## Triple or Nothing

A forum for good logic/math puzzles.

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Actaeus
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### Triple or Nothing

Here's the premise:
There is a carnival game (or casino, doesn't matter.) The player starts by putting \$1 on the table. The dealer flips a coin.
HEADS: Dealer pays \$2 to the table
TAILS: Dealer takes the \$1 and game ends.
Player now chooses whether to play another round. If he doesn't, Player takes all money on the table and walks away. If he does, the coin is flipped again:
HEADS: Dealer triples the pot again
TAILS: Dealer takes all money, game ends.

The expected winnings from each flip is whatever is on the table. Therefore, it is always in the player's best interest to keep playing. However, if the player always chooses to play another round, he *will* eventually lose the original money and walk away with a loss of \$1.
What gives?

Twist: Player is allowed to play multiple games. If Player always plays one round and then ends the game, then repeats, he will win on average \$1 per game. If he plays two rounds each time, he will average \$2. Three rounds, \$3.25. If the player plays 100 rounds each time, even though he only has a 1/1.3e30 chance, he will win \$5.15e47 each time he wins. average: \$4.07e17. If the player always plays an "infinite" number of games (until he loses), he will win \$0 each time.
What gives? Does this involve transfinite numbers? If the player can play an infinite number of games, his winnings/game will approximate the numbers I gave, but will he get an infinite number of heads "once" in an infinite number of games, and win \$infinity? The expected winnings/game increases exponentially, so.....I don't even know where I was going with that.

Summary: What The Fork?
---
EDIT: should this be in Logic Games?

Macbi
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### Re: Triple or Nothing

Spoiler:
When you get Tails, you lose all the money, but you also lose the right to continue playing. This right is worth the right amount to balence the game out.
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Actaeus
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### Re: Triple or Nothing

Spoiler:
Of course! I forgot your money on the table affected future rounds, so expected winnings won't cover it. I believe expected winnings only works if it's out of your pocket.

Cauchy
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### Re: Triple or Nothing

If the player has played n games, he has a (1/2)^n chance of earning 3^n - 1 dollars, and a 1 - (1/2)^n chance of losing \$1, for an expected earning of (3/2)^n - 1. Clearly, as n increases, his expected winnings increase. As is the case with many things, it doesn't go quite as smoothly in the limit. I don't think this has anything to do with an "extra penalty" for getting 0. After all, we could easily see the player still playing even after he loses a flip: the money on the table is now \$0, and if he wins it triples and if it loses it goes to \$0, making the game no different than it was previously. Rather, things simply break down in the limit. It happens a lot in mathematics.
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
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Yakk
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### Re: Triple or Nothing

You cannot play an infinite number of rounds.

Let G(N) be the game strategy of playing N rounds then stopping. Then E(G(N)) (average or expected value) goes to infinity as N goes to infinity.

However, the limits does not cross the E and G function. lim(N->inf) of E(G(N)) != E(G( lim(N->inf)(N) ) )

Partially this is because G(infinity) is a special case. A simple way to extend G(N) to cover infinity is G(infinity) is "play until you lose, no matter what".

Once you turn the problem into symbols and poke at it, there is nothing really strange going on.

...

The second thing to realize is that money isn't linear in value. The E() (expected value) function is actually a poor measure of the return from a game.

Money isn't utility. Suppose you had 10^15 US dollars already -- having 10^18 US dollars wouldn't actually increase your wealth by all that much, because you have already managed to own the entire US economy, and going from 10^15 to 10^18 just reduces what percent everyone else has from a fraction of a percent to a smaller fraction.

Similar effects happen at smaller scales. Your first million dollars is more useful than your second.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

dan
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### Re: Triple or Nothing

You can get rid of the probability stuff and still have essentially the same problem. Say there is a game where, in each round, I give you \$1, and then you choose whether or not to continue. If you continue, the same thing happens again, and if you stop, I take all the money back and the game ends. This is obviously a pretty pointless game. Although it might seem at each stage like you get more money if you keep playing, if you stand back a bit you can see that this isn't actually true at all.

Waterhouse
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### Re: Triple or Nothing

What gives? The question implies the following: how can it be that the gambler, apparently paradoxically, has a chance to win infinite money with a very low risk? The answer is simply that the imagined setup is contrary to the way that the "house" operates in reality.

In any game of chance that is run for profit, the odds are biased toward the house. In this manner the longer a gambler plays, the more money the house wins. In this puzzle, the house is offering three to one payout on a game with a one in two chance of winning, which is clearly biased toward the player.

Another way to see it is that the player could, instead of remaining in the game after winning one round, pocket his winnings, and then immediately start up a new game. In this way, he will be expected to win two dollars half the time and to lose only one dollar the other half of the time.

Such payouts for those odds are never offered in reality.

Actaeus
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### Re: Triple or Nothing

This was not a realistic question....My question was, why is the "best" strategy in terms of payout one that leads to no payout at all? The answer was what Yakk said:
Yakk wrote:You cannot play an infinite number of rounds.

Let G(N) be the game strategy of playing N rounds then stopping. Then E(G(N)) (average or expected value) goes to infinity as N goes to infinity.

However, the limits does not cross the E and G function. lim(N->inf) of E(G(N)) != E(G( lim(N->inf)(N) ) )

Partially this is because G(infinity) is a special case. A simple way to extend G(N) to cover infinity is G(infinity) is "play until you lose, no matter what".

Once you turn the problem into symbols and poke at it, there is nothing really strange going on.

...

The second thing to realize is that money isn't linear in value. The E() (expected value) function is actually a poor measure of the return from a game.

Money isn't utility. Suppose you had 10^15 US dollars already -- having 10^18 US dollars wouldn't actually increase your wealth by all that much, because you have already managed to own the entire US economy, and going from 10^15 to 10^18 just reduces what percent everyone else has from a fraction of a percent to a smaller fraction.

Similar effects happen at smaller scales. Your first million dollars is more useful than your second.

Essentially, the function for expected payout tends to infinity, but the limit is a special case. Or something along those lines.

Yakk
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### Re: Triple or Nothing

Really, it is just that limits to infinity don't always cross functions. Lim(x->infinity)f(g(x)) does not always equal f(lim(x->infinity)g(x)).

The cases where that works are special, rare cases.

Heck, the lim(x->y)f(x) = f(y) is a special case called a continuous function even when y isn't infinity. For it to be true when y is infinity, your function could be called "continuous at infinity", which is not implied by standard continuity.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Cogito
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### Re: Triple or Nothing

The expected winnings in the original game is zero(the chance of winning) times infinity (the payout).

This need not equal zero at all. And if you consider it as a limiting case, then it should be clear the payout isn't zero. It's infinity. So I wouldn't disagree with the original strategy, personally.

Also, Yakk, the non-linear relation between money and utility isn't particular relevant. You can just replace units of money with units of utility.

More generally, it's always dangerous to just state an infinity as opposed to taking it as a limit of finite processes.

Token
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### Re: Triple or Nothing

Cogito wrote:This need not equal zero at all. And if you consider it as a limiting case, then it should be clear the payout isn't zero. It's infinity. So I wouldn't disagree with the original strategy, personally.

The limiting case, I believe, remains undefined. There are two separate limits (one for the probability, one for the payout). Two such limits may only be merged into one when they are both exist and are finite, which is not the case here.
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Cogito
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### Re: Triple or Nothing

In general, if you multiply two limits that equal zero and infinity, then the final result can still be anything.

In the specific case you can merge the two formulae prior to taking the limit and find yourself a well-defined answer.

Expected winnings = 3^n/2^n -1=(3/2)^n-1, as Cauchy pointed out. There's no reason why it wouldn't hold in the limit.

Token
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### Re: Triple or Nothing

Cogito wrote:In general, if you multiply two limits that equal zero and infinity, then the final result can still be anything.

In the specific case you can merge the two formulae prior to taking the limit and find yourself a well-defined answer.

Expected winnings = 3^n/2^n -1=(3/2)^n-1, as Cauchy pointed out. There's no reason why it wouldn't hold in the limit.

Well, there's two reasons you are wrong.

a) No matter how much you insist, you can't combine two limits if one (or both) of them is infinite or undefined. This is exactly what you are doing when you take the limit of (3/2)n-1 as n tends to infinity. Yakk provides an excellent reason for why this doesn't work in his above post.

b) Yes, in general zero times infinity can give you anything - this is what makes it an indeterminate form. This doesn't mean, however, that you can take it to be something that fits your intuition about the problem. In fact, I have it on good authority that when dealing with probability / measure theory, zero times infinity is generally defined to be zero. I think this example illustrates the reason for this quite well.
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Yakk
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### Re: Triple or Nothing

Cogito wrote:Also, Yakk, the non-linear relation between money and utility isn't particular relevant. You can just replace units of money with units of utility.

Well, it does generate an interesting result: at which point should one make a decision to stop?

If you are getting an exponential amount of utility on each stage ... gosh. I am rather not sure that is possible!

If you define Utility to be that which should guild your actions in a the "with probability P, you gain U2, and with probability (1-P) you lose U1. Then you should take the event if P*U2 > (1-P)*U1" case, quite possibly you can demonstrate that total Utility is bounded. In which case, unbounded exponential utility becomes impossible.

It might be a better statement to make that "utility is not probability linear"? Ie, if you have probability P to gain utility U, the utility of that option does not have to be P*U?

Utility, as defined by economics, is interesting in that any strictly monotonic transformation of utility satisfies the properties of utility. Utility is actually just an ordering of preference, it is not a scalar or vector value. Economics uses scalar values to make it easier to understand what the hell is going on: which leads to a number of strange results when you take those scalar values at face value...
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

rhino
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### Re: Triple or Nothing

Yakk: I agree with your suggestion that a bounded utility function seems intuitively correct.

However, utility is not just an order of preference. Suppose we have two agents, say Yakk and Token, whose utility functions are as follows:
Yakk: U(x) = x
Token: U(x) = 1 - 1/(x+1)

Where x is an amount of wealth.
Then Yakk and Token have identical orders of preferences, but will act very differently.

For example: offer them both the chance to toss a fair coin for the price of \$499
If it comes up heads you win \$4999, if tails you win nothing.
Suppose both of them currently have \$499, so Yakk's current utility is 499 and Token's is 0.998.

If Token wins his utility goes up to 0.9998, but if he loses it drops to zero.
If Yakk wins his utility goes up to 4999, but if he loses it drops to zero.

So the expected gain in utility by playing is:
Token: -0.4991
Yakk: +1751

So Yakk should play and Token shouldn't. The (relative) scalar value of utility is important because it measures the strength of a preference for an outcome.

Anyway that's by the by, I agree with both Token and Yakk that you can't just take the value of the "limit strategy" to be +infinity.
For reasons already explained, the strategy of "play forever until you lose" yields an expected gain of -\$1.

There is no real paradox here- this is just a game with no best strategy.

 changed to make the numbers work!

Yakk
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### Re: Triple or Nothing

That would work if people behaved that way. They don't.

Ie, if you have probability P to gain utility U, the utility of that option does not have to be P*U.

You can see this in many situations, including lottaries. The expected return from a lottery ticket is almost always negative. If you assume linear probability of utility, this implies that large sums have super-linear utility compared to small sums, which doesn't line up with human behavior that much...

There is firm theoretical ground for an order-based utility: the approximation of "if A is preferred by B, and B is preferred to C, A is preferred to C" is pretty damn accurate. This requires ordered utility, which can be approximated as a number.

Remember, utility is descriptive not prescriptive.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

rhino
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### Re: Triple or Nothing

Well, a utility that is purely a function of wealth is designed to model rational investors, not emotional individuals.
I expect that the average person would assign greater value to winning \$1000 in a game of chance than to saving an equivalent sum over the course of a year, even though the two are financially equivalent.

In making a utility-based model, we hope that there is a utility function such that "trying to maximise expected utility" lines up well with an investor's actual choices. I know that there are many situations where that is not a good approximation, generally because people act irrationally or assign value to more complicated things than just money. However, to say "what we actually need to consider is the set of all choices that our investor will make in any situation" is missing the point of trying to make a model in the first place. We want something that mathematically tractable and a reasonably good fit, not the actual truth.

If you just mean "investor X prefers \$100 to \$50 and prefers \$50 to \$0" then it is surely obvious that we gain nothing useful from this observation. I discussed this already in my previous post.

Anyway I don't think our disagreement is that fearsome - I don't actually believe everyone makes decisions based on some unconscious utility function.

GreedyAlgorithm
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### Re: Triple or Nothing

Yakk wrote:That would work if people behaved that way. They don't.

Ie, if you have probability P to gain utility U, the utility of that option does not have to be P*U.

You can see this in many situations, including lottaries. The expected return from a lottery ticket is almost always negative. If you assume linear probability of utility, this implies that large sums have super-linear utility compared to small sums, which doesn't line up with human behavior that much...

There is firm theoretical ground for an order-based utility: the approximation of "if A is preferred by B, and B is preferred to C, A is preferred to C" is pretty damn accurate. This requires ordered utility, which can be approximated as a number.

Remember, utility is descriptive not prescriptive.

Er. Either there are several things wrong here or there is a use of the term "utility" to which I'm not accustomed.

Utility is a description of how much an entity values various world states. Given an actual utility function, it's tautological that the entity's best move is to maximize its expected utility. If by taking a certain action you will with probability P gain utility U and with probability 1-P gain utility V, then the expected utility of taking that action is P*U + (1-P)*V.

Wait... are you operating under the assumption that humans come anywhere even remotely close to maximizing, or even trying to maximize, their expected utility and then inferring from that assumption what "utility" must mean? 'cause that's severely missing the point.
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Yakk
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### Re: Triple or Nothing

GreedyAlgorithm wrote:Er. Either there are several things wrong here or there is a use of the term "utility" to which I'm not accustomed.

Utility is a description of how much an entity values various world states. Given an actual utility function, it's tautological that the entity's best move is to maximize its expected utility.

The entities best move is to maximize the utility of that move. If the result of the move is certain, then the utility of the move can be expressed as the utility of the result -- they are interchangeable.

Remember, I'm trying to describe human behavior, not perscribe it.

If by taking a certain action you will with probability P gain utility U and with probability 1-P gain utility V, then the expected utility of taking that action is P*U + (1-P)*V.

That does not describe human behavior.

Wait... are you operating under the assumption that humans come anywhere even remotely close to maximizing, or even trying to maximize, their expected utility and then inferring from that assumption what "utility" must mean? 'cause that's severely missing the point.

Utility, in economics, is an attempt to describe preference among choices by actors. One derives the utility of a set of choices by how humans choose between them.

The not-quite-true claim about utility is that it is an ordering -- ie, if a<b and b<c, then a<c. This holds in most every case tested -- note that there are occasional exceptions!

The claim that utility is a scalar that falls linearly through probabilities is another claim about utility. It, I contend, is very rarely true. Human beings do not pass utilities linearly through probabilities.

So a claim that utility, in the economic sense, passes linearly through probabilities is a false claim that is not talking about human preferences. It is instead talking about some term that has little relation to the economic behavior of human beings.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Cogito
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### Re: Triple or Nothing

a) No matter how much you insist, you can't combine two limits if one (or both) of them is infinite or undefined. This is exactly what you are doing when you take the limit of (3/2)n-1 as n tends to infinity. Yakk provides an excellent reason for why this doesn't work in his above post.

But you have a perfectly good function that describes the expected winnings! I'm not combining indeterminate limits, I'm just combining functions. Certainly (lim x->inf x)*(lim x-> inf 1/x) is zero times infinity. Yet it is also clearly 1.

You can turn any limit into a combination of undefined limits. Yes, if you try to compute the limit of the expected winnings by computing the limits of the payout and the probability, you'll run into indeterminate form. That doesn't mean you can't try a different route.

b) Yes, in general zero times infinity can give you anything - this is what makes it an indeterminate form. This doesn't mean, however, that you can take it to be something that fits your intuition about the problem. In fact, I have it on good authority that when dealing with probability / measure theory, zero times infinity is generally defined to be zero. I think this example illustrates the reason for this quite well.

My intuition did not come into this. I just multiplied the payout with the probability before taking the limit so as to avoid this whole zero times infinity business.

If this is treated differently in probability theory, then I stand corrected, of course. But I'm having trouble picturing it. Give me an urn with an infinite amount of marbles. I reach in and grab a marble. What are the odds of my grabbing a specific marble? 0. How many marbles do you expect me to take out of the urn? Zero times infinity. I think we can agree that this is not zero here.

Yakk
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### Re: Triple or Nothing

Cogito wrote:
a) No matter how much you insist, you can't combine two limits if one (or both) of them is infinite or undefined. This is exactly what you are doing when you take the limit of (3/2)n-1 as n tends to infinity. Yakk provides an excellent reason for why this doesn't work in his above post.

But you have a perfectly good function that describes the expected winnings! I'm not combining indeterminate limits, I'm just combining functions. Certainly (lim x->inf x)*(lim x-> inf 1/x) is zero times infinity. Yet it is also clearly 1.

No, (lim x->inf x) * (lim x->inf 1/x) is not 1.

The lim x->inf of (x * 1/x) is 1, however this statement is not the same as the previous one.

There are situations when you can take (lim(x->a)f(x)) * (lim(x->a)g(x)) and say it equals lim(x->a)f(x)*g(x), but this is not true in general. It is true in some specific cases.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Token
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### Re: Triple or Nothing

Cogito wrote:You can turn any limit into a combination of undefined limits.

I'm afraid you can't. Just as you can't combine limits however you want, you can't just split them up arbitrarily either.

I challenge you to derive (lim(x->a)f(x)) * (lim(x->a)g(x)) = lim(x->a)f(x)*g(x) from first principles when the left hand side is zero * infinity. Of course, you can't, because the left hand side doesn't make sense. And if you can't prove that the equation holds, you sure as hell can't ever use it.

Cogito wrote:My intuition did not come into this. I just multiplied the payout with the probability before taking the limit so as to avoid this whole zero times infinity business.

YOU CAN'T DO THAT.

Just because it's a convenient short cut, doesn't make it a valid technique. Especially when it gives you the wrong answer.

Cogito wrote:If this is treated differently in probability theory, then I stand corrected, of course. But I'm having trouble picturing it. Give me an urn with an infinite amount of marbles. I reach in and grab a marble. What are the odds of my grabbing a specific marble? 0. How many marbles do you expect me to take out of the urn? Zero times infinity. I think we can agree that this is not zero here.

Ah, well, the problem here is not that 0*infinity can equal 1, but that you are trying to create an infinite discrete uniform distribution. If you don't attempt that, the problem goes away.
Last edited by Token on Thu Feb 07, 2008 11:47 pm UTC, edited 1 time in total.
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GreedyAlgorithm
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### Re: Triple or Nothing

Yakk wrote:That does not describe human behavior.

GreedyAlgorithm wrote:Wait... are you operating under the assumption that humans come anywhere even remotely close to maximizing, or even trying to maximize, their expected utility and then inferring from that assumption what "utility" must mean? 'cause that's severely missing the point.

Utility, in economics, is an attempt to describe preference among choices by actors. One derives the utility of a set of choices by how humans choose between them.

The not-quite-true claim about utility is that it is an ordering -- ie, if a<b and b<c, then a<c. This holds in most every case tested -- note that there are occasional exceptions!

The claim that utility is a scalar that falls linearly through probabilities is another claim about utility. It, I contend, is very rarely true. Human beings do not pass utilities linearly through probabilities.

So a claim that utility, in the economic sense, passes linearly through probabilities is a false claim that is not talking about human preferences. It is instead talking about some term that has little relation to the economic behavior of human beings.

Ah, I see. So the way you (meaning economists?) use the term "utility" is to assume that a human has some kind of consistent behavior from which we can infer how much the human likes or dislikes different world states. Or no, not even that. It's to say "this human chooses his actions in a fairly consistent way, and if I assume the human is often trying to maximize something, I can say something about the ordering of world states under his maximization operation - also this generalizes since the result is similar across many humans, and I will call this ordering utility".

Is that about right?

FYI, in the way I use "utility", all of those "not-quite-true" claims are very simple and very obviously true. Here's how I use it, and how I assumed everyone used it: Utility is a measure of the preference a human gives to one world state (as compared to another). Note that this makes no assumptions at all about whether or not humans try to achieve these preferences, which I believe is where the economists' definition fails at describing anything useful. Then again, I'm the one who said that your definition makes that assumption, and I could be dead wrong on that.
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Yakk
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### Re: Triple or Nothing

Note that the "expected" function is just a linear average over probability. Quite often doing a linear average over probability generates gibberish -- it is a particular mathematical operation, whose name happens to result that you get "expected utility", which seems like something someone should maximize.

It is the linear mean utility from a choice.

Ah, I see. So the way you (meaning economists?) use the term "utility" is to assume that a human has some kind of consistent behavior from which we can infer how much the human likes or dislikes different world states. Or no, not even that. It's to say "this human chooses his actions in a fairly consistent way, and if I assume the human is often trying to maximize something, I can say something about the ordering of world states under his maximization operation - also this generalizes since the result is similar across many humans, and I will call this ordering utility".

It is a claim about the ordering of choice. And yes, they call that ordering utility.

As it happens, if your ordering is strong, you can assign numbers (from the reals) to each utility and use that to order them.

This makes it easier to grasp. Note however that at this point any monotonic transformation of utilities results in the same ordering.

In order to have your utility obey linear probability combination, linear summation, or the like, you then have to prove or assume additional things about your utility hypothesis.

The goal of economics is to describe, not dictate, human behavior. It is noted that strong ordering of utility doesn't always work.

Claiming that any one person's preferences of world state can be assigned a number, and then the preference for a distribution of results can be calculated via a linear probability distribution, is a huge massive set of assumptions. What is worse is most of it is non-observational, and hence relatively non-falsifiable -- which sort of makes it less science.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Silas
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### Re: Triple or Nothing

Maybe we need to review what limits are. http://en.wikipedia.org/wiki/Limit_%28mathematics%29

Limits (of functions on R) only exist where functions converge to a Real number. The limit as N approaches infinity of N is nonsense. It doesn't converge. The same is true of the limit as N approaches infinity (this is a misleading way of describing the concept, anyway) of (3/2)^N.

You can set up a formula for evaluating the expected value of the pot after N rounds, but you can only evaluate it for whole numbered N's. Trying to evaluate a function f:W->R at "the limit as N approaches infinity of N" won't work because that limit isn't a whole number.

There's a related question where, if the player starts with a certain finite bank, what strategy of bet-making he should use to maximize the rate by which him bank increases, while holding the probability of ever going bankrupt below a certain level (yes, if you play this game repeatedly, you have a positive probability of never going broke- it's counterintuitive but true). It's not trivial to set up, but if you know what you're about, it's not hard.
Last edited by Silas on Sat Feb 23, 2008 8:44 am UTC, edited 1 time in total.
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### Re: Triple or Nothing

Silas wrote:Maybe we need to review what limits are. http://en.wikipedia.org/wiki/Limit_%28mathematics%29

Limits (of functions on R) only exist where functions converge to a Real number. The limit as N approaches infinity of N is nonsense. It doesn't converge. The same is true of the limit as N approaches infinity (this is a misleading way of describing the concept, anyway) of (3/2)^N.

You can certainly define infinite limits in a rigorous way, just as you can define limits at infinity, and the limit as N approaches positive infinity of (3/2)N is certainly positive infinity, under that definition. This is not the problem. The problem is that there's no reason why, in general, the outcome of a game which is in some sense the "limit" of a sequence of games should be the same as the limit of the outcomes of the finite games.
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### Re: Triple or Nothing

skeptical scientist wrote:This is not the problem. The problem is that there's no reason why, in general, the outcome of a game which is in some sense the "limit" of a sequence of games should be the same as the limit of the outcomes of the finite games.

I'm sorry to be rude. Maybe it's just late, but I cannot discern what, if anything, this sentence means. Can you rephrase it?
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### Re: Triple or Nothing

Silas wrote:
skeptical scientist wrote:This is not the problem. The problem is that there's no reason why, in general, the outcome of a game which is in some sense the "limit" of a sequence of games should be the same as the limit of the outcomes of the finite games.

I'm sorry to be rude. Maybe it's just late, but I cannot discern what, if anything, this sentence means. Can you rephrase it?

Maybe it will be more clear if I use an example. Call the triple-or-nothing game played for exactly n rounds the game Gn. We already know that the expected payoff for Gn is (3/2)n. Now, call the triple-or-nothing game, when played forever, the game G. In some sense, G is the limit as n -> +∞ of Gn, and we already know that +∞ is the limit as n -> +∞ of (3/2)n. However, there's no reason to expect that this means it is in any sense the expected winnings of the game G is infinite.
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### Re: Triple or Nothing

skeptical scientist wrote:
Silas wrote:
skeptical scientist wrote:This is not the problem. The problem is that there's no reason why, in general, the outcome of a game which is in some sense the "limit" of a sequence of games should be the same as the limit of the outcomes of the finite games.

I'm sorry to be rude. Maybe it's just late, but I cannot discern what, if anything, this sentence means. Can you rephrase it?

Maybe it will be more clear if I use an example. Call the triple-or-nothing game played for exactly n rounds the game Gn. We already know that the expected payoff for Gn is (3/2)n. Now, call the triple-or-nothing game, when played forever, the game G. In some sense, G is the limit as n -> +∞ of Gn, and we already know that +∞ is the limit as n -> +∞ of (3/2)n. However, there's no reason to expect that this means it is in any sense the expected winnings of the game G is infinite.

Not that I don't agree with you, but if you're using this argument to 'prove' things about the puzzle in this thread, you can't use the puzzle in this thread to 'prove' your argument

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### Re: Triple or Nothing

I don't have a copy of the book on hand right now, but this problem is discussed in quite some details in Rosenthal's "A First Look at Rigorous Probability Theory" (ISBN 978-9810243227), in case anyone is interested.
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### Re: Triple or Nothing

M.qrius wrote:
skeptical scientist wrote:
Silas wrote:
skeptical scientist wrote:This is not the problem. The problem is that there's no reason why, in general, the outcome of a game which is in some sense the "limit" of a sequence of games should be the same as the limit of the outcomes of the finite games.

I'm sorry to be rude. Maybe it's just late, but I cannot discern what, if anything, this sentence means. Can you rephrase it?

Maybe it will be more clear if I use an example. Call the triple-or-nothing game played for exactly n rounds the game Gn. We already know that the expected payoff for Gn is (3/2)n. Now, call the triple-or-nothing game, when played forever, the game G. In some sense, G is the limit as n -> +∞ of Gn, and we already know that +∞ is the limit as n -> +∞ of (3/2)n. However, there's no reason to expect that this means it is in any sense the expected winnings of the game G is infinite.

Not that I don't agree with you, but if you're using this argument to 'prove' things about the puzzle in this thread, you can't use the puzzle in this thread to 'prove' your argument

No, he was just attempting to use formal notation about the game in the thread as an example.

Let E(G) for a game G be the expected return from the game G.

Take Gn as defined above.
Then E(Gn) = (3/2)n.

In a sense, G is the limn->∞ of Gn.

However, there is no reason to believe that
limn->∞ E(Gn) = E( limn->∞ Gn ) (I dub thee: Formula X)
unless someone wants to provide a proof.

The belief that Formula X holds in general is the reason why someone might find the described pseudo-paradox to be a paradox.

In fact, as noted, you can demonstrate that Formula X does not hold in every case -- as has been done in this thread, with the triple or nothing game. Until someone actually provides a proof for Formula X, there is no contradiction, and no problem.
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Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

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### Re: Triple or Nothing

M.qrius wrote:Not that I don't agree with you, but if you're using this argument to 'prove' things about the puzzle in this thread, you can't use the puzzle in this thread to 'prove' your argument

I wasn't trying to use that example to prove what I was saying above. Someone told me they couldn't understand what I was saying, so I was using that example to try to explain what I was saying, and how it applied here.

The principle that "the outcome of a game which is in some sense the "limit" of a sequence of games is not necessarily the same as the limit of the outcomes of the finite games" doesn't require proof, because it is the absence of an assumption - I was just pointing out that other people seemed to be assuming that the outcome of the limiting game was the limit of the individual outcomes, and that this, if true, would require proof.
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### Re: Triple or Nothing

Maybe it will be more clear if I use an example. Call the triple-or-nothing game played for exactly n rounds the game Gn. We already know that the expected payoff for Gn is (3/2)n. Now, call the triple-or-nothing game, when played forever, the game G. In some sense, G is the limit as n -> +∞ of Gn, and we already know that +∞ is the limit as n -> +∞ of (3/2)n. However, there's no reason to expect that this means it is in any sense the expected winnings of the game G is infinite.

I can only point out that the expected winnings E(G) is, in fact, infinite, under standard definitions. P(you collect your winnings) is zero, but you collect infinite winnings. The winnings are, in fact, bigger than the likelihood of collecting them is small. The problem, as I see it, is that zero probability != impossibility.

Think about what infinite winnings would mean. It'd mean for any value k, it must be possible to increase n far enough that E(Gn) > \$k. And that's obviously true.

I feel kind of bad about coming here and crapping all over people who enjoy paradoxes with my "if you'd just use these definitions you've probably never seen before, it'd all make sense" (skeptical scientist, et al, this isn't you I'm talking about), but... someone is using subtly different terms than me on the internet!
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### Re: Triple or Nothing

Wow. That was, finally, an answer that makes sense to me.
Silas wrote:The problem, as I see it, is that zero probability != impossibility.

Or that 1/∞ != 0, I guess?
Silas wrote:The winnings are, in fact, bigger than the likelihood of collecting them is small.

That kinda makes sense.
I'll let people who actually know probability and calculus continue their discussion, while I hide from all the infinity.

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### Re: Triple or Nothing

Silas wrote:
Maybe it will be more clear if I use an example. Call the triple-or-nothing game played for exactly n rounds the game Gn. We already know that the expected payoff for Gn is (3/2)n. Now, call the triple-or-nothing game, when played forever, the game G. In some sense, G is the limit as n -> +∞ of Gn, and we already know that +∞ is the limit as n -> +∞ of (3/2)n. However, there's no reason to expect that this means it is in any sense the expected winnings of the game G is infinite.

I can only point out that the expected winnings E(G) is, in fact, infinite, under standard definitions. P(you collect your winnings) is zero, but you collect infinite winnings. The winnings are, in fact, bigger than the likelihood of collecting them is small. The problem, as I see it, is that zero probability != impossibility.

But it is actually impossible to collect your winnings if you keep betting till you lose.
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### Re: Triple or Nothing

Sort of - the game G isn't well defined because you haven't said what happens at stage ∞. Alternatively phrased, the game doesn't end if I win on each round, so what are my winnings in that scenario? It doesn't end up mattering, though, since the probability of that happening is zero: If my strategy is "keep playing till I lose", then in 100% of all games I'll end with a net loss of \$1.

Note the difference between "always" and "100% of the time" (the latter is referred to as "almost always"). It isn't impossible to collect your winnings by winning an infinite number of rounds, but that scenario has probability 0.
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### Re: Triple or Nothing

Owehn wrote:Note the difference between "always" and "100% of the time" (the latter is referred to as "almost always"). It isn't impossible to collect your winnings by winning an infinite number of rounds, but that scenario has probability 0.

I'd contest this. Yes, it is possible to win an infinite number of rounds (with probability zero), but I don't see how there's any possible situation that allows you to collect infinite winnings, since by collecting, you have necessarily chosen to stop playing after some finite number of rounds.
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### Re: Triple or Nothing

When we're speaking with formal terms, ∞ mean 'arbitrarily large,' not 'goes on forever.' We can define G differently, but if you do, you shouldn't expect ∞ to have the same properties. G as the case where you never stop satisfies the naive* understanding of ∞, but it doesn't jive with the way we're using ∞ in (3/2).

* as a math term meaning 'informal', not a denigration of your intellect
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### Re: Triple or Nothing

Token wrote:
Owehn wrote:Note the difference between "always" and "100% of the time" (the latter is referred to as "almost always"). It isn't impossible to collect your winnings by winning an infinite number of rounds, but that scenario has probability 0.

I'd contest this. Yes, it is possible to win an infinite number of rounds (with probability zero), but I don't see how there's any possible situation that allows you to collect infinite winnings, since by collecting, you have necessarily chosen to stop playing after some finite number of rounds.

This is what I was talking about with the game G. As the game is phrased, it's incomplete: what winnings do I get if I flip heads an infinite number of times? You could complete the game by saying that after an infinite sequence of heads, I walk away with the infinite amount of money on the table, or you could complete it by saying I win \$10, or nothing. It doesn't matter - the point is that this scenario occurs with probability zero, so your winnings in this case don't figure into the calculation of expected winnings.
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### Re: Triple or Nothing

Actaeus wrote:The expected winnings from each flip is whatever is on the table. Therefore, it is always in the player's best interest to keep playing.

A little late to the party but.. says who?
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