To hedge or not to hedge, that is the question.

One of the decisions that a gambler has to make is if given a winning bet on an outsider, is there an advantage to hedge your bet and take a guaranteed return? Basically hedging means that if you make a pre-match bet, usually on an outsider, and if that player or team plays particularly well, then the odds for that bet will come in. It is now possible to hedge your bet by betting on the opponent, in order to guarantee a win not matter what team wins.

Psychologically, its quite comforting to know that no matter the outcome, we will be taking a collect. It’s a lot more stressful than watching an outsider battle it out. It also takes away the huge disappointment on the off chance that you big outside bet, despite winning well, collapses on the finish line and loses the race which was almost unlosable.

But the question still remains, is it a good way of betting? Sure, it might well help you not lose a few more grey hairs, but mathematically will it help you increase your bank balance?

Some people will say that it really shouldn’t matter, whilst others will say that it helps you increase your bank balance more consistently, which will in turn, help you bet with more money and hence have bigger bet sizes with an increasing bank.

Well, no better way to work it out, then to set up an experiment. And here are the details:

I set up a simulation of 1000 bets, 1000 times. The initial bet was at odds of 3.00, and I assumed that the probability of the bet winning was 35%. Hence here we have a (3.00 x 0.35 – 1) = 5% edge. The bank size I use for the experiment was $1000, and decided to bet with 1/5^{th} Kelly. Hence the initial bet size is equal to:

1000/5 * .05 / (3-1) = $5

Of course this amount will vary as the bank balance increases and decreases over the 1000 bets. So how did it go?

Over the 1000 simulations, the average bank size at the end of the 1000 bets was $1,208.06. So a rather nice 20% increase. 66.7% of the 1000 simulations, or right on two thirds of the time, the bank balance was greater than the initial $1000. So despite the edge, the betting system still lost one in every 3 simulations.

The standard deviation, for those statistically inclined, was 289.81.

Now for the second method which deals with hedging bets. The experiment that I set up for each bet was based on two sides. Each side had to get to 11 points first, and the way the two teams acquire a point is from generating a random uniform number from 0 to 1.

If the random number is less than 0.458527 then the first team scores a point. Otherwise the second team scores a point. As you can probably tell, it is more likely that team 2 scores a point than team 1, and the reason this is, is that the probability of team 1 winning using the above number is exactly 35%. Hence, at odds of 3.00 we have exactly the same scenario as the first experiment.

I then created a rule that if the probability of player 1 winning was greater or equal to 80%, then we hedge our bets. The odds that we bet at are equal to the probability of the opposition.

So for example, if the probability of team1 winning was 85%, then the probability of team2 winning is 15%, and we hedge our bet at odds of 1 / 0.15 = 6.67. The amount we bet, makes it so that the same profit is made on either team.

Sometimes of course, we wont have an opportunity to have a probability of greater than 80%. This being because either the team that we backed did poorly, or perhaps they were tied at 10 all, and it came down to the last point to decide who was the winner.

And the results? Based on 1000 simulations of 1000 bets, the average bank balance at the end was $1,130.65. Thus only a 13% increase in bank balance was achieved compared to the previous method of sticking it out of 20%. Hence we can only come to a conclusion that hedging is an inferior method of gambling.

But that’s not everything. The hedging method recorded a profit in 71.1% of simulations, slightly higher than the first method which recorded 66.7% profits. So perhaps it is more likely to be profitable to hedge the bets, despite an lower than expected profit.

But why is this? The answer is because of the standard deviation. The standard deviation of the hedging method over the course of the simulations was 225. This being significantly lower than the first method of 290.

In other words, the finishing bank of the hedging method was far less variable than the finishing bank of the first method which saw the match play out. This kind of makes sense, because hedging in itself reduces variability in the result.

So hedging may produce less profits, but is more likely to obtain a profit.

Is that all? Well not really. The above simulation assumed that the odds that one could get for hedging was exactly the same as the probability. This is not the case. Even when hedging you are making a bet against the bookmaker, and you lose a little commission in the time being.

Also, it is reasonable to assume, that when you make the hedge bet, that the overlay still exists on the first team that you backed. If there was an overlay before the match on the first team, then there probably is an overlay on the first team at the time of hedging, and hence an underlay on the opponent.

What this means is that the hedge bet that you make, is more than likely to be at an unadvantage. The odds of which you achieve the hedge bet would be lower than that the experiment as outlined above.

Of course this means that the profits in the experiment are also less likely to be as big.

So in conclusion, mathematically it is not a wise decision to hedge your bets. You are essentially betting an underlay, and over the long term the only advantage that you might have by hedging is bragging rights to your friends, and a saving a few grey hairs.

Excellent article.

This was something the missus always asked..”why don’t you back both teams, you can’t lose”.

Most times I’d back the underdog simply because the bookies set the prices so there’s mostly a 0.5 point difference.

I say most of the time because a $5.00 outsider really should lose, both at head to head & at the line.

Betting mostly on the outsiders is a wild ride, but you do win in the end.

I want to make sure I understand this, so let’s continue with the scenario you described:

“The initial bet was at odds of 3.00, and I assumed that the probability of the bet winning was 35%. Hence here we have a (3.00 x 0.35 – 1) = 5% edge. The bank size I use for the experiment was $1000, and decided to bet with 1/5th Kelly. Hence the initial bet size is equal to: 1000/5 * .05 / (3-1) = $5”

and

“The odds that we bet at are equal to the probability of the opposition.”

So we have an initial bet with odds of 3,00, probability 0,35 and stake 0,5% from the bankroll. After that we find odds of 1,54 (1/0,65) for the other team. According to your article, is it more profitable to keep the original bet with the odds of 3,00 or hedge it with the odds of 1,54?

more profitable to keep the original bet than to hedge it

I disagree. Your thinking is probably too complicated.

Mathematically more sophisticated way to approach this problem is to use the bankroll growth rate:

G=(1+x*(o-1))^p*(1-x)^(1-p), where G=growth rate, x=stake %, o=odds, p=probability

Therefore the bankroll growth rate of the first bet is: (1+0,005*(3,00-1))^0,35*(1-0,005)^(1-0,35)=1,000224

Now we have to know what is the bankroll growth rate if we want to hedge our bet. This of course alters the formula of the bankroll growth rate. The new formula is:

G=(1+x*(o1-1)-y)^p*(1+y*(o2-1)-x)^(1-p), where G=growth rate, x=stake % of the first bet, y=stake % of the second bet, o1=odds of the first bet, o2=odds of the second bet, p=probability of the first bet. (The optimum way to determine y=stake % of the second bet is to use the ratio of probabilities and multiply it with the original stake: in this example 0,65/0,35*0,005=0,0093)

Therefore the bankroll growth rate of the hedging option is: (1+0,005*(3,00-1)-0,0093)^0,35*(1+0,0093*(1,54-1)-0,005)^(1-0,35)=1,000259

1,000259>1,000224, meaning it’s more profitable to hedge than keep the original bet.

With this exact method it’s always easy to determine whether to hedge or not. But this is not enough. The method assumes that probabilities are exact. As we know, this is not the case: nobody can give the exact probabilities in sports betting. Therefore it would probably be wiser to hedge even if the bankroll growth rate of the hedge option is smaller than the normal bet option. Actually many of the Sportpunter’s models exaggerates this phenomenon (they are winning but probability estimations are quite inaccurate). This means that if you are following Sportpunter’s models, hedging is even better option than what just pure numbers suggest.

Sorry, I didn’t read your article properly. You were clearly discussing about hedging while live betting, but somehow I missed that. I was talking about prematch hedging where the probabilities won’t change but the odds move to your favor. In your example they change. Now I understand why I didn’t get your article at first time 🙂

nice arguement and stats Elementary Penguin, but I’m not sure you got it right, and not sure I read your reply right either!

The article states that we bet $5 @ 3.00 with a 35% chance. Our bet does exceptionly well. and moves into favourtism, Say 1.20, with 5.00 on the other side. Should we bet on the opposition at 5.00 or should we let the bet play out.

but interesting point you bring up EP, if the prices move pre match, its better to hedge your bets then than actually letting it sit out. I might look into that.