Counterfeit money, guaranteeing a profit

Question

Bob and I found two 50 dollar bills out of nowhere. We know they're either both legitimate or both counterfeit. If they're legitimate, they're worth 50 dollars each, otherwise 0. I get one 50 dollar bill, Bob gets the other 50 dollar bill.

Bob offers to buy my 50 dollar bill for 10 dollars or sell his bill to me for 10 dollars. This is before it's revealed whether they're both legitimate or counterfeit.

An online gambling service lets me make a one-to-one bet that the 50 dollar bills are legitimate or counterfeit. For example, if I bet 60 dollars that the two bills are counterfeit and both turn out to be legitimate, then I lose 60 dollars. But if I bet 60 dollars that the two bills are legitimate and both turn out to be legitimate, I earn 60 dollars.

So my options are:

• Buy Bob's 50 dollar bill for 10 dollars.
• Sell my bill to Bob for 10 dollars.
• Bet that the bills are legitimate for $C$ amount with one-to-one odds.
• Bet that the bills are counterfeit for $C$ amount with one-to-one odds.

If the bills end up being legitimate (we won't know this until afterwards), they're worth 50 dollars each, otherwise 0.

Question: What is my optimal play here, to maximize the amount of money I'm guaranteed to win?

My thoughts: Buy Bob's 50 dollar bill for 10 dollars, and then bet 25 dollars on both bills being counterfeit.

Why bet 25 dollars on both bills being counterfeit? The profit in the two cases:

• Both legitimate: $-10 + 50 - C = 40 - C$.
• Both counterfeit: $-10 + C$.

Theoretically, any value of $C$ such that $40 - C > 0$ and $-20 + C > 0$ will generate a guaranteed profit no matter the outcome, but I want to make the same amount of money no matter the outcome. We don't know what are the respective probabilities that they're both legitimate or both counterfeit.

So let's set the two equations equal: $40 - C = -10 + C$, which implies that $C = 25$. So my strategy guarantees a profit of 15 dollars no matter what if we bet 25 dollars on both bills being counterfeit (and also buy Bob's 50 dollar bill for 10 dollars).

What I'm confused about: In my two equations above, what if instead considering the profit in both cases, we instead use the total amount we end up with:

• Both legitimate: $-10 + 100 - C = 90 - C$.
• Both counterfeit: $-10 + C$.

Setting the two equations equal gets us $C = 50$, which guarantees we end up with 40 dollars total no matter what if we bet 50 dollars on both bills being counterfeit (and also buy Bob's 50 dollar bill for 10 dollars).

So which method is correct, the one considering total profit or the one considering the total amount we end up with?

Answer 1

Question: What is my optimal play here, to maximize the amount of money I'm guaranteed to win?

So which method is correct, the one considering total profit or the one considering the total amount we end up with?

I think the first method is correct, but the second is not, because when we consider the total amount we end up with, we take into account the bill which we already have. So in the case about which you are confused and when both bills are legitimate our profit is not $−10+100−C$ but only $-10+50-C$, that is the same as in the first case.

Answer 2

Suppose you're betting $C$ that the bills are counterfeit. (We'll allow $C$ to be zero or negative, which covers making the opposite bet or not betting at all.) Then if you sold your bill to Bob, your payoff is $10 + C$ (if counterfeit) or $10-C$ (if legit). If you kept your single bill, your payoff is $C$ (if counterfeit) or $50-C$ (if legit). Finally, if you bought Bob's bill, your payoff is $-10+C$ (if counterfeit) or $90-C$ (if legit).

If you're trying to maximize your worst-case total payoff, then you would either sell your bill to Bob and bet $0$ (outcome is $10$), or keep your original bill and bet $25$ (outcome is $25$), or buy Bob's bill and bet $50$ (outcome is $40$).

When you consider profit and not total outcome, you're ignoring the fact that you can improve the worst-case performance of your original bill by increasing your bet that it's counterfeit. This doesn't show up in the profit calculation, and you wind up not betting enough.