Tuesday, October 07, 2008

Expectation of Expectations

Can we quantitatively judge the effect of a debate on the election? One reasonable approach would look at prediction markets. If the value of the security for Obama to win would greatly increase after the debate in the absense of significant other factors, one could conclude that the debate went well for Obama, or at least that he did better than expected.

Intrade, the Irish real-money prediction markets site, went one further and set up a new security 2ND.DEBATE.OBAMA based on the outcome of tonight's presidential debate. In short,

This contract will be settled according to which Presidential Candidate receives the largest increase in value following the Presidential Debate on Tuesday October 7th. The pre-debate value will be calculated for 2008.PRES.OBAMA and 2008.PRES.McCAIN and then compared with the post-debate value for each of these contracts. This contract will expire at 100 if the value of 2008.PRES.OBAMA increases more than the value of 2008.PRES.McCAIN. This contract will expire at 0 if this is not the case.
A similar security for the VP debate paid off zero as there was a slight bump for McCain suggesting Palin did better than expected.

But there is a flaw in the logic for such a security—The market should already take into account the expectation of the debate. In general for any random variable A, the expectation of the expectation of A is the same as the expectation of A, i.e., E(E(A))=E(A).

So, assuming that these securities represent real probabilities, should the value of 2ND.DEBATE.OBAMA before the debate be 50? Not quite. Suppose that there is a 75% chance that the Obama stock goes up 5 and a 25% chance it drops 15. Then the price of 2ND.DEBATE.OBAMA should be 75%. The price of the security indicates who has the least chance of major downside from the debate. But to think one could use this security to predict how well the debate will go makes little sense.

The press release suggests using this security for real-time debate tracking. The price of 2ND.DEBATE.OBAMA should gyrate dramatically on small changes of 2008.PRES.OBAMA. But one could just derive this directly from 2008.PRES.OBAMA and since the later security has much greater liquidity it will likely give better information.

Chris Masse suggested another explanation.

[Intrade] is trying to induce traders into trading more (so as to get more transaction fees), and this new contract offers more upsides for the traders who will bet right. One could double one's initial investment in a matter of 2 days.
In which case all the power to them.


  1. I don't follow your logic. In the example you give, Obama has a 75% chance of having his stock go up after the debate (which, presumably, means he "won" the debate). And the contract is prices at 75. So doesn't a higher price for the contract indeed reflect a higher probability of winning the debate?

  2. I think you have missed the point. It's a matter of mean vs. median. The underlying 2008.PRES.OBAMA and 2008.PRES.McCAIN will price in mean expectations for the debate outcome. The debate market gives predictive information about the median - in other words it gives complimentary information to the underlying about the market-implied probability distribution of future price movements. Now if only we can learn how to usethat information in forecasting.

  3. Actually check out the "X's" on intrade - they're a family of options with different strikes on where the underlyings will be at some specific future date. The prices of the options at different strikes should specify all higher moments of the probability distribution of expected price movements. So yes, there is a very good complementary information there.

  4. It seems that derivatives such as this raise serious manipulability issues.

    Imagine, if you will, that I managed to bet a fairly large amount in a derivative market such as the debate one, that pres.obama will increase after the debate, possibly driving up the price of debate.obama. For concretenss, lets say I put in 60 dollars at a rate of 60, so that I make 40 dollars profit if pres.obama goes up. Then I could simply spend a lot of money on the pres.obama market to change it's price at the appropriate point in time. If the amount I lose in the latter transaction is smaller than what I win from the bet, I have just made a lot of money! In the example, if I need to raise the price by 1 percent to win, I have up to 4000 dollars to invest in pres.obama to affect that increase. Now scale up the 60 dollars above to something more reasonable, and it seems extremely doable!

    I know that the theory is all nice and warm under the assumption that markets are non-manipulable. But that assumption seems wrong if one looks at the history of this market (see e.g. http://www.fivethirtyeight.com/2008/09/intrade-betting-is-suspcious.html), and probably also wrong for the stock market if the manipulators are of too-big-go-fail sizes.

    I suspect more "continuous" derivatives in the stock market are probably not as manipulable. But this example certainly raises the question. Does anyone know if economists worry about these manipulability issues when thinking about derivatives?

  5. I agree with Anon#3.

    This is just a market that trades the volatility of the primary prediction market contract. What you can infer by these contracts are exactly the higher moments of the probability distribution; right now the prediction market gives only the mean.

  6. Panos and Anon#3 are right, but the point Lance is making is that the way Intrade is marketing the security as a gauge of the debate winner is incorrect.

    For example, if the underlying Obama security follows symmetric Brownian motion, then the correct (efficient) price for the debate security would always be 50, regardless of any aspect of the debate.