In an earlier post, [*] I tried to use the Iowa Electronic Market numbers to calculate the probabilities that Davis, Schwarzenegger, and Bustamante would wind up with the governorship when the dust cleared.
The calculation presented was simple, elegant, and wrong. I can only attribute the absence of a flood of emails correcting my logic (indeed, of any emails at all) to the forbearance of my readers. [Where did you say you had studied statistics, Mr. Kleiman? Oh, Harvard. Well, that makes sense then.]
From the propositions that Round I was 70% likely to carry (i.e., that Davis would be thrown out of office) and that Bustamante was 60% likely to win Round II, I calculated that Bustamante was .7 x .6 = .42 likely to become governor, while Schwarzenegger was .7 x .4 = .28 likely to become governor.
The error, which of course you spotted at once, was the elementary one of assuming that two events were independent when they weren’t. As a result, I overestimated Bustamante’s chances and underestimated Schwarzenegger’s. (In addition, the Iowa numbers have changed in Schwarzenegger’s favor since I wrote.)
Round I and Round II are certainly not independent events. Republicans are more likely to vote “Yes” on Round I and for Schwarzenegger in Round II, and Democrats more likely to vote “No” and “Bustamante.” That being so, a big Democratic turnout that helps Bustamante also helps Davis to survive. So the possible cases in which Bustamante wins Round II are disproportionately cases in which Davis survives Round I and Round II is irrelevant, while the cases in which Schwarzenegger wins Round II are disproportionately cases in which it matters.
Using the previous numbers (70% that Round I carries, 60% that Bustamante wins Round II), and assuming that, if Davis survives Round I, it’s 90% likely that Bustamante comes out on top in Round II, then in .3 x .9 = .27 of all cases both things happen: Davis survives, and Bustamante wins what is then a meaningless beauty contest. If in 60% of all cases Bustamante wins Round II, and in 27% of all cases Bustamante wins Round II and Davis survives Round I, then in only 60% – 27% = 33% of all cases does Bustamante become governor. If Davis has a 30% chance and Bustamante a 33% chance, then by subtraction Schwarzenegger must have a 37% chance.
Now the numbers have changed further in Schwarzenegger’s favor. The market has Davis about .33 to survive, and Schwarzenegger about .53 to win Round II. Again assuming that if Davis wins Bustamante is 90% likely to win, that works out to:
Update A reader points me to Tradesports, which has very different odds for Davis and Bustamante than the ones computed above (to find the recall market, type “recall” in the search field:
This seems to be a more efficient market, as measured by bid-asked spreads. Arbitrage, anyone?
(Note: Tradesports also has a Presidential market, which gives Bush about two chances in three to win. )
Bustamante has to do something about the “Bustamante-is-a-racist” meme that Glenn Reynolds, Tacitus, Michelle Malkin, Mickey Kaus, and the Fox News crew have been so skillfully spreading, with so little factual basis. If I were in his shoes, I would say something like “Anyone who thinks that California ought to become part of Mexico again, or that California will become part of Mexico again, has been drinking the wrong brand of tequila. Next question?”
One more update
A long telepone conversation with Guy Gugliotta of the Washington Post led me to realize that I’d been wrong a second time. There is in fact an interpretation of the Iowa results that makes them more or less consistent with the Tradesports results, and that’s that Iowa bettors are figuring that the dependency between Round I and Round II goes both ways. If on Election Day it looks as if Schwarzenegger is going to beat Bustamante, then lots of people (such as me) who would prefer Bustamante to Davis but want at all costs to keep Schwwarzenegger out of office will switch from “Yes” in Round I (and Bustamante in Round II) to “No” in Round I (and Bustamante in Round II). If that were the case, than maybe it’s not so that the chances of Davis being booted are negatively correlated with the chances of Bustamante winning Round II. That doesn’t make my original assumption of independence any less wrong conceptually, but it might not be important numerically.
All of this suggests to me that the Iowa market on California, like the Iowa market on the Presidency, is very badly designed.
And, by the way, Tradesports has now (9/5) flipped, with Davis’s survival below 20% and Bustamante slightly ahead of Schwarzenegger.