# Market of Decisions

We all know about prediction markets:
http://en.wikipedia.org/wiki/Prediction_market

I'm sure that someone has thought of the idea of applying this technique to make decisions instead of just predictions. Here's how I imagine it could work:
1. An objective goal is set for a particular time. For example, if a high school is making a four year budget plan, they might set the goal as: "80% student graduation rate in 2013, with a debt of no more than $100,000 for that year." 2. A pot is chosen, let's say it is a$10,000 pot in this case. This will be used later to reward people for picking a good strategy. It should be big enough to incentivize people to take the time to make good strategies.
3. There is a relatively short, fixed "market period" where anyone (or perhaps just anyone on the board of directors) may propose a strategy to reach the goal, which they bet a certain amount of their own money on, or they may bet their own money on someone else's strategy. The number of strategies is not predetermined.
3a. People may withdraw their bets at any time.
4. At the end of the market period, a strategy is randomly selected from all proposed strategies, weighted by the amount of money attached to that strategy. e.g., if the only two proposed strategies are "increase math funding" with $3000 behind it, and "upgrade the computer lab" with$1000 behind it, then math funding would be increased according to the details of that strategy with a probability of 3/4, and the computer lab would be upgraded according to the details of that strategy with a probability of 1/4.
5. From the end of the market period until the time when the goal was set (2013 in this case), the strategy is adhered to.
6. The result is measured, and money is distributed to the market participants according to how they invested. Suppose a participant invested $x in strategy Z, and suppose the total amount of money invested in Z is$x+$y, and suppose the pot (from step 2) is P. 6a. If the goal is met and Z was the strategy that was used, the participant gets$x+P$x/($x+$y) back. That is, they get back their original investment, plus a proportion of the pot, to reward them for being right. 6b. If the goal is not met and Z was the strategy that was used, the participant gets nothing, to punish them for being wrong. 6c. If Z was not the strategy used, the participant gets their original$x back. We don't know if they were right or wrong so we don't punish or reward them.

The reasoning behind this is, let V denote the event of the goal being met (victory) and let S be a strategy. Also, let T be the total amount of money currently invested in the market, and Ts be the amount of money currently invested in S. The expected value of investing \$x in V is then (in dollars):
Pr(S is chosen) * Pr(V|S) * (x+xP/(Ts+x)) (if the strategy is chosen, and the goal is met)
+ x*Pr(S is not chosen) (if the strategy is not chosen).

Setting this equal to x to find the break-even point, we find
P(V|S)=(x-x*Pr(S is not chosen))/(Pr(S is chosen) * (x+xP/(Ts+x)))
=1/(1+P/(Ts+x))

So if P(V|S) is greater than 1/(1+P/(Ts+x)), it is in the participant's interest to invest, and otherwise it is not. If x is small this simplifies to 1/(1+P/Ts). If the pot is large in comparison to the amount already invested in that strategy, then it is advantageous to invest in the strategy even if there is only a small probability it will succeed. If the pot is small in comparison to the amount already invested, then it is only advantageous to invest in the strategy if there is a very high probability the strategy will succeed. In this way, the amount invested in each strategy in relation to the pot amount should stabilize around a value related to the probability of the strategy succeeding.

So - thoughts? Is this perhaps called something other than a "decisions market" in the standard literature? Refinements? Criticism?

Thanks

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