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calendar   Thursday - August 29, 2013

Bowling Math

A Pig In A Poke


A pig in a poke is a very old expression, used to warn people about getting into situations where they aren’t really sure of all the details. A poke is some kind of bag or box I guess. If you’re buying a pig this way, all you know is that you’re buying a pig. You don’t know if it’s a boar or a sow, or how big the pig is, or whether it even comes with all four legs. All you can hear is the oinking, so you lay your money down and hope your porcine dreams come true. But really, you’re running blind.

This is a lot like the prize allocation in bowling league. Early in the season, everybody votes for one of several payout schemes, and almost everyone only sees the big prize amounts for the winners. Problem is, regardless of the size of the league, there is only one winner. Everybody else is a loser, to some degree. And come the end of the year when the prize money actually gets handed out, it’s almost a guarantee that somebody (in the last place or two) will get their nose out of joint because they feel ripped off. And if they’re mad enough, they don’t come back next year, and the league shrinks. If this happens year after year, eventually the league dies from attrition. And yet every year, the guys figuring out the prize amounts always concentrate on the top prizes first, and the lower placing teams get the leftovers. And then they wonder why teams quit the league.

I’ve heard that the definition of insanity is doing the same thing over and over while expecting different results.

So I set out to fix that, or at least to explore the process to my satisfaction. And I found an odd thing: it seems to matter more to bowlers how much money they get back then what the actual cost is. And that does not make sense at all.

Let’s say you go to a store and see some item you want to buy. Call it an impulse purchase; you’re going to buy this thing in this store, a mystery box, no matter what it costs. You don’t know the price. So you give the cashier a $20, get your box, and she gives you $15 change. Instead of focusing on how much the box cost, you only notice that you got back $15. $15!! The next time you’re in that store, you impulsively buy another mystery box. A pig in a poke. This time, you give the cashier a $50, and she gives you $45 change. Wow, $45!! This is even better! The third time you’re in that store, you’re a bit short on cash, yet you buy another mystery box. You hand over $5, get your box, but you get no change at all. And you feel robbed. Hey, last time I got $45 back!  And somehow you never notice that the box cost $5 every time. That’s how bowler’s minds work when it comes to prize money.

I built a bunch of spreadsheets. Each one is a prize money calculator, based on different payment schemes. At the core, all of them are driven by a fairness algorithm based on the simple Sigma series of Σ 0 to n = n(n-1)/2. In English, that means that I’m cutting up a chunk of the prize money pie into lots of perfectly even slices, one for everybody, and then the rest of the pie is chopped into different small equal bits (I call them “cuts") which are doled out according to a team’s final placement. The higher your team places, the more small bits you get, in addition to your basic slice. Last place gets no extra cuts, second to last gets one, third to last gets two, and so on. This creates a perfectly even stepwise allocation. No matter what place you’re in, the next place down gets awarded one cut less, and the next place up from you gets awarded one cut more. The bigger the value of a cut is, the smaller the basic slices are, and the wider the steps are between adjacent places. Make the cut too big and the losers cry and quit, make it zero and every place gets awarded the same amount, so why bother. Somewhere in between is the perfect spot.


n = number of teams
Basic Slice = (Prize Money - ( (n(n-1)/2 * Cut Value )/n

Prize Money = Income - Expenses (well, duh)


And this works out very well. There is no huge money gap between 1st and 2nd, or 3rd and 4th, or anywhere. Each place gets exactly $Cut Value more than the place below them. And you can even run the numbers in your head, since the number of cuts for any given place is always ( n - Place ), no matter how big or small the league is.

And then I saw the magic. Math-o-magic. The actual cost of bowling for any particular place is akin to that mystery box; the more you put in (weekly dues), the more you get out (prize money for Place X), yet the price of the box (net cost per bowler per game for Place X) never changes. Which completely destroys the thesis of “let’s raise the dues so there’s more money in the prize fund”. As long as the weekly dues are enough to both cover expenses and to give the last place team a prize of at least nothing, then it doesn’t matter whether weekly dues are $15, $20, or $500: the net cost for any Place is always the same. So why not set the dues as low as possible?

The other thing I found was the “I quit” threshold for the prize allocation for last place. And found that it can be avoided by making the Cut Value smaller. And that’s what I really set out to do with all of this, to find the way of keeping everyone (even the losingist losers) happy enough to come back next year.

This is going to be a hard sell. I can drop the price of league by a third, yet the net costs don’t change. And I figured out how to keep teams from quitting over cheap prizes. But like the proverbial monkeys in the cage with the fire hose, bowlers are hide-bound traditionalists, change resistant to the core, and to them “We’ve always done it this other way” is their major argument. Even as it takes them closer to the grave, year after year.

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Posted by Drew458   United States  on 08/29/2013 at 08:36 AM   
Filed Under: • Bowling Blogging •  
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