Lecture 8: Game Theory (homework solutions)
Julia Collins
13th November 2013
The pirate problem
There are five pirates, labelled 1 to 5 with Pirate 5 being the most senior. Pirate 5 has 100
gold coins. He can choose to share some of it with the others. If at least half of them accept
his offer it goes through; otherwise the rest of them kill him. If he dies, Pirate 4 gets the
chance to do the same, and so on, down to Pirate 1. How much money can Pirate 5 afford
to keep? In other words, who does he need to bribe to vote for him, and how much?
The answer is that Pirate 5 can keep 98 gold coins, using 1 coin to bribe Pirate 3 and
another to bribe Pirate 1. Let’s see why this is true by working backwards.
First suppose that all the pirates have been killed except for 1 and 2. In this case, Pirate
2 will keep all the gold, since he will survive whether or not Pirate 1 accepts his offer.
Now suppose that pirates 1,2 and 3 are alive. In this case, Pirate 3 can easily bribe Pirate
1 to vote in his favour, since if Pirate 3 dies then Pirate 1 gets nothing anyway. Pirate 2
is not bribable, since he stands to gain everything if Pirate 3 dies. So Pirate 3 would give
Pirate 1 one gold coin as a bribe, with Pirate 2 getting nothing.
Now suppose that pirates 1,2,3 and 4 are alive. In this case, Pirate 4 can offer Pirate 2 a
bribe of one gold coin to vote for him. This is because Pirate 2 knows that if Pirate 4 dies,
then he would get nothing out of Pirate 3. Pirate 3 isn’t bribable since he would get the
lion’s share of the gold if 4 were to die, and Pirate 1 would only be bribable with two gold
coins since he’ll get one anyway if Pirate 3 is alive. So 4 gets 99 coins, 3 gets nothing, 2 gets
one coin and 1 gets nothing.
Which brings us to the situation with all pirates alive. In this scenario, Pirate 5 needs
two other pirates to vote for him if he is to stay alive. He can bribe pirates 1 and 3 with one
gold coin each, since these pirates know that they would get nothing if Pirate 5 were to die,
and one gold coin is better than nothing. So Pirate 5 can keep 98 gold coins.
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Prisoners
There are 3 prisoners locked into a room. There are 5 hats: 2 white and 3 black. Each
prisoner is given a hat so that they cannot see the colour of their own hat but can see the
colours of the two others. Each minute, somebody is allowed to knock on the door of the cell
and tell the jailor what colour hat they think they have. If they get it right, they go free, but
if they get it wrong they get executed. If all the prisoners are given black hats, how many
minutes will it be before they can figure this out, assuming they are all completely rational
but unable to speak to each other?
First notice that if any prisoner sees the other two wearing white hats, they will know
immediately that their own is black. So if the hats are Black, White, White then somebody
will knock on the door within the first minute.
Now suppose that the hats are Black, Black, White. One of the people wearing a black
hat will think to herself, ”If my hat is white, then within the first minute the person with
the black hat will jump up and knock on the door.” Since this does not happen, the prisoner
knows that their own hat cannot be white and so must be black, knocking on the door in
the second minute.
What if the hats are Black, Black, Black? A prisoner will think “If my hat is white, then
in the second minute, someone will knock on the door to talk to the jailor.” Since this does
not happen, then in the third minute all prisoners will know their hats are black and will
knock on the door.
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