+EV Lottery Musing

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.