In the previous lecture, we saw that if you're going to bet, then you ought to
pick a favorable bet over an unfavorable bet, and even over a neutral or fair bet.
But the next question is, do you want to bet at all?
And there we have to look at other values besides money.
Let's consider this case. Suppose that you can't just bet a dollar.
This game's not for fun. These are serious gamblers.
Your minimum bet is $100,000. And now, if you draw a spade, you win
$1,000,000 net. Cool.
Well what's the expected financial value? One chance in four of picking a spade,
the amount you win net is a $1,000,000. You have to subtract the probability of
losing three out of four because you lose if you pick a heart a diamond or a club
times the amount you bet which is 100,000.
Pretty neat, this bet's got an expected financial value of a $175,000.
Now that sounds like a great bet, but is it really a great bet?
It is favorable and it's only favorable financially.
Imagine that a $100,000 is all the money you've got in the world.
It's what you use to pay your rent with it's all your retirement money and if you
lose it you're going to be out on the street with nothing.
Then should you bet this money? No, absolutely not.
It just doesn't make sense to take that kind of risk with this money if you're
going to be out on the street if you lose because you've got three chances in four
of losing. Sure you might win you might be better
off but you've got an awful lot to lose and that's why you shouldn't make this
bet because there's just too much to lose.
And what does that mean? Think about what it means that there's
too much to lose. It's not that there's too much money to
lose, because you've just calculated the financial value of the bet.
What really is the loss, is security. You're not going to have a place to live.
You're not going to know. How to make ends meet, or buy your next
meal. That's the problem, is there're lots of
other values besides money. And, in these situations where we say
you've got too much to lose. It's those other considerations in
addition to money. That make it, a bad bet.
Now let's look at the other side. In some cases, you don't have too much to
lose, you got too much to win. That is, the amount that you win just
might not add up in the way that you thought it would.
This is the phenomenon called diminishing marginal utility and what that means is
that each increment decreases as you get more and more and more of those
increments. You just don't need that much.
One example is hamburgers. If you're really hungry and you don't
have anything to eat you might want a hamburger.
But what about two hamburgers? Is two hamburgers twice as good as one
hamburger? Not really, I mean, maybe you want two
hamburgers, that would be nice but it's not twice as good.
What about three hamburgers? Is that three times as good as one
hamburger? And what about four or five or six or
seven or eight or nine or ten? What about ten hamburgers?
Are ten hamburgers ten times as valuable to a hungry person as one hamburger?
No. because you're going to get full, before
you ever eat that tenth hamburger. So the marginal utility of a hamburger
goes down. The first one might be very valuable to a
hungry person with nothing to eat. The second 1's a little less.
The third 1's a little less. The fourth one's a little less,
and the difference between nine and ten might be totally negligible, because
they're all going to go bad before you eat all ten of them.
The same thing happens in lotteries. Let's compare two lotteries.
In the first lottery, if you buy one ticket for $100, then you have one chance
in a million of winning $300,000,000. That sounds like a pretty good bet.
I'll let you calculate the actual expected financial value of that bet in
one of the exercises after this lecture. But I bet you can do it in your head
pretty quickly and compare it to this lottery.
If you buy one ticket for $200, then you have a one in 2,000,000 chance of winning
billion dollars. Which of these lotteries has a higher
expected financial value? Well, we'll leave the actual calculations
of the expected financial values to the exercises, but you can probably tell,
that the one with the larger expected financial value is lottery B.
Does that mean that you ought to spend the extra money to play in lottery b
instead of lottery A? Well, that depends, I suppose.
How much more valuable is a billion dollars to you than 300 million?
300 million is a lot of money. I could buy pretty much anything I ever
wanted if I had 300 million dollars. I'm not sure why I would want all that
extra money. It's like that tenth hamburger, I
wouldn't really need it. So, which of these lotteries should you
pick if that's the way you see the relationship between the winnings.
Well even though the expected financial value is greater in lottery b you might
reasonably pick lottery a cause you have a bigger chance of getting the
$300,000,000. And the difference between 300 million
and a billion doesn't matter that much to you.
Not everybody's going to share that value judgement.
But if it doesn't really matter much to you whether you have 300 million or a
billion, then it makes sense to go for lottery A.
And this shows, yet again, that there's more to life than expected financial
value. The other values in life are captured in
the notion of overall value. The actual overall value or sometimes
called utility of an act is simply all of the good and bad effects that the act
actually has, the values of those effects, positive values and negative
values calculated together. And the, expected overall value.
Of the bet, or of an action. Sometimes called the expected utility of
the act is all the good and bad effects of the act multiplied by the probability
of each of those effects. So you bring not just the values into
play, but also. The probabilities.
And if you're really going to decide what you ought to do, you shouldn't be looking
just at money. You should be looking at all the
different values, including security as we mentioned.
Or how much good the money's really going to do you if you've going to have
$300,000,000 anyway. There's lots of other things to consider.
The problem is, how do you calculate that?
How are you going to actually calculate overall value or expected overall value?
Here's an example to show how hard it is, imagine that you have a friend that you
love playin with after school and. You then get an offer, of a job.
And you can make a bunch of money. By doing the job after school.
But then you won't get to play with your friend.
Well, how are we going to figure this out?
Let's calculate the expected overall value.
What's the probability that if you take the job you won't be such good friends
with him anymore, because you won't be playing as much and then you'll lose your
friend? And what's the value of that friend?
You've got to multiply the probability of losing your friend by the value of that
friend. Well, how are you going to do that?
How do you put a dollar value on that friendship?
How do you figure out the probability that your friend's going to leave you and
not be as close anyway as he was before? Well, you just can't do that.
You can't really put it into numbers like that.
There are some economists have very sophisticated techniques for taking
preferences and attaching values to those preferences and then multiplying
probabilities and so I don't want to say it can not be done, but the point here is
just it's a lot more difficult than figuring out whether you ought to make a
bet in a simple little poker game or even whether you ought to put your money into
the lottery. When we start talking about real
decisions about friends and about other values in life, some of them cannot be
reduced to money. And then it's hard to put a number on
them, and it's hard to do the actual calculations.
So I don't want to suggest that any of this decision theory regarding expected
utility is going to actually be applied in real life.
Another reason that's worth mentioning is that third kind of decision that I
mentioned at the beginning of the lecture, namely decisions with ignorance.
Sometimes you don't even know the probabilities and if you don't know the
probabilities, you can't enter the probabilities in the formula for expected
utility or expected financial value or expected overall value.
If you don't know the probabilities, how you going to calculate?
You can't. Now there lots of tricky rules that
people have proposed, for decisions under uncertainty, or decisions with ignorance,
as I call them, but none of those are really conclusive,
and there's big dispute about which one is the right rule.
So a lot of this discussion is going to be inconclusive.
You might have wished that I would come in here and tell you, here's how you
figure out exactly what to do in every area of your life.
Well, I'm sorry, but I just can't do it, and nobody else can either.
Nonetheless, if you think in terms of the various factors that we've been
discussing, Financial value, other kinds of values, probabilities of the different
outcomes. Maybe at least you'll be able to avoid
some of the worst mistakes that people make in decisions.