How Much am I Suppose to Bet?

One of the most important aspects of successfully gambling on any sport is betting the proper amount on a given event (game, fight, race, etc.). Wagering the same amount, regardless of the perceived edge in the lines, is a guaranteed way of going broke in the long run (unless you are using the KISS method and picking more winners than losers). While everyone has there own unique system, allow us to introduce you to the grandfather of proper bankroll management: John Larry Kelly, Jr. and his Kelly Criterion

Quick History

Kelly was a scientist who worked at Bell Labs in the 50′s. He had a PhD in Physics from the University of Texas, and he was a fighter pilot in WWII. The guy was intelligent to say the least.

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The Formula

Kelly created a formula that is now known as the Kelly Criterion or Kelly Strategy. In a nutshell, it allows a gambler to maximize their winnings over a series of wagering events, and it is logarithmic in nature, which means it will maximize your profits by growing your winning exponentially. So lets take a look at it:

f = (bp – q) / b

Where:
f = the fraction of your bankroll to risk on the fight (we will convert this to a percentage at the end)
b = the odds received on the wager (what the bookies are offering and in the form of “b to 1″)
p = the probability of winning (what you calculate the actual odds to be)
q = the probability of losing (“1 – p” to be exact)

Now, we know that may look a little daunting if you haven’t solved any formulas since high school or your freshman year in college. But we are going to walk you through the formula with several examples in order to get the hang of it.

Example 1

For this first example, we’ll look at the Cole Miller vs Matt Wiman fight from Ultimate Fight Night (UFN) 23 on January 22, 2011.

Step 1: Defining what we know:

Let’s start by stating what we know. We have the odds the bookmakers are offering, and we have what we think the odds should be from doing our own analysis. However, those odds are in the form of American betting lines.

On fight night, SportsInteraction was offering odds of +145 on Matt Wiman. This site had set the odds of Matt Wiman winning at +110. This presents a possible betting opportunity on Matt Wiman since he is being undervalued in this fight.

So:
Odds given by the bookmaker: +145
Odds calculated by us: +110

Step 2: Finding b

For the Kelly Criterion formula, we can’t use American Odds, we have to convert those numbers into the proper format for the formula.

For b, we need it in the form of “b to 1″. This is pretty easy to do.

For underdogs: b = (“the bookies line” / 100)
For favorites: b = 1 / (-”the bookies line” / 100)
*Notice that for b in the favorite’s equation, you just need to drop the negative sign in order to get -”The bookies line”. If the line is -200, just turn it into 200, which would look like this: b = 1 / (200 / 100) = .5

Our Wiman line of +145 is an underdog line, so we just plug and play: b = (145 / 100) = 1.45

Step 3: Finding p

p is also pretty easy to find.

For underdogs p = (100 / 100 + “our calculated line”)
For favorites p = (-”our calculated line” / 100 + -”our calculated line”)
*Notice that for p in the favorite’s equation, you just need to drop the negative sign in order to get -”our calculated line”. If the line is -200, just turn it into 200, which would look like this: p = (200 / 100 + 200) = .667

Our calculated Wiman line of +110 is still an underdog line, so we plug and play again: p = (100 / 100 + 110) = .476

Step 4: Finding q

After we get b and p, the last variable we need is q, and it is by far the simplest to figure out. q just tells us the probability of losing, and if we already figured out p, the probability of winning, then q is the probability left over to make up 1 (or 100%).

Therefore: q = (1 – p)

For our example: q = (1 – .476) = .524

Step 4: Putting it all together

We figured out:
b = 1.45
p = .476
q = .524

We just need to plug those into the following equation: f = [(b * p) – q] / b

For our example: f = [(1.45 * .476) – .524] / 1.45 = .115

Our answer, .115, is the fraction of our bankroll to bet on the Matt Wiman fight to maximize our profits in the long run. You can also convert the fraction to a percentage by multiplying by 100 to get 11.5%.

If you have a theoretical bankroll of $1,000 and wanted to bet the maximum Kelly bet, you would have bet 11.5% of your bankroll on the fight, which is $115. This would have netted you $166.75 in profit since Matt Wiman won.

Example 2

Now we’ll look at a the Yushin Okami vs Nate Marquardt fight from UFC 122.

Step 1: Defining what we know:

Let’s start by stating what we know. We have the odds the bookmakers are offering, and we have what we think the odds should be from doing our own analysis. However, those odds are in the form of American betting lines.

On fight night, 5Dimes was offering odds of +175 on Yushin Okami. This site had set the odds of Okaami winning at -140. This presents a possible betting opportunity on Okami as he was being highly undervalued with the sportsbooks having him as an underdog when we thought he should have been the favorite.

So:
Odds given by the bookmaker: +175
Odds calculated by us: -140

Step 2: Finding b

For the Kelly Criterion formula, we can’t use American Odds, we have to convert those numbers into the proper format.

For b, we need it in the form of “b to 1″. This is pretty easy to do.

For underdogs: b = (“the bookies line” / 100)
For favorites: b = 1 / (-”the bookies line” / 100)
*Notice that for b in the favorite’s equation, you just need to drop the negative sign in order to get -”The bookies line”. If the line is -200, just turn it into 200, which would look like this: b = 1 / (200 / 100) = .5

Our Okami line of +175 is an underdog line, so we just plug and play: b = (175 / 100) = 1.75

Step 3: Finding p

For underdogs p = (100 / 100 + “our calculated line”)
For favorites p = (-”our calculated line” / 100 + -”our calculated line”)

Our calculated Okami line of -140 is a favorite line, so we plug and play: p = (140 / 100 + 140) = .583

*Notice that we have simply dropped the “-” from the American line for the favorite’s equation. -”our calculated line” causes a double negative: -”-140″ which is a positive number since the two negative signs cancel out, leaving: 140

Step 4: Finding q

q = (1 – p)

For our example: q = (1 – .583) = .417

Step 4: Putting it all together

We figured out:
b = 1.75
p = .583
q = .417

We just need to plug those into the following equation: f = [(b * p) – q] / b

For our example: f = [(1.75 * .583) – .417] / 1.75 = .345

Our answer, .345, is the fraction of our bankroll to bet on Yushin Okami to maximize our profits in the long run. You can also convert the fraction to a percentage by multiplying by 100 to get 34.5%.

If you have a theoretical bankroll of $1,000 and wanted to bet the maximum Kelly bet, you would have bet 34.5% of your bankroll on the fight, which is $345. This would have netted you $603.75 in profit since Okami won.

Risks and Caveats

Hopefully you noticed in the above two examples that the optimum amount to bet on each fight is an extremely high fraction of your bankroll. The Kelly Criterion is optimized for a perfect world where you know exactly what the odds should be. In the real world, the odds we calculated could be off, we simply don’t know.

What the Kelly Criterion really does is define the upper limit of how much to bet on a given fight. Wagering more than the fraction given by the Kelly Formula will not maximize your profits in the long run and will actually lead to higher potential for Gambler’s Ruin (aka a bankroll of $0). As such, most gamblers will not bet the full Kelly bet.

Instead, it is wise to only bet a fraction of the fully Kelly bet. What fraction you go with is determined by your own risk aversion and how confident you are in your own fight analysis. If you are extremely confident and risk averse, you could bet half of the Kelly bet or “Half-Kelly”. This just involves dividing f by 2.

For Example 1: This would mean a Half-Kelly bet of 5.75% or $57.50 on Matt Wiman.
For Example 2: This would mean a Half-Kelly bet of 17.25% or $172.50 on Yushin Okami.

The higher your risk aversion or lack of confidence in your analysis, the higher you divide the Kelly Bet by. You can do a Third-Kelly bet by dividing f by 3, or a Quarter-Kelly bet by dividing by 4, or if you are really risk averse, dividing by 5 for a Penta-Kelly bet.

The more you divide f by, the less your overall profits in the long-run. The trade off is that you are also diminishing your risk of potential ruin, especially in an uncertain world like MMA.

Now that you understand how the Kelly Criterion works, you can use our trusty Kelly Criterion Calculator to do all of the calculations for you.

Bet at 5dimes

Sources

http://en.wikipedia.org/wiki/Kelly_criterion
http://www.fortunesformula.com/contact.html
http://www.jimgeary.com/poker/letters/KELLY.HTM
http://www.bjmath.com/bjmath/kelly/kelly.pdf