The Olympic drugs prisoner’s dilemma

I woke up this morning with a headache, and the only thing I could think about was Olympic athletes taking drugs.

After all, why do we have such a problem with it.  If elite athletes decide to take drugs and do really cool stuff that we can watch in our own homes isn’t this all good.

People say stuff like “it isn’t in the spirt of the game” or “they are supposed to be role models for kids” but this sounds like a bunch of blah – we want our elite athletes fighting in cages with robot body parts and hyped up on all sorts of drugs right!

Well let’s assume we do – just for fun.  Even in this case there is a clear reason why we may want to try banning drugs – the prisoner’s dilemma faced by athletes.

Assume there is some inherent ranking of athletes, but that the difference in ability from bottom to top in some set is 20%.  Then assume there is a drug that will boost performance by 21%.  Then the “worst” athlete could take the drug and be the winner!  Of course, all the athletes up to the “second-best” know they can improve their position by taking the drug – and they know, as does the “top” athlete, that if someone else is taking the drug the only way to maintain their position is to take the drug also.  As a result, even though the drug taking has a cost (shorter life) it is a dominant strategy.

This is a bit loose, as it is not actually a dominant strategy for the worst athlete – as if everyone else is taking the drug it doesn’t change his position, he just has lower health, so won’t bother with it.  But then we have the potential for a mixed strategy equilibrium with the second-to-worst athlete (as if the bottom athlete doesn’t SWA won’t want to, but if SWA doesn’t the worst athlete will want to, so they will both take drugs with some positive probablity less than 1) – and in the end we aren’t going to worry about this too much 🙂

So say this prisoner’s dilemma holds – we can ensure that the same “outcome” in terms of ranking takes place, and that the athletes have better health than they would get taking drugs, if we just ban drugs and enforce it well.  Drugs in this case are banned from the Olympics because we care about the health of athletes, isn’t that nice!  This illustrates the useful role of regulators and government in the face of issues to do with co-ordination or prisoner’s dilemma types games!

Update:  Now I’ve woken up, this story is dumb.  Better game to get my point across – take two equal athletes that both have a 50% chance of winning.  Taking a drug can make this 100% if the other person is not taking it, or 50% if the other person is, but there is some positive cost to taking it.  In that case both athletes take the drug – both end up in the same position as not taking – and both have worse health than not taking.  Why didn’t I just say that?

However, the idea that drug taking for the best athlete is a strategic complement (when someone else takes it increases the incentive for them to take), but for the worst it is a strategic substitute (when someone else takes it decreases the incentive for them to take), is pretty cool – that makes for a neat game to think about!