I took many fascinating courses in my eleven years at MIT. Among them, my sister Phyllis's favorite title was "Random Signals and Noise". It was actually a highly mathematical, and rather difficult, electrical engineering course.
Another great course was called "Statistical Decision Theory". In its first session, the instructor took great pains to emphasize that the course would address real world decision making, as opposed to being a purely mathematical exercise. To stress this point, he noted that if we talked about tossing a coin, we would be talking about a specific, real, physical coin, which he produced, calling it "the class quarter". He actually passed it around the room so we could all take a look at it, presumably to be sure that it was a standard quarter, and not a trick coin, with, say, two "tails". Despite the room being full of impoverished graduate students, the quarter was returned to the front of the room.
We were then asked to consider a game that could be played with the class quarter, as follows:
Not bad - you might win $100, and in any event, you couldn't lose. We were asked to each write down on a piece of paper what we would be willing to pay to play that game once (we didn't have to write our names on the paper). The instructor collected the pages, and tabulated the numbers on the blackboard. Most of the numbers were $50, or slightly below (many were $48 or $49).
Why $50? Well, this was a graduate class at MIT. All the students knew Calculus and basic probability theory, and were familiar with the concept of "expected value". The "expected value" of the game was $50, since it was a 50-50 chance at winning $100. So in that sense, the value of each play was $50, and that seemed a fair price to pay to play.
But the instructor interpreted our answers as meaning that we had not really absorbed his earlier statement that our class was to address practical, real-world decisions, not mathematical abstractions. He asked us whether this large class of impoverished graduate students were all really willing to risk $48 or more on a single coin flip. I might point out that $48 in 1966 was roughly the equivalent of $325 in 2010 (adjusted for inflation), so this was not a small amount of money. The instructor didn't think the students had really thought this through before submitting their numbers, and he noted that the only way to test this hypothesis would be to play the game for real.
So we would re-submit our "bids" to play the game, and this time, the highest bidder would actually play it, for keeps. There was one small problem. Despite what we thought of as our outrageous tuition payment of $1,700 Note 1 (which now seems like a paltry sum), our course had no budget to pay out the prize if the student won the toss. So in order to be able to play the game for real, the instructor asked each of the students in the class to kick in two dollars for a prize fund. This was not mandatory (you could buy a lunch for 99 cents in those days), but most students participated, and we collected enough money to go ahead with the exercise.
We re-submitted our bids, changing them, of course, given the fact that we might have to play the game for real. I had an advantage, perhaps, over most of the other students. A friend of mine had taken the course and played the game the previous year, and I knew that his winning bid had been around $38 (and he had won the coin toss). I was willing to play at that level, so I bid something like that - perhaps $39 or $40 - I don't remember. The bids were collected, and indeed, most of them had dropped substantially from the previous submissions. Some students bid one dollar, a way of saying, "I don't want to play". But when the high bidder was announced, the winning bid stunned me, and judging from the gasp from the other students, quite a few others were surprised as well. The winning bid was $55.
Our reaction showed that despite all the instructor had said, many of us were still hung up on the idea of "expected value", and so we were surprised that a student had bid higher than $50. It seemed to us to be a losing proposition, as he had an "expected" loss of five dollars. But of course whatever the outcome of the toss, the student could not lose five dollars. The only two possible outcomes would be to lose his $55 bid, or to win $45 (the $100 prize minus the $55 he paid to play).
So he could still win. He was just a gambler, that's all. He was doing nothing different than countless millions do every day in casinos in Las Vegas, Atlantic City, and countless other cities. Every gambler knows that the odds are against him, but that doesn't mean he (or she) can't win. Indeed, years later, I would play the slot machines in Las Vegas, and leave town having earned a fifty percent profit on my money!
What does "expected value" mean, anyway? It's a mathematical construct, but what is its physical interpretation? It's the average value per play that you are likely to get over the long run, if you play the game many times. But that really doesn't have much relevance if you only get to play once.
Our high bidder wrote a check for $55, flipped the class quarter, and lost.
That left the class with a decision to make, namely what to do with the money it now possessed. We voted to have a party at the end of the term, using the money to provide drinks and refreshments. This brought up an aspect of the game that nobody had particularly thought about in advance. Those who had not wanted to contribute $2.00 to the game fund now needed to pony up $3.00 if they wanted to attend the party. In a sense, we had all been betting against the high bidder - it was to our advantage that he had lost. We did allow him to bring a date to the party without kicking in another $3.00. That way, he was only paying $27.50 each for the party (as opposed to $2.00 for most of the rest of us). Maybe he and his date got their money's worth by drinking a lot.
The class continued to be fascinating as the term progressed, introducing the example of "Joe the used car buyer". It exposed me to such concepts as the dollar value of perfect information and the dollar value of partial information (information with only a defined probability of being correct). The techniques I learned in the course were indeed valuable later in my business career, and in my life.
One of the most valuable lessons I learned was to distinguish between a good or bad decision vs. a good or bad outcome. If a decision leads to a bad outcome, it's common for people to say that they made a "bad decision". But a well-considered, carefully analyzed, and hence "good" decision can still have a bad outcome if the probabilities go against you. So a good (that is, well-made) decision can have a bad outcome, and a bad decision can have a good outcome. It's a useful distinction to keep in mind. It can keep you from beating yourself up if a carefully-made decision nevertheless goes against you.
I later heard that the course had been taught one summer to a group of middle-management executives from General Electric, under MIT's Industrial Liaison Program. GE had paid a considerable amount for each executive attending the course, and I don't think additional money needed to be collected from the participants in order to play the game. But what was astounding was that the winning bid in that session of the course was around sixteen dollars. That was the level of risk taking exhibited by at least one group of GE middle-management executives in the late sixties.
By the way, do you want to know more about the fifty percent profit I earned on Las Vegas slot machines? Well, here's the story. In 1983, I was in Las Vegas for the large Comdex computer show. Since I'm not much of a gambler, I actually spent very little time in the casinos - I think I played a bit of blackjack. But in the airport on my way home, waiting for my plane to leave, I noticed a quarter slot tucked away in a corner. I figured what the heck, I'm in Las Vegas, I shouldn't leave without ever trying a slot machine. I put in a quarter, pulled the handle, and won nothing. I then put in another quarter, pulled the handle again, and won three quarters.
I stopped and thought a bit. I had now played the slot machines in Las Vegas, and I was up fifty percent on my investment. If I put in one more quarter, I would likely never again be able to say that. So I pocketed my fifty percent profit (twenty-five cents), and boarded the plane. Note 2
Note 2: August, 2012 update: On Sunday, July 15, 2012, the Boston Globe ran a feature article by Bella English on the front page of the Lifestyle section. It was entitled Jackpot fueled therapistís gambling addiction, and it was about a woman who had won $752,000 playing the slot machines at the casino "Mohegan Sun" in Uncasville, Connecticut. This had fueled a gambling addiction, and in two years, all the money was gone.
This prompted me to write a letter to the Globe, which was printed on July 22, 2012, under the caption, "The trick is knowing when to walk away". It contains the Las Vegas slot machine story recounted above at the end of this blog entry, and ends with the sentences, "The trick is knowing when to walk away. It seems to be easier for some than for others." [return to text]