I am stuck in this question:
A pair of shoes consists of a left shoe and a right shoe, and can be sold together for $ \$10 $. Consider a coalitional game with $a+b$ players: $a$ of the players have one left shoe each, and $b$ of the players have one right shoe each.
Determine the core for each pair of positive integers $(a,b)$. (I have only seen very little examples regarding the problem of finding a core.)
This is called the glove game. When $a=b$ you need to make sure that all of the people with right shoes get the same amount of cash and you need to make sure that all the people in the left get the same amount of cash, this is because if this does not happen the left person who got the least cash can make an alliance with the right person that got the least amount of cash and they can make more money. Prove that when all the right people get the same amount of money and all the left people have the same amount of money the distribution is in the core. (So here there are multiple distributions in the core).
When $a<b$ you need to give $10$ bucks to each of the people with left shoes, we first prove a distribution cannot be in the core if a person with a right shoe gets money. To see this first prove that if more than $a$ people with right shoes get money it is not in the core (This is because you can cut the extra people out of the deal). Now Suppose you have distribution that is in the core, by what we just said at most $a$ people with right shoes get money. That means that there is at least one person who isn't getting any money. You can cut one of the persons with a right shoe from the deal and introduce the person with the right shoe that isn't getting any money to the deal, since he was getting no cash he'll do it for a pay as small as necessary. Therefore this distribution wasn't in the core.
So a distribution that is in the core gives the $10a$ dollars to people with left shoes. The only distribution that works is when each person with a left shoe gets ten bucks, to see any other distribution won't work notice that the person getting the least cash can get a person with a right shoe to work with him for as little money as necessary.
Therefore when $a<b$ the core is unique.
Notice that I didn't prove the things in the core are actually in the core, I can provide extra hints to do this if you require them.