Doping and Uncertainty of outcome

Sports Economic Review 2 (2023) 100002 Contents lists available at ScienceDirect Sports Economic Review journal homepage: www.journals.elsevier.com/sports-economic-review Doping and uncertainty of outcome Kjetil K. Haugen Faculty of Business Administration and Social Sciences Molde University College, Specialized University in Logistics Molde, Norway A B S T R A C T This paper demonstrates, by simple classical game theory, that the claim by Savulescu et al. (2004) of a safer and fairer sport with legalized doping is a highly unlikely outcome. This result, with added arguments related to adverse effects on both demand and supply for the sports product, should hopefully affect the debate on legalization of performance-enhancing drugs. The Nash equilibrium obtained in the analysis predicts more doping, and maybe more importantly, use of more dangerous performance-enhancing drugs. As a consequence, a legalization of performance.enhancing drugs may threaten the actual existence of professional sports markets. 1. Introduction perhaps be summed up as: “What sane parents would send their small and fragile kids to training knowing that they will be fed on steroids?” Doping or use of performance-enhancing drugs (PEDs) is one of sports Unfortunately, this argument did not work. Savulescu and his colleges major problems. Not everybody agrees to this allegation. Oxford phi- have continued their crusade for legalizing drugs in sport. Numerous losophy professor Julian Savulescu (and co-authors), published a paper occurrences in media as well as research publications, Savulescu et al. in 2004, Savulescu et al. (2004) titled “Why we should allow (2005), Savulescu and Foddy (2005), Savulescu (2007), Foddy (2006), performance-enhancing drugs in sport”. The core argument of this paper Savulescu and Foddy (2010), Savulescu et al. (2013), Savulescu (2015), was that legalizing drugs would lead to a less dangerous situation for the Foddy and Savulescu (2014) have been produced up until today. As a athletes, as a list of legal drugs would lead to controlled production as consequence, I have realized that Savulescu's core arguments must be well as medical control for the athletes. Furthermore, Savulescu et al. attacked. Hence, the main point of this paper is to formulate a game argued that a fairer situation would also be the outcome, as the untal- theoretic based argument explicitly demonstrating that the “Savulescu-- ented, genetically unfit or simply poor athletes could use legalized drugs world” not necessarily will lead neither to fairer nor to safer conditions to equate previous performance differences against their talented, for athletes. Actually, as will be demonstrated in subsequent paragraphs, genetically fit or rich rivals. there are good reasons to predict the exact opposite – more doping, and Unfortunately, I observed Savulescu's paper 7 years after its publi- more dangerous doping. cation. However, after reading Savulescu's paper back in 2011, I found out that he had used a paper of mine, Haugen (2004) as a building block 2. Model discussion for his arguments. This paper, Haugen (2004) states what most papers within the “economics of doping” - literature do, that doping is hard and If we study research literature closely, it is almost completely free of almost impossible to fight. Hence, when something is almost impossible contributions looking at potential causality between uncertainty of to fight, Savulescu argued that one might just as well make it legal. Such outcome (UO) and doping.1 However, a noteworthy unpublished paper an argument may work for legalizing heroine, but not necessarily for exists, Bervoets et al. (2016). This paper written by Bervoets, Decreuse drugs in sport. and Faure, entitled “Doping and competition uncertainty” handles, as the My intention with publishing Haugen (2004) was clearly not to title indicates, exactly this potential causality between doping and UO. legalize drugs. Hence, I felt the need to publish a criticism of Savulescu's Unfortunately, the authors introduce a new mechanism into economics of original paper. And I did, in 2011, see Haugen (2011). I did not attack doping, which they name ‘the regularity effect’. As I understand this effect, Savulescu's main philosophical arguments (safer and fairer), but focused it is somewhat about athlete-self-confidence. Taking performance instead on possible adverse effects on the demand and (especially) the enhancing drugs has (at least) two effects. The actual physical effect supply side of sports by legalized drugs. My main argument could which they relate to the maximal level of the athlete, but also (a more) E-mail address: kjetil.haugen@himolde.no. 1 Refer for instance to Rottenberg (1956) for an introduction to this topic. https://doi.org/10.1016/j.serev.2022.100002 Received 27 May 2022; Received in revised form 5 July 2022; Accepted 6 July 2022 Available online 19 July 2022 2773-1618/© 2022 Elsevier Ltd. All rights reserved. K.K. Haugen Sports Economic Review 2 (2023) 100002 mental effect related to their ability to perform at their best in an important competition. This effect of athlete's feeling superior and invulnerable, knowing they have taken the “master-drug” may very well be present. The problem with this added ‘the regularity effect’ is of course that it must be paired with a potential negative effect of similar type. That is, the effect of bad conscience or guilt. Surely, any athlete can get a positive boost of knowing he has an advantage. But, as this boost is achieved through cheating and illegal behaviour, one must also expect a potential negative effect. An effect that very well may outweigh this positive self confidence effect. Anyway, given that one accept this sin- gular added effect, the author show mathematically that the outcome very well may be less rather than more UO, as opposed to Savulescu et al.‘s claim. That is, the playing field is not levelled at all, it ends up more unlevelled than before. A quote from their article is enlightening: “Our results inspire a widespread debate: many observers suggest that authorities should give free access to doping. It is argued that this will level the playing field and restore competition uncertainty. Our model suggests precisely that the opposite will happen. If offered the possibilities of using PEDs, the best athletes will use them maximally, thereby further reducing less talented players' chances of winning. Competitions would degenerate into deterministic events.” Fig. 1. A normal form version of the model. Of reasons discussed above, I find Bervoets et al.‘s model too artificial, cheating is still possible. This means that a game model at least must and will in subsequent sections show, that the effect of a more unlevelled contain three possible drug options – say ND; DI and DL for athlete as well as more unsafe playing field is a quite probable outcome of choice. Here, it is assumed that the players (or agents as they are called following Savulescu et al.‘s suggestions given a reasonable game model; here, to keep the original notation in Haugen (2004)) hence can choose building heavily on standard classical economics of doping literature. to avoid doping ðNDÞ, choose the (now) legal set of doping means ðDL Þ or Before the model is discussed in detail, a couple of quotations from (still cheat) and choose the illegal set of doping means ðDI Þ. the original Savulescu et al. (2004) article may be helpful: Furthermore, we need heterogeneous agents, which is not the case in the original performance-enhancing drug game in Haugen (2004). Q1: On page 666: However, this case is treated in the same article – refer to subsection 2.5 “If we made drugs legal and freely available, there would be no cheating.” in Haugen (2004). Given heterogeneous agents, an assumption referred to as “the magic drug assumption” is relevant. This concept only means that the effect of doping always leads to a secure win if one agent choose drugs and the other not – independent of quality differences between the Q2: On page 668: agents without any drugs. This may seem as a strict assumption. Still, at the very top, quality differences between the very best are marginal, and “By allowing everyone to take performance enhancing drugs, we level the doping may be the tool to make the difference. After all, this is why playing field.” doping is considered to be a problem, is it not? Anyway, given a triplet of choices for the agents, we need to extend the magic drug assumption slightly. A simple and straightforward way of Q3: On page 668: doing this is to state that in this case, the choice of the illegal drug ðDI Þ always beats any other strategy, while the choice of the legal drug ðDL Þ “We should permit drugs that are safe, and continue to ban and monitor secures victory against a no-doping strategy ðNDÞ. drugs that are unsafe.” Given these extended assumptions, the normal form simultaneous Q1 states the obvious. If there are no laws, there is no crime. Still, the complete information game in Fig. 1 in section 3 can be derived. The quotation indicates that the idea is to legalize drugs. Q2 indicates that explanation of the individual pay-offs is left for (this) section 3. competitive balance should improve, or as Rottenberg (1956) named it, uncertainty of outcome, which should increase. As will be demonstrated 3. The model shortly, this is a truth with some modification. Q3 is the important one. It shows the core of Savulescu's proposed changes, although many readers A quick explanation of the pay-offs in the model in Fig. 1 is probably may see an obvious contradiction to Q1. It is not a matter of legalizing necessary. Let us start by examining the diagonal. That is, the strategy drugs, merely making the WADA-list shorter, which of course is a far less combinations fDI ; DI g, fDL ; DL g and fND; NDg. revolutionary suggestion than Q1.2 The heterogeneity of agents is modelled in Haugen (2004) by the However, this is a minor point here. The interesting point, is that drug introduction of a probability density [p, 1 p] and an added assumption testing is still necessary (continue to ban and monitor drugs) and hence, of p > 12.3 Hence, without loss of generality agent 1 is assumed better than agent 2.4 The same assumption is applied here. Then, for the fDI ; DI g strategy combination, agent 1 wins the fight against agent 2 with 2 There is of course obvious reasons why Savulescu et al. can not really sug- gest Q1. Empirical evidence does in fact indicate that doping may be very dangerous – deadly. Many have probably heard about unlucky Tom Simpson 3 (see History.com (2022)) who passed away busily pedalling towards the top of Note also that p ¼ 1 is uninteresting as in that situation, agent 1 will always Mont Ventoux during a stage in Tour de France in 1967, with his pockets filled get a certain win if he chooses to copy agent 2's strategy, That is, he can secure a with amphetamine. It may be more unknown that between 1987 and 1990, 20 win no matter the choice of agent 2. No doping problems in such a situation. 4 young Dutch and Belgian cyclists died from nocturnal heart attacks – see The The term better means here that agent 1 will win more than half of poten- Lancet Haematology (Editorial) (2016). tial/upcoming competitions against agent 2. 2 K.K. Haugen Sports Economic Review 2 (2023) 100002 probability p and earns expected utility of pa when a is the prize of winning. Consequently, agent 2 earns (1 p)a. Both agents use an illegal drug and will face the expected revelation cost of rc. Obviously, also the assumption of equal performance-enhancing drug effect (on both agents) is kept from Haugen (2004). The remaining two elements fDL ; DL g and fND; NDg gets the same prize sharing, but as the drug used now is legal, no possibility of doping revelation exists, and they now only share the prize money [pa, (1 p)a]. The topmost row as well as the leftmost column will by the previous “magic drug assumption” (see section 2) lead to a secure win for the agent choosing the “strongest drug”.5 As a consequence, as all of these strategy combinations contain at least on agent choosing the illegal drug, expected pay-offs are a rc for the winner and 0 for the looser. Here, r is the probability of being exposed for drug use, while c is the exposure- cost. The final remaining strategy combinations, fND; DL g and fDL ; NDg secures a certain win for the agent choosing DL but as the win is achieved by using a legal drug, no expected revelation costs exist. That is, the a rc in the previous case is changed to a. The game is analysed in the next section, section 4. Fig. 2. Best reply for agent 1 given the choice of DI by agent 2. 4. A minimal model analysis agent 1, the following optimization problem must be solved: In order to find Nash equilibria (NEs) in the game from Fig. 1, best replies need to be found.6 In order to perform this task, we adopt the maxfa rc; ð1 pÞa; 0g ¼ a rc (5) basic assumption in Haugen (2004). That is, Obviously, with p 2< 1 2 ;1 >, (1 p)a > 0. Furthermore: (some simple a≫c (1) algebra) Then, it is straightforward to argue that7: a rc > ð1 pÞa ⇒ a rc > a pa ⇒ pa rc > 0 (6) 1 a > rc: (2) which we saw holds above in equation (4). Hence the answer in (5) is 2 correct. Furthermore, shifting to a choice of ND for agent 1, a rc > (1 Also keep in mind the previous assumption given by heterogeneous p)a. But, with the assumption of r, c > 0, then a > a rc, and the opti- agents: mization problem in this case is solved to be: 1 maxfa rc; a; ð1 pÞag ¼ a (7) p> (3) 2 The graphical progress after this two optimizations (using rectangles Let us start with the best reply for agent 1 given a choice of DI for to represent agent 2's best replies) is shown in Fig. 3: agent 2. From basic game theory, the solution to the following optimi- Finally, focusing on the best reply for agent 2 given a choice of DI by zation problem defines the best reply function: agent 1. Then, the following optimization problem is to be solved: maxfpa rc; 0; 0g ¼ pa rc (4) maxfð1 pÞa rc; 0; 0g (8) The answer is obvious, as 1 2 > rc or a rc > 0 (by assumption 1 2 a Now, we already know by (4) that pa rc > 0. Let us use this in- inequality (2)), when p > 12, then pa rc must also be positive. Using formation and at the same time claim that (1 p)a rc > 0. This results ellipses for best replies for agent 1 leads to a graphical representation of in: this result as shown in Fig. 2. Shifting attention to best replies for agent 2 and a choice of DL for ð1 pÞa rc > 0 (9) pa rc > 0 (10) 5 The fDI ; DL g, fDI ; NDg as well as the fDL ; DI g, fND; DI g strategy Adding left and right hand sides of inequalities (9) and (10) together combinations. gives 6 Given a continuous state space, a best reply function in game theory is defined as the optimal choice of action for any player as a function of all other ð1 pÞa rc þ pa rc > 0 ⇒ a pa rc þ pa rc > 0 ⇒ a 2rc > 0 players' possible actions. In this case, with a discrete state space. The best reply 1 (mapping) is defined likewise, but the discrete state space produces a discrete ⇒ a > rc 2 function or a mapping – refer for instance to equation (4) for how one element of (11) the best reply mapping is found in this case. In a two player game, given a continuous state space, best reply functions are continuous, and possible in- This is exactly the starting assumption (2), indicating that the solution tersection(s) between the two best reply functions define Nash equilibria. The to (8) must be (1 p)a rc. Adding the fact that pa > 0 as both p, and a interesting characteristic of such an intersection point, is that if one of the are assumed positive, the best reply for agent 1 given a choice of DL by players deviate from the point, the other will logically come out worse agent 2 can not be ND. (measured in pay-off), which of course is the underlying logic of the Nash Now, we are in a position to find all NEs of the game. We have not equilibrium concept. In this case where best reply functions are mappings, an found all best replies, but as Fig. 4 will show, this is not necessary. intersection of best replies become sub squares of a normal form game with both In Fig. 4, the fully drawn ellipses and rectangles indicate actual an ellipse and a rectangle within the sub square – refer to Fig. 4. 7 (identified above) best replies, while the dotted lined versions denote Refer to Haugen (2004). 3 K.K. Haugen Sports Economic Review 2 (2023) 100002 Fig. 3. Best reply for agent 1 given the choice of DI by agent 2 and Best replies Fig. 4. Partial set of Best replies. for agent 2 given the choices of DL and ND by agent 1. technological development indicate that the difference between talent potential best replies. The point here is simple. No matter which ones of quality becomes sparser and sparser while potency of drugs increase.8 As the dotted lined best replies will kick in, only one NE is possible her. That such, this assumption does not have to be that unrealistic, at least not in a is, both agents choose the illegal drug, or the unique NE fDI , DI g. That is, futuristic perspective. the game theoretic prediction does not give Savulescu et al. any credit. Furthermore, the possible relaxation of the “magical drug assump- Nothing is safer – athletes choose more dangerous drugs. No playing field tion” is performed in Haugen (2004). The results of that relaxation is levelled – the original heterogeneity is preserved and possibly even indicate that other NEs may exist, but they are (refer to Fig. 2.7 in Haugen increased as the resourceful athlete easier can handle a black market. (2004)) quite improbable. Although this analysis is not performed here, it feels very safe to predict similar results. Of course, such an analysis will 5. Conclusions and suggestions for further work be slightly more technical in this case, but far from intractable. It seems safe to conclude that doping in sports is here, and will be The findings in the previous sections indicate that the medicine pre- here, and that the only way of fighting it still is the somewhat boring scribed by Savulescu et al. (2004) does not cure the disease. On the advice of establishing a sensible mix between sanctions, quality of doping contrary, there are good reasons to predict that a system where the tests and prize money. Or, for the more adventurous sport re-designer, number of allowed drugs increase only will lead to an adverse siltation, make sports less doping vulnerable by adding new and many- where athletes will choose more dangerous drugs. There are no reasons dimensional elements as suggested in Haugen (2017).9 to believe that the playing field will level either. On the contrary, the rich and resourceful athletes will most probably end up with more potent (and probably also) slightly less dangerous illegal drugs than their less Declaration of competing interest resourceful and poor competitors. As such, this reorganization of doping seems like a bad idea. Adding previous criticism from Haugen (2011) The authors declare that they have no known competing financial related to adverse effects on customer demand (fans) and talent supply interests or personal relationships that could have appeared to influence does not make their suggestion better. As such, it is tempting to char- the work reported in this paper. acterize this suggestion among the less smart attempts to regulate pro- fessional economic activity. References The obvious problem in their suggestion is actually revealed already in the logical contradiction between Q1 and Q3. If Savulescu et al. had Bervoets, S., Decreuse, B., & Faure, M. (2016). Doping and competition uncertainty. dropped Q3 their suggestion would in fact work. If everything is legal, no Foddy, B. (2006). The ethics of genetic testing in sport. International Sportmed Journal, (3), 216–224. monitoring or WADA is necessary as no doping crimes are possible to Foddy, B., & Savulescu, J. (2014). Using steroids ethically. In D. E. Newton (Ed.), Steroids perform. On the other hand, a good handful of deaths are to be expected. and doping in sport: A reference handbook (pp. 122–124). Santa Barbara, Calfornia The key in endurance sports is of course some drug that kills human ABC-CLIO. ability to feel fatigue. Amphetamine products have this ability, but are easy to find on modern doping tests. However, it does not seem unlikely 8 The idea is simple. In highly professionalized sports, the competing agents that other substances with similar effects may emerge in the future, are in most cases very equal in performance quality. Then, an assumption always substances which may be much harder to reveal in testing and keep on guaranteeing the doper a win may definitely not be unreasonable. Actually, the producing dead athletes. fact that the effect of the drug might produce the tiny necessary advantage is the The model itself may of course be criticized. It is extremely simplified, very essence of the doping problem. and contains (like most mathematical models) some obviously unrealistic 9 The term many-dimensional elements refer to the arguments in Haugen assumptions. The most important candidate is perhaps the “magic drug (2017) related to the potential lack of doping effects in sports where more than assumption”. This assumption means in lay-man terms that if any agent one dimension (performance quality) is needed for success. In running for takes the drug and the other does not, the drug-taker wins with certainty. instance, only a single dimension (running fast) is needed. 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