In general, an investment which costs $p$ to purchase and which yields $q$ at some later date has a return of $r = \frac{q – p}{p}$. For example, a bet which costs 10 today and pays out 12 tomorrow (with certainty) has a return of $r = \frac{12 – 10}{10} = 0.2$. Presented with bets that have equal risk, people tend to prefer bets with higher rates of return.

In order to compare investments which have uncertain payoffs, the expected return is typically used. The expected return is, as the name suggests, the expected value of the return. A bet costing $p$ which pays 1 tomorrow with probability $\pi$ and 0 otherwise has an expected return of $\frac{\pi – p}{p}$. (The variance of the return is $\frac{\pi(1-\pi)}{p^2}$.) A more concrete example would be a bet which costs 8 today and pays out 10 tomorrow with probability 90% and nothing otherwise. This bet has an expected return of $\frac{9-8}{8} = 0.125$. Again, presented with bets that have equal risk, people tend to prefer bets with higher expected returns.

#predictionmarkets #return

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