It might interest you that there’s quite a nice isomorphism between prediction markets and ordinary forecasting tournaments.
Suppose you have some proper scoring rule S(pi) for predictions p on outcome i. For example, in our experiment we used S(pi)=ln(pi). Now suppose the t:th prediction is paid the difference between their score and the score of the previous participant: S(pi,t)−S(pi,t−1). Then you basically have a prediction market!
To make this isomorphism work, the prediction market must be run by an automated market maker which buys and sells at certain prices which are predetermined by a particular formula.
To see that, let C(xi) be the total cost of buying x shares in some possibility i (e.g. Yes or No). If the event happens, your payoff will be xi−C(xi) (we’re assuming that the shares just pay $1 if the event happens and $0 otherwise). It follows that the cost of buying further shares—the market price—is C′(xi).
We require that the market prices can be interpreted as probabilities. This means that the prices for all MECE outcomes must sum to 1, i.e. ∑i∈ΩC′(xi)=1.
Now we set your profit from buying x shares in the prediction market be equal to your payout in the forecasting tournament, xi−C(xi)=S(pi). Finally, we solve for C, which specifies how the automated market maker must make its trades. Different scoring rules will give you different C. For example, a logarithmic scoring rule will give: C(→x)=bln(∑i∈Ωexib).
For more details, see page 54 in this paper, Section 5.3, “Cost functions and Market Scoring Rules”.
It might interest you that there’s quite a nice isomorphism between prediction markets and ordinary forecasting tournaments.
Suppose you have some proper scoring rule S(pi) for predictions p on outcome i. For example, in our experiment we used S(pi)=ln(pi). Now suppose the t:th prediction is paid the difference between their score and the score of the previous participant: S(pi,t)−S(pi,t−1). Then you basically have a prediction market!
To make this isomorphism work, the prediction market must be run by an automated market maker which buys and sells at certain prices which are predetermined by a particular formula.
To see that, let C(xi) be the total cost of buying x shares in some possibility i (e.g. Yes or No). If the event happens, your payoff will be xi−C(xi) (we’re assuming that the shares just pay $1 if the event happens and $0 otherwise). It follows that the cost of buying further shares—the market price—is C′(xi).
We require that the market prices can be interpreted as probabilities. This means that the prices for all MECE outcomes must sum to 1, i.e. ∑i∈ΩC′(xi)=1.
Now we set your profit from buying x shares in the prediction market be equal to your payout in the forecasting tournament, xi−C(xi)=S(pi). Finally, we solve for C, which specifies how the automated market maker must make its trades. Different scoring rules will give you different C. For example, a logarithmic scoring rule will give: C(→x)=bln(∑i∈Ωexib).
For more details, see page 54 in this paper, Section 5.3, “Cost functions and Market Scoring Rules”.