+EV Lottery Musing

Last night a silly enigma occurred to me and I thought I'd share it to see where it goes.

Consider the typical lottery with say 50% return to player. Many here would concur that it's so -EV that to buy a single ticket is at least unwise. Wizard even asserts that even the winner of such a lottery was foolish to buy the ticket.

But... What if it was massively +EV?

Consider a lottery where some sponsoring multi-billionaire decides that the prize value would be 2 x the total value of all tickets bought... I.e. if 1 billion tickets sold for $1 each, the prize would be £2bn. No cap on the total prize fund. In this fictional lottery, it is decreed that the draw will be arranged such that there will be ONE prize in the draw, winner takes all, never ZERO winners and never multiple winners. Also, we will live in a world where there would be no tax implications.

Now, what's the best strategy for this game? Is it foolish to even buy one ticket? What about 10?, 1,000?, 1,000,000?

Is it not 200% RTP on average? Variance is absurdly massive. No Normal distribution?

My first thought was that maybe kelly betting might be appropriate, or maybe we should all sell all our possessions, mortgage to the max extent possible and buy as many entries as possible. But obviously, the risk of ruin would be phenomenal, especially if everyone else did the same.

Maybe the whole world could conspire in some great big Cooperation game and form a syndicate. Hmmm. that would never work: We can't even get 40 folks to conspire.

Thoughts?

Quote: OnceDear

Last night a silly enigma occurred to me and I thought I'd share it to see where it goes.

Consider the typical lottery with say 50% return to player. Many here would concur that it's so -EV that to buy a single ticket is at least unwise. Wizard even asserts that even the winner of such a lottery was foolish to buy the ticket.

But... What if it was massively +EV?

Consider a lottery where some sponsoring multi-billionaire decides that the prize value would be 2 x the total value of all tickets bought... I.e. if 1 billion tickets sold for $1 each, the prize would be £2bn. No cap on the total prize fund. In this fictional lottery, it is decreed that the draw will be arranged such that there will be ONE prize in the draw, winner takes all, never ZERO winners and never multiple winners. Also, we will live in a world where there would be no tax implications.

Now, what's the best strategy for this game? Is it foolish to even buy one ticket? What about 10?, 1,000?, 1,000,000?

Is it not 200% RTP on average? Variance is absurdly massive. No Normal distribution?

My first thought was that maybe kelly betting might be appropriate, or maybe we should all sell all our possessions, mortgage to the max extent possible and buy as many entries as possible. But obviously, the risk of ruin would be phenomenal, especially if everyone else did the same.

Maybe the whole world could conspire in some great big Cooperation game and form a syndicate. Hmmm. that would never work: We can't even get 40 folks to conspire.

Thoughts?

I found some money on the ground years ago and I decided that since this money was expendable money, I'll play Lottery tickets. I was shocked that I "won" less than half the gambled money. Like 45 percent back. Like WTF? Seriously?

Quote: Nathan

I found some money on the ground years ago and I decided that since this money was expendable money, I'll play Lottery tickets. I was shocked that I "won" less than half the gambled money. Like 45 percent back. Like WTF? Seriously?

Nathan, DO NOT hijack this thread. I see you trying to do so in many threads. Just don't.

Quote: OnceDear

Consider a lottery where some sponsoring multi-billionaire decides that...

OK, I'll hijack it instead. Are you smoking something?

Quote:

Maybe the whole world could conspire in some great big Cooperation game and form a syndicate. Hmmm. that would never work: We can't even get 40 folks to conspire.

Thoughts?

We conspire to do what? force a multi-billionaire to set it up? You got some splanin to do. *

*this is an expression that Brits may not know, comes from the old "I love Lucy" tv show, Ricky Ricardo was always saying this to Lucy

Lotteries have been rigged, in America, in the modern age, many of them, nobody got a refund. Lotteries are corrupt mofos, evil only outmatched by incompetence.

Quote: odiousgambit

OK, I'll hijack it instead. Are you smoking something?

I did have some blue cheese for supper.

Quote:

We conspire to do what? force a multi-billionaire to set it up? You got some splanin to do.

No, I was suggesting that if such a lottery did exist, that we get the whole world's population to buy as many tickets as possible and then divide the winnings proportionally. Thus every player doubles his investment with no risk.

The core question was in the poll. Would it be foolish to buy a ticket or many tickets?

Quote:

*this is an expression that Brits may not know, comes from the old "I love Lucy" tv show, Ricky Ricardo was always saying this to Lucy

vaguely familiar. I never thought I love Lucy was particularly funny. But it was a bit before my time. I don't think the humour of it travelled well.

Quote: OnceDear

Nathan, DO NOT hijack this thread. I see you trying to do so in many threads. Just don't.

I was actually pointing out that the Lottery really isn't all that good of a game if you get less than half of your money back.

Quote: onenickelmiracle

Lotteries have been rigged, in America, in the modern age, many of them, nobody got a refund. Lotteries are corrupt mofos, evil only outmatched by incompetence.

Agreed. My musing was a thought experiment in a hypothetical world. More about the logic of taking part in a +EV wager.

I think you would use Kelly here.

Quote: unJon

I think you would use Kelly here.

Help me out here. We all have 100% player advantage. So what do we stake?

100% of bankroll?
Bankroll=All our worldly value ???

Unless we all conspire to share the winnings pro-rata, there are going to be a lot of hungry people.

Quote: OnceDear

Help me out here. We all have 100% player advantage. So what do we stake?

100% of bankroll?
Bankroll=All our worldly value ???

Unless we all conspire to share the winnings pro-rata, there are going to be a lot of hungry people.

Kelly bet = (P(odds + 1) -1)/odds

Odds equals 2 billion - 1. P equals 1 in 1 billion.

Kelly bet = 0.00000005%

Bet a two billionth of your bankroll.

Quote: unJon

Kelly bet = (P(odds + 1) -1)/odds

Odds equals 2 billion - 1. P equals 1 in 1 billion.

Kelly bet = 0.00000005%

Bet a two billionth of your bankroll.

Thanks for that.
I'm not great with 'decimal odds' representation.

The prize, and hence the odds are unknown at the outset because there is no cap on the number of tickets sold. But working with P = 1 in 1 billion is not unreasonable for this exercise.

Using

f=(bp-q)/b where

f = the fraction of the bankroll to bet
b = the decimal odds – 1
decimal odds = return for one unit wager, which is 2e9
so b=2e9-1
p = the probability of winning
q = the probability of losing, which is 1 – p

so we'll use
b=1
p=(1/10^9)
q=1-(1/10^9)

f= ((2 x 10^9-1) x (1/10^9) - (1-1/10^9))/(2 x 10^9)

Which according to google calculator is 5e-10 which agrees with unJon's answer.
I.e. pretty much 0% of my bankroll

So even at 100% player advantage, no-one should play.!!!!!
Except for the one guy who does, and he should bet BIG !?!?

I hope this isn’t too much of a thread hijack...

This thread pretty much is a Kelly thread. And I feel that Kelly is flawed. How do you define “bankroll?”

I have credit card debt. I would assume that means my bankroll is automatically zero.

But, if we are playing a coin flip game and I win $1.50 when I’m right and lose $1 when I’m wrong, I’m going to figure out a way to play it.

There are tons of +ev plays in the horse racing world. I stick to the safer ones for now.

As for this lottery example, I think it needs to be worded differently to get the answers you are looking for. But I may be totally off on this.

Bet whatever limits there are. Although it’s entirely theoretical, there must be some limit, otherwise what would happen if the total amount bet exceeded 50% of all the wealth in the world?

Quote: FinsRule

As for this lottery example, I think it needs to be worded differently to get the answers you are looking for. But I may be totally off on this.

Thanks
I'm just trying to reconcile this.

Quote: OnceDear

So even at 100% player advantage, no-one should play.!!!!!
Except for the one guy who does, and he should bet BIG !?!?

For TomG we could have the restriction that the sponsoring billionaire would only commit to matching the collected ticket money up to say, $1bn with ticket sales capped at $1bn and $2bn total prize.

The best strategy is to buy all tickets. To fail to (almost) achieve that would be pretty disasterous, so best buy none?

There’s just too many logistical factors to make this realistic.

If there was a lottery that had 10,000 tickets available for $1 each and it paid out $20,000, then buy all the tickets.

But it doesn’t scale up. You can’t buy one billion lottery tickets. The machine doesn’t print them out fast enough.

Quote: FinsRule

There’s just too many logistical factors to make this realistic.

Of course. It was a thought experiment. I perceived the enigma tha a +EV 100% player edge opportunity was still a bad bet.

Quote:

If there was a lottery that had 10,000 tickets available for $1 each and it paid out $20,000, then buy all the tickets.

Ah. If you can buy all the tickets, or even half the tickets, it's +EV. But in a world where you are competing to get in first, and have no chance of buying all, then should you try to buy any? This would be a one shot opportunity and I look at scenarios where your personal liquidity would be far too little to make buying all tickets a viable option.

Quote:

But it doesn’t scale up. You can’t buy one billion lottery tickets. The machine doesn’t print them out fast enough.

Not an issue if there were millions of ticket terminals and plenty of time.

I don’t think there is a mathematical answer for the questions you are asking.

Quote: FinsRule

I don’t think there is a mathematical answer for the questions you are asking.

I agree. I really had been eating blue cheese before this enigma popped into my head $:o) I think it's the start of some analysis of the extremes of Kelly betting. I already realised that I've long mis-interpreted how Kelly worked.