Tuesday, August 31, 2010

Prisoner's Dilemma at the swimming pool

At my vacation I was witness of "beach chair reserving behavior". As soon as the pool opens, some guests reserve their beach chairs by putting towels on a couple of beach chairs. Then they go for breakfast or whatever. So, some time later, there are several empty but reserved beach chairs around the pool. Wanting no trouble, people have to sit on the ground. Doing some quick count during the day, I noticed that there were 20 beach chairs and - surprise! - in average only 20 people at the pool. So the system would work pretty well if nobody reserved the chairs and thus getting a good chance to find an empty chair when needed. For all the participants this would be a relief: the beach chair blockers don't have to get up so early in the morning and the others suffer less of beach chair shortage.

This can be modelled as a game theoretic problem, namely the multiplayer Prisoner's Dilemma. The Prisoners Dilemma is named after a fictional story where two suspects are interrogated regarding a major crime. The police have insufficient evidence for a conviction, so they offer the prisoners separately a deal: if one confesses (defects), he goes free (temptation payoff) and the other one gets a high conviction (sucker payoff). However, if both confess, both get punished (punishment payoff). If no one confesses, both get a less severe verdict, in overall the best for everyone (reward payoff).

The following table shows the possible strategies and payoffs (exemplified with the payoffs 0,1,2,3, the higher the better):

Prisoner B stays silent (cooperate) Prisoner B tells (defect)
Prisoner A stays silent (cooperate) (2,2) both get off with small sentence (0,3) Player A gets punished for everything, Player B goes free
Prisoner A tells (defect) (3,0) Player A goes free, Player B gets punished for everything (1,1) both get punished

At the pool, we have the case of a multiplayer Prisoner's Dilemma with the options to reserve a beach chair in the morning or to refrain from this behavior.

Most others do not reserve chairs (cooperate) Most others do reserve chairs (defect)
Player does not reserve chair (cooperate) (2,2) all get a fair chance for a beach chair when needed (0,3) player must sit on the ground, some others enjoy their reserved chairs
Player does reserve chair (defect) (3,0) player has guaranteed beach chair, others are suffering slightly (1,1) only chance to get a chair is reserving in the morning, worse situation than in upper left case

Unfortunately, with a sufficient number of defecting players (people who reserve a beach chair early in the morning) there is no merit in not reserving - you will go without a pool chair then (sucker payoff). So the only feasible strategy is to struggle for any free one in the morning and probably get one chair reserved for your family of 4 people. Another problem is that the people are constantly changing. So even if a cooperative behavior could be agreed on, there might be the arrival of a new bunch of defectors the next day, who would then feel themselves lucky to get all the chairs they desire so easily. There is certainly a tempation to reserve if nobody reserves, because then you have your beach chair guaranteed (otherwise there is still the chance that you get none, if already more than 20 people are at the pool). So, defecting is a stable strategy, although it is in overall worse for the whole group - this a tragedy of the commons.


  1. *gg* So the best strategy would be for the hotel to forbid beach chair reservations.

    Btw: I think you mixed up the 0,3 and 3,0 fields in both diagrams.

  2. yes, as long as there is some superior (hotel, parent, goverment, ...) watching and caring we can resolve such Dilemmas straightforwardly. But often there is none or they don't care...

    Diagrams are corrected now, thx.