Or "how to account for fees"

Over at the EA Forum, bob wrote an article "Don't Interpret Prediction Market Prices as Probabilities".

In his post, he makes 3^[1] cases against prediction market prices generating "fair" prices.

1. Opportunity cost - an incorrect price may be insufficiently incorrect for traders to put right because the reward isn't as appealing as alternative trades
2. Individual Risk-aversion - an incorrect price may be insufficiently incorrect for traders to put right because the risk of loss doesn't compensate for the potential gain
3. Correlation with the real world - an incorrect price may be insufficiently incorrect for traders to put right because in the world where they are wrong, it is especially unpleasant for them

I believe that all three of these are reasonable complaints. I also believe only one of them causes prediction market prices to be materially "wrong" after accounting for some simple adjustments.

Specifically, I think that the market outcomes correlating with the real world does lead to non-fair prices, and I discussed this at length here.

For the other 2 cases, I do not think the effect on prediction markets enough to stop treating prices as ~probabilities. What it means is we have to be more careful about how we interpret prices.

Risk Aversion

In general^[2], fair bets are unappealing to most people. A certain $50 is more appealing to me that a 50-50 shot at $100. This means that the market price of a coin flip might deviate from 50% if no-one is willing to risk $ to put it right. However, as long as the stakes are small enough, it doesn't need to deviate far before someone is willing to start to put the price correctly. In general, the stakes available in prediction markets tend to be small. In my experience, the largest mispricings have generally come where large quantities of money all appear in the same market and there isn't enough liquidity from "smart" money to take the other side. (eg Mayweather-McGregor). This happens infrequently, especially in prediction markets where the $ involved is generally quite small. The volumes involved in prediction markets are generally small enough relative to the wealth of smart money, that I think it is reasonably easy to ignore this issue in the short term.

Opportunity Cost

When placing a bet on a prediction markets there are a number of costs associated with that trade:

1. Time to price the market
2. Fees, vig, margin (including taxes in some juristictions)
3. Time value of money
4. Alternative investment opportunities

For 1/ I think this is relatively easy to deal with. If you see a prediction market where the $ involved is relatively small, you can ignore the prices (to first order). If the money is small enough, it's not worth anyone with good forecasting skills spending any time to figure out the right price, and so the price could be anywhere.

They're the same picture

2, 3 and 4, although they all sound different, I believe are all functionally equivalent. Risky $ at future date t has a risk-neutral price based on bond prices and arbitrage. Aside from issues mentioned earlier (risk-premia / non-risk-neutrality) this should give us the "fair" price we are interested in.

Let's start by talking about a really simple example. A bookmaker who charges an egregious amount of vig. He'll let you bet on either side of a coin-toss (happening immediately to assuage any worries about inflation / alternative investments), but at 60% odds. (ie you bet 60 to win 100 if heads or you can bet 60 to win 100 if tails). If we add up the probabilities here we get (60%+60%) = 120%. Our bookmaker has found himself a nice business.

If someone comes along and looks at the prices posted (60% and 60%) should he conclude that heads is 60% and tails is 60%? Of course not, he should assume that both are ~50%.

Now let's consider a slightly different example. A coin flip taking place in a year, when interest rates are 20%. If I'm offering margin-free odds on this, I would charge ~60% for each side as the simple replicating portfolio of betting on each side is riskless and would pay out the same as a hypothetical par bond with a 20% coupon.

Hopefully this reasoning is enough for people to see that time-value of money and fees are two sides of the same coin.

Making adjustments and long-shot "bias"

What can we say (so far) about a prediction market where a price (after adjusting for fees, inflation, etc) is 60%.

Well, it's generally safe to say that P(event) <= 60%. We hopefully also know something about P(not event) from the market, let's say P(not event) <= 60%. (eg it's a two outcome event and both prices are 50c and the fees are 10c). Then we know that 40% <= P(event) <= 60%. In this market, we might say something along the lines of "we think P(event) ~50%, but we have low confidence in that". 

What if the odds aren't 50-50 thought, does this change things?

Sort of.

Let's go back into fees world and take an example with 95% probability. What would be the fair price for this if I want to have a vig of 5%. There are a few different ways in which I can do this:

• Proportional to probability: multiply each price by 1.05
• Propotional to odds: add 2.5% to each price
• Maintaining the odds ratio: (adjust prices such that the vig is 1.05 and the odds ratio is unchanged)

Our prices could be:

PTP: 99.75% and 5.25%
PTO: 97.5% and 7.5%
MOR: 96.9% and 8.1%

The total vig in all cases is 5%, the odds ratios are 7201, 481, 361. If I do a naive conversion of those prices into probabilities, I might get: (95%,5%), (93%, 7%), (92.3%, 7.3%). Only the first of these would lead to the "correct" prices we started with. However, we can remove the margin from the prices in any way we choose. This is an important point when trying to extract data about low-probability events from prediction markets. Some people just look at this and say "long-shot bias" and move on with their lives, but I have found^[3]I that taking an analytical look at markets and the way in which fees and other costs are affecting the prices can yield more precise probabilities.

1. ^{^}

   He actually structures his post as 4 cases against prediction markets as probabilities, but it's somewhat muddled. For example, in his section $ he is talking about opportunity cost and hedging. His last section on outcomes is similarly equivalent to hedging.

2. ^{^}

   There's a whole discussion which could be had around gambling here, but I'm leaving that aside for now.

3. ^{^}

   I don't want to get into the specifics in great deal here, because the correct adjustments depends substantially on the structure of the market. However, I believe that this sort of reasoning allows you to account for large fractions of the market.