# Prediction Market Mysteries

## **Market Type 1**
**[this part I get, it's added for context purposes]**

Consider a simple market where one can bet on the event D, i.e., the Democratic party wins the 2008 U.S. presidential election. In such a market a bank could without risk accept
$1 in payment for the pair of contingent assets, “Pays $1 if D” and “Pays $1 if not D.” This transaction carries no risk because exactly one of these assets will be worth $1 in the end.
Since the expected dollar value of the asset “Pays $1 if D” is $p(D), if someone is willing to buy this asset for $0.70, we can interpret this willingness as this person saying that the chance that Democrats will win is at least 70%. And a market price of $0.70 can be interpreted as a consensus among potential traders that p(D) ≈ 70%.3 After averaging in their minds over plausible scenarios, traders would have judged that Democrats win in about 70% of such cenarios.

## **Market Type 2**

Now imagine a market like the one above, where people can trade an asset like “Pays $1 if D” for some fraction of $1, but where such trades are called off unless there is also the event C, i.e., Hillary Clinton is the nominee of the Democratic party in the 2008 election. When a trade is called off, the exchange does not happen, and each side instead retains their original assets.
Traders thinking about the prices they are willing to accept here should again average over plausible scenarios, but this time they should only consider scenarios consistent with the event
C, that Clinton is the nominee. A market price of $0.60 here can be interpreted as a consensus that about 60% of these scenarios have the Democrats winning. This would be a consensus that p(D|C) ≈ 60%, i.e., that the conditional probability of Democrats winning, given that Clinton is the nominee, is 60%.

## **Market Type 3**
Another way to get the same sort of conditional probability estimates is to have a market where, instead of trading the asset “Pays $1 if D” for some fraction of the asset $1, people trade the asset “Pays $1 if D and C” for some fraction of the asset “Pays $1 if D.” Since this is trading an asset worth p(D&C) for some fraction of an asset worth p(C), the marketprice-fraction here can be interpreted as an estimate of the ratio p(D&C)/p(C), which is by definition equal to the conditional probability p(D|C). Thus if the Democratic party wanted to know which nominee would give them the best chance of beating Bush, they could compare the market estimate of p(D|C) to similar estimates for other Democratic candidates.

## **Market Type 4**

One can similarly create markets to estimate whether a new policy N is an improvement over the status quo Q in increasing national welfare W. Given a measure of national welfare W, normalized to be between zero and one, markets that trade assets “Pays $W if N” for some fraction of “Pays $1 if N” give a market price estimate of E[W|N]. Just as people considering what fraction of $1 to pay for “Pays $x” will estimate the average value of x across all plausible scenarios, people considering what fraction of “Pays $1 if N” to pay for “Pays $x if N” will average only over plausible scenarios consistent with the event N. When the market estimate of E[W|N] is clearly greater than E[W|Q], speculators are saying that this new policy is expected to increase national welfare.