u(nw_2)+pu(nw_1)[/itex] since
$((1-p)^2+1)\frac{u(nw_1)+u(nw_2)}{2}-(1-p)^2(u(nw_1)+k_1-k_2)-(1-p)u(nw_2)+pu(nw_1)$
$=-\frac{p^2}{2}(u(nw_1)-u(nw_2))-(1-p)^2(k_1-k_2)<0$ (because function u is increasing and w1>w2)

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 risk-averse.

• 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 un-doped 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 $k=\lim_{w_1\to\infty}f(w_1)$

• If the athletes are risk-averse, $k<\frac{1}{2}$.
• If the athletes are risk-neutral, $k=\frac{1}{2}$.
• If the athletes are risk-like, $k>\frac{1}{2}$.
Proof
If the athletes are risk-averse, function u is concave, $f(w_1)< \frac{u(nw_1-c)-\frac{u(nw_1)+u(nw_2)}{2}}{u(nw_1-c)-u(-c)}=g(w_1)$. $\lim_{w_1\to\infty}g(w_1)=\frac{1}{2}$, hence $k<\frac{1}{2}$. Therefor the other two cases are also correct.

As a result, when the athletes are risk-averse, 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 risk-like than when they are risk-averse: the required test system efficiency should be higher.

# Anti-doping policy

From Figure 1, we can derive some implications for anti-doping 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:
Doping tests have not been perfect to provide sufficiently good incentives. Prop. 1 shows that when athletes are risk-neutral or risk-averse, a doping test system with 50% efficiency is sufficient to deter from doping. Moreover, an increase in p lowers the incentives to dope (consider condition (1)). Graphically, we move away from zone D and get closer to zone ND. Payoffs in the un-doped world do not depend on p, so an efficient policy in terms of p is positive for both athletes. Hence, tendency to more efficient doping tests should be accepted by athletes. However, such a policy is costly since efficient tests are very expensive to implement and its results are very uncertain since it may create counter-productive effects, such as incentives to always produce new drugs.
• The number of sports events, n:
A decrease in n leads to a shift of curves Aa and Ab in Figure 1 to the left, thus reducing the size of zone D. Thus a decrease in the number of events reduces the incentives of athletes to use doping: rewards from competition are decreasing with the number of events while the cost of doping is constant. However reducing the numbers of events may be negative for athletes since, in every zone, their payoffs depend positively on n. If the decrease in n needed to prevent from doping is too large, athletes could be finally worse off in the un-doped world than they were at the doped equilibrium. In this case, athletes bear the cost of the anti-doping policy and reducing the number of events.
• The range of prize, w1-w2
The larger is the difference, the higher are the incentives to use doping. Thus, a way to deter from doping is also to decrease the range of the prizes from events. (Graphically, we move closer to zone ND when w1 decreases with given values of w2 and p). Financially athletes should be favorable to this kind of measures since they would gain the same expected payoffs in a completely un-doped world (have no health cost and low efficient anti-doping policy). However, the implementation of such a policy is much more difficult than expected by the model. Prizes include not only financial rewards but also non pecuniary ones such as prestige and the pressure from sponsors and from the media.
• The health cost, c
Informing athletes on the risk of doping induces an increase of the health cost. It can be checked that an increase of c implies a shift to the left of curves Aa and Ab on Figure 1, thus reducing zone D. Prevention measures which increase the health cost c reduce the athletes’ incentives to use doping. Such measures are unambiguously positive for both athletes since their payoffs in the un-doped world do not depend on c. Finally deterring athletes from doping requires a better drug test system, fewer events, lower spread of prizes and more prevention.

# 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, 749-766.
[2]Berivik, G., 1987, The doping dilemma – Some game theoretical and philosophical considerations, Sportwissenshaft, 17, 83-94.