Question

Two children must split a pie. They are gluttons and each prefers to eat as much of the pie as they can. The parent tells one child to cut the pie into two pieces and then allows the other child to choose which pie to eat. The first child can divide the pie into any multiple of tenths (for example, splitting it into pieces that are 1/10 and 9/10 of the pie, or 2/10 and 8/10, and so forth). Show that there is a unique backward induction solution to this game.

Answer 1

This is a zero sum game because one player's loss is another player's gain. There are many ways to cut the pie into uneven pieces. But it is almost impossible to cut the pie in equal halves because there will be atleast the difference of a crumb  between the two pieces. The backward induction of this game will be:

The game:

If Player 1 cuts the piece as equally as he can, Player 2 can even then select the bigger piece (bigger by atleast a crumb). In this case, Player 2 will get a higher payoff of 3. What Player 2 gains, Player 1 loses. So his payoff is -3. If Player 2 selects smaller piece, the case is reversed. Player 2 gets -3, and Player 1 gets 3.

On the other hand, if Player 1 cuts the cake into uneven pieces, and Player 2 picks the bigger piece, her gain is larger, because she gets a larger piece than if it were cut equally. So her payoff now is 5. Player 1 gets -5. If Player 2 selects the smaller piece, she will get a payoff of -5 and Player 1 will get 5.

Backwrd induction:

Looking at the backward inuction, Player 1 knows that whether he cuts the pie into equal pieces or uneven pieces, Player 2 will pick the bigger piece because her payoff is higher (3>-3 in case of even pieces and 5>-5 in case of uneven pieces). But for Player 1, his own payoff is greater in case of equal pieces (5>3). So he will cut the pie into equal pieces, as much as he can.

So, (Even, Big) is the unique equilibrium of this game with payoff (-3, 3).

That is, Player 1 will cut the pie into as equal pieces as he can, and Player 2 will choose the bigger piece.