## A Multiple-Worlds Lottery

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- DataPacRat
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### A Multiple-Worlds Lottery

Elsewhere, I've been having a conversation about the Multiple Worlds Interpretation of quantum mechanics, and what it would take to maximize the number of future timelines in which a person continues to live. To have something resembling a concrete model to discuss, I've ended up putting together something that's pretty close to a mathematical word puzzle; and I'm wondering whether anyone in this forum might have any thoughts into it.

Assumption the first: A catastrophe is coming, within the next decade. There's a 10% chance it will happen in one year, 10% it will happen in two years, and so on.

Assumption the second: The catastrophe requires resources to survive, abstracted as money. There's a 90% chance that it requires $1,000 to survive, a 9% chance that it requires $10,000 to survive, a 0.9% chance that it requires $100,000 to survive, and so on.

Assumption the third: There is an extremely limited budget to work with: $10 per month.

Question: What real-world actions could be taken with that budget to maximize the odds of surviving any given level of catastrophe? (More precisely - how can the number of future timelines in which the protagonist survives be maximized?)

Possible solution 1: Buy lottery tickets. If a ticket provides one-in-a-million odds of providing $500,000, then that sets a lower bound for surviving $100k-level catastrophes of one-in-a-million.

Possible solution 2: Compound interest investment. At 1% to 4% interest, then by the 8th year, the protagonist reaches $1,000. If the catastrophe is of the $1k level, and it happens on the 8th, 9th, or 10th year, that will be enough to survive, giving around 30% odds of survival. (Which ignores the possibility of the bank/investment failing, which is theoretically the risk that the interest pays for.)

Some other possible solutions: Casinos, buying a bitcoin and playing SatoshiDice.

So... any insights?

Assumption the first: A catastrophe is coming, within the next decade. There's a 10% chance it will happen in one year, 10% it will happen in two years, and so on.

Assumption the second: The catastrophe requires resources to survive, abstracted as money. There's a 90% chance that it requires $1,000 to survive, a 9% chance that it requires $10,000 to survive, a 0.9% chance that it requires $100,000 to survive, and so on.

Assumption the third: There is an extremely limited budget to work with: $10 per month.

Question: What real-world actions could be taken with that budget to maximize the odds of surviving any given level of catastrophe? (More precisely - how can the number of future timelines in which the protagonist survives be maximized?)

Possible solution 1: Buy lottery tickets. If a ticket provides one-in-a-million odds of providing $500,000, then that sets a lower bound for surviving $100k-level catastrophes of one-in-a-million.

Possible solution 2: Compound interest investment. At 1% to 4% interest, then by the 8th year, the protagonist reaches $1,000. If the catastrophe is of the $1k level, and it happens on the 8th, 9th, or 10th year, that will be enough to survive, giving around 30% odds of survival. (Which ignores the possibility of the bank/investment failing, which is theoretically the risk that the interest pays for.)

Some other possible solutions: Casinos, buying a bitcoin and playing SatoshiDice.

So... any insights?

- Forest Goose
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### Re: A Multiple-Worlds Lottery

I don't think I fully understand what it is your discussing; I don't see the link to many worlds and QM, nor do I understand what exactly is being discussed in terms of resources (you say resources are abstracted as money, but seem to be discussing actual money). Are you, essentially, asking what type of investments will maximize returns over a time period with unknown end point?

Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.

- DataPacRat
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### Re: A Multiple-Worlds Lottery

Forest Goose wrote:I don't think I fully understand what it is your discussing; I don't see the link to many worlds and QM, nor do I understand what exactly is being discussed in terms of resources (you say resources are abstracted as money, but seem to be discussing actual money). Are you, essentially, asking what type of investments will maximize returns over a time period with unknown end point?

The original discussion was, generally, assuming MWI is true, what are the practical consequences? Putting aside thermodynamic miracles (eg, Boltzmann brains) as being so unlikely that it's unreasonable to ever expect experiencing one, that leaves the possibility that at any given moment, it's possible that /all/ your futures lead to you ending up dead. (For example, if a gamma-ray burst is already on its way to Earth.) Which brings up the question - is it possible to make a choice which would change the odds enough to allow someone to survive such a situation? (Which gets into the whole issue of whether free will can exist in MWI, but that's a whole other kettle of fish.) For example, if I were to be struck with a disease requiring a million dollars of drugs and surgery to cure, then there are very few future timelines in which I have that much money to spend on that - implying that the only future timelines in which I survive would be ones in which I win a lottery, or something similar. To whatever degree the atoms of my brain are arranged so that my mind is more likely to buy a lottery ticket, that's the portion of timelines I would survive. If I were to make arrangements, in advance, so that my mind becomes more likely to take the actions required for my survival, then the greater the number of future timelines I would survive in. (I'm taking it as somewhat granted that having 1-in-10 odds of surviving is to be preferred to having 1-in-100 odds of surviving.)

I'm trying to work out a reasonably simple model to toy with and explore possible behaviours, which covers the main features of this outlook - measurable odds of requiring lots of wealth, and extremely limited starting wealth to build from. My previous post is the best model I've come up with so far, but I'm quite willing to chuck it if anyone comes up with a better approach.

- Forest Goose
**Posts:**377**Joined:**Sat May 18, 2013 9:27 am UTC

### Re: A Multiple-Worlds Lottery

MWI has to do with quantum things, though. If you flip a coin, classically modeled, you have a 50% chance of H or T, but that doesn't entail anything about the outcomes of an actual physical coin flip* - especially if you are factoring in atoms in your brain. I guess I just don't see the quantum link to any of this**; it really seems like you're asking how to maximize your money, in a traditional sense, as quickly as possible with a scifi-ish premise. I'm not saying that in a dismissive sense, that's a perfectly reasonable and interesting question***, I just want to make sure I'm not missing anything.

*I'm not really comfortable treating classical probability models as if they were hypothetical ontological models of relative quantum probability.

**Whatever the laws of nature; if your model says that "having more money" increases survival, then you will always survive more with more money. Nothing you've suggested for increasing your money seems to rely on QM either.

***While true, I'm also not sure I see how this is really a mathematics question, it seems more a question about investing - the real meat of the question seems to hang on what investments actually exist, not any question about the behaviour of mathematical models of possible investments.

*I'm not really comfortable treating classical probability models as if they were hypothetical ontological models of relative quantum probability.

**Whatever the laws of nature; if your model says that "having more money" increases survival, then you will always survive more with more money. Nothing you've suggested for increasing your money seems to rely on QM either.

***While true, I'm also not sure I see how this is really a mathematics question, it seems more a question about investing - the real meat of the question seems to hang on what investments actually exist, not any question about the behaviour of mathematical models of possible investments.

Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.

- DataPacRat
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### Re: A Multiple-Worlds Lottery

Forest Goose wrote:MWI has to do with quantum things, though. If you flip a coin, classically modeled, you have a 50% chance of H or T, but that doesn't entail anything about the outcomes of an actual physical coin flip* - especially if you are factoring in atoms in your brain. I guess I just don't see the quantum link to any of this**; it really seems like you're asking how to maximize your money, in a traditional sense, as quickly as possible with a scifi-ish premise. I'm not saying that in a dismissive sense, that's a perfectly reasonable and interesting question***, I just want to make sure I'm not missing anything.

*I'm not really comfortable treating classical probability models as if they were hypothetical ontological models of relative quantum probability.

**Whatever the laws of nature; if your model says that "having more money" increases survival, then you will always survive more with more money. Nothing you've suggested for increasing your money seems to rely on QM either.

***While true, I'm also not sure I see how this is really a mathematics question, it seems more a question about investing - the real meat of the question seems to hang on what investments actually exist, not any question about the behaviour of mathematical models of possible investments.

I've started treating the generally-used version probability as a composite, made up of two things - mere mental uncertainty about already-existing physical facts, multiplied by the quantum uncertainty about events which could happen in a variety of ways depending on quantum happenstance. Eg, once a coin is flipped and in the air, then to a very large degree, its near-future behaviour is already determined due to ordinary Newtonian physics; but if I go to random.org and arrange for a quantum-based number generator, then even once I've pushed the button to generate a number, it's still uncertain what will happen right up to the moment I see the result. Or, put another way, with the coin flip, it's possible to assume that nearly all the timelines issuing from that point will have the same result; while with the quantum-based generator, half the future timelines will be heads and half will be tails.

If a proposal to gain wealth involves more classical uncertainty, such that it's quite possible that all of the future timelines involved end up in failure, that would be a poorer choice than a proposal that involves quantum uncertainty, in which even a small fraction of the future timelines involved end up with success.

As for limiting it to real-world investments, that's mainly to avoid the spherical cows problem of "Oh, all you need is something with a compound interest rate of 1,000%...".

### Re: A Multiple-Worlds Lottery

I think there's problems with trying to deal with probabilities in many worlds, because everything (that can happen) happens with certainty. I think there's also some concerns with possible infinities of worlds where trying to think about the proportion of worlds with feature x is gonna spit out 0% despite some worlds with feature x being possible.

Everything is quantum though, we just don't notice at larger scales typically because of the probabilities involved, so I'm not too worried about that objection to this.

Everything is quantum though, we just don't notice at larger scales typically because of the probabilities involved, so I'm not too worried about that objection to this.

- DataPacRat
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### Re: A Multiple-Worlds Lottery

Dopefish wrote:I think there's problems with trying to deal with probabilities in many worlds, because everything (that can happen) happens with certainty. I think there's also some concerns with possible infinities of worlds where trying to think about the proportion of worlds with feature x is gonna spit out 0% despite some worlds with feature x being possible.

Everything is quantum though, we just don't notice at larger scales typically because of the probabilities involved, so I'm not too worried about that objection to this.

While it's true that in a large enough sample of MWI worlds, any self-consistent arrangement of stuff can be found, it's still worthwhile to deal with significant numbers of worlds which can be divided up between those in which a particular arrangement exists, and in which it doesn't. (The relevant term appears to be 'measure', such as http://plato.stanford.edu/entries/qm-manyworlds/#3.6 .) And, where possible, to estimate how large the two groups are, relative to each other. (Which may or may not get into issues of infinities, but to at least try to keep our results sane, I'm hoping we can treat the numbers of worldlines as being a large-but-finite number, so that, for an analogy, there are half as many even numbers as there are whole numbers.)

- gmalivuk
- GNU Terry Pratchett
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### Re: A Multiple-Worlds Lottery

Whatever metaphysical significance you might place on quantum versus classical randomness, I'm not sure how any of it changes the details of your original question. You still just want to maximize your expected return, and we still can't say which method is a better way of doing that without more details on what investment methods are available and so on.

- DataPacRat
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### Re: A Multiple-Worlds Lottery

gmalivuk wrote:Whatever metaphysical significance you might place on quantum versus classical randomness, I'm not sure how any of it changes the details of your original question. You still just want to maximize your expected return, and we still can't say which method is a better way of doing that without more details on what investment methods are available and so on.

I'm not sure that simply 'maximizing expected return' is /quite/ the goal here.

Say I'm looking at a typical lottery ticket, deciding whether or not to buy it - costing $1, with a one-in-a-million chance of winning $500,000. By simple utility calculation, the net value of the ticket is roughly -$0.50, meaning that if the goal is merely to maximize expected return, it shouldn't be purchased. However, for a great many catastrophes, the $1 purchase price won't make much difference; and for a measurable number of catastrophes, the $500,000 prize will result in roughly one-in-a-million timelines where I survive, compared to zero-out-of-a-million timelines where I survive if I don't buy the ticket.

- Forest Goose
**Posts:**377**Joined:**Sat May 18, 2013 9:27 am UTC

### Re: A Multiple-Worlds Lottery

Those aren't really different, though; you're just restating that you have a 1 in 1,000,000 chance of winning $500,000. Too, you're only considering those timelines in which having more money lets you survive, so the only thing relevant to anything is: maximizing outcomes that make you more than $x - but since there are no hard limits, it's essentially maximizing expected returns. Finally, you aren't providing any details or constraints; this isn't really a mathematics (, or quantum physics) question, it's a question about what someone should do with their money.

Everything is quantum, but that doesn't mean that the classical models we use in those situations have any relevance to this. An actual coin flip is a quantum event, but the quantum nature has to do with all sorts of physical interactions and other stuff - any two coin flips are going to be extremely different quantum systems - the classical model of a coin flip is not the same thing nor a result of the same principles. That everything is quantum doesn't let us treat classical probabilities, metaphysically, like they were quantum (the physical odds, as a quantum system, that my lotto ticket wins are probably not the same as the classical probability - nor am I sure it even makes clear sense to talk about a generic lottery scenario as the systems making them up are not going to be anything alike as quantum mechanical systems).

Everything is quantum though, we just don't notice at larger scales typically because of the probabilities involved, so I'm not too worried about that objection to this.

Everything is quantum, but that doesn't mean that the classical models we use in those situations have any relevance to this. An actual coin flip is a quantum event, but the quantum nature has to do with all sorts of physical interactions and other stuff - any two coin flips are going to be extremely different quantum systems - the classical model of a coin flip is not the same thing nor a result of the same principles. That everything is quantum doesn't let us treat classical probabilities, metaphysically, like they were quantum (the physical odds, as a quantum system, that my lotto ticket wins are probably not the same as the classical probability - nor am I sure it even makes clear sense to talk about a generic lottery scenario as the systems making them up are not going to be anything alike as quantum mechanical systems).

Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.

- DataPacRat
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### Re: A Multiple-Worlds Lottery

Forest Goose wrote:Finally, you aren't providing any details or constraints; this isn't really a mathematics (, or quantum physics) question, it's a question about what someone should do with their money.

That's a fair enough point.

One part of the discussion I'm hoping to have involves comparing the various types of strategies, to get at least a rough idea of what sort of real-world investments would be appropriate. One approach could be to just spend the entire budget on whatever form of lottery promises the highest payout, regardless of the odds; another could be to use one strategy to acquire a certain amount of 'seed capital', until a threshold was reached, before changing to something with a different risk/reward profile (eg, betting on horse racing and then going into day-trading); another might be to focus entirely on the 90% of timelines requiring the lowest amount of wealth, and ignoring the timelines requiring anything more than that.

I don't know enough about the risk/reward profiles of nearly any such activities to even have a good idea what the minimum amount required for investment is, or maximum feasible output in any given period.

(I'm also not quite sure what the protocol on this board is if my initial post really was in the wrong sub-forum.)

### Re: A Multiple-Worlds Lottery

It seems to me if you're thinking in terms of MWI that you're gonna do all the options anyway, since it's not really reasonable to conclude that things are classical right up until just after you made whatever investments and only then do things start branching a la MWI.

Although to be fair there are a fairly large number of interpretations of MWI that tend to be called MWI, even if most of them are 'the' MWI, so some flavours might be more ok with that than others.

In any case, I don't think this is something you're gonna get a solid answer on. Increasing your potential wealth in some circumstances could be a bad thing anyway (perhaps theres more risk involved in things you get up to in your leisure time, or perhaps money leads to you being held hostage, or various other negative events that while not necessarily likely, are potentially comparable to winning the lottery in terms of odds).

And yeah, everything being quantum doesn't mean you can take classical models and treat them quantum-ly (well ok, sometimes this is how we get our quantum models, but that's physics handwaving rather than doing it right), it just means that if you really wanted to you could come up with a quantum model that'd give the right probabilistic answers if you were super careful and had the right initial conditions and such. That's a very big 'if' though, so in practice it's entirely impractical when classical approximations work almost always.

Although to be fair there are a fairly large number of interpretations of MWI that tend to be called MWI, even if most of them are 'the' MWI, so some flavours might be more ok with that than others.

In any case, I don't think this is something you're gonna get a solid answer on. Increasing your potential wealth in some circumstances could be a bad thing anyway (perhaps theres more risk involved in things you get up to in your leisure time, or perhaps money leads to you being held hostage, or various other negative events that while not necessarily likely, are potentially comparable to winning the lottery in terms of odds).

And yeah, everything being quantum doesn't mean you can take classical models and treat them quantum-ly (well ok, sometimes this is how we get our quantum models, but that's physics handwaving rather than doing it right), it just means that if you really wanted to you could come up with a quantum model that'd give the right probabilistic answers if you were super careful and had the right initial conditions and such. That's a very big 'if' though, so in practice it's entirely impractical when classical approximations work almost always.

- gmalivuk
- GNU Terry Pratchett
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### Re: A Multiple-Worlds Lottery

All any of that means is that your utility isn't linear. If it means surviving where you otherwise wouldn't, then winning half a million dollars might be worth more to you than 500,000 times the worth of one dollar to you. There's nothing especially quantum about that. If you need to come up with $1000 by tomorrow or you're homeless, and you currently just have a dollar to your name, then because the difference between one dollar and zero is *so* much less significant than the difference between having an apartment and being homeless, it may be a completely rational decision to buy a lottery ticket with a 0.05% chance of paying out $1000, even though the expected return is less than the cost of that "investment".DataPacRat wrote:gmalivuk wrote:Whatever metaphysical significance you might place on quantum versus classical randomness, I'm not sure how any of it changes the details of your original question. You still just want to maximize your expected return, and we still can't say which method is a better way of doing that without more details on what investment methods are available and so on.

I'm not sure that simply 'maximizing expected return' is /quite/ the goal here.

Say I'm looking at a typical lottery ticket, deciding whether or not to buy it - costing $1, with a one-in-a-million chance of winning $500,000. By simple utility calculation, the net value of the ticket is roughly -$0.50, meaning that if the goal is merely to maximize expected return, it shouldn't be purchased. However, for a great many catastrophes, the $1 purchase price won't make much difference; and for a measurable number of catastrophes, the $500,000 prize will result in roughly one-in-a-million timelines where I survive, compared to zero-out-of-a-million timelines where I survive if I don't buy the ticket.

- DataPacRat
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### Re: A Multiple-Worlds Lottery

gmalivuk wrote:All any of that means is that your utility isn't linear. If it means surviving where you otherwise wouldn't, then winning half a million dollars might be worth more to you than 500,000 times the worth of one dollar to you. There's nothing especially quantum about that. If you need to come up with $1000 by tomorrow or you're homeless, and you currently just have a dollar to your name, then because the difference between one dollar and zero is *so* much less significant than the difference between having an apartment and being homeless, it may be a completely rational decision to buy a lottery ticket with a 0.05% chance of paying out $1000, even though the expected return is less than the cost of that "investment".

Non-linear utility - now there's an approach I hadn't thought of, and which could be highly relevant to this whole idea. Might you happen to have any decent references or tutorials, preferably ones that are easily applicable to the topics discussed in this thread so far (and preferably ones that will fit in my nearly non-existent budget)?

- gmalivuk
- GNU Terry Pratchett
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### Re: A Multiple-Worlds Lottery

Not offhand, and in any case you'd still need to decide for yourself what the utility is. I first encountered the idea in the opposite direction, to explain why lotteries are bad but insurance is good. A rough function for that is u=log(m), so that for example every doubling of your money is worth the same to you even if the amount of money is different. But this is the opposite of what you want.

Once you have a utility function, though, then use that in any expected value calculations instead of money itself.

Once you have a utility function, though, then use that in any expected value calculations instead of money itself.

### Re: A Multiple-Worlds Lottery

Quantum mechanics doesn't add anything to this problem, so let's all just ignore that and focus on the math. Here is the problem boiled down:

1. There is a deadline coming in [1,2,3...10] years, with equal probability.

2. To win you must have a certain amount of money by the deadline. The amount required is $10^(3+n) with probability .9*(10^-n) for n >= 0.

3. You have an income of $10 per month ($120 per year).

You have two investment strategies:

a) Buy lottery tickets. Each ticket costs $1 and has a 1 in 1 million chance of winning $500,000.

b) Invest the money at 1% to 4% interest.

Goal: Maximize the chance of winning.

So first off, I calculate that in 10 years using strategy (b) with optimal returns, you make only $1471. So you will only win if the goal is $1000 (90% chance). You will have enough money for this goal after 8 years, so you have a 0.3 * 0.9 = 0.27 chance of winning.

Next is strategy (a). I'm going to upper bound this by assuming we get all $120*10 = $1200 up front and put it all into lottery tickets, and assuming that any winning ticket means we win the challenge. The probability of 1200 tickets winning is less than 1200/1,000,000 = 0.0012. So a pure (a) strategy loses significantly to (b).

We should be able to do better with a mixed strategy. Once we have $1000 with strategy (b), the rest of the money doesn't matter. So we can put this into lottery tickets to give us a small chance of winning if the goal is higher than $1000. Intuitively, I think we would maximize our chances if we spend all of our excess money over $1000 at the end of each year. This means we can buy 129 tickets at the end of year 8, 163 at the end of year 9, and 162 at the end of year 10 (the difference in years 9 and 10 is do to left over change). Of course, if one of the tickets does win, we should probably change our strategy for the next year.

I'm not going to try to calculate the exact expectation of winning with this strategy because there are too many details and it's only going to be slightly better than a pure (b) strategy, but I think that will be the best strategy.

In the problem he has described utility is a piecewise constant function of money, so I don't think you can just ignore money.

1. There is a deadline coming in [1,2,3...10] years, with equal probability.

2. To win you must have a certain amount of money by the deadline. The amount required is $10^(3+n) with probability .9*(10^-n) for n >= 0.

3. You have an income of $10 per month ($120 per year).

You have two investment strategies:

a) Buy lottery tickets. Each ticket costs $1 and has a 1 in 1 million chance of winning $500,000.

b) Invest the money at 1% to 4% interest.

Goal: Maximize the chance of winning.

So first off, I calculate that in 10 years using strategy (b) with optimal returns, you make only $1471. So you will only win if the goal is $1000 (90% chance). You will have enough money for this goal after 8 years, so you have a 0.3 * 0.9 = 0.27 chance of winning.

Next is strategy (a). I'm going to upper bound this by assuming we get all $120*10 = $1200 up front and put it all into lottery tickets, and assuming that any winning ticket means we win the challenge. The probability of 1200 tickets winning is less than 1200/1,000,000 = 0.0012. So a pure (a) strategy loses significantly to (b).

We should be able to do better with a mixed strategy. Once we have $1000 with strategy (b), the rest of the money doesn't matter. So we can put this into lottery tickets to give us a small chance of winning if the goal is higher than $1000. Intuitively, I think we would maximize our chances if we spend all of our excess money over $1000 at the end of each year. This means we can buy 129 tickets at the end of year 8, 163 at the end of year 9, and 162 at the end of year 10 (the difference in years 9 and 10 is do to left over change). Of course, if one of the tickets does win, we should probably change our strategy for the next year.

I'm not going to try to calculate the exact expectation of winning with this strategy because there are too many details and it's only going to be slightly better than a pure (b) strategy, but I think that will be the best strategy.

Once you have a utility function, though, then use that in any expected value calculations instead of money itself.

In the problem he has described utility is a piecewise constant function of money, so I don't think you can just ignore money.

- gmalivuk
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### Re: A Multiple-Worlds Lottery

I never said anything about ignoring money. The expected value is still a function of money, there's just the intermediate utility function in there as well. So instead of E[m], you have E[u(m)]. See, money's still right there.Derek wrote:In the problem he has described utility is a piecewise constant function of money, so I don't think you can just ignore money.Once you have a utility function, though, then use that in any expected value calculations instead of money itself.

- DataPacRat
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### Re: A Multiple-Worlds Lottery

Derek wrote:Quantum mechanics doesn't add anything to this problem, so let's all just ignore that and focus on the math. Here is the problem boiled down:

1. There is a deadline coming in [1,2,3...10] years, with equal probability.

2. To win you must have a certain amount of money by the deadline. The amount required is $10^(3+n) with probability .9*(10^-n) for n >= 0.

3. You have an income of $10 per month ($120 per year).

You have two investment strategies:

a) Buy lottery tickets. Each ticket costs $1 and has a 1 in 1 million chance of winning $500,000.

b) Invest the money at 1% to 4% interest.

Goal: Maximize the chance of winning.

So first off, I calculate that in 10 years using strategy (b) with optimal returns, you make only $1471. So you will only win if the goal is $1000 (90% chance). You will have enough money for this goal after 8 years, so you have a 0.3 * 0.9 = 0.27 chance of winning.

Next is strategy (a). I'm going to upper bound this by assuming we get all $120*10 = $1200 up front and put it all into lottery tickets, and assuming that any winning ticket means we win the challenge. The probability of 1200 tickets winning is less than 1200/1,000,000 = 0.0012. So a pure (a) strategy loses significantly to (b).

We should be able to do better with a mixed strategy. Once we have $1000 with strategy (b), the rest of the money doesn't matter. So we can put this into lottery tickets to give us a small chance of winning if the goal is higher than $1000. Intuitively, I think we would maximize our chances if we spend all of our excess money over $1000 at the end of each year. This means we can buy 129 tickets at the end of year 8, 163 at the end of year 9, and 162 at the end of year 10 (the difference in years 9 and 10 is do to left over change). Of course, if one of the tickets does win, we should probably change our strategy for the next year.

I'm not going to try to calculate the exact expectation of winning with this strategy because there are too many details and it's only going to be slightly better than a pure (b) strategy, but I think that will be the best strategy.

Thank you /very/ much for taking the time and effort to go through the analysis. The conclusions reached seem to apply even if the lottery only loses, for example, 2% per ticket (eg, SatoshiDice) or less (Blackjack), instead of the 50% of a typical lottery ticket.

I think the next step in my figuring will be to try twerking the simple model to more closely approximate reality, and try to figure out if the same conclusion still applies. For example, I could try making the amount of money required to win be more of a continuous function rather than set in discrete steps, and try to figure out what balance between investment and tickets provides the best returns.

Does anyone have any thoughts on possible ways to improve the model, and the consequences thereof?

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