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I am researching about betting arbitrage and I always like to understand why formulas solve for certain things when based on their variables in the equation, don't seem like they would solve for said thing. In this case, I am wondering about a formula for betting arbitrage. I understand how to determine the market-derived implied probability (including a vig) of an outcome with fractional odds or decimal odds and the intuition behind it.

What I cannot understand is the formula of how to determine how much to bet on each outcome to guarantee yourself arbitrage. (given an event where the implied probability of both possible outcomes sums to less than 100%).

Given a game between 2 teams where their odds provide implied probabilities under 100% (some websites also call this the arbitrage percentage), there is an arbitrage opportunity available.

Can someone please explain to me the intuition of how the formula of how much to bet on each team to obtain the available arbitrage makes sense given its variables? Formula below:

Stake for Team A = (Total Stake * Team A Arbitrage Percentage)/Total Arbitrage Percentage

Rodrigo de Azevedo
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user60519
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  • Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Shaun Dec 23 '22 at 16:54
  • Please ask one question at a time. – Shaun Dec 23 '22 at 16:54
  • thanks, it has been updated. In terms of what I have tried, it is hard to explain as thinking of how the variables interacting would create the stake has not worked out for me. – user60519 Dec 23 '22 at 17:04
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    That's better. For more context, try answering the following questions in an [edit]. What are you studying? What text is this drawn from, if any? If not, how did the question arise? What kind of approaches (to similar problems) are you familiar with? What kind of answer are you looking for? Basic approach, hint, explanation, something else? Is this question something you think you should be able to answer? Why or why not? – Shaun Dec 23 '22 at 17:06
  • Take a look at this – Rodrigo de Azevedo Dec 25 '22 at 22:11

1 Answers1

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Suppose a certain sports event has 3 possible outcomes: Team A wins, Team B wins, or draw. You find the best available odds for betting on each of those outcomes. The intuition is: bet an amount on Team A that guarantees you'll win $\$100$ if Team A wins. Then bet on team B enough so that you'll win $\$100$ if team B wins. And same for draw.

Now you've placed 3 bets and you will win $\$100$ no matter how the sports results goes. But how much money have you spent placing those bets? If the total is $\ge \$100$ then you've broken even or lost money; this is not an arbitrage opportunity. If the total is $< \$100$ then you're guaranteed to make money, since you'll definitely win $\$100$ back.

The formulas on your website just come down to calculating a) how much money you need to bet on each outcome, and then b) deciding whether you'll make or lose money if you try it.

Note in the example above I assumed there are 3 outcomes, but in real life you would bet on ALL possible outcomes. For many sports matches there would be only 2 outcomes, but for some events there could be 4+ possible results, and you'd need to bet on each of them. Once again, the core intuition is to bet on all possible results so that you'll be guaranteed to win exactly $\$100$ no matter what, and then if you can pay $< \$100$ to place that combination of bets then it's an arbitrage.

Theo Bendit
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David Clyde
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  • Thanks for response and I do understand the idea around arbitrage betting. The question I have though is what is (if any) the intuition behind that formula I put on my post that tells the bettor how much to stake on each team given that there is an opportunity for arbitrage.

    Why do those variables in their variation give the amount to bet on a team (given arbitrage available)? The variables individually don’t seem like they would give us the bettor stake.

     – user60519 Dec 25 '22 at 13:57
  • @user60519 Betting more of your stake that you bet on outcome $A$, makes the negative the total payout if outcome $B$ happens. If you want to maximise the minimum payout, balancing the two outcomes, you'll want the payout in either outcome to be equal (any adjustment either side will reduce the payout in one of the possible outcomes). This is what you get from these stakes. If you want $$100$ payout regardless, you put each arbitrage percentage (in dollars) on each outcome. But this is working backwards; you don't aim for a certain payout, instead you have a certain amount of money to bet. – Theo Bendit Dec 25 '22 at 16:17
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    @user60519 It does give you a proportion: $100$ divided by the total of the arbitrage percentages gives you the proportion that your money can grow, when betting so that both outcomes pay the same. To get this optimal guaranteed payout, you just need to spend whatever money you have in these proportions. So rather than spending $A%$ dollars on $A$ and $B%$ dollars on $B$, for a total of $$(A+B)$, to get $$100$, you spend $A/(A+B)$ of whatever money you have on $A$, and $B/(A+B)$ of whatever money you have on $B$, and get the maximum guaranteed payout for your stake. – Theo Bendit Dec 25 '22 at 16:17