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## Question

Alice, Bob, and Charlie order a pizza for \$10. It turned out that Alice and Bob were very hungry, eating 6 slices a piece, and Charlie just had 2 slices.

They want to split the pizza cost fairly, but they are subject to a practical issue. No one has coins to make change, so they have to round how much each owes to the nearest dollar.

What’s the fair way to split the bill?

What if the pizza was \$11?

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If we lift the assumption that A and B pay the same amount, there is another option.

A and B use "Rock, Paper, Scissors" [we know that they have no coins] or other method to determine a loser. That person [say A] will pay \$1 more.

So the payment is [A, B, C] pays [5, 4, 2]. THis takes care of the problem of C paying less for the \$11 pizza than the \$10 and the total offset is the same

A over pays by 6%, B underpays by 15% and C overpays by 27%.

The total absolute offset = 6% + 15% + 27% = 48% .... which happens to be the totla absolute offset for the [5,5,1] payment scheme described above.

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Instead of simply rounding up or down, consider how much the percentage change would be for each person compared to what they would pay [if coins were available].

consider Charlie
If the bill is even [\$10], then he must pay an even number of \$\$
If the bill is odd [\$11] then he must pay an odd number of \$\$

For a \$10 check
The actual amount due from AB and C is [4.29, 4.29, 1.43]

you could charge Chrlie 0, \$2 or \$4
Leaving the others to pay \$5, \$4 or \$3

[4,4,2] would give A and B a 7% break and C would overpay by 40%

[5,5,0] would have A &B overpay by 17% and C would get a 100% break

Looking at the differences, [4,4,2] yields the lowest absolute % offset of 54% (7 + 7 +40).

[5,5,0] would yield an absolute offset of 112% (6,6,100)

So for a \$10 check, the fairest split is \$4 from A & B and \$2 from C

For \$11
The choices are [5,5,1] and [4,4,3]

[5,5,1] over charges A&B by 6% and give C a 36% break
total offset = (6+6+36) = 48%
[4,4,3] gives A&B a 22% reduction and overcharges C by 75%
total offset = (22+22+75) = 119%

So the fairest split is \$5 from A and B, \$1 from C
Edited by dgreening

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Instead of simply rounding up or down, consider how much the percentage change would be for each person compared to what they would pay [if coins were available].

consider Charlie

If the bill is even [\$10], then he must pay an even number of \$\$

If the bill is odd [\$11] then he must pay an odd number of \$\$For a \$10 check

The actual amount due from AB and C is [4.29, 4.29, 1.43]

you could charge Chrlie 0, \$2 or \$4

Leaving the others to pay \$5, \$4 or \$3

[4,4,2] would give A and B a 7% break and C would overpay by 40%

[5,5,0] would have A &B overpay by 17% and C would get a 100% break

Looking at the differences, [4,4,2] yields the lowest absolute % offset of 54% (7 + 7 +40).

[5,5,0] would yield an absolute offset of 112% (6,6,100)So for a \$10 check, the fairest split is \$4 from A & B and \$2 from CFor \$11

The choices are [5,5,1] and [4,4,3]

[5,5,1] over charges A&B by 6% and give C a 36% break

total offset = (6+6+36) = 48%

[4,4,3] gives A&B a 22% reduction and overcharges C by 75%

total offset = (22+22+75) = 119%

So the fairest split is \$5 from A and B, \$1 from C

So with a more expensive pizza, Charlie pays less even though the value of each slice increases?

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I would say that for distribution to be fair the average cost per slice (CPS) paid should be as close as possible. For example if the cost of pizza was \$7 the solution would be easy and they would split the cost 3+3+1 for \$.50 per slice paid by all.

For \$8 pizza, the extra dollar has to come from someone and it should be either A or B, but not C. If it came from C then A and B would still be paying \$.50/slice while C would be paying \$1/slice or twice as much as A and B. So 4+3+1 or 3+4+1 distribution for \$8 pizza seems fair and one of the three pays \$.67/slice while others are still paying \$.50/slice

For \$9 pizza, it should be 4+4+1 for the same reason.

Now, for \$10 pizza the options are

1. 5+4+1 with CPS at \$.83 / \$.67 / \$.50
2. 4+4+2 with CPS at \$.67 / \$.67 / \$1

The absolute difference between the highest and the lowest CPS is the same in both cases and is \$.33. However, by taking the lowest CPS as the base and measuring the premium paid by others as a percentage compared to the base, option 2 becomes preferred - Charlie pays 50% premium (\$1 over \$.67) compared to option 1 when either Alice or Bob have to pay 66% premium (\$.83 over \$50).

For \$11 pizza, it's either

1. 5+4+2 with CPS at \$.83 / \$.67 / \$1, or
2. 5+5+1 with CPS at \$.83 / \$.83 / \$.50
3. 4+4+3 with CPS at \$.67 /\$.67 /\$1.5

Again, using the analysis above option 1 is preferred.

So, basically I arrived at the same result as dgreening, but using a slightly different "measure of fairness". I think there isn't an absolute best answer here. It all depends on what criteria is used to establish what's fair.

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If we allow for rock, paper, scissors or something similar, we have introduced a random element to the split. So why stop there? They can play a kind of rock, paper, scissors, lizard, spock, fork, fire. They assign to each sign a unique number from 0 to 6 inclusive, then they all show a sign simultaneously. They add up the corresponding numbers, divide by 7, and take note of the remainder R. For the \$10 pizza, A pays 4, B pays 4, C pays 1. The last dollar is paid by A if R is 0 or 1, paid by B if R is 2 or 3, and paid by C otherwise. A perfectly fair split. For the \$11 pizza, A pays 5, B pays 5, C pays 2. The \$1 discount is given to A if R is 0 or 1, given to B if R is 2 or 3, and given to C otherwise. Another perfectly fair split.

I feel that the problem is cheated if we allow for random elements. Since there would be no "fair" way to charge A and B differently without a random element, we're left with 4,4,2 and 5,5,1 for the \$10 and \$11 pizza respectively. The "paradox" that C would pay less for the more expensive pizza is acceptable. But it ultimately comes down to our definition of fair.

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For \$10 or \$11 cost of pizza, pay \$12 (including tip).

Both Alice and Bob pay \$6 each to make the \$12. Charlie pays \$1 each to Alice and Bob.

Everyone "pays" \$1 per slice and the vendor is happy too! • 1

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