The cake is a lie!

Probability Level 3

Suppose a group of n n people want to split a group of whole cakes among each other. Each cake is of one type only e.g. vanilla, chocolate, strawberry, etc.

Each person also has a value function associated with each type of cake, which denotes how much each cake is "worth" to each player. The sum of all the value functions for a given person is 1 1 , and every person values nothing at 0 0 . Note that a value function must be ≥ 0 \geq 0 and ≤ 1 \leq 1 .

For example, person 1 might value vanilla at 0.05 0.05 , while person 2 values vanilla at 0.8 0.8 . Person 1 might value chocolate at 0.77 0.77 , while person 2 values chocolate at 0.00002 0.00002 .

In addition, a person values a fraction f f of a cake at a fraction f f of the value function he/she associates it with. For example, given the above value functions, person 2 would value half a vanilla cake at 0.4 0.4 , while person 1 values 1 7 \frac{1}{7} of a chocolate cake at 0.11 0.11 .

The total value a person receives is the combined values of all the cakes he/she has, based on his/her own value functions. For example, if person 1 receives a vanilla cake and a chocolate cake, his total value would be 0.05 + 0.77 = 0.82 0.05 + 0.77 = 0.82 . If person 2 receives half a chocolate cake and half a vanilla cake, her total value would be 0.00001 + 0.4 = 0.40001 0.00001 + 0.4 = 0.40001 .

Is it possible to divide the cakes among all n n people such that each person receives at least a total value of 1 n \frac{1}{n} ?

Source: AMST 2015 Power #1b
It depends on the value functions It depends on something else not listed here It depends on the number of people It depends on the size of the cakes Never possible Always possible

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Steven Yuan by Steven Yuan , 20, USA

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1 solution

Giving each person 1 n \frac{1}{n} of all the cakes satisfies the conditions

. Let x i x_i be the value function of some person for the i i th cake. Then ∑ x i = 1 \sum x_i = 1 .

Now since each person gets 1 n \frac{1}{n} of all the cakes, Total value is ∑ 1 n x i = 1 n ∑ x i = 1 n \sum \frac{1}{n}x_i = \frac{1}{n} \sum x_i = \frac{1}{n}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Exactly what I did!

Vikram Venkat - 6 years, 3 months ago

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