You're right, and I apologise for that.  I don't know if you saw the run-up to increasing the risk from 0.25% to 0.5%, but I announced it in this thread giving 24 hours' notice, and had the site chat announcing the change automatically every 10 minutes in the run up to the change.

I'll do the same for any future changes, too; give notice both here and in the on-site chat.

You don't get the same thing by investing less.

There was an example earlier: A: invest 500 at 1% or B: 125 at 0.25%.

You'd think they were the same, but they're not.  After a few thousand losses, A has lost nearly 500 coins and B has lost nearly 125 coins.  That's not the same at all...

It's here:

https://just-dice.com/nakowa3.txt.bz2

Not really.  I think he deposited 50 BTC, lost it all, deposited another 50 BTC, lost 46 BTC, then took the remaining 4 BTC balance up from there.  There were lots of withdrawals and deposits along the way.  I think the "from 4 BTC" rumour started from the fact that his balance was 4 BTC at one time.

Lifetime charts for each account:

His stats from this session:

user 161188 -- wagered: 1482007.05642997; bets: 15360; balance: 9614.07834057; profit: 1163.78674773; losses: 7689; wins: 7671

And in detail:

mysql> select count(*), sign(profit), chance/1e4 from bets where uid = 161188 group by chance, sign(profit) order by chance;
+----------+--------------+------------+
| count(*) | sign(profit) | chance/1e4 |
+----------+--------------+------------+
|        2 |           -1 |       48.5 |
|        3 |            1 |       48.5 |
|     7686 |           -1 |       49.5 |
|        1 |            0 |       49.5 |
|     7661 |            1 |       49.5 |
|        4 |            1 |       50.5 |
|        3 |            1 |      98.99 |
+----------+--------------+------------+
7 rows in set (0.06 sec)

mysql> select 100 * 7661 / (7661 + 7686 + 1);
+--------------------------------+
| 100 * 7661 / (7661 + 7686 + 1) |
+--------------------------------+
|                        49.9153 |
+--------------------------------+
1 row in set (0.00 sec)

49.9153% of his "49.5%" bets won.  There was a single bet of 0 BTC, which lost:

bet #143904532: 0 BTC @ 49.5% hi: lucky:168905 profit:0.00000000

Here's a chart of the site's profits for the day, with the expected profit (minus 20k, to bring them closer together!) overlayed.

The random walk goes horizontally, whereas the expectation goes up at 30 degrees to the horizontal.  How?  And why?

It's something he told me via IM when he first invested:

Wed 26th June 2013
(10:10:57 PM) squiggles: btw, this investment is not from me, [redacted], in fact, I myself have more than 6 digits of bitcoin, it’s pointless for me to earn more coins.

Suppose 99 investors invest 1 BTC each with 0.99% risk and one brave investor invests 1 BTC at 1% risk.

The 1% (highest risk) investor gets only 1% of all bets, except for the most dangerous max profit bets, of which he gets 100%.

So by being the only investor willing to risk the fully Kelly, he gets to experience very high variance because the lower risk investors take an unfair share of all the sub-max profit bets.

That's the problem with your proposal for me, Chris.

Here's another analogy:

--
A and B both invest 100 BTC.  Someone wants to bet 0.1 BTC.  Both investors stand to gain 0.05 BTC.

Now investor B decides to double his risk by investing another 100 BTC.

So now if someone bets 0.1 btc,
Investor A can win ±.33 and investor B can win ±.67.

This doesn't seem fair.  Investor A is still okay with betting at this risk level.  Why does investor B take some of his bet?
--

In both cases investor B has doubled his risk, and as a result 'takes' some of investor A's action.  In both cases he has diluted the other investor's share by risking more.  In your analogy he did it by doubling his Kelly factor, and in mine he did it by doubling his investment.  In both cases he's doubling his exposure, and so doubles his reward.

Fair enough.  Here it is again taking that into account:

>>> a=500
>>> for i in range(20): a *= (1-0.0025); print "%.2f" % (a*100/500,),
99.75 99.50 99.25 99.00 98.76 98.51 98.26 98.02 97.77 97.53 97.28 97.04 96.80 96.56 96.31 96.07 95.83 95.59 95.36 95.12, ..., 0

>>> a=125
>>> for i in range(20): a *= (1-0.0100); print "%.2f" % ((a+375)*100/500,),
99.75 99.50 99.26 99.01 98.77 98.54 98.30 98.07 97.84 97.61 97.38 97.16 96.94 96.72 96.50 96.29 96.07 95.86 95.65 95.45, ..., 75

If the whale keeps winning when they invest 500 at 0.25%, they end up losing everything.  If they invest 125 at 1%, they end up only losing the 25% they invested...

Yes, because bets vary in size...

Forcing the 1% investors to take all the variance so the 0.25% investors don't have to isn't fair on the 1% investors.

Investing at 0.25% allows you to risk less of your investment per roll.  It doesn't change the fact that most bets aren't near the maximum.  It allows you to select your own risk of ruin as you see fit.

Because you would get less variance.

Suppose a whale bets max bet and wins 20 times.  If you invest 500 BTC at 0.25%, you'll still have 95% of your bankroll left:

>>> a=500
>>> for i in range(20): a *= (1-0.0025); print "%.2f" % (a*100/500,),
99.75 99.50 99.25 99.00 98.76 98.51 98.26 98.02 97.77 97.53 97.28 97.04 96.80 96.56 96.31 96.07 95.83 95.59 95.36 95.12

But if you invest 125 BTC at 1%, you'll have only 82% of it left:

>>> a=125
>>> for i in range(20): a *= (1-0.0100); print "%.2f" % (a*100/125,),
99.00 98.01 97.03 96.06 95.10 94.15 93.21 92.27 91.35 90.44 89.53 88.64 87.75 86.87 86.01 85.15 84.29 83.45 82.62 81.79

Your risk of ruin is much greater if you invest at 1% than if you invest at 0.25%.

You're right.


I don't think so.  Leverage would be if I was to consider your bankroll to be N times bigger than you've actually deposited.

But that's not what this is.  What you're doing here is deciding what percentage of your investment you want to risk at most, and then, when the player decides to only bet X% of the maximum, you get to risk X% of the most you're willing to risk.

I think you're reading my table wrong.  I've deleted all but the relevent parts in the quote above.

On the first bet, the bankroll is 200 and the max bet is 1.25

I'm using dot as a decimal point, not a thousands-separator in case that's the issue.

People who run 100% kelly get 4 times the action of people running 25% kelly, yes.  By definition, pretty much.

A says "I'm willing to risk up to 0.25 BTC".  B says "I'm willing to risk up to 1 BTC".  We sum those two, and present to the gambler "the site is willing to risk up to 1.25 BTC".

If the gambler comes back and says to the site "I'll take 50% of what you're offering", we say to each investor "he's taking 50% of your offer".

I don't see how any other way would be fair.

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.

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.

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*.

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.