## Sunday, October 4, 2015

### Corporate prediction markets aggregate random behavior

Alex Tabarrok linked to a (gated) paper on how "corporate prediction markets work well" as Tabarrok puts it. Robin Hanson in comments says that it depends on what you mean by "worked" — the companies in question discontinued the prediction markets. Anyway, I found an ungated older version [pdf, broken, now here] (it actually points to Koch Industries as the anonymous Firm X in the final paper) and had a look.

Anyway, one thing it points to as evidence of functioning markets are a couple of figures where (scaled) price of the security is roughly equal to the (scaled) payoff:

Here's the accompanying text:
Figures [1 and 2] graph the future value of securities, conditional on current price for binary securities at Google [and] Koch ... respectively. The prices and future values of binary securities range from 0 to 1, and trades are divided into 20 bins (0-0.05, 0.05-0.1, etc.) based on their trade price.The average trade price and ultimate payoffs for each bin are graphed on the x and y-axes, respectively. A 95% confidence interval for the average payoff is also graphed, along with a 45-degree line for comparison. ...
Google and Koch’s markets appear approximately well-calibrated. Both markets exhibit an apparent underpricing of securities with prices below 0.2, and an overpricing for securities above that price level, but this is slight, especially for Koch.
So following the $\hat{p}_{i} = \hat{p}_{j}$ with scaled $p_{i}$ (price in period $i$) indicated with a hat and scaled $p_{j}$ (payoff in period $j$) is some indication the market is forecasting well? That's odd because in the maximum entropy picture, that's exactly what you get with random market exchanges. If we go back to the maximum entropy asset pricing equation:

$$\text{(5) }\; p_{i} = \frac{\alpha_{i}}{\alpha_{j}} \frac{\partial U/\partial c_{j}}{\partial U/\partial c_{i}} p_{j}$$

Or more transparently for our purposes, equation (4) right before it

$$\text{(4) }\; p_{i} = \frac{c_{i}}{c_{j}} p_{j}$$

If we scale $p_{i}$ and $p_{j}$ we can remove the pricing kernel so that

$$\hat{p}_{i} = \hat{p}_{j}$$

The only assumption is that there are a lot of time periods between $i$ and $j$. In the Google example, there were 10 time periods (dimensions $d$), so we should expect deviation from the previous formula that is on the order of

$$\varepsilon \sim 1 - \frac{d}{d + 1} \approx 9 \%$$

which is about what we see. The Koch Industries markets had 58 periods which means about 2% error, but that market only had 57 participants as opposed to Google's 1,465 which would add roughly errors of 13% and 3%, respectively (using the $1/\sqrt{N}$ heuristic). Adding these in quadrature, we get 13% for Koch and 9% for Google for a back-of-the-envelope error calculation. Since these data are scaled, that means about 0.1 for both markets (after rounding). I added the error bars in orange to the data:

Works pretty well for back of the envelope! Systematic low prices are indicative of non-ideal information transfer.