Monday, January 10, 2011

Muffin Cutting and The Power of the Knife

Every day, I eat half of a Costco muffin for breakfast (and so does my wife). She gets up earlier than I do, and for this reason, she gets to decide on how the muffin is divided. By the time I go to eat my half muffin, she has already begun eating her "half." Aside from cutting my own muffin, the only choice I have is to accept the sliver she left me.

This is a modification of what is known as the ultimatum game: Given a cake (or muffin), one player decides on how to divide it. The other accepts or rejects.

In the standard ultimatum game, both get none of the muffin if the second rejects. Rejection is an off-the-equilibrium path threat that is not credible. After all, more is better (or assumed to be), so just a crumb from the muffin will be enough to prevent the second player from rejecting the deal. If this is how our muffin game worked, my wife should leave just a crumb.

There are a couple of important differences between the muffin-cutting game and the ultimatum game. First, my wife gets to eat her share regardless of my decision, so saying "I reject this half muffin" has no bite because she'll just eat what she cut anyway.

Second and more importantly, our daily muffin-cutting game is a dynamic game. If I reject the half muffin today, she'll have to eat what she meant to leave for me tomorrow. Moreover, rejection of the deal means that I get the power of the knife tomorrow, and I get to keep that power as long as my division of the muffin is accepted by my wife.

This dynamic aspect of our muffin-cutting game keeps the power of the knife in check. The knife cannot cut too deep because that will leave too great of an incentive to take the knife away.

Our daily muffin-cutting routine has gone on for about four months, and I have only had to reject a couple of times to make sure I get my fair share of the muffin. Almost always, I end up getting (almost) half of the muffin and I don't have to do the cutting. In equilibrium, she gets a little more than half of the delicious muffin, and I don't have to cut it in half. It's a good deal for both of us.

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