Eliezer, your main point is correct and interesting, but the coin flip example is definitely wrong. The market’s beliefs don’t affect the bias of the coin! The map doesn’t affect the territory.

The relevant FINANCE question is ‘how much would you pay for a contract that pays $1 if the coin comes up heads?’. This is then the classic prediction market type contract.

The price should indeed be ten elevenths. Of course, you don’t expect to make money buying this contract, which was exactly your point.

What WILL be true is that the expected change in the price of the contract from one period to the next will be zero. This need not mean that it goes up 50% of the time, but the expected value next period (in this case) is the current price.

The first proof that I know of this was done by Paul Samuelson in 1965, in his paper ‘Proof that Properly Anticipated Prices Fluctuate Randomly’.

Russ, I think that if you take the example literally, the price would be 91%, not 50%, and you wouldn’t expect to make money.

Eliezer, the PS definitely clarifies matters.

Although I also think the example is actually instructive if taken literally too. In particular, if you see nine heads in a row, each additional head means you expect a higher chance of heads next flip. But you do not expect an increase in the price of the contract that pays $1 if heads comes up. THAT still has an expected price change of zero, even thought we expect more heads going forward.

In other words, future EVENTS can be predictable, but future PRICES cannot.