How much would you pay to play this fair coin flipping game?

[docnet]

Homework Statement

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Relevant Equations

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I feel like there is something missing in my solutions, because the answers are coming out weird.

(a)

The reward for getting a head is $2^n$ dollars, the number of flips $n$ it takes to get the first heads.

the probability to get a head in the $n$th flip is $\frac{1}{2^n}$

so the expected value of the game is

$$E(X)=\frac{1}{2}\cdot 2+\frac{1}{4}\cdot 2^3+\frac{1}{8}\cdot 2^3+\frac{1}{16}\cdot 2^4+\cdots$$

$$E(X)=\sum_{i=1}^\infty 1 \Rightarrow \text{ the sum diverges}$$

Who would want to pay an indefinite amount of money to play this game...?

(b)

$$E(X)=\sum_{i=1}^{40} 1 = 40$$

By this logic, I would pay no more than 40 dollars to play this game

The probability of going more than 40 rounds is

$$P(X>40)=\sum_{i=41}^\infty\frac{1}{2^i}\approx 0$$

Yet, the expectation value drastically differs from the value in (a)

(c)

$$E(X)=\sum_{i=1}^{40} 1 + \sum_{i=1}^\infty \frac{1}{2^n} =40+1 =41$$

I'd pay 41 dollars to play this game

(d)

no one has that much monay. that's too many zeros.

It turns out this is famous problem called St. Petersburg Paradox. - it seems to be a rare example of when probability theory isn't rational for predicting the outcome, since the player is probably only going to win a modest amount of money in this game.

 

[Office_Shredder]

I think all of your answers are correct here.

 

[Orodruin]

Who would want to pay an indefinite amount of money to play this game...?

People asking questions like this expecting students to base their answer on expectation value only are doing students a disfavor. The value function for someone playing the game is generally not going to be linear* and even if it were, the variance plays a large psychological role for people’s willingness to gamble.

* For example, once you have more money than you could possibly spend, there is no added value in earning more.

 

[PeroK]

* For example, once you have more money than you could possibly spend, there is no added value in earning more.

Tell that to Elon Musk, Jeff Bezos etc.

 

[Orodruin]

Tell that to Elon Musk, Jeff Bezos etc.

Even so, I do not believe for a second their value functions are linear. Logarithmic perhaps

 

[Tazerfish]

Tell that to Elon Musk, Jeff Bezos etc.

Fair, some people are keen on accumulating for the hell of it or for power, fame or their ego.
Some people want to be the winner of capitalism, to control armies of works if not armies of soldiers.
Although the value of more money never drops to zero, it gets real close.

Going from living on the street to 40k a year (stable housing, secure water, food and healthcare) is probably much more valuable than going from 40k to 1m a year unless you really get off on the power.

Also, for different theories of personal value/decision making, you can have a look at Daniel Kahnemannn'sProspect Theory.