The Combinatorial Approach
So far, businesses that have taken advantage of prediction markets have made direct predictions of events relevant to their decision making, but as the above discussion implies, it may be far more useful to have prediction markets make forecasts that are contingent on the outcomes of particular decisions. We have already seen that conditional prediction markets can be implemented by unwinding transactions in the event that a condition is not met. In the corporate context, a prediction market contract might be used to forecast the corporation’s stock price at some point in the future, but only if the corporation decides to build a particular factory. If the corporation ends up not building the factory, all transactions in the market are refunded. Another market might predict the corporation’s stock price in the event that the corporation does not make a particular decision. Even with conditional markets structured in this way, reaching meaningful decisions requires comparing the two (or more) conditional predictions. An alternative approach to conditional markets is to combine mathematically the prices of two or more tradable contracts to generate a single prediction. An early use of conditional markets takes this approach. Joyce Berg and Thomas Rietz, who help run the Iowa Electronic Markets, decided in the 1996 A problem with this approach to conditional markets is that to the extent that the price of any individual tradable contract has some amount of noise, the division of the prices of two separate tradable contracts has the potential to aggravate the noise. Consider, for example, figure 7.1, which uses tradable IEM contracts to calculate Bush’s conditional vote share a year before the 2004 election, during the first week of November 2003. Not much happened that week, but the conditional vote shares as calculated by this method changed considerably and in different ways for different candidates. It seems implausible, for example, that Hillary Clinton’s expected performance against Bush, had she run and won, would have fallen from 68 percent of the electorate one day to 53 percent the next day, a massive shift by political standards. These numbers seem hard to reconcile with reality and with the relative stability of political campaigns generally as illustrated by market predictions of their results. The high volatility of the shares shown in figure 7.1 is probably in part a result of relatively low market liquidity, and indeed conditional probabilities stabilized somewhat later in the election season. Given relatively large bid-ask spreads and the possibility of asymmetric information, there often will be little incentive for market participants to correct slight mispricings in a prediction market tradable contract. Small mispricings in each of two tradable contracts can combine to create greater deviations when the two tradable contracts are combined mathematically. The problem, of course, becomes greater when the prices of three or more tradable contracts must be considered. For example, in order to determine how much better one candidate can be expected to fare than another in the general election, one would need to subtract the derived conditional estimate for one from that of another. It seems likely that other market designs might alleviate the problem to some degree. For example, with the market scoring rule, it will more often be in the interest of a market participant to correct a slight mispricing in a tradable contract, putting aside the time cost of entering into a transaction, because someone making such a correction would not be accepting the underlying risk associated with the contract as a whole. Similarly, market subsidies would give greater incentives to correct mispricing, so market participants will have strong incentives to identify situations in which the relationship between two or more tradable contract prices does not correspond with perceived reality and correct the mispricings. Research has also focused on the development of market designs for contexts in which market participants might want to make predictions about complex combinations of large numbers of variables. Robin Hanson, for example, has described how to create a market maker that would allow anyone to bet in a “state space” representing relationships among a number of variables, and he argues that in theory it should cost no more to fund an automated market maker to allow trades in the entire state space than to fund automated market makers limited to each variable.8 Lance Fortnow and coauthors have similarly described a prediction market design that allows participants to trade in tradable contracts based on logical formulas expressed in propositional logic.9 For example, if A and B represent events that might or might not occur, one could bet not only on A or on B but on “A and B” and “if A then B,” as well as more complex combinations. An advantage of these approaches is that the market maker would automatically ensure coherence among the variables, because it would not create an additional tradable contract for every combination. For example, someone betting on “if A then B” might automatically be given shares on “A and B” and shares on “not A.” In the appropriate proportions, this combination is equivalent to the conditional bet, as long as there are just enough “not A” shares so that the nonoccurrence of condition A leads to an exact refund of the total investment. From a user interface perspective, users would not need to know the difference. The principal difficulty with these approaches is that the necessary calculations would increase exponentially in the number of variables, making a market allowing bets about any relationships among a large number of variables potentially infeasible with today’s computers. These systems, however, may work well if limited to a dozen binary variables or so, enough for many practical applications. At least until better systems for allowing conditional betting and betting on other relationships between variables are fully developed, it may be hazardous to place too much emphasis on the difference between the price of two conditional tradable contracts. This is true even if the unwinding approach to conditional markets is used. The danger is especially high in cases in which the price difference is relatively small in comparison to the prices the two tradable contracts. Suppose, for example, that a conditional prediction market of any type is used to gauge the price of Apple Computer stock at some point in the future if an additional water cooler is installed on the third floor of 1 Infinite Loop, and another such market is used to predict the price if the water cooler is not installed. The connection between the water cooler and the price of Apple stock is likely to be so attenuated that market participants will have little incentive to take it into account in their models. Any difference between the price of the two tradable contracts is likely to represent random noise. Assessing alternative decisions using conditional markets will thus likely work best when the condition seems likely to have a significant effect on the variable of interest. At the very least, any subsidy the market sponsor has provided will be better spent, because the more important the variable of interest, the greater the effort market participants will make o model its effect. There may, however, be some techniques that could be used to make conditional market predictions more reliable. For example, a prediction market might be used to assess the difference in the assessments of two other markets making conditional forecasts based on two different possible decisions. At least with this prediction market, participants would focus directly on the effect of the decision. Though a large amount of market noise might make the difference in predictions an unreliable indicator of market sentiment at any one time, because noise is largely unpredictable, a prediction of the difference between two tradable contracts’ prices might be a reliable indicator of the effect of the relevant decision. A similar approach would be to use conditional prediction markets to forecast the change in stock price in the five minutes after an official decision was announced. If the relevant decision merely involved a water cooler, then, because the stock price would be unaffected, both prediction markets would anticipate zero effect on Apple stock or both might reflect a very small stock price increase attributable to the general tendency of shares to rise, but there would be virtually no difference between these markets. If, however, the decision reflected an issue that might have an effect of a few cents on stock price, such as whether to cancel a particular computer line, then the conditional prediction markets might make slightly different decisions. Market participants would not need to worry much about the overall health of Apple but instead would focus on the extent to which the decision might change Apple’s profitability. The result is that the market subsidy would be better targeted. This system does introduce the danger of a new kind of manipulation: a person who credibly commits to buying or selling stock whenever the decision happens to be made. It might not be difficult for someone with modest capital to commit to buying or selling enough shares at a particular point in time to change the market price for a short period of time. One solution is to add some uncertainty to the length of time that determines the measurement of stock price change that is used to resolve the prediction market payouts. The average interval would need to be long enough that manipulating the market in this way would be prohibitively expensive. Even a wealthy investor cannot change stock price over a long period of time without losing a great deal of money. A somewhat more complex alternative would be for the conditional prediction markets to be forecasting the change in another prediction market anticipating the future stock price in the minutes after the announcement. This second prediction market could be structured as a deliberative market in which the time frame used to assess predictions is short, though somewhat uncertain (see Chapter 4). The advantage of this approach is that manipulation of this second market would be much easier to counter, because arbitrage would not require purchase or sale of equity interests in the firm as a whole. In addition, the second market might predict change in the actual stock market over a longer period of time, with greater uncertainty about the exact measurement point, making it difficult to commit credibly to manipulating this market.
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