Revisit: Incan Gold and game theory

[I've been on business travel this week, so in the absence of original material, I'm reposting an article from last spring when I was first discovering Incan Gold.]


We had a family session of Incan Gold this afternoon [original post 16 April 2011].  An interesting development came up when my wife Kathy and I had bailed out of an expedition, and only my two sons Liam and Corey remained to explore the ruins.  One instance each of three different monsters had been turned up, which meant that there was a very real possibility that a second monster of one type would appear and scare the remainder of the party out of the ruins at any point.  But then an artifact showed up, and a very interesting stand-off ensued.  By the rules of the game, if there are two or more people in the expedition, neither gets the artifact, and it stays on the card.  In a subsequent turn, if exactly one of the remaining two people decides to return to his tent, he gets all treasure left on cards from previous turns - including the coveted artifact.  If both players turn back, neither gets the artifact, and the round is over.  If both continue on, both continue to share discovered treasure but risk encountering a monster and losing everything.

What followed was an almost comical staring contest between the two of them to try to figure out whether the other was going to stay or return, and therefore whether to return (in hopes that the other was staying, which would leave the artifact to the returning player) or stay (and keep any subsequent treasure for oneself).

The decision to turn back or to continue is simultaneous among remaining players, so the result is a fairly classic game theory problem, in which the outcome of a decision depends upon an opponent's simultaneous unknown decision.

Own decision  Opponent decides to stay  Opponent decides to go
Stay          Turn over another card    Opponent gets artifact
Go                  Get artifact          Nobody gets artifact


Since "Turn over another card" is mutually risky or mutually beneficial but in no case advantageous for one player over the other if both players stay, then game theory would conclude that the only logical decision would be to go.  But if both players decide to go, then neither gets the artifact.

The piece that's missing in my decision table above, however, is that if either player stays, another card will be turned over, to the risk or benefit of the player(s) staying.  So there might be an advantage to staying if a player perceives a potential treasure greater than getting the artifact.  But that's really unlikely, in fact, so the stand-off will typically end up in both players going back and neither getting the artifact. Having said that, however, the game actually plays unpredictably, and perceived risk and reward tend to rule over cold logic.

We've really come to like this risk management game.  I'm apparently way too conservative, however.  I came in last today, and Corey (10) beat us all.  (I seem to recall that he ended up with the artifact more than once, by the way.)