Sometimes crowds are stupid.

Chad Orzel of uncertain principles reminds us that "The 'point spread' for a football game is set at the level required to get equal numbers of bets on the two teams." Furthermore this is not the consensus of "experts". And since gambling serves as a form of entertainment for a lot of people, there are probably a lot of people that bet on the team they like, not the team they think will beat the spread. So you can't even say that the "wisdom of crowds" applies in situations like this. Sometimes crowds are stupid, especially when they have an emotional stake in the matter. (And not just in sports betting! Consider the current housing bubble, or the dot-com bubble.)

And it's not entirely clear that bookies even always act in the way Orzel says they do; they act to maximize their expected profits, which turns out to not be the same. (See Steve Levitt's paper for more detail.) Not surprisingly, this means there's a winning strategy for betting on football, or at least it looks like there is -- bet on home underdogs, or so Levitt says. (They tend to be undervalued.) I'd do it, but I'm pretty risk-averse when it comes to my own actual money, as opposed to other people's expected-value money.

How much is it worth to win at Jeopardy!?

Most weekday evenings at 7pm I watch the television show Jeopardy! One day I'll be on the show, if I actually get around to auditioning and then I get lucky enough to get picked. For now, I just scream "how could you not know that!" from the comfort of my living room. I bet all the contestants do that.

For those of you who aren't familiar with the game, it is a game where three players competing against each other answer trivia questions (although Jeopardy! has this silly trope where your answer has to be "in the form of a question", that's entirely irrelevant here). If they get the questions right, they gain money; if they get them wrong, they lose money. Those players who have a positive amount of money after the first sixty questions (which occur in two rounds of thirty; each round has six categories of five questions, worth varying amounts) get to participate in "Final Jeopardy!". Most players end up with a positive amount of money, since they know themselves well enough to only attempt to answer questions which they think they know the correct answer to. The players are told the category of the question; they then can wager any or all of their money that they'll get it right. Then the question is stated and they have thirty seconds to write down the answer.

Fans of the show have put together a Jeopardy! archive. There's a wagering calculator available online that takes into account common wagering strategies. But this does not take into account the following facts:

• only the player who wins the game gets to keep their money (probably an average of $20,000 or so); the second- and third-place players get $2,500 and $1,000 respectively. (Incidentally, the show's host, Alex Trebek, seems to refer to in-game totals as "points", which I suspect is tied to this;


• perhaps wagering strategy should depend on how likely one thinks one is to get the question right, and how likely one thinks one's opponents are


• the player who wins get to come back the next day; the others don't.

The first two here, I plan to address in a later post. It's the third one I want to talk about right now.

So first we must answer the question -- what's the probability of winning one's second game, given that you've won the first? Of winning one's third game, given that you've won the first two? It's obvious that a defending champion is probably better than the average player (because they've won at least one game), but how much better?

Fortunately, the Jeopardy! Archive can help us answer that. It would be enough to know what proportion of champions win one game, two games, three games, and so on. The archive compiles an impressive set of statistics for each season, but this is not one of them, so I have to do it myself. There's a natural cutoff point in the data -- for a long time Jeopardy forced its champions to leave after five games, but they don't do that any more, since the beginning of the 2003-04. (This led to Ken Jennings' historic 75-game run in 2004.) I originally planned to go back to the beginning of that season, but they only go as far back as the beginning of the Ken Jennings run, near the end of that season.

Since June 2, 2004, there have been:
155 one-game winners
61 two-game winners
21 three-game winners
6 four-game winner
9 five-game winners
1 six-game winner
1 eight-game winner
1 nineteen-game winner
1 74-game winner

Now, if there was no effect like the one I just mentioned, you'd expect there to be three times as many one-game winners as two-game winners, two-game winners as three-game winners, and so on. It actually seems like if there is such an effect, it's not that strong. I attribute the big peak at five games to psychology; it's probably hard to win a sixth game because you're going into some sort of "unknown" territory. Notice that only four players -- Ken Jennings, David Madden, Tom Kavanaugh, and Kevin Marshall -- have won their sixth game.

So I'll assume that there is no "memory effect" -- that if you win today, you have a one-in-three chance of winning tomorrow. This seems believable -- the categories can be very different from day to day -- but I've never seen this analysis before. (It wouldn't surprise me if other Jeopardy! hopefuls have done it, though, because they seem to be That Sort Of People.)

Thus, when wagering in Final Jeopardy, one should wager as if the prize is not just the money you're going to win -- but one and a half times that much, since you can expect to win half a time more. The average champion is a one-and-a-half game winner.

But there are two problems:
- how do you use that information? Does the amount of money one expects to win really affect proper wagering strategy?
- more importantly, you only get to play at Jeopardy! once. I think the rules say that; in any case, I've never heard of somebody who's played twice on the Alex Trebek version of the show. (There are a few cases of people who played on the Alex Trebek version and on some prior version.) So anything you say about "expected value" is meaningless! What's the point of an operation that talks about the average amount you expect to win if you don't get to play long enough for that average to take effect?

iPhone bets part 2, and aggregating customer service information

Yesterday I posted betting on the iPhone, about an online bookmaker that was taking odds on various circumstances surrounding the release of the iPhone.
Wired has "Ready for an IPhone? Tips to End Your Existing Cell Contract", which is what it sounds like. Their tips include "odds of success", such as:

Pawn it off
Don’t want your contract anymore? Find someone who does. Websites like Celltradeusa.com specialize in connecting thousands of people together for the express purpose of transferring the financial responsibilities of cell contracts from one person to another. As long as the recipient meets the minimum qualifications (credit check, etc.) you can transfer the plan over without getting hit with the early termination fee.
Odds of success: 2-to-1

but I think they just made up these odds. It would actually be interesting to know these sorts of things about various companies' customer service. Some of the tips they offer are more of a gamble -- for example, abuse the system by using up lots of minutes while you're roaming, if roaming is free for you. They offer 20-to-1 odds on that. Before I tried that, I might like to know "1000 people say they've tried this; it worked for 50 of them". It would be interesting to see a web site that collected such information -- there are many problems that a lot of people have and having data on how they solved them could be useful.

However, there's always the issue that I'll call the "asymmetry" of word of mouth. If you expect a plan like that not to work, you'll tell lots of people if it does work and not that many people if it doesn't work. It's kind of like how you hear about the actors that made it (because they're on your TV all the time) but you don't hear about the ones that didn't make it (although you do see them if you go to a restaurant, because they're waiting tables to make ends meet). So any information on such a web site would have to be taken with a shaker of salt.

betting on the iPhone

You can bet on everything these days!

BetUS.com -- which appears to be mostly a sports betting site -- is giving odds on various iPhone-related events. (I came to this via Marginal Revolution.)

I can't get inside, but livescience.com (the first link above) claims that BetUS.com is offering the following odds:

Consumers are reported camping out waiting for an iPhone—3/1

At first glance, I'd take this bet. People camp out now for product launches, it's What They Do in this consumer culture. And the sort of people who do that are, to some extent, Apple's target market. However, the iPhone is being released at 6pm local time on Friday. And the iPhone is expensive -- $500 just for the physical device, and then depending on who you believe somewhere around $80 for the service -- so you've got to think that maybe the people buying them will have jobs. (I'm sure there's a Steve Jobs joke in here somewhere, but I can't find it.)

Apple’s stock jumps at least 10% in value in regards to the price on 6/30/07—1/2

Technically, this can't happen. Why? Because June 30 is a Saturday. Stocks don't trade on Saturdays. But assuming they mean the next trading day after the release -- that is, Monday, July 2nd -- this would be an interesting disproof of the efficient market hypothesis. This hypothesis claims that the price of a traded asset -- such as Apple stock -- reflects all the knowledge that's available about the company.

On the other hand, Apple has been trading around 125 lately; it was at 90 as recently as mid-April. Either a lot of information about Apple has suddenly come out, or investors are just crazy. Or both.

Consumers pay at least three times the original price ($1,500) on ebay - 2/1

Hard to call. Did consumers learn from when people tried to flip PS3s and Xboxes last winter? Sure, some people pulled it off, but a lot got stuck with them.

iPhone spontaneously combusts—150/1

I hope this is a joke.

Judging from the little information I have, though, and the fact that the odds are simple integer ratios, I'm guessing that these odds don't move, but are set by BetUS.com. I was expecting something like tradesports.com or intrade.com, in which people can buy and sell "contracts" on various events -- these are rapidly emerging as an interesting means of predicting the probability of various "complicated" events, where one can't come up with a simple model to make a decent guess at the probability of an event. We expect that, if people are willing to pay $25 for a "contract" that pays out $100 if people are reported camping out waiting for an iPhone, then if we could repeat this experiment over and over again, one time out of four there would be people camping out. (The question of what this even means is kind of tricky, though, because there aren't going to be three more iPhones. Tonight I prefer the interpretation of complicated probabilities like these in terms of wagers, but that could always change.)

edit, 5:08 pm: People are already camping out. Reuters reports that as of this morning, there were four people in line outside the Apple Store on 5th Avenue in Manhattan.