Isner and Mahut in a prisoners dilemma

Daniel Rothschild
June 23rd, 2010
Culture

So the longest Wimbledon match in history is currently underway. As I type, Isner and Mahut are tied 56-56 in the last set; they’re tied 2-2 in previous sets.

According to reports (and common sense), the two men are exhausted and grimacing with each serve (though they seem to have rallied when they crossed the 10-game threshold). My guess is that they arms will be completely shot by this for the next few days. As a result, whoever wins this match will be at a severe disadvantage going into the next round, and will probably be so exhausted that he will lose. In other words, assuming winning the tournament is the goal and one that’s much more prized than merely advancing to a second round, it’s a classic prisoners’ dilemma. The dominant strategy by each player is to keep playing, even though every game further diminishes his likelihood of winning the next round, and therefore taking them out of contention for an overall win. In other words, every additional set that Isner and Mahut play decreases their chances of making it to the third round and thus reduces their combined expected utility.

Would there have been a way for them to have reached an optimal solution earlier on? Obviously sportsmanship precludes any kind of (spoken) match fixing, but mightn’t there be a rule change allowing a set to stop before it becomes a death match? For instance, before a match begins, players could agree to opt out of the normal rule and set a limit on the maximum number of sets they’ll play in a final, fifth match before the need to win by two games is eliminated. Setting this rule from the beginning of a match allows the players to circumvent the prisoners dilemma; whoever wins the match is more likely to win the tournament than in the absence of such a rule.

Efficiency concerns are seldom used in sports; we prefer the epic struggles and Pyrrhic battles. But that doesn’t obviate a little sideline thought experimentation.

Steve Horwitz hits some dingers

donaldkagan

This is not a prisoners dilemma, it is a Pyrrhic battle. These two things don't really have any overlap at all.
donaldkagan

This is not a prisoners dilemma, it is a Pyrrhic battle. These two things have no overlap.
http://twitter.com/merrillaaron Aaron Merrill

Actually, advancing even one round can have a monetary advantage. Payouts increase every round. http://wiki.answers.com/Q/What_is_the_wimbledon…

Point taken though.
danrothschild

Donald,

I thought about that before I posted, and I dashed the post off without thinking too much about it. My thought was that both players going into a first round match were eyeing a big prize at the end, and intermediate prizes as relatively insignificant. As it exists under the current rules T > R, because T is some expected value of the utility of winning the tournament (not this round), and R is nothing. P and S are also nothing. So T > R >= P >= S (that was how I was thinking about it); you're right, that's not technically a prisoners dilemma. I was thinking that the only way to “win” is to win the tournament, and the only way to win the tournament is to constrain play in early rounds.

I was mostly interested in thinking about what rules might lead to more efficient outcomes in the future. And I dashed off my post without fully thinking it through (this is, after all, a blog). Thanks for correcting me.
donaldkagan

Anything for a Rothschild.
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