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unJon
unJon 
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November 27th, 2018 at 7:32:55 AM permalink
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.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
OnceDear
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OnceDear
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November 27th, 2018 at 9:35:45 AM permalink
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 Ė 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 Ė 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 !?!?
Take care out there. Spare a thought for the newly poor who were happy in their world just a few days ago, but whose whole way of life just collapsed..
FinsRule
FinsRule
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November 27th, 2018 at 10:58:48 AM permalink
I hope this isnít 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 ďbankroll?Ē

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 Iím right and lose $1 when Iím wrong, Iím 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.
TomG
TomG
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November 27th, 2018 at 10:59:29 AM permalink
Bet whatever limits there are. Although itís 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?
OnceDear
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OnceDear
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November 27th, 2018 at 11:16:24 AM permalink
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?
Take care out there. Spare a thought for the newly poor who were happy in their world just a few days ago, but whose whole way of life just collapsed..
FinsRule
FinsRule
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November 27th, 2018 at 12:55:35 PM permalink
Thereís 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 doesnít scale up. You canít buy one billion lottery tickets. The machine doesnít print them out fast enough.
OnceDear
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OnceDear
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November 27th, 2018 at 1:06:44 PM permalink
Quote: FinsRule

Thereís 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 doesnít scale up. You canít buy one billion lottery tickets. The machine doesnít print them out fast enough.

Not an issue if there were millions of ticket terminals and plenty of time.
Take care out there. Spare a thought for the newly poor who were happy in their world just a few days ago, but whose whole way of life just collapsed..
FinsRule
FinsRule
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November 27th, 2018 at 1:11:08 PM permalink
I donít think there is a mathematical answer for the questions you are asking.
OnceDear
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OnceDear
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November 27th, 2018 at 1:22:06 PM permalink
Quote: FinsRule

I donít 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.
Take care out there. Spare a thought for the newly poor who were happy in their world just a few days ago, but whose whole way of life just collapsed..

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