## Another gambling game

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Xias
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### Another gambling game

If this has been proposed before, I'm not sure what it's called so I wouldn't know how to search for it.

Say you are one of 1,000,000 people offered the opportunity to play a game. Entry is \$100, and once everyone either pays or drops out, a pot of \$1,000,000 is divided among everyone who chose to pay. So, if everyone chooses to participate, they all get \$1 and lose \$99. If 10,000 play, they all get \$100 and break even. If you're the only person to play, you get the full million and win big.

Do you pay, or abstain?

What if you knew that all other players were perfect logicians (and that this was common knowledge, etc.)? What if you have no knowledge of the other players?

Is there some threshold cost of entry where you would play, but abstain if the cost was \$.01 more?

Yat
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### Re: Another gambling game

If all other players are perfect logicians, then
Spoiler:
they will all come to the same conclusion : "Either I play and everybody plays, and we all lose \$99, or I don't play, and nobody plays, and we are all even. I don't play."
Now, I am not a perfect logician, I am just human, so my reasonning can be different from the one above. I play, and get the million.
Now, about the threshold, I will play for up to \$1,000,000 (not included), because above \$1 this reasonning will be the same, and below \$1, the only difference is that everybody will play, but there will still be gain for everybody.

Yat
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### Re: Another gambling game

Spoiler:
mixed strategies: if the perfectly logical strategy is to play with a probability p, then there are 1,000,000p players to split the \$1,000,000, so the gain is 1/p-100.
As this happens with probability p, that makes p(1/p-100), that is 1-100p.
That means that if p is below 1% then this is a win. It also mean that you can get as close to \$1 as you want if you make p small enough, which does not make much sense.

Snark
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### Re: Another gambling game

Basicallly what Yat said:
Spoiler:
Without proof, I'd guess that you'd play with p = .01 (or the largest p possible that is less than .01 which is close enough as makes no difference).
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Barstro
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### Re: Another gambling game

I believe that a group of gamblers would look for an individual clue to determine what to do; some random event that has a 1% occurrence of happening and paying the \$100 if it does happen. This would allow for the everyone to statistically break even without passing up the opportunity to win.

Poor ability to explain aside, this is what happens in poker.

Spoiler:
If a player has been betting his hand a certain way, and it didn't pan out, he can either fold of bluff. Players don't want to have a pattern on whether or not they bluff, or else other good players will recognize it and will statistically call their bluffs more often. Instead, the player will decide what percentage the bluff is worth, and base it on the next card. [With a hand like this, I should fold 75% of the time. I see four cards and one of them is a heart. If the next card is a heart, I will bluff/raise, if it is not a heart, I fold.]

A politician would explain to those 1,000,000 people the inherent flaw as pointed out by Yat that it would be foolish for everyone to play. After scaring them all away, the politician would they buy the only ticket.

ThirdParty
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### Re: Another gambling game

First, use your perfect reasoning ability to deduce the true moral theory.

If the true moral theory entails that it's reasonable to act as you want other people exactly like you to act, even if you have no actual influence on their actions, then:
Spoiler:
Let P be the probability that each individual plays. The group's profit is 1000000*(1-(1-P)^1000000) - 100*P*1000000, and you're as likely to get a share of it as any other member of the group, so this is what you should aim to maximize.

Assuming I haven't messed up my math somewhere, this is maximized at about P=0.0000092. (Less than that and there's too much risk that nobody will win the million; more than that and your group wastes money on redundant tickets.) So you should randomize in such a way that your likelihood of playing is about one in a hundred thousand. (Flipping nine coins and rolling three six-sided dice and only playing if you get all "heads" and "six"es would be about right.)

If you end up playing and winning a hundred thousand dollars or so, you should then give it away to the morally-appropriate charity. Your net profit, if you're among one of the lucky one-in-a-hundred-thousand, is the satisfaction of having done the right thing.
On the other hand, if the true moral theory entails that it's reasonable to act in the way that benefits you most, even if harms others more than it benefits you, then:
Spoiler:
Let P be the probability that each individual plays. Each individual's expected profit is P*1000000/(P*1000000) - 100*P, which is positive at P<0.01 and negative at P>0.01.

So if you think you're among the 1% most risk-seeking members of the population of perfect logicians, you should play. If you think you're among the 99% most risk-averse members of the population of perfect logicians, you should not play. Either way, your expected net profit is zero.

Barstro
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### Re: Another gambling game

ThirdParty wrote:So if you think you're among the 1% most risk-seeking members of the population of perfect logicians.[/spoiler]

Does logic not transcend risk-taking?

Or do you mean risk to be;
low-risk = guaranteed 5% gain
high-risk = 50% chance of 50% loss and 50% chance of of 60% gain?

dudiobugtron
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### Re: Another gambling game

Spoiler:
The problem with the perfect logician part of the puzzle, as Yat pointed out, is that your best course of action depends on what you think the other people will do, and what you think the other people will do is the same as what you think you should do (because of common knowledge, etc). This ends up with a bit of an irresolvable situation. Obviously everyone will come to the same conclusion as you; which means no one should play. However, if you know no-one else is playing, then you should definitely play - since that an easy \$999,900. However, everyone else will come to the same conclusion, which means they actually will play after all. But then it's stupid to play, so you should not play. But then neither will anyone else. etc...

However, in real life:

Barstro wrote:
ThirdParty wrote:So if you think you're among the 1% most risk-seeking members of the population of perfect logicians.[/spoiler]

Does logic not transcend risk-taking?

Or do you mean risk to be;
low-risk = guaranteed 5% gain
high-risk = 50% chance of 50% loss and 50% chance of of 60% gain?

Since people are different, they have different expected values from a set amount of money.
For example, people may have different tax rates on their lottery winnings. Also, the 'opportunity cost' of losing or not gaining the money will be different for different people. (Eg: someone who can spend the money on a life-saving operation for themselves would put more value on it than someone who could only put it to best use by re-painting their house. Also, someone who needs \$100 for food and rent would suffer a greater loss upon losing the \$100 than someone who already had all their basic needs covered.) Some people may even just enjoy making 'bad' bets, or get benefit from merely participating in the betting process.

A perfect logician with access to all of this information about themselves would almost certainly make a different decision from another perfect logician with different background circumstances. (Of course, in real life, perfect logicians with access to all the information would of course already be multi-billionaires, or running the planet, etc..., depending on what they wanted to do; in any case, they probably wouldn't choose to spend their time participating in this lottery!)
Last edited by dudiobugtron on Wed Oct 24, 2012 1:06 am UTC, edited 1 time in total.

Vytron
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### Re: Another gambling game

Spoiler:
Personally, I'd think of it as in a greed-to-risk ratio. I.e., if you don't have much greed at all, you'd be fine by winning some money, so you'd pick up a ticket with probability 0.01 - ε, as above.

However, if you're greedy, some small win isn't worth the hassle, what if 9999 guys do that and they each win one cent, what was the point?

So an strategy that plays with less probability increases outcome. Perfects logicians wouldn't do it, because they no longer have that chance of winning a penny, and their chances of winning nothing doubles, but, normal people?

Suppose you only play if you double your investment, then, you set p to 0.005 - ε, this way, you are much more likely to not play, but if you do, you win about \$200 and 4 cents! (+\$100.014 of profit.)

However, this escalates quickly, the above had ~4999 people playing, and it can be argued that more greedy people had a better expected outcome, because their possible winnings is 10004 times higher than if they had a higher value of p (they win 100.04 instead of 0.01). Greedy people that don't play, either wouldn't have played anyway, or would have won 1 cent instead, so it's worth it.

How low can p be and keep doing this? For instance, if you set p to 0.001 - ε, then about 999 people will play, but they will win \$901. Compared to a previous group, their expected winnings are 9 times higher (they win 901 instead of 100.04), and their chances of not playing are only 5 times worse, so it should increase their expected value?

I'm going to stop here in case it's nonsense, but it seems people are better off just making sure that as few people as possible play, and hope they're one of them.
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ThirdParty
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### Re: Another gambling game

Barstro wrote:do you mean risk to be;
low-risk = guaranteed 5% gain
high-risk = 50% chance of 50% loss and 50% chance of of 60% gain?
Yes, that's right. See dudiobugtron's post for why some perfectly-rational people would choose the low-risk option while other perfectly-rational people would choose the high-risk option.

Xias
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### Re: Another gambling game

I dig the responses, guys. This was something I was playing with and the first spoiler in dudiobugtron's post is what I came to as well. Kind of a paradox.

Here's a thing though:

Spoiler:
If everyone plays with probability 1/x, the expected share (if you play) would be x.

I think that's rather interesting.

phlip
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### Re: Another gambling game

Even given superrationality, I disagree with the superrational solutions posted so far...
Spoiler:
Say every one of the million people chooses to play with probability p. Then we would, between them all, expect a total of 108p dollars be paid in entry fees, and a total of 106(1 - (1-p)1000000) paid out in winnings (as exactly a million dollars is paid out total except when every single one chooses not to play). Since all players are identical and EV is linear, that means that each player's EV is (1-(1-p)1000000) - 100p.

Doing some manipulations, and with help from W|A, this is maximised at p ~= 9.21×10-6, ie expecting about 9-10 people to participate out of the group. If you increase p from here, then the fact that you'd have to split your winnings with more people outweighs your increased chance of actually playing; if you decrease p from here, the reverse is true.

Obviously the best solution is for the players to collude, arbitrarily pick one player to pay up, and then split the profits, as that guarantees the smallest amount of money paid in to get the full payout. However, this requires superrationality and trustworthiness... as once a player has been arbitrarily picked, the symmetry that superrationality relies on breaks down, and it's now in that one player's best interest to renege on splitting the winnings.

However, without superrationality:
Spoiler:
The strictly-right answer is to try to guess what the other players are going to do... if you can't predict it absolutely, take the Bayesian route and assign probabilities according to your confidence that predictions are accurate based on the information you do have. Then you can run the maths to determine your payout probability curve, and then determine whether that payout is "worth it" according to your own utility curves and risk-averseness. In broad terms, this essentially boils down to the obvious "if everyone else is playing, you shouldn't; if every one else isn't, you should", with some vaguely-defined ideas about how to calculate the threshold in the grey area in the middle. Notably this means that if everyone is doing the superrational plan above (or if you can convince everyone to do the superrational plan) then you should break from the pack and go in (so the superrational plan isn't a rational equilibrium). It also means that a mixed strategy is rarely the right option, unless it's right near the edge, and your risk curves kick in... either the payout is worth it or it isn't.

The problem comes in the "everyone's a perfect logician, this is common knowledge" case, as everyone is doing the above calculations too, and the resulting system doesn't have a Nash equilibrium, so "what a perfect logician would do" is ill-defined. And so the logicians have no predictions for what the others will do, and the question breaks down as they have no way of predicting whether playing is a good idea or not.

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enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};
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ThirdParty
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### Re: Another gambling game

phlip: I agree completely with your solution, but I don't see how it differs from the one I posted above.

Yat
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### Re: Another gambling game

phlip wrote:Even given superrationality, I disagree with the superrational solutions posted so far...
Spoiler:
Say every one of the million people chooses to play with probability p. Then we would, between them all, expect a total of 108p dollars be paid in entry fees, and a total of 106(1 - (1-p)1000000) paid out in winnings (as exactly a million dollars is paid out total except when every single one chooses not to play). Since all players are identical and EV is linear, that means that each player's EV is (1-(1-p)1000000) - 100p.

I don't get it. how can there be something1000000 in an individual player's EV ? The case where nobody plays is simply included in the case where this individual player doesn't play. Either a player abstains and gets nothing, or he plays, and gets his share of the million. This (1-p)1000000 seems totally irrelevant to me. What am I missing ?
I would say the total paid out in winnings is 1000000 no matter what, because the only case where the million is not distributed is not relevant.
EDIT: Now I get it... I was going for the product of the probability of playing by the expected gain, which is stupid.

phlip
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### Re: Another gambling game

ThirdParty wrote:phlip: I agree completely with your solution, but I don't see how it differs from the one I posted above.

It differs in the fact that no I totally didn't miss your post, shut up.

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enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};
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mward
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### Re: Another gambling game

Two points which don't appear to have been made so far:

(1) If more than 1% of the people play, then playing is a net loss. But well over 1% of the population appear to be willing to bet when the expected return is negative (just look at the number of lottery players there are!). So, if the other players are "normal people", then
I would expect there to be well over 10,000 players and therefore would not play.

(2) If all the other players are perfect logicians, then the logical thing to do is to form a syndicate: pick one of us at random to play and everyone else abstain. A randomly-selected group of 10,000 of us will each contribute one penny towards the stake. We share out the winnings equally between us: earning a guaranteed total of either \$0.99 or \$1.00 each.

Is there some threshold cost of entry where you would play, but abstain if the cost was \$.01 more?

As the cost of entry gets lower, I would expect more and more people to play. So I would not expect the return to become positive until the cost of entry gets below \$1.00. So I would be willing to play for \$0.99 but not willing for \$1.00.

With the 999,999 fellow perfect logicians: we would be willing to play for any cost up to \$999,999.99 (earning a total of \$0.00000001 each: i.e. we all have a 1/1,000,000 chance of winning one penny). We would not be willing to play for \$1,000,000.

dudiobugtron
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### Re: Another gambling game

mward wrote:Two points which don't appear to have been made so far:

...

(2) If all the other players are perfect logicians, then the logical thing to do is to form a syndicate: pick one of us at random to play and everyone else abstain. A randomly-selected group of 10,000 of us will each contribute one penny towards the stake. We share out the winnings equally between us: earning a guaranteed total of either \$0.99 or \$1.00 each.

phlip wrote:
Spoiler:
Obviously the best solution is for the players to collude, arbitrarily pick one player to pay up, and then split the profits, as that guarantees the smallest amount of money paid in to get the full payout. However, this requires superrationality and trustworthiness... as once a player has been arbitrarily picked, the symmetry that superrationality relies on breaks down, and it's now in that one player's best interest to renege on splitting the winnings.

XTCamus
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### Re: Another gambling game

OK I must admit that the perfect logicians angle confuses me in general, and in particular in this case. But why not then, at least for the case of imperfect humans, determine to the nearest \$.01, the best answer empirically? Ask this question, gather the data, shut up and calculate, then play accordingly and win? (Now, I would not bother to actually do so just to solve a riddle, but to me it seems that a non-theoretical approach is the best way to solve this puzzle! Well, at least in theory.)