Last month a local restaurant group, Chow foods---among whose restaurants is one of our favorite Sunday breakfast spots, The Five Spot---ran a contest/charity event: "Chow Dow." The game: guess the value of the Dow Jones Industrial Average at the close of the market on October 29th, 2009. The closest bet under the closing value which did not go over the value would be the winner. The prize was the value of Dow in gift certificates to the Chow restaurants: i.e. approximately $10K in food (or as we would say in Ruddock House at Caltech: "Eerf Doof!" We said that because it fit nicely with another favorite expression, "Eerf Lohocla!", this later phrase originating in certain now obscure rules enforced by administrative teetotalers.) I love games like this, and I especially love games where the rules are set up in an odd way. Indeed what I found amusing about this game was that, as a quick check of the rules on the Chow website showed, you could enter your guesses at anytime up until October 28th. Relevant also: maximum of 21 bets per person with a suggested donation of $1 per guess. So what would your strategy be optimizing your probability of winning, assuming that you are going to enter 21 times?
Below the fold: my strategy, the amazing power of the X-22 computer, and....chaos!
So the first thing I wanted to figure out was to get an estimate of how many people would be entering the contest. I mean, the number one rule of being a physicist is that you should always take the limit (the second rule is that symmetry is your friend.) If I was playing against one other player, then my odds would have been fantastic and so I would certainly play. However if I was playing against the entire city of Seattle, well, the odds would not be good (another Caltechism, this time about certain mating abnormalities at the school: "The odds are good, but the goods are odd.") For our Sunday breakfast at the 5 spot I dutifully counted the number of customers who left and what percent submitted a bet (this could easily be obtained due to certain gambling laws requiring the entrants put their entries into a box at the front of the store.) Amusingly this percentage was very high for our first visit and then declined significantly with each visit (given the structure of the rules I would have thought the opposite.) Given this percentage, the size of the restaurants, I obtained an estimate for the number of entries (entry envelopes had three entries per envelope.)
So given this estimate for the number of players, I then decided to make some model of their behavior. Well what would you guess if you were playing? I assumed that most people had a general idea about the current day's value of the Dow and that they would then guess some range around that value. Indeed I decided that I would even given them the benefit of the doubt and assume that they would make an estimate of the Oct 29th closing price based upon the CBOE Volatility Index (aka the Vix). The Vix gives some indication of the expected volatility of the market and one can use it to extrapolate what sort of range of prices the market might expect for a given set period of days forward. So I modeled an entrant on a particular day as someone who guessed a value according to the distribution they would use if they had used the closing price of the Dow and the Vix on that day. If I assumed the bets came in uniformly over the the month of betting, then I could obtain a probability distribution for my competition. So I had my model of the other players and an estimate of the number of players.
Next step: well how do I model the actual outcome? Again, I assumed that the prices would be distributed normally around the closing value on Oct 28, with volatility given again by the Vix on that day properly normalized (I have been told, but do not know whether it is true, that shorter term moves are normally distributed, not log-normally distributed as seems to occur long term.) So given my model of my competition and this distribution for a given distribution of bets even a naive monte carlo simulation cold quickly figure out your odds of winning. Throw in a little computing power and some gradient based optimization, I could then calculate an "optimized" set of bets and calculate my odds. I calculated that my odds were not great, but that I had an expected value greater than zero.
In the meantime, because, as should be obvious if you've read this far, I am a naive novice in all things finance, I emailed someone who I thought might have some good insights into the contest: Rocky from the Onehonestman blog. Rocky told me some things to look out for, including the fact that Oct 29th was the day the GDP number (prelim) was announced. Because of this I made the judgment that the volatility on that day would probably be significantly higher than normal and so adjusted the distribution on that day. And, on Oct 28th, Rocky also sent me his own predictions for the close of the Dow (9507.90, 9728.64, 9822.52, and 9955.41) calculated using the super duper X-22 computer which you discussed with great reverence in this vintage film:
Chaos is always responsible!
So on Oct 28th I dutifully went to the 5 spot and ordered a $9.76 sirloin steak (the promotion being every entry the closing price of the Dow on Oct 28, suitably adjust a few decimal places) which was delicious, and submitted my own 17 bets along with Rocky's 4 X-22 generated masterpieces. See the best thing about contests like this is that you get a steak, so even if you lose, you still are full and happy.
The next day, after closing at 9,762.69, the Dow surged (techincal term) 200 points to 9,962.58. My bets were not even close, my closest being 88.61 under the correct value. But Rocky's X-22 had gotten a guess within seven points! So there was still some chance that my entry and Rocky's X-22 would be the winner.
Batedly, thus, I have waited by my phone this last week, in hopes that seven points was close enough to score the prize. Unfortunately, yesterday I checked the Chow foods website and an announced winner, Mark Metcalf, was declared. No word yet on how close he got, nor whether he made a model of his competitors, but I suspect that perhaps the fact that the Dow went up nearly EXACTLY 200 points might have had something to do with his guess :)
So yeah, theoretical physicists can be pretty bad at stuff like this, and an old X-22 kicked my butt. But what would you have done differently?
(One valid answer to what would you have done differently is to say "don't play" but I've been told by the Mrs. Pontiff that not playing is "unsportsmanlike." I really need to make her watch War Games more often, I guess.)
This reminds me of the time I tried a similar tactic with PowerBall. They publish an Excel spreadsheet with the winning numbers from every PowerBall drawing ever performed. I used some statistical reasoning to make a prediction of what the numbers would be on a given day figuring the system isn't truly random (certain numbers come up more often than others). On the day I chose to test my reasoning, not a single number I picked came up. But as my father-in-law likes to say, "you've got to be in it to win it."
With that said, the time you spent on this could have been spent writing the statement you need to give me for the Quantum Times and that you promised me about a week ago. Hence, I will soon be contacting Mrs. Pontiff about restricting your Chow until you finish it.
So the first thing I wanted to figure out was to get an estimate of how many people would be entering the contest.
Thanks for the shout out! In the interest of full-disclosure, the X-22 actually had the exactly correct answer, however, I did not provide it to you for the same reason that the DoD doesn't provide exact GPS coordinates to civilians.
I'd also note that the contest explicitly excluded entries from residents in NY, RI and FL (either to exclude "ringers" such as James Simons and myself, or because Rhode Island residents have especially clairvoyant abilities.) [See Paragraph #6 at http://www.chowfoods.com/DowChow/index.aspx ]
Hey, Lazlo got the girl, the RV, and 32.4 percent of the prizes!
I suspect day of the week probably has more to do with the DOW than anything else.
It's interesting that you assumed (1) that the other bettors were sophisticated enough to use a measure of stock market volatility like the VIX, and yet (2) would submit their bets uniformly over the month, rather than at the last minute when they would have the most information about the outcome...
Actually, I thought that was the story you were going to tell, that you had dramatically underestimated the number of bettors because they almost all waited until after the Oct 28 close to enter their predictions.