Pirates and Non-ergodicity

A while back a coworker shared with me a great article by Peter Livesey entitled “Pirates with PhDs”. In the article Peter shared an interesting yet accessible thought experiment that for me illuminated Game theory and Nash equilibriums in a way that Economics courses in college never had.

The non-intuitiveness of the pirate puzzle’s results also brought to mind another concept the author Nassim Taleb has referenced multiple times in his writings: non-ergodicity.

That is, the puzzle as presented excludes a consideration of time. The pirates are simply interested in maximizing their return from a single transaction. But, our lives consist of many transactions. And, a strategy that seems irrational for a single transaction is perhaps more rational when applied across a lifetime of transactions.

So, how might time affect the result of the pirate puzzle?

In a response to his initial article, Peter explored one possible way in which time might be applied. That is, what if the pirates played the game over and over, and each time the order of the pirates was randomized?

The results are pretty interesting, and Nash equilibria seem to exist for that scenario. Summarily, death avoidance takes on a greater role, and the results of a single game become even more lopsided, with a single pirate receiving all of the gold. But, over time each pirate is likely to end up with approximately the same amount of gold.

But, how else might time apply? And could some application of time to the scenario help explain the outcomes Peter observed playing the game with a group of friends, or how our instincts compel us to act (i.e. punish those who are too greedy, and act generously to avoid being killed).

An alternative incorporation of time

In Peter’s time scenario lopsided results in a single game are more tolerable because every pirate eventually has the opportunity to be the pirate who receives all the gold. There is an egalitarianism that arises from economic mobility.

But what if the pirates’ position in the lineup is static?

I am not an economist, but it seems to me that in this scenario no Nash equilibria exist.

With a static lineup the results for every game would match the result of a single game. No pirate can meaningfully change their earning potential. And no rational pirate would continuously participate in a game where they receive no gold.

Time and our intuitions

It is unclear from Peter’s experiment what the expectations were regarding how many games would be played and what the order would be from game to game. But, it should not be surprising that the theoretically optimal result for a single game fared poorly.

Life is obviously more nuanced. But, I think even a simplistic model that incorporates time supports our instinct to punish selfishness and unfairness, and reward and exhibit generosity.

We seem built to respond to situations in a way that is repeatable. And being biased towards a longer view would unsurprisingly be better for our longevity.