Can you solve this PE interview brainteaser?

Imagine you have a special dollar. You can play a game with me. I'll toss a coin and if you call it right, you have two dollars; if not, you take home only one dollar. You can keep playing if you call correctly, doubling your money each time How much would you pay for the special dollar?

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I'd think about this way - at every moment in time your expected value of flipping the coin again is $0.50 - let me explain, lets say you call four tosses right in a row, at that point the dollar is worth $16, if you flip again your Expected Value = 32 * 50% + $1 *50 = $16.5 - so you can take $16 in cash or take an action whose expected value $16.5. At that point you should flip it again if you are a straight value maximizing machine. You have a 50% chance of winning $64 and a 50% chance of winning $1, EV is 32.5. The complicate part is two fold: (1) if you flip the coin an infinite number of times, you will eventually get $1 and (2) there is a certain point where common sense comes into play. If you called it right ten times in a row, you would be faced with the choice between pocketing $1,024 and taking an action with an EV of 1,024.50 but where you had a 50% chance of losing 1,023. At that point, I think most people would say they would rather take the money than keep playing. The trick is to define upfront how many times you plan to flip. Lets say that is 5 times - which means you have 3.1% (.5^5) of winning $32 and a 96.9% chance of winning $1. The EV of flipping 5 times is $1.97. Interesting, the EV of flipinng 100 time is ~$2 (.5^100 x 2^100 + (1-.5^100) x 1. So I would pay something like $1 under the theory that buying something for half of what it is worth is usually a good trade - the challenge is obviously if you do a deal where you can't lose (like paying $1) you might not properly understand the game ;)

 
Best Response

This question is referred to as the St. Petersburg Paradox - you can read up more on it its Wikipedia page

In theory, you would be willing to pay an infinite amount of money to play since the expected value of playing is also infinite. It's important to state in an interview that while this is the case, you obviously wouldn't pay an infinite amount of money given how risky the game is. As stated above, to minimize risk you should not pay anything above one dollar. If you pay anything above a dollar then your expected odds of getting your money back fall below 50%.

 
RX:
This question is referred to as the St. Petersburg Paradox - you can read up more on it its Wikipedia page

In theory, you would be willing to pay an infinite amount of money to play since the expected value of playing is also infinite. It's important to state in an interview that while this is the case, you obviously wouldn't pay an infinite amount of money given how risky the game is. As stated above, to minimize risk you should not pay anything above one dollar. If you pay anything above a dollar then your expected odds of getting your money back fall below 50%.

Thing is that in that paradox you win the pot while in this question you lose all your money until that point. That makes the question completely different than the paradox.

So mathematically this question has an expected value of 1 for infinite times and little more than 1 for any finite amount of times. I would say I will play 1 time and I will pay less than 1.5 dollar for it because at 1.5 I am indifferent

 

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