Push Lines - Include in probabilities?

[MattyKGB]

Consider a single bet that has 2% chance of an even money win, 1% chance of loss and 97% chance of push.

Would you say the edge on this bet is +1% or +33%? I'd say it's +1%.

[sharpasitgets]

Consider this hypothetical parlay. We have 5 push games, each with a win probability of 58%. Given this, our expected rate of return is 20 / (1/.58^5) = 31%

However, this rate of return does not occur 100% of the time, when pushes occur we receive a discounted rate of return. So we need to know what the total reduction is given all possible push outcomes.

To do this, first define the total push sample space. Then calculate the rate of return and the associated probability for each discounted rate occurring.

To make it easy, assume a push probability for all games is 25%

Sample Space
5 Games push = 0.000977*0=0
4 Games push = (5) (0.00293) (-.376) =-0.005508
3 Games Push = (10) (0.008789) (-.486) = -0.04271
2 Games Push = (10) (0.026367) (-0.025) = -0.00659
1 Game Push = (5) ( 0.079102) (.131) = 0.051812

Therefore the rate of return for the push sample space is -0.003% and one can expect this discounted rate when at least 1 push occurs.

Therefore:
Calibrated rate of return=Discounted rate + (Expected Rate of Return*(1-Probability of Push 1+))
Calibrated rate of return=-0.003 + (0.310 * (1-0.762) ) = %7.05

[MattyKGB]

Sharp - I don't understand your example. Do these games have 58% win prob PLUS 25% push prob? If so, that's unrealistic. Or does the 58% already include some portion of the 25%?

[sharpasitgets]

58% represents our winning probability when a game does not push.

To calculate the winning probability of push games use the following general form.

In case of dog.
Probability of dog win/probability of no push

In case of fav:
Probability of fav 2+/probability of no push.

In this hypothetical assume the spread is +1 with a win probability of (58%) given a push probability of (25%) we can deduce a fair money line of +129.

[MattyKGB]

Ok so you are drawing the same equivalency that BTS and PLP are. That is the crux of the issue.

[MattyKGB]

So here's the answer then.

Compared to an assumption that the push probability gets allocated proportionally to win & loss, the push makes a big difference.

Compared to an assumption that the push probability gets split equally between win & loss, the push makes a small difference.

[ProlinePlayer]

Consider a single bet that has 2% chance of an even money win, 1% chance of loss and 97% chance of push.

Would you say the edge on this bet is +1% or +33%? I'd say it's +1%.

I don't think that the method here is a matter of right or wrong but more a case of preference of several legitimate ways of doing this.

But in your example, if I was playing this and would win 2 out of 3 settled bets at even money, I would consider that I had a very large edge. Not a marginal 1% edge.

PLP

[MattyKGB]

1% is a huge edge when it's on a low-variance bet like this one

[MattyKGB]

Let's change the example a little.

Suppose you pay $100 for a proposition with a 2% chance of returning $200, a 1% chance of returning $0 and a 97% chance of returning $100.01. Now what's your edge?

[sharpasitgets]

I would consider that a 2% edge. However, this hypothetical is not a push example. All possible outcomes are settled.