More Pies to Divide

In the last post, Dividing the Pie, Redux, I used an iterative method in an attempt to derive a fairness-(C)-optimal payout schedule for the unshifted, 13-player tourney first introduced in Getting the Most from an Afternoon.

My earlier idea, from Dividing the Pie, was that this wouldn’t work because the was fairness (C) is defined would always prefer a winner-takes-all payout scheme. But this was not the case. I determined that the best payout scheme was 35/30/35.

In this post, I’ll use more or less the same method in an effort to understand, better, just what’s going on.

As a convention, I’ll report results in this form:

L4 50/30/20: 53.71
L3 25/25/25/25: 34.75

The L4 on the first line indicates that the result is from a simulation with the luck parameter set to 4, as it was for all of the runs in the last post. The boldface indicates a run with a full million trials, and hence with a rather small confidence interval for the estimate of fairness (C). In this case, the 95% confidence interval is 53.63 to 53.78.

The second line, not in boldface, shows a result from a much shorter runs with 50,000 trials with the luck parameter set to 3. In this case, the confidence interval is from 34.53 to 34.97. The fourth number indicates that a fourth place, the runner up in the consolation, was paid.

Later on, I’ll find occasion to make some slightly longer runs with 100,000 trials, which I’ll put in italics.

In the previous exercise, I determined that the best allocation for L4 was:

L4 35/30/35: 48.46.

This looks suspiciously egalitarian to me. Without violating my guideline that all payouts should be multiples of 5%, there’s no perfect three-way split. So to get one, let’s pay a fourth place. Never mind, for the moment, that four places is more than most tourney directors would pay for a 13-player event.

Now, this introduces even greater diversity to the small group of winners in terms of their mean skill. The mean Z scores for the top four places, in order, are:

1.057; 0.492; 0.697; and 0.227

It defies most people’s fairness (A) expectation that such differing skill levels should get the same reward, but that’s apparently what the fairness (C) method shows:

L4 25/25/25/25: 45.16
L4 30/25/25/20: 46.56

So, instead of running away to a winner-takes-all payout, the method has run away to an equally unhelpful even distribution.

Our first indication that we wouldn’t run away to the winner-takes-all result was that that payout was worse that some of the others:

L4 100: 61.34

Perhaps the system’s failure to reward the better players more richly comes is the result of the high degree of chance in individual results. So let’s look at some results from runs with a lower luck factor:

L1 50/30/20: 11.01
L1 100: 8.70

This example shows that the results of the iterative method are not necessarily optimal. starting with the first line, we get:

L1 50/30/20: 11.01
L1 50/25/25: 9.07

But at this point, single changes don’t lead to a better score. Since we know, however, that  the winner-takes-all allocation is better than either of these, let’s look at successive runs in which both of the other places give up money to the champion:

L1 60/20/20: 8.99
L1 70/15/15: 8.92
L1 80/10/10: 8.98
L1 90/5/5: 8.83

There’s a minor hiccup I attribute to the variability of the short runs. But the trend is pretty clear. Once it’s allowed to make two changes at once, an iterative procedure leads to the winner-takes-all result, as I once predicted it would.

So what happens with intermediate degrees of luck?

L3 25/25/25/25: 34.81
L3 100: 44.67

Since the winner-takes-all option isn’t better than the share-the-wealth option, I predict that it’s the even split option that will exert a gravitational attraction to an iteration starting with a fairly standard scheme for paying four places in a consolation. That’s what happens:

L3 40/25/25/15: 36.70
L3 35/25/25/15: 36.05
L3 30/25/25/20: 35.44

Here’s what happens with luck equal to two:

L2 25/25/25/25: 22.55
L2 100: 26.13
L2 40/25/25/10: 22.97
L2 35/25/25/15: 22.90
L2 30/25/25/20: 22.81

The same thing happens – the payout schedule is dragged toward the share-the-wealth option. This time a bit more hesitantly – I’d have to use more iterations to sharpen the numbers if it wasn’t quite so obvious what’s happening. At luck = 1.5:

L1.5 25/25/25/25: 16.24
L1.5 100: 17.18

The numbers are pretty close, here, so I expect it would be hard to find a trend. Let’s ratchet down the luck one more time to see if we can find another case where the winner-takes-all prevails:

L1.25 25/25/25/25: 13.21
L1.25 100: 12.91

Finally, one that should move the other way, but oh-so-slowly. Let’s find one with a little more room to run. For these runs with luck = 1.15 I’ll use 100,000 trials instead of 50,000 in hope that makes things clearer:

L1.15 25/25/25/25: 11.93
L1.15 100: 11.00: 11.00
L1.15 40/20/20/20: 11.76
L1.15 55/15/15/15: 11.58
L1.15 70/10/10/10: 11.42
L1.15 85/5/5/5: 11.33

So, what’s the moral of the story? By this time, this post is running so long that nearly everyone who started reading the post has probably decided that life is too short to finish it. So, with apologies to the conscientious few who read these words, I’ll save the conclusion for the next post.