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Mathematical analysis

Posted By: Don Schlesinger
Date: 31 Jul 01, 5:37 pm

In Response To: Winning by hitting 52% (fezzik)

I wrote to SW about this very topic a while back. Here is some math to back up fezzik's analysis. I think you'll find it quite an eye-opener. Of course, getting the 14 to 5, instead of 13 to 5, may be difficult.

First, some background. Although you might decide to bet less, we'll assume, simply for the purpose of comparison, that we'd like to make the Kelly-optimal wager in each situation. You also need to know that, if that wager is represented by f*, then, as a percentage of your total bank, f* = EV/A, where EV is your positive expectation for the bet and A is the ratio of the payoff of a winning bet to a losing one (see Thorp's "The Mathematics of Gambling, p. 129).

We'll examine three scenarios of three kinds of bets each: two separate straight bets, a parlay at 13 to 5, and a parlay at 14 to 5, for a bettor with 55%, 53%, and 57% handicapping ability. The bank will always be $2,000, so that fair comparisons can be drawn. There are some very interesting conclusions!

The 55% player is somewhat of the "norm." This is a solid handicapper with considerable skill. If he bets $110 on each of two games (we're laying 11 to 10 for straight bets), he can expect to win both 30.25% of the time, split 49.5%, and lose both 20.25%. The EV is +$11, or a 5% ROI (11/220), but the Kelly-optimal bet is 5.5%, because that EV is divided by 10/11, which is to say it is multiplied by 11/10.

So, we need a $2,000 bank to properly bet $110 on each game.

Now suppose we can bet a two-team parlay receiving 13 to 5. We'll win 30.25% of them, but lose 69.75% of them. Since we win $13 or lose $5, for each $5 invested, we see that the ROI, or EV, is 8.9%, but the optimal bet is 8.9%/13/5 = 3.42% of bank. For the same $2,000 bank, we must now bet a more conservative $68.46, for an expected dollar win of $6.09, which isn't so great, compared to the straight bets.

But what if the parlay pays 14 to 5? Well now, ROI is 14.95%, and the f* is 14.95/14/5 = 5.34% of bank. We may bet $106.79 on the parlay, and with EV of +14.95%, we expect to win $15.96, which is now superior to our $11 straight-bets expectation!

With risk the same, the parlay now becomes the more intelligent wager!

Let's go through the same analysis for a weaker handicapper (53%) and then a stronger (57%) one.

The straight bet EV is reduced considerably to just 1.3%, and the f* is just 1.43% of bank. We now may bet only $28.60 on each of the two games, and we expect to win the magnificent sum of 74 cents!

For the 13 to 5 parlay, we now win just 28.09% of them, so ROI is a mere 1.12% and the f* shrinks to an invisible 0.43% of bank. Our bet is $8.65 (!) and we expect to win 9.7 cents. Fuhgeddaboudit!

The 14 to 5 changes everything. ROI is now 6.74% and f* is 2.41% of bank. We may bet $48.16 and our dollar EV is $3.25. Quite an improvement.

Finally, let's consider the handicapper of great skill; he of the 57% success rate.

Optimal bets for each of the straight wagers jump to 9.7% of bank, with ROI a creditable 8.82%. So, we bet $194 on each of the two games, expecting to win $34.21.

At a 13 to 5 rate, the weaker parlay sees us win 32.49% of the time, with ROI of 16.96% and f* = 6.52% of bank. We bet $130.49 and expect to win $22.13 -- not as good as the straight bets.

But get a look at 14 to 5! The ROI is a huge 23.46%, with f* of 8.38% of bank. So, we may bet $167.59 on the parlay, expecting to win $39.32, which, again, beats the straight wagers, for the same risk.

In fact, risking less than with the straight bets and wininng more is a pretty neat trick, don't you think?

Don

Messages In This Thread

Winning by hitting 52% -- fezzik -- 30 Jul 01, 10:50 pm

Mathematical analysis -- Don Schlesinger -- 31 Jul 01, 5:37 pm

Very nice, but what about... -- Flipper -- 31 Jul 01, 6:03 pm

One more option -- James -- 31 Jul 01, 6:19 pm

-105 vig -- fezzik -- 31 Jul 01, 6:37 pm

You can get +100 for NFL bets (nt) -- StevieY -- 31 Jul 01, 7:04 pm

Has anyone figured out -- James -- 1 Aug 01, 12:25 pm

Lots of bad bettors -- StevieY -- 1 Aug 01, 3:20 pm

A possible way -- David Matthews -- 1 Aug 01, 4:57 pm

I doubt it -- James -- 2 Aug 01, 9:31 am

Volume :-) (nt) -- Colin Caster -- 1 Aug 01, 6:05 pm

+100 NFL -- lepto -- 13 Aug 01, 3:02 am

Backchannel, please. (nt) -- Colin Caster -- 13 Aug 01, 8:10 pm

No, calculations are incorrect -- Don Schlesinger -- 31 Jul 01, 8:25 pm

Not surprised. Thanks. -- James -- 1 Aug 01, 10:20 am

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