
Title:
0522 Game Theory Solution

Description:

So I predict that most people say they would hold, and why is that?

Well, under the identity function, sort of the arithmetic function,

$3 million is 3 times better than $1 million, and so half of $3 million is 1.5 times better

than $1 million.

So the gamble is more.

But that's only true if $3 million really is 3 times better than $1 million.

For most people, that's not true.

That going from $100 to a $1 million is a big, big jump.

Going from $1 million to $3 million is a smaller jump than that,

even though arithmetically it's more,

in terms of what you can do, it seems like less,

and that doesn't mean that people are irrational in any way.

Instead what it means is that for people, the value of money is not a linear function.

Rather it's something more like a logarithmic function,

meaning if you have a certain amount of money,

if you double that money,

you don't get twice as much value out of having that money,

rather you just get 1 increment more of having that money.

So let's try again.

I'm going to input the math module, and now I'm going to ask,

what's the best_action starting from $100 in my pocket,

but valuing money with logarithmic function rather than with the identity

or linear function.

Now my best_action tells me that what I should be doing is holding.

That corresponds to my intuition. That that's the right thing.

I can also ask, well, what if I had $10 million already,

then would I take the bet, assuming my value of money is still logarithmic,

and best_action tells me that yes, I should.

If I have $10 million, now I'm starting to look at money as more closely linear again.

I'm at this stage where logarithmic function is approximately linear locally.

If I've got $10 million, I could say, yeah, I'm risking my $1 million,

but that's no big deal. I've already got $10.

It's a good bet because if I win, I get 3 or 0that's 1.5 on average,

and that's more than 1, so I'm willing to take that bet,

and I don't mind not gaining the additional $1 million.