### extrinsic bias in the prediction market

People have proposed using price signals from prediction markets to estimate the odds of certain events. On Intrade right now, you can buy contracts for the two outcomes of the 2012 US Presidential Election. Each contract expires at $10 if the event occurs or $0 if it doesn’t. For example, “Barack Obama wins” contracts are $6.33 a pop right now, while “Mitt Romney wins” contracts go for $3.65. On the page, these are taken directly as probabilities, because it is assumed that the gamble is zero-sum.

Specifically, if \(p\) and \(\bar{p}=1-p\) are respectively the probabilities of two complementary events, and \(a\) and \(b\) are respectively the prices of contracts on them, which can be bought and sold freely, then no-arbitrage imposes that \(-a-b+10 = 0\) and statistical no-arbitrage imposes \(-\bar{p}a +p(10-a) = 0\) and \(-pb +\bar{p}(10-b) = 0\). Solving indeed gives the prices \(a=10p\) and \(b=10\bar{p}\).

However, this isn’t the end of the story.

The prediction market isn’t a closed system. Event outcomes are correlated with other payoffs outside of it. For instance, the election outcome has personal income tax consequences for certain individuals. While playing in the prediction market has no expected gain or loss, its contracts can diversify just such an external payoff to reduce its variance.

Suppose the total tax exposure of an Obama presidency is \(T_o\) and of a Romney presidency \(T_r\), and the probabilities of the two winning are respectively \(p\) and \(\bar{p}\), then the expected payoff is \(-pT_o -\bar{p}T_r\) while the variance is \(p\bar{p}(T_o-T_r)^2\).

Without loss of generality, assume \(T_o > T_r\). Then we can reduce the variance of the payoff by buying “Obama wins” contracts at normalized price \(q=a/10\). Let’s say we buy an amount worth \(N\) if expired in the money. The payoff becomes \(-T_o +(1-q)N\) with probability \(p\) and \(-T_r -qN\) with probability \(\bar{p}\). If the contracts are priced for no arbitrage as before (\(q=p\)), the expected payoff is \(-pT_o +p\bar{p}N -\bar{p}T_r -\bar{p}pN = -pT_o -\bar{p}T_r\) as before. However, the variance is \(p\bar{p}(T_o-T_r-N)^2\), which is a decrease for any \(N\in (0,2(T_o-T_r))\), with the aggregate (i.e. hedged) payoff becoming completely deterministic for \(N=T_o-T_r\). This is the point of maximal utility gain for a risk-averse hedger. One ends up “pre-paying” a portion of the potential additional tax burden in exchange for immunity from the election outcome.

The fact that hedgers exist and are biased in one direction means that the normalized price of a contract may no longer be exactly the probability of its expiring in the money. The imbalance in the market caused by risk aversion should create precisely an insurance premium to be added to the price of “Obama wins” contracts. Of course, the reality is more complicated, since not all individuals have homogeneous tax burdens under the outcomes. If the number of risk-averse hedgers is small, then the no-arbitrage assumption may still approximately hold.