## Lottery

A forum for good logic/math puzzles.

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sq_ocelot
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Location: USA

### Lottery

I had a neat idea for a lottery:

For \$1, a person can buy a ticket (any number of tickets per day, any number of tickets for the duration of the lottery).

They can then put their ticket in one of five banks: A, B, C, D, or E.

At the start of each day, the number of tickets in each bank is posted.

After one week, the bank that has the fewest number of tickets in it wins, and the total money from the whole game is divided up among the people in that bank.

A winner, then is guaranteed to win at least 5 times what he put in the winning bank.

How do you think this would work? What would a good strategy be?
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zombiefeynman
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### Re: Lottery

It seems like this is a prisoner's dilemma-ish puzzle. What is beneficial to one is detrimental to a lot of other people, and if enough people do it it's detrimental to everyone. If you can get a bunch of friends to put 3 in [high bank] and 1 in [low bank] you should all make money. I think.

sq_ocelot
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### Re: Lottery

Well, then all you would do is preserve Bank X's status as the most full, what if the low bank you all invested in doesn't stay the lowest? Then you're all out. But I guess that's what a lottery's all about. Even if you win, you'd all get around a dollar more (assuming that there is not a crazy difference in the amount of tickets in each bank).

I'm curious how this would work. Should we make a poll and try it?
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Silknor
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### Re: Lottery

A simple strategy would be to invest the same amount in each bank. Excluding the possibility of a tie between all 5 banks (in which case you'd get your money back), this always produces a positive return. Of course in the real world, the total distributed would be some percentage of the total invested, not the entirety invested.
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The Big G in Charge
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### Re: Lottery

But you'd have to spend a lot of money that would be lost to potentially stack up other banks to make one bank lower than the others. I think that's a kind of defeating strategy.

But putting a ticket in each bank could be effective. You'd win no matter what as far as I can tell- unless everybody did the exact same thing, then it's pointless because you'd get the same money back.

floodslayer
Posts: 28
Joined: Sat Mar 21, 2009 12:17 am UTC

### Re: Lottery

Since this is a zero-sum game the expected result is always what you put in (I think). The game might be even more interesting if the odds were less than 1 (for example, if the organizer of the lottery keeps the contents of the smallest bin, or if ties result in nobody winning). Any thoughts on what the results might be in these cases? I know in the first case your odds are bounded below by 0.8. Not sure if these are acceptable odds for normal casino games or state lottos, but if not you could always increase the number of bins.

For a large lotto, I think the organizers would prefer to pay large sums to a small number of players, rather than small sums to many winners. This is because the second case is more difficult to administer/has a higher cost associated with distributing winnings.

quintopia
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### Re: Lottery

This is similar to a real problem I encountered today.

Suppose you have a hidden prize, and a group of people who will form teams to search for it. The fraction of the people who are on your team is the probability your team will find it, but the prize is always divided evenly among all the players on the team. I think that it's an interesting problem because of the fact that your expected winnings are the same no matter what size your team is, so deciding your team size is entirely based on other factors. What would you do?

6453893
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Location: Australia

### Re: Lottery

If you bought the same number of ticket in each lottery, you would be guaranteed to win C(Nt/Nx), where Nt is the total number of tickets, Nx is the number of tickets in the winning lottery, and C is the number of tickets you invested in each lottery. As far as I can see, Nt is always at least five times the size of Nx, and thus larger than 5C, which is your total spending. Of course, if everyone plays this strategy, it all comes crashing down.

EDIT: Of course, Silknor said all of that more succinctly and ages ago. I scrolled too fast and missed his post, along with the few after it. Sorry Silknor.

It is interesting though to factor in what percentage of the money would be kept by the lottery. If they kept the money of the smallest bank, say NE: (NA+NB+NC+ND)-4NE must be equal to or greater than NE to make a profit.
Last edited by 6453893 on Sun Apr 12, 2009 9:11 am UTC, edited 2 times in total.

GreedyAlgorithm
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### Re: Lottery

quintopia wrote:This is similar to a real problem I encountered today.

Suppose you have a hidden prize, and a group of people who will form teams to search for it. The fraction of the people who are on your team is the probability your team will find it, but the prize is always divided evenly among all the players on the team. I think that it's an interesting problem because of the fact that your expected winnings are the same no matter what size your team is, so deciding your team size is entirely based on other factors. What would you do?

Join the largest team, to maximize utility, 'cause of diminishing marginal utility of money?
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scikidus
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### Re: Lottery

It seems that the flaw in this lotto system is if everyone invests equally in each bank.

What if people can only invest in, say, 4 of the 5 banks?
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BoomFrog
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### Re: Lottery

If everyone is perfectly logical no one will play because the expected return is zero. If this game existed in reality I would watch the results each day but not play until the last day. I'm guessing there would be some patterns, probably the low bank each day would become the high bank the next day. If that was the case then on the last day I'd invest evenly in all but the lowest bank from the day before.
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quintopia
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### Re: Lottery

If everyone was perfectly logical, they would all wait until the eleventh hour to put their money in, and none would get any info from any previous days.

Also, why would perfectly logical people avoid a game with expected return zero? Things with zero expected return make good investments, since the risk is low (you don't expect to lose). If there is an alternate investment with positive expected return, they would divert their money to that, but in the absence of that, playing a game with zero expected return is just as good as holding onto the money in the long run.

Token
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### Re: Lottery

quintopia wrote:Things with zero expected return make good investments, since the risk is low (you don't expect to lose).

This is not true. Even investments with positive expected return can be high risk.
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quintopia
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### Re: Lottery

eh, true. The point is, if you have enough money, you're gonna tend to not lose it, except as much as is lost to inflation. Yet, there's a small chance you'll increase your money, and, again if you have enough money and play long enough, when you hit that point you can stop with a profit.

hocl
Posts: 82
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### Re: Lottery

quintopia wrote:If everyone was perfectly logical,
If everyone were logical

scikidus wrote:What if people can only invest in, say, 4 of the 5 banks?
If investing equally is guaranteed to give a profit, then doesn't randomly picking one bank have a positive expected value? Bit of a paradox.

quintopia wrote:This is similar to a real problem I encountered today.

Suppose you have a hidden prize, and a group of people who will form teams to search for it. The fraction of the people who are on your team is the probability your team will find it, but the prize is always divided evenly among all the players on the team. I think that it's an interesting problem because of the fact that your expected winnings are the same no matter what size your team is, so deciding your team size is entirely based on other factors. What would you do?
Pick a team with people you like. That way, you have the added utility of seeing your friends get money, too. Or pick a group of idiots, so you can scam them out of their shares.

Yat
Posts: 136
Joined: Tue Feb 03, 2009 2:05 pm UTC

### Re: Lottery

hocl wrote:If investing equally is guaranteed to give a profit, then doesn't randomly picking one bank have a positive expected value? Bit of a paradox.
Yes it would seem so... but after giving it a few thoughts, I think I see where the flaw is.

First, I will assume that if one of the banks is empty or if two banks have the same number of tickets, then the game is void, everybody takes his money back, so these cases give an expected value of 0.

Then I will simplify the problem to a game with only 2 banks. The logic will be exactly the same, and it will be much easier to do the math. Let N be the total number of players. If N<3, it is obvious that the game will be void, so what happens with N=3...

The probability that all players chose the same bank is 1/4. The game is void.
If two different banks have tickets, then one bank has 2, the other one has 1. If we think that you have a probability 1/2 to win in these cases, then this makes the expected value positive. But that is not the case : as only one out of three players wins, we can guess that the probability of winning is 1/3.

To make this clear, let's just consider the state of the game before you make your choice. As your choice does not influence the other player's, then we can consider they have already made their choice, and you just need to add your ticket in one bank. The probability that both other tickets are in the same bank is 1/2. In this case, you lose, because your ticket will make the bank the bigger one.
If both tickets are in the same bank, then you have a probability on 1/2 to chose this bank and induce a void game, and a probability of 1/2 to chose the other one and win 3.

That makes :
void game (0): 1/4
won game (3): 1/4
lost game (-1) : 1/2

This confirms that if the game is not void, your winning probability is 1/3, and the expected is 0.

If N gets bigger, the probability of making a bank lose by adding your ticket gets smaller, but the expected number of tickets in the smallest bank gets closer to N/2. You can calculate what happens with any value of N, the limit being having exactly a probability of 1/2 to win 2, the expected value being 0.

eaglef2
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### Re: Lottery

Think about this, if n players use the omni-invest strategy, assuming that if a tie between any number of banks the money is split between them, then this strategy fails. The only way you win money is to grow some balls and choose a real strategy, to put it bluntly.

Since the banks don't seem to be making money from this lottery, I think that the bank with the least amount of tickets should get all the money invested to it and the investors split the money from the other non-winners. This would discourage a tie because that's less money overall for the winners, and it would discourage the omni-invest strategy because if everyone does this, then all the money is lost.

And I believe that the game works by that you are no longer the investor/voter, but your money is. Seeing as with 10\$ you have 10 "votes" as to which bank you want to lose, not win, if you invest all 10\$ in one bank, you have lowered their chance of winning, but also increased the amount of votes you have towards that bank. Its almost a quantum state, in that a ticket represents a lose and a win at the same time.
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