(Written for Arbital in 2016.)

Betting on one-time events

Consider evaluating, in June of 2016, the question: "What is the probability of Hillary Clinton winning the 2016 US presidential election?"

On the propensity view, Hillary has some fundamental chance of winning the election. To ask about the probability is to ask about this objective chance. If we see a prediction market in which prices move after each new poll — so that it says 60% one day, and 80% a week later — then clearly the prediction market isn't giving us very strong information about this objective chance, since it doesn't seem very likely that Clinton's real chance of winning is swinging so rapidly.

On the frequentist view, we cannot formally or rigorously say anything about the 2016 presidential election, because it only happens once. We can't observe a frequency with which Clinton wins presidential elections. A frequentist might concede that they would cheerfully buy for $1 a ticket that pays $20 if Clinton wins, considering this a favorable bet in an informal sense, while insisting that this sort of reasoning isn't sufficiently rigorous, and therefore isn't suitable for being included in science journals.

On the subjective view, saying that Hillary has an 80% chance of winning the election summarizes our knowledge about the election or our state of uncertainty given what we currently know. It makes sense for the prediction market prices to change in response to new polls, because our current state of knowledge is changing.

A coin with an unknown bias

Suppose we have a coin, weighted so that it lands heads somewhere between 0% and 100% of the time, but we don't know the coin's actual bias.

The coin is then flipped three times where we can see it. It comes up heads twice, and tails once: HHT.

The coin is then flipped again, where nobody can see it yet. An honest and trustworthy experimenter lets you spin a wheel-of-gambling-odds — reducing the worry that the experimenter might know more about the coin than you, and be offering you a deliberately rigged bet — and the wheel lands on (2 : 1). The experimenter asks if you'd enter into a gamble where you win $2 if the unseen coin flip is tails, and pay $1 if the unseen coin flip is heads.

On a propensity view, the coin has some objective probability between 0 and 1 of being heads, but we just don&#...