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There are two possible outcomes for a sporting event — Team A wins, or Team B wins. How can I calculate the maximum profit given the following arbitrage opportunity?

  • Sportsbook 1 (Outcome Team A wins) — Odds 2.17 (I plan to bet \$100 and they will refund \$50 as part of a promotion if it loses)

  • Sportsbook 2 (Outcome Team B wins) — Odds 1.70 (No promotion here, but how much do I bet here to ensure maximum gain regardless of the outcome of the event).

If I bet \$100 and Team A wins, I stand to win \$217 (for a total profit of $117). If they lose, I only lose \$50.

How much do I need to bet on the second sportsbook to ensure that I can take full advantage of the arbitrage opportunity regardless of which team wins. There is no promotion for the second sportsbook.

Rodrigo de Azevedo
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zerof
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  • What happens when you write out the payoff structure, assuming a $$100$ bet on the promotion-carrying $A$ and an $$X$ bet on $B$? – lulu Feb 05 '22 at 12:36
  • I tried to figure out how much I need to bet on Sportsbook 2 to get my loss back from Sportsbook 1 if Team A loses, but I didn’t figure out how to calculate the optimal bet assuming each Team is equally likely to win. – zerof Feb 05 '22 at 12:37
  • Again, just write out the payoff structure. Assume an $$X$ bet on $B$ (vs a $$100$ bet on $A$). What do you make if $A$ wins? Well, that's obvious. You make $$117-X$. Now, what do you make if $B$ wins? Set those expressions equal to each other. – lulu Feb 05 '22 at 12:40
  • Note: in the computation I sketched the actual probability of one or the other team winning is entirely irrelevant. Thanks to the promotion, this is a strict arbitrage. You will make exactly the same profit regardless of which team wins, so you are indifferent to the actual probabilities and you are indifferent to the outcome of the game. – lulu Feb 05 '22 at 12:44
  • Take a look at this. – Rodrigo de Azevedo Feb 05 '22 at 13:05
  • Sorry, I still do not follow on what I need to do given my situation. I don’t know what my total investment should be because of the $50 refund I get on Sportsbook 1 for P(Team A) to win – zerof Feb 05 '22 at 13:37
  • Once again: Fix a $$100$ bet on $A$. Now, set a variable beet $$X$ on $B$. Work out the two payouts (depending on who wins) as a function of $X$. You should be able to do that in your head. I already told you that your profit/loss in the case that $A$ wins is $117-X$ so you just have to figure out the profit/loss if $B$ wins. Now set those two functions equal and solve for $X$. That's all you have to do. – lulu Feb 05 '22 at 14:59

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