Krigman Even-Money Equivalent Wagers and Risk of Ruin

[jfdinneen]

For those who wish to compare their average performance to that of other handicappers, I would strogly recommend reading Alan Krigman's excellent piece, "Edge, Volatility, and Risk of Ruin in Gambling (http://www.bjmath.com/bjmath/Betsize/winwayz/tmpweb.htm) ".

In an effort to encourage its usage, I have provided the required Excel formulae below for calculating the even-money equivalent stake, win probability and risk of ruin.

Handicapper Statistics
Initial Bankroll: $1,000.00
Final Bankroll: $2,000.00
Session Length: 250
Decimal Odds (Avg): 2.50
Wager(Avg): $35.00
Win Probability (Avg): 43.00%

Krigman Even-Money Equivalents
Initial Bankroll: $1,000.00
Final Bankroll: $2,000.00
Session Length: 250
Decimal Odds: 2.00
Wager: $43.40
Win Probability: 53.02%
Risk Of Ruin: 6.12%

EvensEquivalent_Wager=SQRT(SUM(POWER(PRODUCT(Wager ,SQRT(ABS(PRODUCT(POWER(-SUM(Decimal_Odds,1),2),SUM(Win_Probability,-1), Win_Probability)))),2),POWER(PRODUCT(Wager,SUM(PRO DUCT(Win_Probability,SUM(Decimal_Odds,1)),-1)),2)))

EvensEquivalent_WinProbability=SUM(0.5,PRODUCT(0.5 ,PRODUCT(SUM(PRODUCT(Win_Probability,SUM(Decimal_O dds,1)),-1),1/SQRT(SUM(POWER(SQRT(ABS(PRODUCT(POWER(-SUM(Decimal_Odds,1),2),SUM(Win_Probability,-1), Win_Probability))),2),POWER(SUM(PRODUCT(Win_Probab ility,SUM(Decimal_Odds,1)),-1),2))))))

EvensEquivalent_RiskOfRuin=PRODUCT(SUM(POWER(PRODU CT(SUM(1,-EvensEquivalent_WinProbability),1/EvensEquivalent_WinProbability),PRODUCT(SUM(Initia l_Bankroll,Final_Bankroll),1/PRODUCT(PRODUCT(EvensEquivalent_Wager,1/Wager),Wager))),-POWER(PRODUCT(SUM(1,-EvensEquivalent_WinProbability),1/EvensEquivalent_WinProbability),PRODUCT(Initial_Ba nkroll,1/PRODUCT(PRODUCT(EvensEquivalent_Wager,1/Wager),Wager)))),1/SUM(POWER(PRODUCT(SUM(1,-EvensEquivalent_WinProbability),1/EvensEquivalent_WinProbability),PRODUCT(SUM(Initia l_Bankroll,Final_Bankroll),1/PRODUCT(PRODUCT(EvensEquivalent_Wager,1/Wager),Wager))),-1))

Enjoy!

John

[SchagFactorToWin]

Thanks for the link.

I couldn't understand how to calculate variance, however. Well, I know how to calculate variance but I couldn't follow what variance he was measuring (from a horse racing POV). What do you think?

[jfdinneen]

SchagFactorToWin,

The additional formulae you require for use with Krigman's spreadsheet are as follows:
Unit_Expectation=SUM(PRODUCT(PRODUCT(SUM(Decimal_O dds,-1),1),Win_Probability),PRODUCT(-Wager,SUM(1-Win_Probability)))
Unit_Variance=SUM(PRODUCT(POWER(SUM(PRODUCT(SUM(De cimal_Odds,-1),Wager),-Unit_Expectation),2),Win_Probability),PRODUCT(POWE R(SUM(-Wager,-Unit_Expectation),2),SUM(1-Win_Probability)))
Unit_StandardDeviation=SQRT(Unit_Variance)

Note: assumes wager = 1 for calculation of unit equations.

John

[jfdinneen]

SchagFactorToWin,

A minor correction to Unit_Expectation calculation:
From
Unit_Expectation=SUM(PRODUCT(PRODUCT(SUM(Decimal_O dds,-1),1),Win_Probability),PRODUCT(-Wager,SUM(1-Win_Probability)))
to
Unit_Expectation=SUM(PRODUCT(PRODUCT(SUM(Decimal_O dds,-1),Wager),Win_Probability),PRODUCT(-Wager,SUM(1-Win_Probability)))

John

[SchagFactorToWin]

How did you get the risk of ruin of 6.12% in your example? I plugged your numbers in and didn't get that.

Also, what is the relationship between the Handicapper Statistics and the Krigman Equivalents above?

Finally, Session Length doesn't appear in any of your formulas. Shouldn't it?

[jfdinneen]

SchagFactorToWin,

See attached graphic. These input values give the correct result.
Please confirm?

John

[jfdinneen]

SchagFactorToWin

The Even-Money Equivalents (i.e wager=$43.40, win_probability=53.02%, and decimal_odds=2.00) generate the same expectation and variance as the original handicapper statistics (i.e wager $35.00, win_probability=43.00%, and decimal_odds=2.50), thereby providing a means of direct comparison with other handicappers and of calculating the even-money equivalent risk of ruin.

Session length is used in calculating the survival-based risk of ruin.

John

[SchagFactorToWin]

Git it now. Thanks. I'll play with it this afternoon, I know my bank is not set properly. I think this will help.

[jamey1977]

Thanks for the link.

I couldn't understand how to calculate variance, however. Well, I know how to calculate variance but I couldn't follow what variance he was measuring (from a horse racing POV). What do you think?
What is this dude talking about? Restricting plays to even money horses. ? You have to win what 6 out of 10 to be profitable. 60 plays a month on a circuit and win between 35 and 40 ? For a lousy 15 to 20 percent. R.O.I.? It can be done but. You have to have a good month. So much crap happens. Track is off, speed horse gets fried in a duel. Or the horse simply tanks for the day. I tried this.I failed miserably. But it can be done. But luck is always involved. A few bad calls, bad photo finishes and bad races and you're at 30 wins out of 60 plays. R.O.I. Zero.

[Robert Goren]

The session length of 250 is far too small for any practical application.

[SchagFactorToWin]

The session length of 250 is far too small for any practical application.

The session length is an input box- you can make it any size you want.

[lansdale]

For those who wish to compare their average performance to that of other handicappers, I would strogly recommend reading Alan Krigman's excellent piece, "Edge, Volatility, and Risk of Ruin in Gambling (http://www.bjmath.com/bjmath/Betsize/winwayz/tmpweb.htm) ".

In an effort to encourage its usage, I have provided the required Excel formulae below for calculating the even-money equivalent stake, win probability and risk of ruin.

Handicapper Statistics
Initial Bankroll: $1,000.00
Final Bankroll: $2,000.00
Session Length: 250
Decimal Odds (Avg): 2.50
Wager(Avg): $35.00
Win Probability (Avg): 43.00%

Krigman Even-Money Equivalents
Initial Bankroll: $1,000.00
Final Bankroll: $2,000.00
Session Length: 250
Decimal Odds: 2.00
Wager: $43.40
Win Probability: 53.02%
Risk Of Ruin: 6.12%

EvensEquivalent_Wager=SQRT(SUM(POWER(PRODUCT(Wager ,SQRT(ABS(PRODUCT(POWER(-SUM(Decimal_Odds,1),2),SUM(Win_Probability,-1), Win_Probability)))),2),POWER(PRODUCT(Wager,SUM(PRO DUCT(Win_Probability,SUM(Decimal_Odds,1)),-1)),2)))

EvensEquivalent_WinProbability=SUM(0.5,PRODUCT(0.5 ,PRODUCT(SUM(PRODUCT(Win_Probability,SUM(Decimal_O dds,1)),-1),1/SQRT(SUM(POWER(SQRT(ABS(PRODUCT(POWER(-SUM(Decimal_Odds,1),2),SUM(Win_Probability,-1), Win_Probability))),2),POWER(SUM(PRODUCT(Win_Probab ility,SUM(Decimal_Odds,1)),-1),2))))))

EvensEquivalent_RiskOfRuin=PRODUCT(SUM(POWER(PRODU CT(SUM(1,-EvensEquivalent_WinProbability),1/EvensEquivalent_WinProbability),PRODUCT(SUM(Initia l_Bankroll,Final_Bankroll),1/PRODUCT(PRODUCT(EvensEquivalent_Wager,1/Wager),Wager))),-POWER(PRODUCT(SUM(1,-EvensEquivalent_WinProbability),1/EvensEquivalent_WinProbability),PRODUCT(Initial_Ba nkroll,1/PRODUCT(PRODUCT(EvensEquivalent_Wager,1/Wager),Wager)))),1/SUM(POWER(PRODUCT(SUM(1,-EvensEquivalent_WinProbability),1/EvensEquivalent_WinProbability),PRODUCT(SUM(Initia l_Bankroll,Final_Bankroll),1/PRODUCT(PRODUCT(EvensEquivalent_Wager,1/Wager),Wager))),-1))

Enjoy!

John

Hi John,

Although I'm an admirer of yours, I'm somewhat leery of the approach suggested here. If you're adapting this from the Krigsman piece, it's worth mentioning that this was intended to apply to blackjack which, as I know you realize, like other casino games, is a mathematical system, i.e., it has no 'reality' component, unlike horse-racing and, say, poker. Such games can be simmed with great precision and accuracy.

Krigsman mentions the influence of blackjack authority Don Schlesinger in his article. I have had a number of exchanges with Schlesinger on the viability of calculating the kind of key facors you mention, i.e. SD, ROR, etc., and he was extremely skeptical about doing so for horserace handicapping, citing not only the obstacles of the number of factors, but also the danger of averaging, stating the one needs to know, as precisely as possible, one's mathematical advantage for each bet.

If you would like to take it up with him, I could see if is willing to address the issue, since his main area of concentration is blackjack.

Cheers,

lansdale

[jfdinneen]

...stating that one needs to know, as precisely as possible, one's mathematical advantage for each bet....
lansdale,

Thanks for the kind words.

The original goal was to draw attention to using a "Standard Normal Form" when comparing betting profiles. As you know, a distribution of wagers can be summarized (minimum) with a measure of height (average) and a measure of width (variance) and is independent of how those wagers were originally generated. This exercise does not prescribe how an edge/overlay handicapper should calculate an upcoming bet (which you correctly point out). If there was any confusion as to the intent, I apologize.

John