# What is matched betting?

In my previous post, I found a curiosity: an asset with different prices in two different betting exchanges. This can be used to turn a guaranteed profit, and in this post I show you how! Consider the question: “Will Donald Trump be president at year-end 2019?” Just like betting on sports, you can bet on the answer to this question. On PredictIt, the market estimates the probability of “Yes” at 90-91%. But on BetFair, the market estimates the probability at 94-95%! We can exploit this difference in beliefs to turn a guaranteed profit, through the magic of matched betting!

Because the PredictIt market thinks it’s less likely that Trump will still be president, they’re willing to offer a better return on “Yes”, so we’ll buy some “Yes” at PredictIt. Conversely, BetFair offers a better return on “No”, so we’ll buy some “No” at BetFair. But how much should we buy? First, I’ll show you the magic numbers for how to make a guaranteed profit, and then I’ll show you how to calculate them. Here’s what we should buy:

• \$850 of “Yes” at PredictIt.
• \$58.38 of “No” at BetFair.

If we purchase this, how much profit will we make? The cost is \$850+\$58.38 = \$908.38, but what’s the revenue? The revenue depends on whether Trump is still president at the end of the year.

• Case A: If Trump is still president, we make our revenue from PredictIt. We bought 934 shares, at 91¢ each, and each yields \$1. This gives us a revenue of \$934.
• Case B: If Trump is not still president, we make our revenue from BetFair. We bought \$58.38 at odds 16.00, making revenue of 16 * \$58.38 = \$934. (Plus 8¢ change due to a rounding error.)

In either case, the revenue is \$934. Minus our costs, our guaranteed profit is \$25.62. Magic! But where did those magic numbers \$850 and \$58.38 come from?

First, we fix one of the numbers. It turns out that PredictIt only allows you to bet up to \$850 in one market, so we fix our PredictIt purchase at that maximum. Then we choose an amount to bet at BetFair such that the profit in either case is the same. We can formulate some equations to solve:

``````P  := profit we'll make in either case
Bb := amount to bet at BetFair
Bp := amount to bet at PredictIt
Ob := odds at BetFair
Op := odds at PredictIt

P = Bp*Op - (Bp + Bb)     // profit in Case A (Trump is still president)
P = Bb*Ob - (Bp + Bb)     // profit in Case B (Trump is not still president)

Bp*Op = Bb*Ob             // combine simultaneous equations
Bb = Bp*Op / Ob           // solve for how much to bet at BetFair
Bb = 850 * (1/0.91) / 16  // plug in the real-world numbers
Bb = 58.38                // this is how much to bet at BetFair``````

In the world of betting, this technique is called matched betting. In the world of trading, this technique is called arbitrage. Arguably, your profit is derived from the service you provide of transferring information from one market to the other.