Transcript Honest is the Best Policy

Honesty is the Best Policy
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.
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Excursions in Modern Mathematics, 7e: 3.3 - 1
Honesty is the Best Policy
Amy divides the pizza into three pieces, X, Y,
Z, and each player’s valuation of each slice is
X
Y
Z
Amy
33.3%
33.3%
33.3%
Tommy
9%
45%
46%
Tammy
9%
45%
46%
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Excursions in Modern Mathematics, 7e: 3.3 - 2
Honesty is the Best Policy
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.
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Excursions in Modern Mathematics, 7e: 3.3 - 3
Honesty is the Best Policy
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.
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Honesty is the Best Policy
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?
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Honesty is the Best Policy
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)
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Honesty is the Best Policy
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 LoneDivider 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.
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Honesty is the Best Policy
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.
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Excursions in Modern Mathematics, 7e: 3.3 - 8
Honesty is the Best Policy
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.
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Excursions in Modern Mathematics, 7e: 3.3 - 9