What Would You Pay? A Thought Experiment

[gordonj005]

This interesting thought experiment was brought up by my philosophy TA last week and I thought I'd pass it along in hopes for a lively discussion.

The game is as follows. You flip a coin.

H TH TTH TTTH ...
$2 $4 $8 $16

If you flip heads, the game is over and you get the coressponding prize money. The question is, how much would you be willing to pay to play this game? Seeing that winning $2 is equally as likely as winning 4$, and winning 4$ is equally as likely as winning $8, ad infintium.

Any thoughts?

 

[zoobyshoe]

I don't play coin toss games cause I saw "No Country For Old Men".

 

[D H]

Seeing that winning $2 is equally as likely as winning 4$, and winning 4$ is equally as likely as winning $8, ad infintium.

Winning $2 (probability=1/2) is twice as likely as winning $4 (probability=1/4), and so on. There's still a problem with this game: What is the expected value?

 

[Hepth]

I would pay $2.

 

[256bits]

If you flip heads, the game is over and you get the coressponding prize money. The question is, how much would you be willing to pay to play this game? Seeing that winning $2 is equally as likely as winning 4$, and winning 4$ is equally as likely as winning $8, ad infintium.

Winning $2 (probability=1/2) is twice as likely as winning $4 (probability=1/4), and so on. There's still a problem with this game: What is the expected value?

If you bet $4 you are betting that the head will come up on the second toss.

If head is non first toss you lose.
If tail is on first toss and tail is on second toss, you lose.
I head is on second toss you win

Wouldn't that be 1 out of 3 to win?

For an $8 bet.
You lose if
H
TH
TTT

You win if TTH
chance of winning is 1/4

etc.

 

[D H]

If you bet $4 you are betting that the head will come up on the second toss.

If head is non first toss you lose.
If tail is on first toss and tail is on second toss, you lose.
I head is on second toss you win

Wouldn't that be 1 out of 3 to win?

For an $8 bet.
You lose if
H
TH
TTT

You win if TTH
chance of winning is 1/4

etc.

You are doing probability incorrectly and are interpreting the game incorrectly.

Here's how the game works. You say something like "I'll pay $10 to play that game". Suppose the person running the game accepts. You flip a coin. Get heads on the first toss and you get $2 back, for a net loss of $8. Heads on the second toss represents a net loss of $6, on the third, $2. Live past the third toss and you are making a net profit. Live past the 20th toss and you will have won over a million, minimum.

Suppose the person running the game says "Not enough" to your paltry offer. You offer more and more, but the answer always remains "Not enough." At what point should you claim "Too much" and walk away? (Answer is further down.)Here's how the probabilities work. First off, the probabilities for each of the distinct outcomes must sum to one. Look at what you have: 1/2+1/3+1/4+1/5+... That is the divergent geometric series. The sum is not one. So right off the bat something is wrong.

What you did wrong was to erroneously apply the principle of indifference. For example,

If head is non first toss you lose.
If tail is on first toss and tail is on second toss, you lose.
I head is on second toss you win

Wouldn't that be 1 out of 3 to win?

Here you are assuming that the three events (a) heads on the first toss, (b) tails on the first and second tosses, and (c) heads on the second toss are equiprobable events. Your 1 out of 3 would be correct if these events were equally probable. But they aren't. The probability of heads on the first toss is 1/2. For tails on the first and second tosses the probability is 1/4. This is also the probability for the case for tails on the first toss and heads on the second toss. The probability of winning the $4 payoff is 1/4.One way to determine how the amount one should pay to play a game is to compute the expected payoff from the game. For a discrete game such as this, the expected return is the sum of the payoffs weighted by their corresponding probabilities. For this game, the expected return is

\sum_{n=1}^{\infty} $1 \cdot 2^n\cdot\frac 1 {2^n} = \sum_{n=1}^{\infty} $1

You should be willing to pay any finite amount to play in this game!However, there is no gambling house in the world that could cover the potential payoff for this game. In the real world there would be some upper limit beyond which the game would end because you just won the house.

So, to make the game a bit more realistic, let's say that you win $2 if you first flip heads on the first toss, $4 for the second, $8 for the third, and so on. Flip 20 tails in a row and you are paid $1,048,576; game over. Now how much would you be willing to pay to play?

 

[PAllen]

In a PM to Evo I raised that the cute feature of this game is the infinite expectation combined with that fact that really relying on this in the real world would be absurd. Besides DH reasoning (which would say pay no more than $20 to participate in the modified game), a general feature of pure reliance on expectation is asymmetry of consequence. The consequence of losing all of your money is often much worse than the real gain in your life of some large amount of money. This depends on your circumstances, but expectation simply ignores asymmetry of consequence.

 

[DaveC426913]

The consequence of losing all of your money is often much worse than the real gain in your life of some large amount of money. This depends on your circumstances, but expectation simply ignores asymmetry of consequence.

Which is why there is no mathematically correct answer, and the thread is entitled "What would you pay?"

 

[Ryan_m_b]

I would offer £4, just to make the game interesting for me but also not to waste money.

Can I twist the question slightly: How much would you be willing to accept from a gambler to play the game if you represent the house?

 

[PAllen]

There is, conceptually, a way to formalize asymmetry of consequence. That is simply that you should be computing expectation of f($), where f codifies your personal posititve/negative consequence. A very generic feature of f(), for example, is diminishing returns - a million on top of a billion is enormously less consequential than a million on top of zero. On top of this, the shape for negative $ will not match that for positive dollars. The utility of expectation for 'every day' bets is then formally understood as nothing more than that linear approximation of a sufficiently small region of any smooth function works well.

 

[D H]

The consequence of losing all of your money is often much worse than the real gain in your life of some large amount of money.

I didn't bring utility functions and all that into my analysis per the KISS principle (Keep It Short and Simple). But yes, people do tend to be risk averse, particularly when the risk is large.

There is a flip side to this: People can be risk-affine when the perceived risk is small. A good number of people would be more than willing to pay more than $20 to play my modified version of the game. They'd line up by the droves. Evidence: Modify my game a bit more and you get the government-run lotteries.

 

[lostcauses10x]

Love this type of question that is supposedly math but in reality coming across such a game:
The person offering the game and the "chance" to win money, is out to take your money.
I know this: so would keep my money in me pocket.

 

[fluidistic]

I'd pay 0.5 cent so I can't pay (cash at least) if I lose, though I can win some money.

 

[D H]

Can I twist the question slightly: How much would you be willing to accept from a gambler to play the game if you represent the house?

The unmodified game? You'd be a fool to take any finite amount.

Now if I could make one slight modification to the game, say one flip per day, a billion or so should do it. That gives me almost a month to find people who, for a modest set of fee (and with a billion dollars on hand, even a million is a modest fee), would supply me a bunch of completely new and relatively untraceable identities, buy me some cool hideouts around the globe, and develop an escape plan should the mark manage to flip tails on each of the first 29 days.

Nobody said I had to follow the rules.

 

[AlephZero]

Which is why there is no mathematically correct answer, and the thread is entitled "What would you pay?"

It's not a question about math. It's a question about human behaviour.

My answer to the question would depend how many times I expected to play the game (for both the original or D.H.'s modified/truncated version).

If I knew I was playing only once, the answer would be $2, because that is the maximum amount where I can't possibly lose anything. I offered less than $2 I would expect the other player to walk away rather than play. I wouldn't offer more than $2 because I'm not a gambler by nature.

If I was playing repeated games, it would be worth a bigger stake each time, to keep the person was running the game interested in playing longer by letting him/her make lots of small wins and occasional big losses. It might even be worth raising the stakes as the game progressed, to give him/her a "chance" (against the odds!) to even the score.

 

[Moonbear]

I know I'm risk aversive when it comes to gambling. I'd pay $2. I can't really see any reason to pay more if you can win all the higher prizes regardless of how much you spent on the game.

Perhaps you meant the question to be, "How much could someone ask to pay to play and you'd still be willing to try?" I probably still wouldn't play for more than $2, but you might get more people to play if charged $4 or $8 considering they play lottery tickets for that much.

 

[DaveC426913]

Which is why there is no mathematically correct answer, and the thread is entitled "What would you pay?"

It's not a question about math. It's a question about human behaviour.

Uh. Isn't that what I just said?

 

[PAllen]

I know I'm risk aversive when it comes to gambling. I'd pay $2. I can't really see any reason to pay more if you can win all the higher prizes regardless of how much you spent on the game.

Perhaps you meant the question to be, "How much could someone ask to pay to play and you'd still be willing to try?" I probably still wouldn't play for more than $2, but you might get more people to play if charged $4 or $8 considering they play lottery tickets for that much.

You could interpret 'willing to pay' as if there were an auction on the right to play once.

What is amusing is that the same theory that correctly says all casino gambling and all lottery games (unless the winning amount rolls unclaimed a number times) are foolish, claims that you should be willing to bid any amount to play this game once.

 

[AlephZero]

Uh. Isn't that what I just said?

Sorry, data transmission error. The message I "received" was something like "why is the OP asking how much you would pay on a rational basis, when there is no mathematically correct answer?". Which is not what you wrote!

 

[nucl34rgg]

The game host should pay me at least 51 cents to play his game, of course! (Since 1+1+1+...=-1/2) lol ;) kidding...

 

[D H]

The game host should pay me at least 51 cents to play his game, of course! (Since 1+1+1+...=-1/2) lol ;) kidding...

50 cents. Be fair!

I was tempted to post this answer myself some time ago.

 

[EricVT]

This is discussed here:

http://www.physics.harvard.edu/academics/undergrad/problems.html

If anyone is interested. It's toward the bottom, Week 6.

 

[nucl34rgg]

I think the answer is that you should play the game if you have to pay 4 dollars or less. (If you pay 2 dollars, you are guaranteed to always at least break even. If you pay $4, you have a 50-50 shot at at least getting your money back or making more. If you pay anything over $4, you have less than a 50% chance to win, so you shouldn't play.

 

[larrybud]

Since the question is what I would pay, I would pay zero, so that any winnings would be pure profit.

Frankly, I think the question (and the game rules) are both poorly worded.

 

[FlexGunship]

I would pay $1.99. Guaranteed to win, baby!

 

[D H]

Since the question is what I would pay, I would pay zero, so that any winnings would be pure profit.

And what if the person running the game said "No thanks" to that offer? What is the top amount you would be willing to pay to play? Keep in mind that the expected value is infinite. Also keep in mind that the only way to obtain this expected value is to flip an infinite number of tails in a row, which is something you will not be able to do in a finite amount of time. (And you will presumably die in a finite amount of time.)

 

[FlexGunship]

And what if the person running the game said "No thanks" to that offer? What is the top amount you would be willing to pay to play? Keep in mind that the expected value is infinite. Also keep in mind that the only way to obtain this expected value is to flip an infinite number of tails in a row, which is something you will not be able to do in a finite amount of time. (And you will presumably die in a finite amount of time.)

Most forms of gambling (and lottery) are just taxes on stupid people. If you're not literally enjoying the act of gambling, then the value of the game might be negligible to some.

In my opinion, this is a boring game... I would elect to low-ball.

 

[PAllen]

One way to look at games of chance, and, in general, all forms of gambles in your life, is to view all the gambles in your life as part of an overall game. Then, even though expectation is an average of an infinite number of trials, and strictly doesn't apply to individual decisions, by taking advantage of winning expectation gambles as they present themselves, over your life you may come out ahead.

This leads to an important corollary: If the chance of winning is too small, there cannot be enough comparable opportunities in your lifetime. Thus, there is a threshold probability below which expectation is irrelevant. This can be quantified if you can guess the number of similar opportunities you might have in your life. For example, for a lottery, you can compute the most likely outcome of the number of plays you can reasonably make in your life, of tickets that have developed a winning expectation (as happens sometimes when a lottery round has no winner, and the prize rolls over). You find that for this finite game, your lifetime winnings are almost certain to be less than your cost. Even if you include other types of opportunities, you conclude that such a lottery play is still not rational.

This can be combined with observations about reality check (as DH first pointed out, the stated game is unimplementable); and also that you really need to think about your personal utility function of dollars, not just raw dollars.

In my view, then, choices like this are fully quantifiable in principle, the practical difficulty being lack of definition of things like 'lifetime similar opportunities' and 'personal utility function for money'.

This is not a game like poker, where guessing human behavior is paramount; nor like the running of lottery (though guessing mass behavior here is pretty trivial). For playing in a lottery or this game, there is no human behavior component involved.

[Edit, putting all this together for me, I would be willing to bid up to $10 to play this game].

 

[D H]

Most forms of gambling (and lottery) are just taxes on stupid people.

Everyone gambles many times a day. They just don't know it. Hopping into the car to drive to work, run errands, or go out on the town is a gamble. You might be killed or injured by someone with failed brakes or by some fool on a cell phone. Every election of how to allocate one's retirement account is a gamble. It doesn't matter whether its the riskiest REIT that can be found or a supposedly safe money market account.

 

[DaveC426913]

Everyone gambles many times a day. They just don't know it. Hopping into the car to drive to work, run errands, or go out on the town is a gamble. You might be killed or injured by someone with failed brakes or by some fool on a cell phone. Every election of how to allocate one's retirement account is a gamble. It doesn't matter whether its the riskiest REIT that can be found or a supposedly safe money market account.

I see a qualitative distinction between
- performing activities that one need to do accomplish things in one's daily life, knowing those activities carry a risk of failure, and
- taking a risk purely for the thrill of the possible win.

Strictly using the term gambling, I would apply it to the latter but not the former.

 

[PAllen]

I see a qualitative distinction between
- performing activities that one need to do accomplish things in one's daily life, knowing those activities carry a risk of failure, and
- taking a risk purely for the thrill of the possible win.

Strictly using the term gambling, I would apply it to the latter but not the former.

Where would you put investing in a mutual fund? Starting your own business? Objectively, the latter has a relatively low probability of success, but a favorable expectation (or so you judge).

 

[nonequilibrium]

I have an objection to the reasoning that the expected gain is infinite.

Rather, the expectation value E(X) is infinite. However, one must ask "how does E(X) acquire its usual meaning of the expected gain?". The answer is: "due to some limit theorem". These limit theorems (e.g. law of large numbers) requires that E(X) is finite. Consequently, the math doesn't seem to tell us anything about the expected gain.

 

[moejoe15]

I'd bet 4$. You have a 50/50 chance of losing 2$ on the first toss. If you don't lose you have a 50/50 chance of winning 4$ (8-4=4) on the next. You also will keep doubling up for every consecutive tails you throw after the first one. If you throw 3 tails in a row you have 16$ and a 50/50 chance to double it on the next toss, as well as after every additional tails thrown.