How do prediction markets work?

Prediction markets for binary questions let people vote by betting on one of the two outcomes. In finance speak this is called offering a binary option derivative.

The idea is simple: Say you can bet money on a fair coin landing on head. What return needs to be promised to you, such that you are incentivised to take a bet? Say that you bet an amount so small that a loss would not hurt. The answer then is that any return of just over 2x can overcompensate the chance of loss and is a bet that should be taken.

In general, when the real probability is $P_A(t)$ the return should be greater than $1/P_A(t)$ to take the bet. A prediction market has a current market probability $P_{A,m}(t)$. To make sense and fulfill its job properly, it will offer bets on A at a return of $1/P_{A,m}(t)$ and bets on B at a return of $1/P_{B,m}(t) = (1-P_{A,m}(t))^{-1}$. There are three options:

1. If $P_{A,m}(t) \approx P_A(t)$ it is just about not worth it to take a bet

2. If $P_{A,m}(t) < P_A(t)$ one should bet on A, because the returns are higher than needed to expect a win on average

3. If $P_{A,m}(t) > P_A(t)$ one should bet on B, because the returns are higher than needed to expect a win on average (for B, we are indeed in the second situation)

Note that these strategies are only valid on average, which requires preserving capital to take many bets in this way instead of only a few to beat even the worst of bad luck.

Prediction markets are often explained using the notion of trading assets A and B that will be redeemable for $1 in the future if A or B, respectively, turn out to be the realized outcome, and will be worth zero (i.e. cannot be redeemed) otherwise. This is an equivalent notion that can be useful. The market prices of these assets are equal to their expected value assuming the market probabilities, which computes simply to the probabilities themselves (instead of their inverses used for the fair returns). Naturally, the fair prices are the true probabilities. Point 2. above can be then read as "the market price of the A asset is cheaper than its fair price, so one should buy it" and Point 3 as "A is expensive, therefore B is cheaper than its fair price"

Since the price of the A asset and the B asset are the respective probabilities, two shares of each asset together are worth exactly one dollar. Indeed, holding such a pair will always pay out exactly one dollar as one of them will be worth that much and the other will be worth zero. There is no risk or randomness here: a pair of both shares is exactly equal to a dollar and can be redeemed or minted for it at any time.

This is already everything a prediction market needs to do for the binary options. The trouble starts when we consider how to actually run such a market in practice.

Say a trader wants to bet on A. As we have seen, the market needs to promise a certain payout in case A wins. In that case the market loses and the trader wins. The opposite is true when B wins. One interesting way to see this is to acknowledge that we can use x dollars to mint x pairs of A and B assets and give x A assets to the trader, taking the current dollar value $x P_A(t)$ of it as cash, meaning we have effectively spent the complement $x - x P_A(t) = x (1-P_A(t)) = x P_B(t)$ for buying x B assets at their market prices - meaning we are betting on B, taking the other side of the bet.

Centralized markets that want to avoid taking large risks have several options here. For example, they might prescribe a "bad" price they are already pretty confident about is wrong, meaning that they prescibe odds such that traders will lose on average. A way to do this is to have very good insights into the problem and charge high fees (spreads). Having better insights than the rest of the market may work for rent-seeking sports betting platforms, but are deeply unfair (not to say a scam) and are not interesting for using prediction markets to actually learn about probabilities of difficult questions. Another option that generalizes to decentralized markets is to not have alpha, but just charge very large fees. Of course a risk remains that these fees are still too small to avoid net losses of the protocol.

Another approach is to not take positions as the protocol. This can be accomplished by matching traders of opposite opinion directly, for example with a limit order book. Contro accomplishes direct matching in a way that is most convenient for users and most sensible for getting good predictions.