How to fairly do tip pooling with uneven proportions of contributions? Imagine a restaurant where the servers each have shifts of the same length and pool the same proportion of their tips (e.g. 50%).
It's easy to calculate how much each server should receive from the pool at the end of the shift.  You divide the pool by the number of contributors, and they each get the same amount.
My question is whether it's possible to fairly pool tips when the servers have shifts of different lengths and where each server gets to elect ahead of time what proportion of tips to pool, from 0% to 100%.
For instance, you might have a pool like this:
| Server      | Shift (hrs) | Total tips | % pooled |
| ----------- | ----------- | ---------- | -------- |
| Alice       | 4           | $80        | 50%      |
| Bob         | 8           | $100       | 25%      |
| Charlie     | 8           | $120       | 50%      |
| Diane       | 6           | $80        | 25%      |

Is there some way to weight each server's contribution based on shift length and proportion of tips pooled and come up with a distribution method that allows each contributor to receive a weighted-average payout?
(For context, I'm trying to work out a payment model for an app that allows workers to do tip pooling, and I'd like to determine whether they can specify what proportion of their tips to pool, or whether it has to be a static percentage.)
 A: So the sole purpose of the pool is to redistribute tip money among the servers? If you are taking any money out for other purposes, such as tipping other staff who are vital to the restaurant's operation but don't get the chance to graft the customers themselves, then I would certainly recommend that all servers be required to submit the same percentage of their tips. Anything else would be unfair.
But if the pooled money is only redistributed to the servers, then consider the following:
Assign to each server a "stake", which is the percentage of their tips that they contribute, times the hours that they worked. Sum all the stakes to get the total stake. The amount of the pooled tips that each server receives is the ratio of their stake to the total stake.
In your example, let me add another column for the stakes:
| Server      | Shift (hrs) | Total tips | % pooled | Contribution | Stake          |
| ----------- | ----------- | ---------- | -------- | ------------ | -------------- |
| Alice       | 4           | $80        | 50%      | $40          | 50% x 4 = 2    |
| Bob         | 8           | $100       | 25%      | $25          | 25% x 8 = 2    |
| Charlie     | 8           | $120       | 50%      | $60          | 50% x 8 = 4    |
| Diane       | 6           | $80        | 25%      | $20          | 25% x 6 = 1.5  |
| ----------- | ----------- | ---------- | -------- | ------------ | -------------- |
| Total       |             |            |          | $145         |           9.5  |

Now $\dfrac {\$145}{9.5} \approx \$15.26$, so the servers receive:
| Server      | Stake | Return                |
| ----------- | ----- | --------------------- |
| Alice       | 2     | 2   x $15.26 = $30.52 |
| Bob         | 2     | 2   x $15.26 = $30.52 |
| Charlie     | 4     | 4   x $15.26 = $61.04 |
| Diane       | 1.5   | 1.5 x $15.26 = $22.39 |

The total is $\$144.47$. You can toss the remaining $53¢$ to the bellhop ;-). Or maybe just save it for tomorrow's pool. (And it is up to you whether you want to round to cents or dollars, or whatever.)
To consider it's fairness, note that Alice contributed the same percentage as Charlie, but worked only half the hours, so she gets only half the pay-out he does. And Bob worked the same number of hours as Charlie, but only contributes half the percentage, so he also only gets half the pay-out as Charlie. So in both cases, the proportion works out.
Now it might seem unfair that Alice loses money on the deal while everyone else gains. But no matter how this is done, it is a zero-sum game. Someone has to lose for anyone else to win. The only reason for participating is because the another day Alice might be the one collecting few tips, so she pays in now in hopes of future benefit. Overall, the idea is to even out the fluctuations, so that everyone has a dependable flow of income.
However, if someone regularly out-performs others, they have no incentive to participate. If Alice is attentive to her customers while Bob spends most of his time in the back goofing off, then Alice will be the loser every time, and is just subsidizing Bob's laziness with her hard work. So she would be wiser to keep her money to herself.
