Yesterday’s post had a riddle about 10 very logical pirates trying to fairly split up 100 Pieces of Eight (PoE). Today we give the solution, which is a great example of backwards induction. What we do is first imagine an easier situation, then solve for more complicated solutions using that. Specifically, we’ll first assume we have 1 pirate, then 2, and so on, up to 10 (or however many we have on board!) For ease of notation, we’ll name the lowest ranking pirate on the ship #1, the second lowest #2, and so on. So if there are 8 pirates, #8 is the captain (and only gets 1/2 of a vote).
One pirate– This pirate would get 100 PoE, and be pretty happy about that.
Two pirates– #2 is doomed. He can offer #1 all 100 PoE, but pirate logic says #1 is happier with 100 PoE and a dead crewmate.
Three pirates- #3 is in a great situation. If he dies, so does #2, so he can keep all 100 PoE for himself, get 1.5 votes (#2 would rather live with 0 PoE than die with 0 PoE), and live.
Four pirates- #4 needs two votes. There’s no way #3 will vote his way, but if he offers #1 and #2 a single PoE each, he’ll make it with 2.5 votes, and 98 PoE.
Five pirates- Now we’re cooking. #5 also needs two votes. The cheapest pirate vote he can buy is #3 with 1 PoE. Then he can offer either #1 or #2 two PoE. This is tricky because the riddle is very sensitive to initial conditions (more on that later). Let’s say that both #1 and #2 have an expectation of 1 PoE. In any case, #5 gets to keep 97 PoE.
Six pirates– This guy needs to buy 3 votes. Again, #5 is useless, #4 only costs 1 PoE , and we need to buy 2 of the remaining three pirates, each of which costs 2PoE. Again, using expectations, each of the pirates #1,2,3 expects 4/3 PoE. #6 keeps 95 PoE.
Seven pirates– We give #5 a single PoE, and two of the bottom 4 pirates 2 PoE (so again, they expect 1 PoE), keeping 95 for himself.
Eight pirates– #6 gets 1 PoE, 3 of the bottom 5 pirates gets 2PoE (so the expectation is 6/5 PoE), #8 keeps 93 PoE.
Nine pirates- #7 gets 1 PoE, 3 of the bottom 6 pirates gets 2PoE (so the expectation is 1 PoE), #9 keeps 93 PoE.
Ten pirates- #8 gets 1 PoE, 4 of the bottom 7 pirates gets 2 PoE (expectation is 8/7 PoE), #10 keeps 91 PoE.
So there’s our answer, but tomorrow we come back with ridiculous numbers of pirates.