Just-Dice.com : Invest in 1% House Edge Dice Game

[GOB]

Investor lowRisk: 10 000 BTC @ 0.25%, Investor highRisk: 10 000 BTC @1%.

Plan A: my Assumption which is actually flawed.
Plan B: GOB's implementation.
Plan C: An Alternative that is slightly modified version of GOB's.

Computation: A - difficult
B: Actually quite simple.
C: A bit more difficult than B.

Max Profit:
A: 25 + 100 = 125
B: 0 + 100 = 100
C: 25 + 100 = 125

Case 1: Bet 0.25% of house roll.  Bets 50 BTC @ 49.5%
A: LowRisk = ±10, HighRisk = ±40
B: LowRisk = ±25, HighRisk = ±25
C: LowRisk = ±25, HighRisk = ±25
A isn't exactly fair for risk-averse investors.  They're okay with risking 25 BTC.

Case 2: Bet 100 BTC @49.5%
A: LowRisk = ±20, HighRisk = ±80
B: LowRisk = ±0,  HighRisk = ±100
C: LowRisk = ±25, HighRisk = ±75

Case 3: Bet 125 BTC @49.5%
A: LowRisk = ±25, HighRisk = ±100
B: Bet too high
C: LowRisk = ±25, HighRisk = ±100  - OOOPS it's 100 here
B - Without lowRisk helping out, the bet can't go as high.
Note for C that low risk can only win up to 25 btc.

Great summary.

It is true that with scenario B you don't get as high of a max profit. I knew that going in and I don't see it as a negative-- I think investors should be paid fairly/what makes sense/etc., and max profit will be what it'll be. The goal is not necessarily to maximize max profit, after all.

Despite that I see the value in your scenario C. However, I think what I posted a couple posts above still stands. In Case 3, without investor HighRisk being willing to risk 100BTC on each roll, site Max Profit would have been 25, and that bet might never have been made. Thus, LowRisk is profiting, or benefiting, or deriving value, from risk Highrisk was willing to provide.

However, now that I write that out, I'm realizing that is true of this site in general, in concept: the reason my tiny investment is able to profit is because a bunch of other people have invested an additional 40kbtc that draws bettors to the site.... hmmmm. I'm stumped.

Can I ask you a favor? Could you make a Case 4: Bet 110 BTC @49.5%? Specifically for Plan C? i.e. would LowRisk make 10 or 25 btc?

Case 4: bet 110 BTC @49.5%
Break into two bets.  50 btc and 60 btc.
lowRisk ±50 , High Risk ±75

Okay, and the problem with Investment B, is what if there are multiple tiers.  Like 5 or even 3.  And what if the ratios were different.  It wouldn't scale and the top tier (most risk) investors would not actually have a bigger bankroll than the next lower tier.  You understand what I'm trying to say.  Its a great concept but what if the high risk doesn't even have enough money to go above the max profit for the low risk investors.  They wouldn't be attracting any whales.  SO it is the low risk investors helping high risk.

Um, I think there's an error in your case 4. You have LowRisk at ±50, but their max profit was 10,000 * 0.25% = 25btc

Yeah, that's a good point. I do understand what you're saying.

[Peter R]

Mean and Variance of a Gambler's Profits after a Sequence of Bets on Just-Dice.com

[This supersedes my earlier work and corrects several errors.  But the "probabilty cloud" charts I was plotting were still correct.]

Let

    B = bet size
    P = gamber's profit
    p = probability of a win
    E = house edge

JD betting is a Bernoulli process.  There are two outcomes: win or lose.

	GAMBLER'S PROFIT TABLE
WIN		LOSE
probability = p		probability = 1-p
P = B(1-E-p)/p		P = -B

Gambler's expected profit:

     <P> = (profit if he wins)x(probability of winning) + (profit if he loses)x(probability of losing)

            = B(1-E-p)/p  x  p     +     -B  x  (1-p)
 
            = -E B             (of course!)

Gambler's expected variance:

      <(P - <P>)^2> = (B(1-E-p)/p    -    -E B)^2   x p        +        (-B    -    -E B)^2    x    (1-p)

                            = B^2 (1-E)^2 (1-p) / p

QED!

Lastly, as he makes additional bets both his expected profits and his expected variance add over his sequence of bets.  So one can calculate over time where on a probabilty cloud chart the winnings or loses would fall.

QED

This corrects my earlier claim that variance does not add for Bernoulli processes.  Variance always adds given the definition of variance  

[broolstoryco]

If you pick 0.25% and I pick 1%, then I make (or lose) 4 times as much as you.  Simple as that.  

It's not as simple as that though.

If you both invest 100 BTC, and a whale constantly bets to win the max profit, and loses, then the 1% guy wins more than 4 times that of the 0.25% guy:

After 100 or so whale losses, the 1% guy's profit is almost 6 times that of the 0.25% guy.  His share of the bankroll keeps increasing relative to the more timid guy.

The converse is also true.  If the whale wins, then the 1% guy loses less than 4 times as much as the 0.25% guy, and after around 100 whale wins has lost around 3 times as much as the 0.25% guy.  His share of the bankroll keeps decreasing relative to the more timid guy.

So whats the problem here?

[GOB]

If you pick 0.25% and I pick 1%, then I make (or lose) 4 times as much as you.  Simple as that.  

It's not as simple as that though.

If you both invest 100 BTC, and a whale constantly bets to win the max profit, and loses, then the 1% guy wins more than 4 times that of the 0.25% guy:

       A (   p/l)        B (   p/l)     roll    max    ratio
-------- (------) -------- (------) -------- ------ --------
100.2500 (0.2500) 101.0000 (1.0000) 200.0000 1.2500 4.000000
100.5006 (0.5006) 102.0100 (2.0100) 201.2500 1.2606 4.014981
100.7519 (0.7519) 103.0301 (3.0301) 202.5106 1.2714 4.030050
101.0038 (1.0038) 104.0604 (4.0604) 203.7820 1.2822 4.045206
101.2563 (1.2563) 105.1010 (5.1010) 205.0642 1.2931 4.060451
101.5094 (1.5094) 106.1520 (6.1520) 206.3573 1.3042 4.075785
101.7632 (1.7632) 107.2135 (7.2135) 207.6614 1.3153 4.091208
102.0176 (2.0176) 108.2857 (8.2857) 208.9767 1.3265 4.106721
102.2726 (2.2726) 109.3685 (9.3685) 210.3033 1.3379 4.122325
...
127.4046 (27.4046) 262.5266 (162.5266) 387.0141 2.9170 5.930639
127.7231 (27.7231) 265.1518 (165.1518) 389.9311 2.9438 5.957197
128.0424 (28.0424) 267.8033 (167.8033) 392.8749 2.9708 5.983919

After 100 or so whale losses, the 1% guy's profit is almost 6 times that of the 0.25% guy.  His share of the bankroll keeps increasing relative to the more timid guy.

The converse is also true.  If the whale wins, then the 1% guy loses less than 4 times as much as the 0.25% guy, and after around 100 whale wins has lost around 3 times as much as the 0.25% guy.  His share of the bankroll keeps decreasing relative to the more timid guy.

       A (   p/l)        B (   p/l)     roll    max    ratio
-------- (------) -------- (------) -------- ------ --------
 99.7500 (-0.2500)  99.0000 (-1.0000) 200.0000 1.2500 4.000000
 99.5006 (-0.4994)  98.0100 (-1.9900) 198.7500 1.2394 3.984981
 99.2519 (-0.7481)  97.0299 (-2.9701) 197.5106 1.2289 3.970050
 99.0037 (-0.9963)  96.0596 (-3.9404) 196.2818 1.2184 3.955206
 98.7562 (-1.2438)  95.0990 (-4.9010) 195.0633 1.2081 3.940449
 98.5093 (-1.4907)  94.1480 (-5.8520) 193.8552 1.1979 3.925778
 98.2631 (-1.7369)  93.2065 (-6.7935) 192.6574 1.1878 3.911192
 98.0174 (-1.9826)  92.2745 (-7.7255) 191.4696 1.1777 3.896691
 97.7724 (-2.2276)  91.3517 (-8.6483) 190.2919 1.1678 3.882275
 97.5279 (-2.4721)  90.4382 (-9.5618) 189.1241 1.1579 3.867943
...
 78.4426 (-21.5574)  37.7237 (-62.2763) 116.7439 0.5776 2.888855
 78.2464 (-21.7536)  37.3464 (-62.6536) 116.1662 0.5733 2.880154
 78.0508 (-21.9492)  36.9730 (-63.0270) 115.5929 0.5691 2.871500

Chriswen, this looks like your scenario C, so looks like you're right!

[dooglus]

C) Bets that pay more than 1%, neither of us participate (if Doog allows investors to choose >1% personal max profit)

No.  Every investor bankrolls every bet, in proportion to the size of their investment times their chosen risk.

If there are two investors, both investing 100 BTC, A risking 0.25% and B risking 1%, then the max profit is 1.25 BTC for the first roll.  0.25% of 100 from A, and 1% of 100 from B.

If the player decides to play to win half of the max profit, then A and B both risk half of what they're willing to risk.  A risks 0.125 BTC and B risks 0.5 BTC.

This "variable risk" idea isn't designed to let the timid players avoid taking any part in the big max-profit bets.  It's designed to allow them to select different fractions of Kelly.  A week ago everyone was risking up to 1% of their bankroll when a max bet happened.  Today everyone is risking 0.5% of their bankroll to max bets.  Once this feature is in place, we'll have both happening at once.  But there's no option to say "I don't want to bankroll big bets at all".  Nobody in their right mind wants to be bankrolling more than their fail share of big bets.  The variance is too high.  It's much safer to take a hundred bets of 1 BTC each than a single bet of 100 BTC, given that the edge is the same for both.

[GOB]

C) Bets that pay more than 1%, neither of us participate (if Doog allows investors to choose >1% personal max profit)

No.  Every investor bankrolls every bet, in proportion to the size of their investment times their chosen risk.

If there are two investors, both investing 100 BTC, A risking 0.25% and B risking 1%, then the max profit is 1.25 BTC for the first roll.  0.25% of 100 from A, and 1% of 100 from B.

If the player decides to play to win half of the max profit, then A and B both risk half of what they're willing to risk.  A risks 0.125 BTC and B risks 0.5 BTC.

This "variable risk" idea isn't designed to let the timid players avoid taking any part in the big max-profit bets.  It's designed to allow them to select different fractions of Kelly.  A week ago everyone was risking up to 1% of their bankroll when a max bet happened.  Today everyone is risking 0.5% of their bankroll to max bets.  Once this feature is in place, we'll have both happening at once.  But there's no option to say "I don't want to bankroll big bets at all".  Nobody in their right mind wants to be bankrolling more than their fail share of big bets.  The variance is too high.  It's much safer to take a hundred bets of 1 BTC each than a single bet of 100 BTC, given that the edge is the same for both.

That makes sense! Thanks for clarifying. I'll go back and edit my post with a note.

[Peter R]

Hi Doog.  Sorry to keep beating this horse, but can you confirm whether this is true:

Investor A and B deposit 100 BTC each.  

Investor A selects 0.25% and Investor B selects 1%.  

A whale makes the max bet.  Investor A risks 0.25 BTC on this roll and Investor B risks 1 BTC.  Should the whale win, Investor A would be left with 99.75 BTC and Investor B would be left with only 99 BTC.  

A smaller fish comes along and makes a bet to win max_profit / 10.  Investor A now risks 1/10th * 0.25% of 99.75 and Investor B risks 1/10th * 1% of 99 BTC.

In other words, they gain or lose in proportion to what they risk per roll.  

If each investor could hypothetically react instantly to maintain their balance at 100 BTC, then Investor A would earn an income stream identical to Investor B, just scaled by a factor of 4.  

And on and on...

[dooglus]

If you pick 0.25% and I pick 1%, then I make (or lose) 4 times as much as you.  Simple as that.  

whale constantly bets to win the max profit, and loses, then the 1% guy wins more than 4 times that of the 0.25% guy:

whale wins, then the 1% guy loses less than 4 times as much as the 0.25% guy, and after around 100 whale wins has lost around 3 times as much as the 0.25% guy

So whats the problem here?

The only problem: Peter said "I make 4 times as much as you" and that's not true, because the two investments change in size *relative to each other*.

[xypos]

Where can you choose the risk?

[dooglus]

Hi Doog.  Sorry to keep beating this horse, but can you confirm whether this is true:

It sounds true to me.  The only reason I don't earn exactly 4x what you earn is because we don't adjust our investments to be equal to each other after every roll.

I risk 1% and win, my investment grows more than yours, since you only risked 0.25% and won.  Our bankrolls were equal before the bet, but mine is now slightly bigger than yours.  (101 vs 100.25).  If we both immediately divested our profits, bring the rolls back to 100 each before the next bet, then I'd always win (or lose) 4x as much as you.

[broolstoryco]

C) Bets that pay more than 1%, neither of us participate (if Doog allows investors to choose >1% personal max profit)

No.  Every investor bankrolls every bet, in proportion to the size of their investment times their chosen risk.

If there are two investors, both investing 100 BTC, A risking 0.25% and B risking 1%, then the max profit is 1.25 BTC for the first roll.  0.25% of 100 from A, and 1% of 100 from B.

If the player decides to play to win half of the max profit, then A and B both risk half of what they're willing to risk.  A risks 0.125 BTC and B risks 0.5 BTC.

This "variable risk" idea isn't designed to let the timid players avoid taking any part in the big max-profit bets.  It's designed to allow them to select different fractions of Kelly.  A week ago everyone was risking up to 1% of their bankroll when a max bet happened.  Today everyone is risking 0.5% of their bankroll to max bets.  Once this feature is in place, we'll have both happening at once.  But there's no option to say "I don't want to bankroll big bets at all".  Nobody in their right mind wants to be bankrolling more than their fail share of big bets.  The variance is too high.  It's much safer to take a hundred bets of 1 BTC each than a single bet of 100 BTC, given that the edge is the same for both.

I think it would make more sense as follows:

Your first scenario would amount to the same, ie A risks 0.25 BTC, B 1 BTC.

However, in the second case (bet size 0.625), A would again risk 0.25 BTC and B would risk 0.375

Third case, player bets 0.25, A and B would each risk 0.125.

This way only the large bets get asymmetrically filled, while the smaller bets remain the way they are filled now. In cases where whales such as nakowa play, investors would be able to have different exposure levels. However, with smaller bets, things would stay as they are now.

[organofcorti]

Mean and Variance of a Gambler's Profits after a Sequence of Bets on Just-Dice.com

[This supersedes my earlier work and corrects several errors.  But the "probabilty cloud" charts I was plotting were still correct.]

Let

    B = bet size
    P = gamber's profit
    p = probability of a win
    E = house edge

JD betting is a Bernoulli process.  There are two outcomes: win or lose.

	GAMBLER'S PROFIT TABLE
WIN		LOSE
probability = p		probability = 1-p
P = B(1-E-p)/p		P = -B

Gambler's expected profit:

     <P> = (profit if he wins)x(probability of winning) + (profit if he loses)x(probability of losing)

            = B(1-E-p)/p  x  p     +     -B  x  (1-p)
 
            = -E B             (of course!)

This is only for one bet though. I think you'll need to re-derive variance including 'n' for number of bets. Sorry, bud.

[mechs]

Hi Doog.  Sorry to keep beating this horse, but can you confirm whether this is true:

It sounds true to me.  The only reason I don't earn exactly 4x what you earn is because we don't adjust our investments to be equal to each other after every roll.

I risk 1% and win, my investment grows more than yours, since you only risked 0.25% and won.  Our bankrolls were equal before the bet, but mine is now slightly bigger than yours.  (101 vs 100.25).  If we both immediately divested our profits, bring the rolls back to 100 each before the next bet, then I'd always win (or lose) 4x as much as you.

I understand the max bet part, but what if a player bets 1BTC and wins (way below a max bet).  Do the 0.25% and 1% guy participate equally based on portion of the bankroll or does the 1% person get 4X more than the 0.25% investor?

[Oleander]

It's not as simple as that though.

If you both invest 100 BTC, and a whale constantly bets to win the max profit, and loses, then the 1% guy wins more than 4 times that of the 0.25% guy:

       A (   p/l)        B (   p/l)     roll    max    ratio
-------- (------) -------- (------) -------- ------ --------
100.2500 (0.2500) 101.0000 (1.0000) 200.0000 1.2500 4.000000
100.5006 (0.5006) 102.0100 (2.0100) 201.2500 1.2606 4.014981
100.7519 (0.7519) 103.0301 (3.0301) 202.5106 1.2714 4.030050
101.0038 (1.0038) 104.0604 (4.0604) 203.7820 1.2822 4.045206
101.2563 (1.2563) 105.1010 (5.1010) 205.0642 1.2931 4.060451
101.5094 (1.5094) 106.1520 (6.1520) 206.3573 1.3042 4.075785
101.7632 (1.7632) 107.2135 (7.2135) 207.6614 1.3153 4.091208
102.0176 (2.0176) 108.2857 (8.2857) 208.9767 1.3265 4.106721
102.2726 (2.2726) 109.3685 (9.3685) 210.3033 1.3379 4.122325
...
127.4046 (27.4046) 262.5266 (162.5266) 387.0141 2.9170 5.930639
127.7231 (27.7231) 265.1518 (165.1518) 389.9311 2.9438 5.957197
128.0424 (28.0424) 267.8033 (167.8033) 392.8749 2.9708 5.983919

After 100 or so whale losses, the 1% guy's profit is almost 6 times that of the 0.25% guy.  His share of the bankroll keeps increasing relative to the more timid guy.

The converse is also true.  If the whale wins, then the 1% guy loses less than 4 times as much as the 0.25% guy, and after around 100 whale wins has lost around 3 times as much as the 0.25% guy.  His share of the bankroll keeps decreasing relative to the more timid guy.

       A (   p/l)        B (   p/l)     roll    max    ratio
-------- (------) -------- (------) -------- ------ --------
 99.7500 (-0.2500)  99.0000 (-1.0000) 200.0000 1.2500 4.000000
 99.5006 (-0.4994)  98.0100 (-1.9900) 198.7500 1.2394 3.984981
 99.2519 (-0.7481)  97.0299 (-2.9701) 197.5106 1.2289 3.970050
 99.0037 (-0.9963)  96.0596 (-3.9404) 196.2818 1.2184 3.955206
 98.7562 (-1.2438)  95.0990 (-4.9010) 195.0633 1.2081 3.940449
 98.5093 (-1.4907)  94.1480 (-5.8520) 193.8552 1.1979 3.925778
 98.2631 (-1.7369)  93.2065 (-6.7935) 192.6574 1.1878 3.911192
 98.0174 (-1.9826)  92.2745 (-7.7255) 191.4696 1.1777 3.896691
 97.7724 (-2.2276)  91.3517 (-8.6483) 190.2919 1.1678 3.882275
 97.5279 (-2.4721)  90.4382 (-9.5618) 189.1241 1.1579 3.867943
...
 78.4426 (-21.5574)  37.7237 (-62.2763) 116.7439 0.5776 2.888855
 78.2464 (-21.7536)  37.3464 (-62.6536) 116.1662 0.5733 2.880154
 78.0508 (-21.9492)  36.9730 (-63.0270) 115.5929 0.5691 2.871500

In this example, you're setting the max bet to 1% of the total bankroll (20,000 on the first roll).  But this is wrong!  The .25% people are willing to risk .25% of their 10,000, or 25.  The 1% people are willing to risk 1% of their 10,000, or 100.  The max bet on the first roll is 125, not 200.

[wachtwoord]

Hi Doog.  Sorry to keep beating this horse, but can you confirm whether this is true:

It sounds true to me.  The only reason I don't earn exactly 4x what you earn is because we don't adjust our investments to be equal to each other after every roll.

I risk 1% and win, my investment grows more than yours, since you only risked 0.25% and won.  Our bankrolls were equal before the bet, but mine is now slightly bigger than yours.  (101 vs 100.25).  If we both immediately divested our profits, bring the rolls back to 100 each before the next bet, then I'd always win (or lose) 4x as much as you.

I understand the max bet part, but what if a player bets 1BTC and wins (way below a max bet).  Do the 0.25% and 1% guy participate equally based on portion of the bankroll or does the 1% person get 4X more than the 0.25% investor?

I'm hoping and expecting the latter.

[mechs]

Hi Doog.  Sorry to keep beating this horse, but can you confirm whether this is true:

It sounds true to me.  The only reason I don't earn exactly 4x what you earn is because we don't adjust our investments to be equal to each other after every roll.

I risk 1% and win, my investment grows more than yours, since you only risked 0.25% and won.  Our bankrolls were equal before the bet, but mine is now slightly bigger than yours.  (101 vs 100.25).  If we both immediately divested our profits, bring the rolls back to 100 each before the next bet, then I'd always win (or lose) 4x as much as you.

I understand the max bet part, but what if a player bets 1BTC and wins (way below a max bet).  Do the 0.25% and 1% guy participate equally based on portion of the bankroll or does the 1% person get 4X more than the 0.25% investor?

I'm hoping and expecting the latter.

Then this really is not so much a set your own kelly % more more a set your own leverage.

[Peter R]

This is only for one bet though. I think you'll need to re-derive variance including 'n' for number of bets. Sorry, bud.

I think it was correct, but perhaps there was ambiguity with my notation [you cited the expected profit for a single bet while in the last paragraph I said that you add variance and expected profits over all bets to get the "probability cloud" graphs].

Try this (my proof above still holds):

B = Bet size (per bet not cumulative bet volume over N bets)
P = Gambler profit (per bet)
E = House edge
p = Probability of a win

Gambler's expected profit per bet:

    <P> = -E B            

Gambler's expected variance per bet:

    V = <(P - <P>)^2> = B^2 (1-E)^2 (1-p) / p

Over N bets (to get the probability cloud graphs) we add up the variance for each individual bet:

    Total Variance = V(bet 1) + V(bet 2) + ... V(bet N)
    Expected profit = <P>(bet 1) + <P>(bet 1) + ... + <P>(bet N) +

Peter

[chriswen]

If you pick 0.25% and I pick 1%, then I make (or lose) 4 times as much as you.  Simple as that.  

It's not as simple as that though.

If you both invest 100 BTC, and a whale constantly bets to win the max profit, and loses, then the 1% guy wins more than 4 times that of the 0.25% guy:

       A (   p/l)        B (   p/l)     roll    max    ratio
-------- (------) -------- (------) -------- ------ --------
100.2500 (0.2500) 101.0000 (1.0000) 200.0000 1.2500 4.000000
100.5006 (0.5006) 102.0100 (2.0100) 201.2500 1.2606 4.014981
100.7519 (0.7519) 103.0301 (3.0301) 202.5106 1.2714 4.030050
101.0038 (1.0038) 104.0604 (4.0604) 203.7820 1.2822 4.045206
101.2563 (1.2563) 105.1010 (5.1010) 205.0642 1.2931 4.060451
101.5094 (1.5094) 106.1520 (6.1520) 206.3573 1.3042 4.075785
101.7632 (1.7632) 107.2135 (7.2135) 207.6614 1.3153 4.091208
102.0176 (2.0176) 108.2857 (8.2857) 208.9767 1.3265 4.106721
102.2726 (2.2726) 109.3685 (9.3685) 210.3033 1.3379 4.122325
...
127.4046 (27.4046) 262.5266 (162.5266) 387.0141 2.9170 5.930639
127.7231 (27.7231) 265.1518 (165.1518) 389.9311 2.9438 5.957197
128.0424 (28.0424) 267.8033 (167.8033) 392.8749 2.9708 5.983919

After 100 or so whale losses, the 1% guy's profit is almost 6 times that of the 0.25% guy.  His share of the bankroll keeps increasing relative to the more timid guy.

The converse is also true.  If the whale wins, then the 1% guy loses less than 4 times as much as the 0.25% guy, and after around 100 whale wins has lost around 3 times as much as the 0.25% guy.  His share of the bankroll keeps decreasing relative to the more timid guy.

       A (   p/l)        B (   p/l)     roll    max    ratio
-------- (------) -------- (------) -------- ------ --------
 99.7500 (-0.2500)  99.0000 (-1.0000) 200.0000 1.2500 4.000000
 99.5006 (-0.4994)  98.0100 (-1.9900) 198.7500 1.2394 3.984981
 99.2519 (-0.7481)  97.0299 (-2.9701) 197.5106 1.2289 3.970050
 99.0037 (-0.9963)  96.0596 (-3.9404) 196.2818 1.2184 3.955206
 98.7562 (-1.2438)  95.0990 (-4.9010) 195.0633 1.2081 3.940449
 98.5093 (-1.4907)  94.1480 (-5.8520) 193.8552 1.1979 3.925778
 98.2631 (-1.7369)  93.2065 (-6.7935) 192.6574 1.1878 3.911192
 98.0174 (-1.9826)  92.2745 (-7.7255) 191.4696 1.1777 3.896691
 97.7724 (-2.2276)  91.3517 (-8.6483) 190.2919 1.1678 3.882275
 97.5279 (-2.4721)  90.4382 (-9.5618) 189.1241 1.1579 3.867943
...
 78.4426 (-21.5574)  37.7237 (-62.2763) 116.7439 0.5776 2.888855
 78.2464 (-21.7536)  37.3464 (-62.6536) 116.1662 0.5733 2.880154
 78.0508 (-21.9492)  36.9730 (-63.0270) 115.5929 0.5691 2.871500

Chriswen, this looks like your scenario C, so looks like you're right!

It's scenario A.

[chriswen]

Hi Doog.  Sorry to keep beating this horse, but can you confirm whether this is true:

It sounds true to me.  The only reason I don't earn exactly 4x what you earn is because we don't adjust our investments to be equal to each other after every roll.

I risk 1% and win, my investment grows more than yours, since you only risked 0.25% and won.  Our bankrolls were equal before the bet, but mine is now slightly bigger than yours.  (101 vs 100.25).  If we both immediately divested our profits, bring the rolls back to 100 each before the next bet, then I'd always win (or lose) 4x as much as you.

I understand the max bet part, but what if a player bets 1BTC and wins (way below a max bet).  Do the 0.25% and 1% guy participate equally based on portion of the bankroll or does the 1% person get 4X more than the 0.25% investor?

I'm hoping and expecting the latter.

Then this really is not so much a set your own kelly % more more a set your own leverage.

Yes, this is what I realized after looking at scenario A.

[dooglus]

I think it would make more sense as follows:

Your first scenario would amount to the same, ie A risks 0.25 BTC, B 1 BTC.

However, in the second case (bet size 0.625), A would again risk 0.25 BTC and B would risk 0.375

I disagree.

Think of it like this:  A and B are both running their own sites.  How much does each risk?

When someone wants to bet the maximum, A risks 0.25 BTC and B risks 1 BTC.  We agree.

Then someone wants to bet half as much.  I think each should get half of what they did on the max bet:
A:0.125, B:0.5

You're suggesting that A should get some of B's action.  Why's that?  How is it fair that A gets to take some of B's share, only on the relatively safe bets?  The safer the bet, the more A gets to take of B's action.  We reward B's willingness to run at full Kelly by giving him 4x the action that A gets, on every bet.  Everyone prefers lots of small bets to a few big ones.

B wants to risk 4x what A risks.  So that's what happens.  It's just as if every investor was running their own separate game.