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.

Then there are those who expect not only to have a non-negative real return, but at least a market return. This is the expectation that during the consumption period, we will get at least as much value as we “stored” proportionally to the fraction of output we produced in the economy at the time of “storage.” For example, if technology improved, we would expect that our “stored” widget version 1, would be returned to us as the much improved widget version 10. This seems greedy yet fair. In contrast, the expectation for non-negative real return seems rather tame now, doesn’t it? Until you consider that the economy might actually contract. What if, due to whatever circumstances, the excess producers during the consumption period do not produce enough value to proportionally divide among those who “stored” value an amount that is more than what they stored? This could be by will or by necessity. What if all we had were widget version 0.5? Would it still be plausible to insist on non-negative returns? It would not.

In fact, following a period of great excess in production, especially one that prematurely fulfilled useful work for some time into the future (until new technology and demand develop), the right view is that the nominal amount of stored value is irrelevant — it is not exchangeable.

Now the second assumption. Capital markets are a great invention. They solve the centralized allocation problem in a distributed way. We assume that rational actors freely acting in their own self interests will produce a solution that allocates capital in the “optimal” way. Perhaps it isn’t a global optimal (requires cooperative strategies), but a more basic question is what planning problem do they solve in the first place? Let us give individual actors the benefit of the doubt that they can do planning and are perfectly rational. Nevertheless, it seems difficult to imagine that any would do planning much beyond their own lifespans. That would not be rational. Thus, capital markets are essentially solving an allocation problem based on each individual’s preferences for maximizing their own utilities during their lifespans. This has great implications. First, it puts an upper bound on the planning period for the allocation problem at the longest remaining lifespan among individuals. This is like 80 years or so. That is pretty myopic. It seems like a long time, but for many decisions, especially concerning resources, it is very myopic.

It gets worse. That was just an upper bound. Inputs into capital markets are weighted by the amount of money committed to a trade. It is a given that older (working) people have greater wealth, and as a consequence their inputs as expressed by capital commitment are weighted more in the allocation decisions. That is exactly what you don’t want, because a shorter remaining lifespan is now correlated with larger weight in decisions. The net result is that the capital allocation decisions are being carried out for optimality over very short planning periods, possibly only on the order of 10 or 20 years at most. It isn’t even optimal for most people alive any more.

So this is a fundamental problem in capital markets. Under the law of one price for open markets, only one view can be ultimately carried out, and it is one that is too myopic, and even myopic for most people. The only way to correct for this is to weight the inputs properly, and that would require external intervention to enforce a political planning decision.