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