Recall from my last post (here)

I offer you the following bet:

I will flip a coin.

If HEADS you get 1 dollar and we end there.

If TAILS I flip again

If HEADS you get 2 dollars and we end there.

If TAILS I flip again

If HEADS you get 4 dollars and we end there.

If TAILS I flip again

The expected value is infinity.

Would you pay $1000 to play this game?

Everyone who responded said NO. Most gave reasons similar to what I have below.

This is called The St Petersburg Paradox. Not sure it's a paradox, but it is odd. The concrete question of *would you pay $1000 to play* might be a paradox since most people would say NO even though the expected value is infinity. See here for more background.

Shapley (see here) gives a good reason why you would not pay $1000 to play the game, and also how much you should pay to play the game (spoiler alert: not much). I will summarize his argument and then add to it.

1) Shapley's argument: Lets say the game goes for 40 rounds. Then you are owed 2^{40} dollars.

The amount of money in the world is, according to this article around 1.2 quadrillion dollars which is roughly 2^{40} dollars.

So the expected value calculation has to be capped at (say) 40 rounds. This means you expect to get 20 dollars! So pay 19 to play.

2) My angle which is very similar: at what point is more money not going to change your life at all? For me it is way less than 2^{40} dollars. Hence I would not pay 1000. Or even 20.

*Exercise*: If you think the game will go at most R rounds and you only wand D dollars, how much should you pay to play? You can also juggle more parameters - the bias of the coin, how much they pay out when you win.

Does Shapley's discussions *resolve* the paradox? It depends on what you consider paradoxical. If the paradox is that people would NOT pay 1000 even though the expected value is infinity, then Shapley resolves the paradox by contrasting the real world to the math world.

A more tempting variant:

ReplyDeleteflip #1) if HEADS you get 1 dollar and the games ends here, sorry

if TAIL I flip again ("goto flip #2")

flip #2) if HEADS you get 2 dollars and the games ends here OR we revert to flip #1 (you decide)

if TAIL I flip again

flip #3) if HEADS you get 4 dollars and the games ends here OR we revert to flip #2 (you decide)

if TAIL I flip again

flip #4) if HEADS you get 8 dollars and the games ends here OR we revert to flip #3 (you decide)

if TAIL I flip again

... and so on (the flips are like a "stack")

You're allowed to play the game only once in your life ... it will cost you $1000? Do you wanna play?

Ignoring the amount of time it takes to play, in this version you get to choose how much money you want so you should play no matter the cost. Considering the time it takes to play you may spend your whole life playing this boring game depending on the cost.

DeleteNo, "flip #1) if HEADS you get 1 dollar and the games ends here, sorry" means you don't just get to choose how much money you want.

Delete@=(v)( ')> No, if at flip #1 (and you can come back to it after some previous tosses) you get Head you receive $1 (no matter what you paid) and the game is over; and you can play only once in your life.

DeleteThe idea of this variant is: if you're lucky enough your luck will increase ... but is the game worth $1000? Or what is the maximum amount you'll pay to play it?

I think there's actually a somewhat deeper question here that is not entirely addressed by simply observing that there is a maximum payoff in the real world. Namely, under what circumstances does it make sense to use expectation maximization to guide our behavior?

ReplyDeleteWhen people calculate expected value, there is often a tacit assumption that the scenario will be repeated often enough that there will be a high probability of realizing the expected payoff. If this assumption is violated then it's not clear that maximizing expectation is a sensible thing to do. Would you rather that I give you a one-time opportunity to win $100 million by rolling a 100-sided die, or would you rather that I give you $900,000 outright? I think most people would opt for the guaranteed $900,000 and I don't think they're being irrational.

Determining the expected value of a game is an important first step, but it is not the only consideration. It is also important to ask yourself what your real goals are. The Wikipedia article on the Kelly criterion has a nice illustration of this point. Suppose you are given $25 and may place even-money bets on a coin that lands heads 60% of the time. You can bet any fraction of the amount of money you currently have, up to a maximum of $250. You have time to place about 300 bets. What do you do? Expectation maximization tells you to bet your entire bankroll every time, but this strategy virtually guarantees that you will go broke. But if you bet 20% of your bankroll each time then you are highly likely to come out far ahead at the end of the session.

If I knew that I'd get $10 for doing a questionnaire about a book, I would probably do it.

ReplyDeleteIf I knew that two lucky winners would get $500 each for doing the questionnaire and my "expected value" was $10, I likely would not do it.

I would do both of those. But usually when they offer a chance to win something, they don't tell you the expected value, and it is probably pretty low. But, I agree the distribution of the payoffs is relevant in general.

DeleteTo answer your question #1, I like Shapley's argument. It matches my intuition that games like these are a bad idea. One doesn't need to go to an "all the money in the world" extent, merely the extent of the finances of the "house" (the organization offering the game). This is lovely because it also can finesse the problem of dealing with infinitely long, but very low-valued (locally) tails whose sums diverge.

ReplyDeleteAnd I very much agree with anonymous on really hating giving windfalls to lucky winners. Drug development is sometimes like that: lots of people work along similar lines, and the one bloke who figures out the trick (or the group whose guess pans out) makes a fortune, and no one else does, even though they all did really good science. (Apparently Pfizer ran multiple Covid-19 vaccine development programs in parallel, so they finessed this problem, although calling brute force "finesse" is esthetically problematic.)

If I had infinite money to play infinitely long and an infinite lifetime and a time machine to return to when I started with my winnings then I would play.

ReplyDeleteMany such paradoxes are resolved simply by observing that one's utility does not scale linearly with the amount of money one has.

ReplyDeleteIn my opinion: More fundamental than the supply of money (which can easily be changed by switching currencies) or utility (which is subjective) is the requirement for potentially infinite number of repetitions (to realize the expectation). This directly assumes infinite time, which is not necessarily "supplied" by modern physics. I would be interested in others' opinions.

ReplyDelete