Coin Problem

Problem:

Suppose you can bid on a coin of an unknown value between 0 and 100. If your bid placed is greater than the value of the coin, then you win the coin. Furthermore, if you win the coin, then you are guaranteed to be able to sell it to your friend, a coin collector, for 1.5 times the coin's value. How much should you bid for the coin?

Solution:

Let X be the value of the coin, where X ~ unif(0,100).

Let Y be your bid.

Your return, R, is 1.5X - Y if Y > X, or -Y if Y ≤ X

Your expected return, E(R), is given by the following formula:

E(R) = E(1.5X - Y | Y > X)*P(Y > X) + E(-Y)*P(Y ≤ X)

E(R) = -Y + 1.5*E(X | Y > X)*P(Y > X)

To gain some intuition into how these terms work, here are some numerical examples:

Bid Y = 0

In this case, P(Y > X) is 0, because 0 is the lower bound of the coin's value.

So E(R) = 0.

 

Bid Y = 1

E(R) = -1 + 1.5*E(X | X < 1)*P(X < 1)

The probability of X < 1 is 1/100. If we know that X < 1, then the random variable X | X < 1 is distributed uniformly between 0 and 1. Thus, E(X | X < 1) = 1/2.

So E(R) = -1 + 1.5*0.5*0.01 = -0.9925

 

Bid Y = 2

E(R) = -2 + 1.5*E(X | X < 2)*P(X < 2)

The probability of X < 2 is 2/100. If we know that X < 2, then the random variable X | X < 2 is distributed uniformly between 0 and 2. Thus, E(X | X < 1) = 1.

So E(R) = -2 + 1.5*1*0.02 = -1.97

 

Bid Y = 10

E(R) = -10 + 1.5*E(X | X < 10)*P(X < 10)

Following a similar pattern, E(R) = -10 + 1.5*5*0.1 = -9.25

 

Bid Y = 50

E(R) = -50 + 1.5*E(X | X < 50)*P(X < 50)

Following a similar pattern, E(R) = -50 + 1.5*25*0.5 = -31.25

 

Bid Y = 80

E(R) = -80 + 1.5*E(X | X < 80)*P(X < 80)

Following a similar pattern, E(R) = -80 + 1.5*40*0.8 = -32

 

Bid Y = 100

E(R) = -100 + 1.5*E(X | X < 100)*P(X < 100)

Following a similar pattern, E(R) = -100 + 1.5*50*1 = -25

This leads us to the finding that E(X | Y > X) = Y / 2; additionally, P(Y > X) = Y / 100.

Therefore, our general formula for our expected return is:

E(R) = -Y + 1.5*(Y/2)*(Y/100), or:

E(R) = 0.0075Y² - Y

It turns out that under the rules of this game, there is no bid that returns a positive expected return. To see this, we can find the optimum of the return function:

E(R)' = 0.015Y - 1 = 0

Y* = 1 / 0.015 = 66.66667

However, this optimum Y* is a local minimum rather than a local maximum, because the expected return function is convex; E(R)'' = 0.015 > 0. Indeed, E(R) when Y = Y* is -33.33333.

In our numerical examples above, we checked the boundary conditions of Y = 0 and Y = 100, and thus have confirmed that for this game, Y = 0, or not placing a bet at all, yields the highest return.

*    *    *

Problem:

Suppose the return multiplier on the resale value of the coin differs from 1.5. What multipliers would cause you to change your bidding strategy?

Solution:

Let a represent the multiplier, and assume a > 0. Our expected return formula is modified to the following:

E(R) = -Y + a*(Y/2)*(Y/100)

E(R) = (a/200)*Y² - Y

We can first identify the values of a under which we would expect any positive returns; or, solve for a such that:

(a/200)*Y² - Y > 0

Y((a/200)*Y - 1) > 0

a > 200/Y

In our most extreme bid of Y=100, this inequality indicates that a > 2 is the requirement for this game to yield any positive return. This explains why a = 1.5 indicated that the best strategy was not to place a bid.

Now, assuming a > 2, what is the optimal Y to bid? Once again, we consider the optimum Y* of the return function:

E(R)' = (a/100)*Y - 1 = 0

Y* = 100/a

However, once again, this is a local minimum, since E(R)'' = a/100 > 0. So we must check the boundary conditions.

When Y = 0, E(R) = 0. When Y = 100, E(R) = 50a - 100. With a > 2, this indicates that bidding 100 is always the strategy that yields the highest expected return.

Intuitively, this makes sense, as a bid of 100 guarantees that you will win the coin and that you will receive the payout. As the payout multiplier increases above 2, the expected payout dwarfs the fixed cost of placing the bid.

November 6, 2024

2016 felt unfair because he lost the popular vote. This time, the popular vote aligned with the electoral results. That feels fair to me.

I can't say it was a surprise.

With this outcome, we know there won't be another insurrection. Because only one side would have done that, and they won. No one is claiming voter fraud.

I'm tired.

Over 72 million people voted for him. I and most other people in the circles around me are in the minority. I can't delude myself into thinking that all 72 million people are all idiots and that I know better than all of them.

You can't run a campaign on vibes. Brat ain't enough.

We survive. We take it day by day. We did it before, and we'll do it again.

The grocery stores are still open. The electricity is still running. The checks are still clearing. The hot water is still flowing. It is not the apocalyptic hellscape the left said it would be.

When Obama won in 2008, the community by which I was surrounded--conservative evangelical Christianity--decried the advent of socialism, the collapse of democracy, and even the ascendance of the anti-Christ. In retrospect, 2009-2016 were exceedingly politically tame.

Even if we lose democracy, we won't be killed or slaughtered. We are not in physical danger. Millions and even billions exist and survive under much more oppressive governments.

It may turn out worse for others. But I don't have the energy to think or care beyond the (minimal) impacts to me personally, because that's the only way I can maintain any sense of hope.

And I hope more than anything else that SNL does not do "Hallelujah" again.