The Expected Return on a Bet in a Prediction Market

The return or rate of return is the standard way of characterizing the benefit associated with making an investment. The return on an investment is calculated by dividing an investment’s net payoff by the cost of the investment. A bet on a prediction market is a kind of investment. Comparing bets in terms of their returns allows us to ignore irrelevant details and focus on the relative benefits of bets.

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.

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