TOULMIN ESSAY

The Multi-Player Performance-Enhancing Drug Game Kjetil K. Haugen1, Tamás Nepusz2, Andrea Petróczi3*

1 Faculty of Economics, Informatics and Social Science, Molde Univeristy College, Molde, Norway, 2 Department of Biological Physics, Eötvös Loránd University, Budapest,

Hungary, 3 Faculty of Science, Engineering and Computing, Kingston University, Surrey, United Kingdom

Abstract

This paper extends classical work on economics of doping into a multi-player game setting. Apart from being among the first papers formally formulating and analysing a multi-player doping situation, we find interesting results related to different types of Nash-equilibria (NE). Based mainly on analytic results, we claim at least two different NE structures linked to the choice of prize functions. Linear prize functions provide NEs characterised by either everyone or nobody taking drugs, while non-linear prize functions lead to qualitatively different NEs with significantly more complex predictive characteristics.

Citation: Haugen KK, Nepusz T, Petróczi A (2013) The Multi-Player Performance-Enhancing Drug Game. PLoS ONE 8(5): e63306. doi:10.1371/journal.pone.0063306

Editor: Rodrigo Huerta-Quintanilla, Cinvestav-Merida, Mexico

Received December 14, 2012; Accepted April 1, 2013; Published May 17, 2013

Copyright: � 2013 Haugen et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: The authors have no support or funding to report.

Competing Interests: The authors have declared that no competing interests exist.

* E-mail: [email protected]

Introduction

Doping in competitive sport is a peculiar phenomenon. The

need for performance enhancement is emerging from the desire to

maximise or even expand human capacities [1], and by doping to

gain competitive edge in a situation where athletes’ performances

are judged on two levels simultaneously: athletes compete against

the opponents in situations where typically only one can win and

are also automatically entered into a quest for breaking records

which opens up the competitive arena including all from the past.

From the array of substances with performance enhancing

properties, a wide range represents fully acceptable means, whilst

a defined set is prohibited by some authorities. In general terms,

and for the purpose of this paper, the term ‘‘doping’’ refers to the

latter category.

From the system’s point of view, the current detection-based

anti-doping policy does not automatically eradicate the use of

prohibited substances, but rather presents a barrier with a

quantifiable risk of being caught. It is easy to see that such a

system leads to two primary strategies employed by the athletes: i)

compliance, driven by respect for the rules, desire to compete

clean or fear of being caught and ii) circumvention, i.e. outwitting

the system by using not-yet-known or undetectable substances,

masking or simply betting on chances of not being selected for

testing. Just before the creation of the World Anti-Doping Agency,

then-International Olympic Committee president Juan Antonio

Samaranch voiced his opinion that doping should be acknowl-

edged and allowed as long as it is safe [2]. The fierce reaction by

the public and sporting community has led to tightened rules,

creation of the World Anti-Doping Agency (WADA) and an ever

increasing demand for investment into testing [3]. We will show

that a situation where all athletes end up using doping is possible

even under the current prohibiting-detecting anti-doping system,

putting their health further in risk by using drugs and masking the

use at the same time, increasing the drug intake and the

possibilities of side effects from individual drugs and drug

interactions.

In order to design effective anti-doping measures, gaining

insight into the driving forces behind doping behaviour is vital.

Whilst the doping decision is very complex involving moral,

economical and health considerations, theoretically this complex-

ity can be distilled into a simple decision situation where pros and

cons are weighed against each other in the context of unknown but

assumed choices of the opponents. Based on the assumption that

eradicating doping from sport requires a significant change in this

decisional balance; by formulating and formally solving multi-

player doping games, we aim to make a contribution to developing

a better understanding of doping decisions.

In this paper, first we review the literature pertaining to the

game theoretical approach to sport and in particular, doping. We

set up a model for a doping game involving more than one pair of

opponents. This extends the classic doping game to a multi-player

situation, which resembles many actual doping situations better

than a game restricted to two players. We then report analytical

solutions for n~3 and n~4 cases complemented with two important propositions. Finally, we discuss both theoretical and

practical implications of the multi-player extension of the doping

game and offers directions for future research.

Relevant Literature on Economics of Doping

For athletes entering a high level sport competition, it is a

requirement to abide by the rules as set by the relevant governing

body. Philosophers would argue that at this point, the question

whether the athlete ought to follow these rules or whether they

have a reason to do so is contested and depending on the outcome,

a decision is made about the action which follows. With regard to

doping, athletes must decide whether they feel that they are under

obligation (normative) or have a compelling reason (rationality) to

obey the rules and refrain from using a prohibited performance

enhancing substance.

Homo Economicus: Pay-offs and Sanctions Lay explanation for doping rests on the assumption that for

those who engage in doping practices ‘‘winning is everything’’,

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hence they use prohibited methods to ensure this outcome. This

approach assumes that the choice is purely rational-economical

and responds to externally imposed incentives and deterrents [4].

As long as the perceived advantages from using doping constitute a

far better scenario than any scenario with no doping, factoring in

the risk of being detected and its consequences, the only logical

action is to dope.

Contrary to the detection-sanction based deterrence methods,

economical models recognise the importance of the prize

structure, considering both benefits and costs. Prize structures

can also be manipulated so the monitoring cost is kept low [5].

These models [6–12] suggest that eradicating doping would

require changes in the external factors, such as increased dis-utility

(including the chance of being detected and its consequences),

decreased utility (reduced pay-off) or some combination of the two.

In reality, it is unlikely that such change will be effectively

implemented. Based on a review of economical models of doping,

[13] posit that rank order contests such as sport competitions with

highly skewed prize structures inevitably lead to undesirable

practices (i.e. doping) players may employ in order to enhance

their chances to finish in positions with high pay-off. A follow-up

study with empirical data from thirteen different athletic events

reinforced the assumption that increase in competition increases

doping [14], and consequently lead to a quest for sophisticated

detection and a requirement for equitable sanctions.

The nature of doping makes policing difficult and leads to an

imperfect but costly monitoring system, where externally imposed

sanctions may inadvertently motivate doping use by indicating

that doping is widespread, hence the need for harsh sanctions [15].

Game-like Situations In addition to the mainstream doping research, a distinct

direction (economics of doping) evolved around game theoretical

approaches, focusing on different combinations of the normative

determinants. Early theoretical exploration of the doping dilemma

focused on typical game strategies athletes may engage in, starting

from 2-player symmetric and asymmetric games [16,17].

An empirical cross-sectional investigation, using ranked out-

comes of best, next best, next worst and worst, concluded that

‘‘ruthless’’, winning-at-all-cost-type athletes clearly dominated the

doping user group but drug users were also found among

Naessian-type athletes who value the process more than the

outcome [18]. Although very tentatively, a possibility to change

games upon some personal experience was also suggested. This

evolutionary characteristic is in keeping with Berentsen and

Lengwiler’s [19] cyclical dynamics of mostly honest or fraudulent

games among heterogeneous players. Subsequent game theoretical

models (e.g., [20], [21]) have shown that the likelihood of doping is

the function of i) intensity of the competition, ii) the efficiency of

the detection system, iii) sanctions if caught, iv) distribution of the

prizes and v) health costs.

Although several authors have discussed the obvious extension

into a multi-player setting, few formal attempts on modeling or

solving multi-player games exist in sports economics research. [8]

suggested that athletes with a win-at-all-costs attitude are likely to

find doping use the optimal strategy, regardless of the number of

other athletes who dope. Expanding on the previous work, [17]

also noted that a tendency toward doping use exists in n-player

games. Whilst this body of work draw attention to an important

element (i.e. sport competition is more often than not is an n-

player game), none of these offered analytical or numerical proof.

Berentsen and Lengwiler’s doping evolution game [19] expands

the 2-player setting to a multi-player model. However, despite the

multi-player setting of heterogeneous (weak and strong) players,

decisions are made in pairs. Hence the model avoids explicit multi-

player game formulation. The closest attempt to formally

investigate multi-player effects is found in Strulik’s model [22].

In order to avoid the complexity of a ‘‘real’’ multi-player

formulation, the utility received by player i in season t (according to the authors notation) does not depend directly on other players’

actions in season t. Other players’ actions are built-in into the variables such as the fraction of players taking doping in season t (ht).

Considering the existing models to date, sports economic

research appears to be lacking formal analysis of multi-player

doping games. The aim of this paper is to address this gap by

extending the classical doping game theoretical model to a multi-

player game.

Model Set-up

For this model, we consider athletes as those who are in the

Registered Testing Pool (RTP) by being identified as elite

performer by their respective governing bodies (thus subject to

doping testing at any time including sport events) or based on their

ranking at the sport events. The RTP includes international level

athletes being tested by their international sport federations, and

international and national level athletes being tested by national

anti doping organisations, or in some cases, national governing

bodies of sport, National Olympic Committees and Regional Anti-

Doping Organizations. In this respect, there is no distinction

between amateur or professional sport. Consequently, we consider

prize as the chance of winning vs. the chance of being caught and

sanctioned. However, we recognise the fact that in real life, both

affects the athlete’s ability to benefitting from sponsorships,

endorsement deals and prize money.

Assumptions and Basic Modeling We start with the performance-enhancing drug game of [21],

also discussed by [23], with the aim of adding multi-player

capabilities. We try to stay as close as possible to the original

model, only doing minimal necessary enhancements in order to

investigate the multi-player case.

We assume the following:

1) n athletes compete in a certain (one-shot) sports event.

2) Each athlete has a binary choice between taking perfor-

mance enhancing drugs (D) or keeping ‘‘clean’’ (ND). 3) All players are (at least initially) assumed equally good. This

‘‘cloning’’ assumption is merely a direct extension of the

original assumption of [21], and it will be discussed in some

more detail in the section titled Conclusions and suggestions

for further research.

4) Drug testing is routinely performed on all athletes, leading to

a certain probability of exposure of ri ,i[f1, . . . ,ng, given that drugs have been consumed. Here, we make a slight

generalization compared to [21], allowing for different

exposure probabilities between athletes. In practice, one

could for instance assume either that previous history has

trigged different testing programs between different athletes

(countries) or simply different doping prevalence between

different athletes of groups of athletes (countries).

5) The cost of exposure is again extended similarly as in 4)

above to ci ,i[f1, . . . ,ng. This extended assumption opens up for the model-wise ability to investigate different group

specific costs related to drug exposure. It may for instance

seem reasonable to assume that certain countries ‘‘forgive’’

dopers easier and earlier than others. Hence, variable ci ’s

Multi-Player Doping Game

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may introduce a certain difference in cultural views on

performance-enhancing drugs.

6) Drug tests are assumed perfect in the sense that anyone who

have not consumed drugs will not be exposed through a test.

7) There is only one drug available, and the effect of the drug is

assumed to be equal for all athletes i[1, . . . ,n. This assumption is also a strict copy of the original assumption,

it can of course be relaxed at later stages.

8) The given drug is assumed effective (with certainty) in the

sense that if any athlete on drugs compete against any

‘‘clean’’ athlete, the outcome is a certain victory in the sports

competition for the drug consumer.

So far, no really important assumptions have been made

compared to the original model [21]. We have opened up for

athlete specific exposure probabilities ri as well as exposure costs ci . Still, these assumptions are merely direct extensions to the original model.

However, the utility structure of this model needs some careful

remodelling. In the original model, a binary type of pay-off

structure was defined. In the multi-player case, it seems obvious

that a ‘‘winner takes all’’ pay-off structure is neither realistic nor

practical. Hence, it seems evident that a more flexible pay-off

structure is necessary. Let us furthermore define:

9) ai is the utility (w0) received by athletes through their ranks in the sports competition. That is, a1 is the utility received by the first prize winner, a2 is the utility received by the runner- up and so forth.

10) Furthermore, it seems reasonable to assume:

a1§a2§ � � �§an§0 ð1Þ

This multi-player model opens up for some different and

interesting options regarding the interpretation of solutions. In

the original model with 2 players, a concept like doping prevalence

was neither very interesting nor very realistic to focus on. In the

case of a multi-player game, the prevalence becomes both relevant

and tractable.

Let us now formally define decision variables for the players/

athletes:

di~ 1 if and only if athlete i chooses the D{strategy 0 if and only if athlete i chooses the ND{strategy

� ð2Þ

Given the definition of the binary decision variables di , the sumPn i~1 di may directly be interpreted as the group prevalence.

Hence we define

Xn i~1

di~d ð3Þ

where d is the prevalence, i.e. the number of athletes choosing to take drugs. With n denoting the total number of athletes, n{d then constitute the number of athletes not taking drugs.

Given the above assumptions 3) and 8), there will be two groups,

one group taking drugs and another group keeping clean. The first

d athletes (ranked first, second, . . . ,d th) take the drug, while the n{d ‘‘clean’’ athletes will occupy ranks dz1,dz2, . . . ,n. We

refer to this two groups as the D-group and the ND-group, respectively. Again, given assumptions 3) and 8), all drug-takers

are equally good and all ‘‘clean’’ athletes are equally bad, leading

to the following probability densities for the two groups:

P( Player i has ranking j D Player i [ D{group )~ 1 d

if jƒd

0 otherwise

( ð4Þ

P( Player i has ranking j D Player i [ ND{group )~

0 if jƒd

1 n{d

otherwise

( ð5Þ

Pay-off Functions Now, given assumptions 1) - 10) as well as derived equations (4)

and (5), the pay-off functions of the players can be defined. It will

prove convenient to define pay-off functions for the D-group and the ND-group separately. We define:

PDi ~E( pay-of f f or athlete i in the D -group ) ð6Þ

and

PNDi ~E( pay-of f f or athlete i in the ND -group ) ð7Þ

PDi and P ND i can be expressed using straightforward expected

value calculations as follows:

PDi ~ 1 d : Pd j~1

aj {ri ci if dw0

0 if d~0

8>< >: ð8Þ

and

PNDi ~ 1

n{d : Pn

j~dz1

aj if dvn

0 if d~n

8< : ð9Þ

It is straightforward to show by applying equations (8) and (9) that

the original 2-player performance-enhancing drug game [21]

emerges as a special case of the above model. The content is left

for Supporting Information File S1, section 1.

Analytical Results We performed a full scale computation (finding all possible NEs)

for the three-player case (n~3) while we settled for a partial analysis for the n~4-case. The reason is partly due to the overwhelming amount of analytic work involved, but more

importantly, our analytic efforts lead us to formulate and prove

two propositions which answer quite many relevant questions

related to the behaviour of the multi-player performance-

enhancing drug game.

Multi-Player Doping Game

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Results for the n~3-case are shown in Tables 1, 2 and 3. The corresponding derivations and the results for the n~4 case are included in the Supporting Information File S1.

As a simple decoding scheme is used to describe various Nash-

equilibria in Tables 1, 2 and 3, some further explanation is

necessary. A letter describes the behaviour of one player. Each

letter may be D (doping with certainty), F (plays fair with

certainty), m (mixed strategy) or * (any pure or mixed strategy). For

instance, DDF describes a strategy where two players cheat and

the third plays fair all the time, while mmD is a strategy where two

players play mixed strategies and the third cheats all the time. The

order of letters does not matter, mmD is the same as mDm or

Dmm. Note that there is a fine distinction between m and *: m

denotes a player playing a mixed strategy with a specific

probability of doping, while * denotes a player that can take

drugs with any probability without any effect on its own payoff.

Furthermore, Tables 1 and 3 contain Ad ~

Pd i~1 ai

d , i.e. the

average prize received by a player that cheated if there are d

cheaters in total. �AAd ~

Pn i~dz1 ai

n{d , i.e. the average prize remaining

for a fair player if there are d cheaters is also introduced. Note that �AAd also is a function of n and a more correct notation could be �AAnd . However, to avoid notational confusion, we stick to this simplification.

As can be readily observed from Tables 1 and 3, certain rc-

thresholds Ad { �AAd{1 � �

compose various regions of different NEs.

Similar rc-thresholds are found in the four-player case, leading to

a ‘‘regionalization’’ that is very similar to the three-player case, but

the number of regions is one larger and some of the conditions are

more complex to allow for cases when the prize function is not

strictly concave or strictly convex (unlike the three-player case,

where every prize function is concave, convex or linear). A more

comprehensive discussion and derivation of the four-player case is

to be found in the Supporting Information File S1.

Comparing Tables 1 and 3 indicate a striking difference. Apart

from the (obviously) undetermined rc~A2{ �AA1 case, either everybody plays fair or everybody cheats if the prize function is

linear, while more complicated Nash equilibria may appear if the

prize function is concave or convex. The question of whether this

simple NE characterization also is present in the n-player case led

us to formulate (and prove - refer to Supporting Information File

S1, section 2 for the proof) the following:

Theorem 1 Given an instance of the n-player doping game, a common rc product for all players, and a linear prize structure (i.e.

a common b~a1{a2), the Nash-equilibria are as follows:

1. Everyone cheats when rcv n{1

2 b.

2. Everyone plays fair when rcw n{1

2 b.

3. Any pure or mixed strategy when rc~ n{1

2 b.

Possible practical implications of Theorem 1 are discussed in the

Conclusions section.

One obvious point to look at when analysing a multi-player

version of the performance-enhancing drug game is how NEs

evolve as the number of players change. Furthermore, as

predictability (normally) is viewed as important related to doping

regulation, general knowledge related to when (under what

parametric conditions) will NEs be pure or mixed might prove

valuable in a practical context. These questions (and others) are to

some extent answered by the following theorem (refer to

Supporting Information File S1, section 2):

Theorem 2 Given an instance of the n-player doping game and a common rc product for all players, the sufficient and

necessary condition for the ‘‘k players cheat and n{k players play

fair’’ pure strategy to be a Nash-equilibrium is as follows:

Akz1{ �AAkƒrcƒAk{ �AAk{1

where Anz1 and �AA{1 are defined to be {?. As mentioned above, one obvious thing to look at when

analysing a multi-player game, and comparing it to a two-player

version, is how NEs change when the number of players change.

Theorem 2 may be applied more or less directly to shed some

interesting light on this question. Let us define Ln as the length of

the interval An{ �AAn{1,A1{ �AA0 � �

; that is,

Table 1. Summary of the three-player case - non-linear prize function.

rc Cases of Nash-equilibria

Less than A3{ �AA2 DDD

Exactly A3{ �AA2 DDD, DDF if b1vb2 or DD* if b1wb2

Between A3{ �AA2 and A2{ �AA1 DDF, Dmm, mmm

Exactly A2{ �AA1 mmm with p~0:5

Between A2{ �AA1 and A1{ �AA0 DFF, Fmm, mmm

Exactly A1{ �AA0 DFF, FFF if b1vb2 or FF* if b1wb2

Greater than A1{ �AA0 FFF

doi:10.1371/journal.pone.0063306.t001

Table 2. Exact probabilities of mixed Nash-equilibria for the three-player case. bi is defined as ai{aiz1, i.e. the prize difference between rank i and rank iz1.

Case Probabilities

mmm p1~p2~p3~3

rc{b1

b2{b1 {1

Dmm p1~1, p2~p3~6

rc{b1

b2{b1 {3

Fmm p1~0, p2~p3~6

rc{b1

b2{b1 {2

DD* 0ƒp3ƒ1p1~1, p2~1;

FF* 0ƒp3ƒ1p1~0, p2~0;

Note that DD* and FF* appears only if the prize function is convex, i.e. b1wb2 . doi:10.1371/journal.pone.0063306.t002

Table 3. Summary of the three-player case - linear prize function.

rc Cases of Nash-equilibria

A2{ �AA1 Less than DDD

A2{ �AA1 Exactly ***

A2{ �AA1 Greater than FFF

doi:10.1371/journal.pone.0063306.t003

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Ln~A1{ �AA0{Anz �AAn{1 ð10Þ

This interval contains NEs which are either mixed or pure with

a mixture of player behaviour. Hence, outside this interval either

everybody plays fair or everybody cheats. If we can identify the

behaviour of Ln as n increases it could be meaningful in a regulative setting. Let us investigate the contents of Ln closer.

A1 is by definition the maximal prize amax, while �AAn{1 is the minimal prize amin. The remaining two elements of Ln are both

equal to the average prize, �aa~ 1 n

Pn i~1 ai . Hence, equation (10)

can alternatively be expressed as:

Ln~amaxzamin{ 2

n

Xn i~1

ai ð11Þ

In order to judge Ln’s monotonicity, it turns out to be easier to assume a normalized prize structure. That is, we convert the given

prize function to a version where Pn

i~1 ai~1. Given this

assumption, equation (11) can be further simplified to:

Ln~amaxzamin{ 2

n ð12Þ

Assuming that amax and amin are constants, the only variable in this equation is n. When n increases, 2=n decreases, {2=n increases, thus Ln increases. But since Ln is the length of the unpredictable region with mixed NEs or pure NEs with mixed

player behaviours, the length of the predictable region decreases.

Therefore, when n increases, Ln increases, thus the length of the predictable region decreases.

Conclusions and Suggestions for Further Research

Conclusions Based on our results, one obvious conclusion to draw is the

extent of complexity increase in NE structures observed simply

when moving from a two-player to a three-player game. The

contents of Table 1 shows clearly that almost any possible NE can

be a game prediction. This constitutes a major difference from

existing game theoretic doping research - see for instance [21],

[20] or [23]. This is perhaps not surprising as such, but we did not

predict such a complexity increase before conducting this research.

In any case, some caution should be taken when generalizing

doping research based on two-player models when practical

regulative means are considered.

Furthermore, the results derived through Theorem 1 provides

valuable insight. A popularized version of the theorem could be to

say that if the prize function is linear and rc is constant (homogeneous athletes and doping tests), the n-player game behaves just like the two-player game. By itself, this is an

interesting result, but we should be cautious in drawing policy

implications too far. It could be tempting to say; then it is easy to

fight doping, both in 2-player as well as in any-player competi-

tions, by making all drug testing as well as punishment

standardized and change all prize functions to be linear. If that

can be achieved, then merely a single doping test on one athlete is

enough. Surely, this is way too far drawn. As Figure 1 indicates,

real world prize functions are not linear, and they should not be

linear either - see for instance [24] and [25]. The point is very

simple, demand for sport is (typically positively) related to the

effort that the athletes put into competing. Fans want to see

athletes trying harder, an egalitarian prize function does not lead

to optimal performance, and hence, the problem of fighting

doping can be said not to be independent of total sports demand.

As such, as noted by other authors, an optimal level of doping may

exist [21]. Still, as our results indicate, standardizing doping tests

as well as punishments (rc kept constant) makes sense as it reduces complexity in analysing doping games.

The popular version of Theorem 2 as depicted in equation (12)

is interesting. This equation tells us that the region of unpredict-

ability increases with the number of players in the game. To some

extent this could indicate that keeping the number of players low

actually is a good idea. Again, we should take care in pulling the

policy implication string too long, but to fight doping, being able to

predict athletes’ choices regarding doping seems evident. As such,

a smaller number of competitors should (at least theoretically) be

better than a larger number.

A practical example might be interesting to judge. Modern cross

country skiing has undergone a major structural change the latter

years. To a great extent, the classical competition structure of

interval starts have been replaced by an increasing number of

common (mass) start events. The actual reason for this changer is

of course labelled on demand and TV-viewers - who presumably

prefer to see a competition where the first man crossing the finish

line is the winner. The fact that the time spent in mass start events

is significantly shorter is of course also judged positive when

competing to get TV-time. In an interval start event, even though

you principally compete with the same number of competitors,

your actual number of real competitors (based on your perfor-

mance quality compared to the rest of the heat) is in practice way

below the total number of participants. However, in a mass start

event, the number of possible winners increase dramatically,

meaning that you efficiently compete with a much larger group.

Hence, based on this argument, if doping is to be controlled, this

development is perhaps not the best. Still, again, this is a totality,

and doping neither could nor should be fought regardless of

possible adverse demand effects. This underlines the complexity of

the task at hand.

Finally, the role of the prize function becomes evident when a

doping game is extended from a two-player to an n-player game. Our results in Table 1 show clearly that the shape of this function

is an important determinant for existence of various Nash

equilibria. To some extent, this is obvious, as a two-player game

can only contain a linear prize function. The possibility of dividing

prizes egalitarian or not opens up when the participation number

is larger than 2. Still, we feel that the results we have derived (even

if they are not perfectly complete) warrant greater caution from

sport officials when it comes to how prize functions are designed.

To some extent, other research points at a similar problem - see

for instance [26] and [27].

Suggestions for Further Research Handling multi-player games is in general far more difficult,

both model- and solution-wise compared to 2-player games.

Definitely, this is the situation we have faced conducting this

research. As a consequence, we have carefully chosen to make

assumptions which may seem crude compared to reality.

Assumptions 2), 6), 7) and 8) (see the section on Assumptions

and basic modeling), as well as the inherent not explicitly

mentioned assumptions of information completeness and non-

existent prize re-awards [5], are all examples of conditions that

may seem far from reality.

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Assumption 2) - homogeneity of player’s ability is a prime

candidate. Most two-player research - see e.g. [7,12,21] tend to

make such an assumption initially and then at later stages relax it.

From a practical point of view such an assumption may at least to

a certain extent seem sensible in a 2-player setting, but far more

dramatic in a multi-player setting. However, relaxing such an

assumption - in a multi-player setting - provides significant added

challenges in the modeling as well as in the analysis stage. The

reason is relatively straightforward to explain. In a multi player

heterogeneous player situation, players may naturally be separated

through a common knowledge probability distribution. Such a

probability distribution must be 2-dimensional in this setting, both

in prize as well as player dimensions. As a consequence, any given

player must have a probability distribution for all possible prizes

and all prizes must be won by at least on player. This fact

introduces two probabilistic (norming) constraints which turns out

to create significant modelling problems. In addition, the effect of

doping must be included as well, leading to a situation where the

model itself must account for all possible drug allocations and

probabilistic consequences for all possible combinations of players,

prizes and drug taking habits.

Unfortunately, this is not the only problem involved in relaxing

the homogeneity of player’s ability assumption. Note that the share

existence of a common knowledge ability 2-dimensional proba-

bility distribution actually means that all players must know (and

agree) on such a distribution and not only that, they must also

know and agree on all possible (individual) drug effects. So, to sum

up: relaxing this assumption leads, in our opinion, to an obvious

question related to the inherent complete information assumption.

In practice one would expect that an individual player would know

more on his abilities as well as the effect of taking drugs than his

opponents. As a consequence, a game of incomplete (and

asymmetric) information becomes even more appropriate. This

is, as we see it, obviously a candidate for further research.

Similar types of (of course not identical) problems emerge if

other assumptions are to be relaxed. For instance a relaxation of 7)

- introducing more than one drug and/or individual drug effects -

creates totally different modelling environments. This also holds

for an introduction of prize re-awards. Prize re-awards [5] relates

to the practical fact that if a player is caught in doping activity

certain costs emerge. Such costs are (at least in principle)

accounted for in our model through the ci parameters. However, a doping verdict also has other consequences, namely that of prize

re-awards - all other (not yet caught in doping) moves up one place

on the prize-list. Surely, such mechanisms holds potential for

significant game theoretic consequences, as reported in 2-player

research [5,7,9], but we again feel that this subject needs projects

of its own.

Figure 1. The ‘‘prize function’’ of the PGA tour [28]. doi:10.1371/journal.pone.0063306.g001

Multi-Player Doping Game

PLOS ONE | www.plosone.org 6 May 2013 | Volume 8 | Issue 5 | e63306

We have focused on modelling and solving a multi-player

doping game. We have found interesting results related to the

shape of prize functions and some results related to NE behaviour

as a function of n. Obviously, we have not solved all possible

modelling issues and it is relatively easy to criticise our models as

being crude and unrealistic. Whilst it is true, we would welcome

further investigations relaxing our assumptions and gaining even

more insight into the mystery of doping.

Supporting Information

Figure S1 The original pay-off matrix from [21].

(TIF)

Figure S2 Nash equilibria in the three-player doping game as a

function of the value of rc. Each Nash equilibrium is represented

by a box. Nash equilibria with the same rc product are in the same

column. rc increases from left to right. An arrow points from one

Nash equilibrium into another if changing rc transforms a Nash equilibrium into another one.

(TIF)

Figure S3 Nash equilibria in the four-player doping game as a

function of the value of rc. Each Nash equilibrium is represented by a box. Nash equilibria with the same rc product are in the same column. rc increases from left to right. An arrow points from one Nash equilibrium into another if changing rc transforms a Nash equilibrium into another one.

(TIF)

File S1

(PDF)

Author Contributions

Conceived and designed the experiments: KKH TN AP. Performed the

experiments: KKH TN. Analyzed the data: KKH TN. Wrote the paper:

KKH TN AP.

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