Here’s another fun problem the solution to which illustrates fundamental concepts in game theory. (Other game theory problems and posts are listed under the game theory tag.) This problem is not hard to grasp but it takes a lot of words to set it up:
Ten cannibals live on an island. Their culture is very hierarchical and highly values rationality. To make the hierarchy explicit they’ve named themselves 1, 2, 3, …, 10. The order of the hierarchy corresponds to their number-name. Number 1 is dominant, 2 is second, and so on down to the lowly number 10.
A week ago they ate the last humans on the island, and they aren’t interested in the other flora or fauna available. The cannibals are now very hungry.
A moment ago a human named “Meathead” showed up, his sailboat having blown off course. Being the dominant cannibal, number 1 has the first option of eating Meathead. No other cannibal is permitted to touch Meathead before number 1 makes a decision as to whether or not he eats Meathead himself.
If number 1 eats Meathead he will get drowsy and fall asleep. (We assume cannibals always sleep after eating and at no other time. So these cannibals are very tired, as well as hungry.) If number 1 eats and sleeps, the number 2 cannibal has an opportunity to eat the number 1 cannibal while his defenses are down. But if number 2 eats number 1 he himself will fall asleep and he may be prey to number 3, and so on to number 10. If a cannibal eats the next dominant one he risks being eaten himself by his immediate subordinate during his slumber.
However, number 1 may choose not to eat Meathead. In that case, number 2 gets to make a decision about eating Meathead. If he eats Meathead he will sleep. If he sleeps he may get eaten by number 3 (and number 3 may then be eaten by number 4, and so forth). If number 2 passes on Meathead then number 3 gets a crack at Meathead…and so on.
Just to be explicit, cannibal number n may only eat Meathead (if given the option) or eat cannibal number n-1 (and only if n-1 is asleep). Cannibal n is not permitted to eat any other cannibal.
Number 1 gets the first move. What does he decide? What are the decisions of the other cannibals? Which cannibal, if any, eats Meathead? Which other cannibals, if any, eat? Which cannibals, if any, are eaten? Remember, these are rational cannibals. In fact, you can assume they all know the others are rational. (If you like, assume common knowledge of rationality.) That is, each will eat if (s)he can do so risk free. None will eat if it means certain death. I’ll post analysis on Monday and discuss an application in a subsequent post.