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Excursions in Modern Mathematics, 7e: 3.3 - 1Copyright © 2010 Pearson Education, Inc. Tommy, Tammy, and Amy are to share a pizza. One half of the pizza is tomato, the other half is bacon. Amy likes bacon and tomato equally. Tommy and Tammy like bacon a little, but they prefer tomato. Their exact valuation of the two will be given in the table to follow. As always, we assume each player has no information regarding the other players’ preferences. Honesty is the Best Policy

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Excursions in Modern Mathematics, 7e: 3.3 - 2Copyright © 2010 Pearson Education, Inc. Amy divides the pizza into three pieces, X, Y, Z, and each player’s valuation of each slice is Honesty is the Best Policy XYZ Amy33.3% Tommy9%45%46% Tammy9%45%46%

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Excursions in Modern Mathematics, 7e: 3.3 - 3Copyright © 2010 Pearson Education, Inc. Tommy and Tammy now must place their bids. According to the rules, they should include any slice that they value worth 1/3 of the pizza. Thus, if they are playing honestly, Tommy and Tammy will each include slices Y and Z in their bids. Y and Z are the C-pieces, X is the U-piece. Since there are two C-pieces, Tommy and Tammy will each get a C-piece. Honesty is the Best Policy

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Excursions in Modern Mathematics, 7e: 3.3 - 4Copyright © 2010 Pearson Education, Inc. Y and Z are each “fair” to Tommy and Tammy, so we allocate these pieces randomly. Suppose Tommy receives slice Y and Tammy receives slice Z. Amy then receives slice X. Amy’s slice is worth 33.3% Tommy’s slice is worth 45% Tammy’s slice is worth 46% This is a fair division since each member received at least 1/3 of the value. Honesty is the Best Policy

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Excursions in Modern Mathematics, 7e: 3.3 - 5Copyright © 2010 Pearson Education, Inc. Notice that, as far as Tommy is concerned, Tammy got a (slightly) better deal than he did. Tommy thinks “I shouldn’t have included Y in my bid. If I claimed that all the value was in piece Z, then Tammy would have ended up with slice Y and I would have ended up with slice Z.” Is Tommy correct? Honesty is the Best Policy

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Excursions in Modern Mathematics, 7e: 3.3 - 6Copyright © 2010 Pearson Education, Inc. Tommy’s belief that he would have been better off only bidding for Z is incorrect, and here’s why: Tommy and Tammy have the same value systems, so if Tommy would be better off only bidding for Z, then Tammy also would be better off only bidding for Z. (This is a consequence of the rationality assumption: If Tommy is smart enough to devise a strategy, then Tammy smart enough to devise the same strategy) Honesty is the Best Policy

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Excursions in Modern Mathematics, 7e: 3.3 - 7Copyright © 2010 Pearson Education, Inc. In this case, slice Z is the only C-piece. Therefore, X and Y are each U-pieces, and we are in the second case of the Lone- Divider Method. Amy is now given one of the two U-pieces. Suppose Amy takes piece Y. X and Z are combined, and Tommy and Tammy enter into a “Two Person Divider Chooser game” to divide the combination of X and Z. Honesty is the Best Policy

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Excursions in Modern Mathematics, 7e: 3.3 - 8Copyright © 2010 Pearson Education, Inc. Now, since Tommy and Tammy have the exact same value systems, Divider-Chooser will give Tommy exactly half of X and Z, and Tammy exactly half of X and Z. How much are these pieces worth in terms of the whole pizza? X and Z combined are worth 9% + 46% = 55% of the whole pizza. Tommy and Tammy each ended up with half of that, so they each ended up with 27.5% of the whole. Honesty is the Best Policy

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Excursions in Modern Mathematics, 7e: 3.3 - 9Copyright © 2010 Pearson Education, Inc. Tommy and Tammy tried to play dishonestly in an attempt to get a “more than fair share” (46% instead of 45%). They each ended up with 27.5%, which is less than their fair share. Moral: Tommy and Tammy should have played honestly. Honesty is the Best Policy

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Two players: The Divider-Chooser Method

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