Question

There are 2 players: Albert and Barbara. Albert has 100 cards which he distributes in 2 buckets. After that, he leaves 1 bucket for himself, and offers Barbara the other one. Barbara saw how Albert distributed the cards, but does not know which of the buckets Albert is offering her. Valuation of 1 card by Albert - 1 pound, valuation of Barbara -1.5 pounds. Barbara offers the price for the bucket offered to her, and Albert decides to accept the offer or refuse. They both want maximum payoff. If one player is indifferent, then he chooses an action that will benefit the other player.
a) In one bucket there are 100 cards, and in the other 0. Albert knows how many cards are in the boxes, but Barbara does not. What price should Barbara offer?
b) What is the best card distribution for Albert?

Answer 1

a) If there are 100 cards in one bucket and 0 in the other bucket. The bucket with 100 cards is worth 100 pounds to Albert and worth 150 pounds to Barbara.

Barbara knows there is one bucket with 100 cards and one with 0 cards and will be willing to offer 150 pounds for the bucket with 100 cards and 0 for the bucket with 0 cards but she doesn't know which one Albert is offering. Since they both will be trying to maximize their payoff, if Barbara offers anything more than 0, Albert will just keep the bucket with 100 cards. (For example suppose Barbara offers 75 pounds for the bucket, Albert will just keep the bucket with 100 cards, and take the 75 and he will have a total of 175, and give the bucket with 0 cards to Barbara)

Therefore Barbara will offer 0 in this case.

b) With above example in mind, Barbara will always be willing to pay according to her valuation of the bucket which has the lower number of cards. For example if Albert splits the cards as 60 in one bucket and 40 in another bucket, Barbara will be willing to pay at max 40*1.5 = 60 pounds for the bucket knowing that Albert will give him only the bucket with 40 cards even if she offers any more than that.

Best card distribution for Albert will be one where he gets the highest value.
Suppose he keeps X cards in 1 bucket and 100-X in another bucket. We can assume the bucket with X is the one with the lower or equal amount of cards i.e. X is less than or equal to 50 (as anyway for example X=40 or X=60 will give the same result of one bucket with 40 cards and another with 60, we are just providing a variable x to the one with the lower number of cards)

So If Albert puts X cards in the bucket with lower amount of cards and 100-X in the other one, Barbara will be willing to pay 1.5*X for the bucket that Albert offers her.
So Albert now has 1.5X+100-X = 100+0.5X

So Albert will have his best card distribution when he has as many cards in the bucket with lower amount of cards as possible such that he receives the highest amount possible from Barbara who values each card higher than him.

So he will split the bucket as 50 cards in one and 50 in another so that Barbara offers him 75, and his total valuation will be 75+50 = 125 (75 from barbara and 50 from his cards)

Edit: If they know each others' valuations, Barbara will pay as less as possible and Albert will accept if he gets greater than or equal to X which is the number of cards in lower bucket. The question specifically mentions that if one player is indifferent, he will choose the option that benefits the other. So albert will be paid 1 pound per card in the lower bucket and he will accept . In this case Barbara will offer him 50, Albert will get a total valuation of 100 pounds (50 cards and 50 From barbara) and Barbara will have a total valuation of 75 pounds (from the 15 cards)

Note: He cannot increase it any further as if he puts more than 50 in one bucket, he will have to put less than 50 in the other bucket and Barbara would thus be willing to pay less than 75 for the other bucket with lower number of cards

Hope it helps. Do ask for any clarifications if required.