In the topic Rule 32B3 question, Steve Immitt sets forth the following prize distribution problem:
Overall prizes:
1st $2375
2nd $1149
3rd $536
4th $306
5th $230
Under-2400 prizes:
1st $1379
2nd $766
(The prizes could be a little less Insane than these, but the problem wouldn’t go away. We’ll stick with the above as a valid example.)
The top eleven players finished as follows:
5.0
4.5
4.0
4.0 (under 2400)
4.0 (under 2400)
4.0 (under 2400)
3.5 (under 2400)
3.5 (under 2400)
3.5 (under 2400)
3.5 (under 2400)
3.5 (under 2400)
How to distribute the prizes? The first two are easy:
5.0 wins $2375 (clear first)
4.5 wins $1149 (clear second)
What’s next? Under rule 32B3, the four players tied at 4.0 combine and split the highest four remaining prizes. These prizes include some of the under-2400 money, since some players in the group are under 2400:
Each 4.0 wins $747 (one-quarter of $536+$306+$1379+$766)
The remaining prize is split among the five players tied at 3.5:
Each 3.5 wins $46 (one-fifth of $230)
What’s wrong with this picture? If the three under-2400 players at 4.0 had each scored just 3.5, then the higher-rated 4.0 would be eligible only for the $536 third prize. He would have won less by outperforming three under-2400 players than he did by tying with them!
I call this an anti-monotonicity. Roughly speaking, a function is monotonic if it goes up when it looks as though it should go up, and down when it looks as though it should go down. Here, just the opposite happens.
Rule 32B3 is also replete with discontinuities – situations where a small change in the prize structure results in a huge change in the prize(s) won by one or more individual(s). These discontinuities have been documented in other threads.
So, if 32B3 is severely deficient in the continuous and monotonic departments, how does it fare in the awarding-class-prizes-to-class-players department? Well, let’s see.
A couple of posters in that other thread referred to “blue currency” and “pink currency”. I prefer a simpler, far more logical, approach. To figure out where the class prizes went, simply calculate all the prizes two ways: (a) as if there were no class prizes, and (b) with the class prizes included. The difference, player by player, is the amount of class money that went to that player.
Column A is each player’s prize, if there had been no U2400 prizes. Column B is each player’s prize, including the U2400 prizes:
[code]
Player: A: B: Difference:
5.0 2375 2375 0
4.5 1149 1149 0
4.0 268 747 479
4.0 (under 2400) 268 747 479
4.0 (under 2400) 268 747 479
4.0 (under 2400) 268 747 479
3.5 (under 2400) 0 46 46
3.5 (under 2400) 0 46 46
3.5 (under 2400) 0 46 46
3.5 (under 2400) 0 46 46
3.5 (under 2400) 0 46 46[/code]
Thus, $479 of under-2400 money went to a player rated 2400+.
What went wrong? Here is the wording of 25B3, the Stupidest Rule in the Rulebook:
If winners of class prizes tie with winners of place prizes, all the cash prizes involved should be summed and divided equally among the tied winners …
This wording has been around for a long time. The above version comes from the 1st edition (Morrison, 1975) of USCF’s Official Rules of Chess. It is repeated verbatim in the 2nd (Morrison, 1978), 3rd (Redman, 1987), and 4th (Goichberg, 1993) editions. In the 5th (Just) edition, the concept is generalized to include class winners in multiple classes:
If winners of different prizes tie with each other, all the cash prizes involved shall be summed and divided equally among the tied winners …
Now, pardon me, but isn’t there a conflict of philosophy here?
The whole idea of class prizes is that a class player can win more money for the same score than a player outside the class. For example, a B player in a single-section five-round event might well win the under-1800 prize with a score of 3-2, whereas a master or expert with that score will probably go home empty-handed.
You can claim that class prizes are unfair if you want to, and you will have a valid point. But please make that point elsewhere, such as in a new thread. Let’s keep this thread focused on the best way to implement the idea of class prizes, which is to allow a class player to win more than a non-class player for the same score.
It makes no sense, then, that “all the cash prizes … be summed and divided equally” among class and non-class players alike. This goes against the whole grain of class prizes.
Here is a better way:
A. First award the place prizes, without regard to the class or “under” prizes. In other words, figure out what the place prizes would be if there were no class prizes.
B. Next, award the class or “under” prizes to the players in those classes. If some players already receiving a prize in step A would now have their prize increased, get the increase from the class prizes, rather than tinkering with the place prizes already calculated. Observe the limit-one-prize-per-player rule, awarding any leftover amount to other player(s) in the class.
C. If there are class or “under” prizes in multiple classes, award prizes in the more inclusive categories before those in the less inclusive categories. For example, figure the under-2200 prizes before the under-2000 prizes.
How would this work in our Insane example at the top of this post? First, award the place prizes:
5.0 2375
4.5 1149
4.0 268
4.0 (under 2400) 268
4.0 (under 2400) 268
4.0 (under 2400) 268
3.5 (under 2400) 0
3.5 (under 2400) 0
3.5 (under 2400) 0
3.5 (under 2400) 0
3.5 (under 2400) 0
Now throw in the U2400 prizes:
5.0 2375
4.5 1149
4.0 268
4.0 (under 2400) 268 + 715
4.0 (under 2400) 268 + 715
4.0 (under 2400) 268 + 715
3.5 (under 2400) 0
3.5 (under 2400) 0
3.5 (under 2400) 0
3.5 (under 2400) 0
3.5 (under 2400) 0
(Each 4.0 under 2400 gets one-third of the $2145 under-2400 prize fund.)
But this violates the limit-one-prize-per-player rule. Each of these three players is entitled to at most $268 or $715, whichever is larger, i.e. $715. The excess $268 is thus removed from each of these three players’ winnings. BUT we don’t touch the already-awarded place prizes. Instead, the excess $804 ($268 times 3) comes from the under-2400 fund, and is then redistributed to the five under-2400 players tied at 3.5. Each gets one-fifth of $804, or $161:
5.0 2375
4.5 1149
4.0 268
4.0 (under 2400) 268 + 715 - 268
4.0 (under 2400) 268 + 715 - 268
4.0 (under 2400) 268 + 715 - 268
3.5 (under 2400) 0 + 161
3.5 (under 2400) 0 + 161
3.5 (under 2400) 0 + 161
3.5 (under 2400) 0 + 161
3.5 (under 2400) 0 + 161
Voila! All prizes have been awarded, and 100% of the under-2400 prize fund has gone to the players under 2400.
And, gone are the discontinuities, the anti-monotonicities, and the general stupidity of the Stupidest Rule in the Rulebook.
Bill Smythe