Heads-up tourneys and variance

[jay_shark]

Here is the link if you're interested . If you feel the need to ask questions , please do so and I'll reply back .

http://forumserver.twoplustwo.com/showfl...part=1&vc=1

[TNixon]

That's perfect. Wonder why that post didn't show up in my searching. Figured it should have shown up when I was searching this forum for "variance" at least. [img]/images/graemlins/smile.gif[/img]

Your figures, along with the results from a quick-n-dirty simulator I threw together, show pretty much what I expected to see, that 10 buyins gives you a fair chance of going bust, but not so much that I'm afraid of busting out my last $20 before catching back up a little.

10-25 buyins, based on how much ROR you're willing to accept, sounds *much* more reasonable than the "absolute minimum of 40 buyins for a 70% winrate" that other article reccommended.

[img]/images/graemlins/smile.gif[/img]

In fact, according to my simulator, even a 55% winrate gives a decent chance (71%) of surviving through a string of 250 games, although on average you only have 28 buyins at the end of that 250 games when you do make it through.

I've only run stats on games from one of the two computers I mainly play on, but I'm about 68% on the $2s, and 63% on the $5 (over an admittedly small sample, 50 $2 games and 90 $5 games). I'll have to check the other computer to be sure, but I'm fairly positive it's the $10s where I was getting clobbered. (48% over 64 games on this one, which would lead me to believe that I'm adjusting *extremely* poorly to the players at the higher buyin)

[TNixon]

More interesting results from my simulator:

I added the concept of stepping up and stepping down, added in the actual levels from Full-Tilt, just to get a feel for what a correct step-up and step-down strategy might be, and the results are extremely interesting.

In all the following simulator runs, I start with $20, which is enough for 9 attempts at full-tilt's $2 HUSNGs. I allow stepping up through the $50 HUSNGs, and assume a fixed win rate (which I know is absolutely wrong, since your win rate is likely to be smaller at higher levels, especially when you first step up). But the win-rate I've been running with is a pretty small 58%, which I don't feel is unreachable for most people who care enough to read this forum. [img]/images/graemlins/smile.gif[/img]

Each run is 10k attempts at a string of 250 games, with a survival rate and a finish bankroll rate, which results in an equity calculation. (percentage survival multiplied by the average ending bankroll)

The step-up and step-down multipliers are how many buyins at the next and previous level you have to have before stepping up and stepping down. So, if you're currently at $5 + $0.25 with $80, and a stepup and stepdown multiplier of 10, you'll move up when you have $105, and down when you have $21.50.

<font class="small">Code:</font><hr /><pre>
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 5.00 SD: 10.00
Survived 6955 times with 426.75 avg remaining
296.804880 equity
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 8.00 SD: 10.00
Survived 7769 times with 240.48 avg remaining
186.830201 equity
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 10.00 SD: 10.00
Survived 8027 times with 161.66 avg remaining
129.764105 equity
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 13.00 SD: 10.00
Survived 8128 times with 102.65 avg remaining
83.437791 equity
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 15.00 SD: 10.00
Survived 8246 times with 85.94 avg remaining
70.865849 equity
</pre><hr />

So, fixing the stepdown value at 10, and varying the stepup, larger stepup values (waiting longer to stepup) give you a higher chance of surviving, (12% higher going from a stepup of 5 to 15), but at a HUGE opportunity cost. Clearly, you'd be far better off stepping up early rather than late, and simply redepositing if you went broke.

Now the other end, varying the stepdown:
<font class="small">Code:</font><hr /><pre>
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 5.00 SD: 5.00
Survived 5382 times with 633.26 avg remaining
340.821326 equity
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 5.00 SD: 8.00
Survived 6537 times with 490.59 avg remaining
320.700775 equity
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 5.00 SD: 10.00
Survived 7007 times with 422.73 avg remaining
296.204438 equity
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 5.00 SD: 13.00
Survived 7537 times with 306.03 avg remaining
230.654902 equity
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 5.00 SD: 15.00
Survived 7883 times with 236.34 avg remaining
186.310536 equity
</pre><hr />

While the previous results indicated that stepping up earlier was better, because of the very slow increase in survivability compared to the decrease in equity, the results here are almost the opposite, in a couple respects.

Stepping down late (waiting until you only have 5 buyins at the previous level is pretty late) gives the highest equity, but with a pretty significant failure rate. When stepping down that late, 47% of the time you'll go broke over the 250 games, and 53% of the time, you'll end up with an average $640 bankroll.

Stepping down sooner increases your survivability, just like stepping up later does, but the overall equity cost seems to be much smaller. While you pay a heavy price for waiting longer to step up, it appears the price for stepping down sooner is much less. To me, it seems worth sacrificing a small amount of equity for significant increases in survival rate.

In fact, it's very encouraging that the strategy I was already using (step up early and step down later) appears to be ideal for me, given my level of acceptable risk. In fact, SU:5 SD:10 is *very* close to what I was already doing anyway without actually attaching explicit numbers onto the transitions, and a 70% survival rate over 250 games seems perfectly acceptable to me, especially given the fact that it will leave me with around 20x my original startup on average. And if I have to redeposit, well, it will have to be more than $20 anyway, so I can re-figure a strategy from there.

[img]/images/graemlins/smile.gif[/img]

However, it's quite obvious that stepping up with 5 buyins and down with only 10 is not a good overall long-term strategy. When running with 5000 games instead of 250, waiting longer to step up and stepping down sooner *increases* your equity up to a point, rather than decreasing it, since your survivability has such a big influence over so many games. From there, it's just a matter of finding the right balance between stepping up and stepping down. Interestingly enough, it still appears to be better to step up and step down fairly early. Right around SU:10 and SD:20 seems to be a maximum, at least given a winrate of 58%.

But while it's not a good long-term strategy, it certainly does seem to be a good way to get a leg-up for pretty cheap, with a fairly minimal risk of failure, given that I'm starting with $20 in all the above scenarios.

[img]/images/graemlins/smile.gif[/img]

Oh, and just in case anybody's curious, leaving all the factors the same except for the initial buy-in, and making that $50 instead of $20, which is a much more reasonable number to start with, gives very similar trends, but *much* increased survivability rates, and bigger final bankrolls. I ran with $20 because that's about what I have in full-tilt right now, but here are the 250 game runs starting with $50:

Varying step-up:
<font class="small">Code:</font><hr /><pre>
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 5.00 SD: 10.00
Survived 8720 times with 653.82 avg remaining
570.132111 equity
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 8.00 SD: 10.00
Survived 9339 times with 498.47 avg remaining
465.520792 equity
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 10.00 SD: 10.00
Survived 9523 times with 385.32 avg remaining
366.936872 equity
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 13.00 SD: 10.00
Survived 9817 times with 212.85 avg remaining
208.956978 equity
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 15.00 SD: 10.00
Survived 9877 times with 158.25 avg remaining
156.300737 equity</pre><hr />
And varying step-down:
<font class="small">Code:</font><hr /><pre>
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 5.00 SD: 5.00
Survived 6904 times with 903.74 avg remaining
623.943522 equity
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 5.00 SD: 8.00
Survived 8301 times with 731.22 avg remaining
606.986914 equity
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 5.00 SD: 10.00
Survived 8780 times with 648.21 avg remaining
569.124702 equity
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 5.00 SD: 13.00
Survived 9335 times with 516.80 avg remaining
482.430339 equity
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 5.00 SD: 15.00
Survived 9545 times with 440.76 avg remaining
420.704468 equity
</pre><hr />

But over 5000 games, the difference in final bankroll is very small, but the survivability difference is quite large:
<font class="small">Code:</font><hr /><pre>
Trying 10000 times over 5000 games with $20.00. WR: 0.58 SU: 10.00 SD: 20.00
Survived 7823 times with 24260.49 avg remaining
18978.979982 equity
Trying 10000 times over 5000 games with $50.00. WR: 0.58 SU: 10.00 SD: 20.00
Survived 9598 times with 25040.24 avg remaining
24033.626326 equity
</pre><hr />

Amazing what a difference that $30 makes. Obviously in the long term, most failures happen pretty early.

Anyway, enough geeking out for one day. Thanks for the help everyone.

[img]/images/graemlins/smile.gif[/img]

[SkyReVo]

Nice post TNixon but there's one thing I don't understand:

Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 5.00 SD: 10.00
Survived 6955 times with 426.75 avg remaining
296.804880 equity

You say you'll go up a limit if you got 5 Buy-ins for it and go down with 10 Buy-Ins. Let's say you got 55$ which is 5x $11 so you step up to them. But it also is 10x Buy-in for the lower limit (5.50) which forces you (if we use your rules) to step down after losing just one SNG?

Also I wanted to ask whether you could do that with
10000 times / 5k games / $180. WR: 0.60 SU: 10 SD: 15 for me?
Would be pretty happy about it.

[dboy23]

The article quoted in the OP is complete trash but the program results are the nuts. If that is java, is there anyway I could get the source for it? I will pm you me email if you say yes.

I'd love to somehow find a way to see these results graphically. I will find a way to do that and take screenshots.

[Kharlog]

I'd also like to see the source code. PM me if you're willing to show it [img]/images/graemlins/smile.gif[/img]

[SkyReVo]

[ QUOTE ]
I'd also like to see the source code. PM me if you're willing to show it [img]/images/graemlins/smile.gif[/img]

[/ QUOTE ]

#2 please! I think this could be interesting for nearly everyone visiting the HU forum.

[stillnotking]

[ QUOTE ]
I'm having a little bit of trouble understanding why this would be true.

[/ QUOTE ]

Congratulations, this means you have an IQ in the triple digits. The article is complete nonsense. Anyone who thinks HU SNGs, with binary outcomes, would have higher variance than any other form of poker is simply an idiot.

In cash games the variance increases as the number of players decreases, while in freezeout tournaments the opposite is true. That's perhaps a counterintuitive result, but it's obvious with a little reflection, and even more obvious if you have some empirical data to look at. Which makes me think the author of that article did neither.

[TNixon]

[ QUOTE ]
You say you'll go up a limit if you got 5 Buy-ins for it and go down with 10 Buy-Ins. Let's say you got 55$ which is 5x $11 so you step up to them. But it also is 10x Buy-in for the lower limit (5.50) which forces you (if we use your rules) to step down after losing just one SNG?

[/ QUOTE ]

Actually, that's exactly what it means. After realizing that was the case for a lot of steps, I reconfigured the simulator so that the stepdown is specified in buyins at the current level, rather than the previous one, and the results indicated that's is exactly what you should do, for the best results: Play the highest buyin you can for whatever percentage of your bankroll you're willing to risk at once, even if it's for only one game. That's why when varying SU vs SD, the "sweet" spots all seemed to have a step-down value of twice the step-up, because so many of the steps are 2x increments.

Everything I've ever read on the issue claims you should step up with enough buyins to play a few (where "a few" varies from 5-10 or so) at the higher level before having to step down, which does make sense in some respects, but it seems pretty clear that statistically speaking, every variation from this "play the highest buyin you can" rule, is -EV.

Of course, statistics aren't everything, and there are other considerations too.

The primary consideration is that you just probably won't have as high a win rate at the higher limit, especially when you first move up. But it almost makes more sense to me to take a shot here and there, when your bankroll allows, rather than waiting until you have enough to stay there a while. If any necessary adjustments while playing against players at any particular level is gained after X games, then it really doesn't matter if X happens all at once, or scattered in the middle of lower buyin games, and it only takes a minor winning streak to shoot you solidly into the next buyin, where you would have been playing two or three times as many games at the lower level to get to the same spot.

There are potential psychological issues as well. For example, I can say from personal experience that stepping up multiple times and losing, then having to spend time at the lower level to build back up again can get a little frustrating. For example, right now I'm playing the 10s, even though yesterday I was playing mostly 50s and 30s, thanks to being caught by a super-nasty losing streak that started out with me losing an ungodly number of times when I got all my chips in as a 70-90% favorite, and bottomed out with me making really stupid frustrated moves. [img]/images/graemlins/smile.gif[/img] It's tough to go back to the lower level once you've had some success at the higher levels.

But, again, that's not really any different from stepping up with enough leeway for 5 games, and losing 3 or 4 in a row. It's still the same number of games played, they're just spread out more.

So from there, it's really just a matter of picking a number you're comfortable with. If you're starting with a small, very easily replacable roll, and are confident that you're actually a winner (which you're almost certainly not if you're just starting out with HU tourneys, but in my case I had already played somewhere in the neighborhood of 300 or 400 games at lower levels, with a fair amount of success, before I started trying to build from my last $27), then playing a third of your bankroll is not at all unreasonable. You just have to pull back more and more as your roll gets less and less replacable. For example, at $20 I was playing a third, around $100 or $150 or so I started playing a fourth, but when I hit $400 I was only playing a sixth. (unfortunately, I'm sitting at $100 right now thanks to yesterday's shennanigans, but just finished climbing back through the 10s, and ready to hopefully win a few 20s)

Another interesting result is that if you have a winrate anywhere near 60%, there seems to be very little reason to play with more than 10 buyins. A winrate of 58% gives good odds of survival even playing only 6 buyins, across 2000 games, and the equity penalty from there get pretty stiff, with minor increases in survivability paid for by large decreases in final bankroll. For example, here are some results with a win rate of 58%, and a buyin cap of $200, meaning you'd never play higher than $200 no matter how big your roll got:

<font class="small">Code:</font><hr /><pre>
Trying 10000 times over 2000 games with $50.00. SU: 10.00 SD: 10.00
Survived 9686 times with 28559.43 avg remaining
27662.665487 equity avg exit: 360 latest: 1555

Trying 10000 times over 2000 games with $50.00. SU: 20.00 SD: 20.00
Survived 9858 times with 14795.61 avg remaining
14585.512685 equity avg exit: 305 latest: 1013
</pre><hr />

Playing with 20 buyins gives a 2% greater chance of surviving over 2000 games, but *half* the final bankroll, on average, when you do. That's a pretty stiff penalty.

Here are the results for 6 buyins over 2000 games just for comparison:
<font class="small">Code:</font><hr /><pre>
Trying 10000 times over 2000 games with $50.00. SU: 6.00 SD: 6.00
Survived 8630 times with 33545.79 avg remaining
28950.015961 equity avg exit: 376 latest: 1971
</pre><hr />
Over 2000 games, there's a *very* small equity gain, for a fairly hefty drop in survival. Of course, over a smaller number of games (150-200), playing with 6 or 4 or even 3 buyins is a huge win, but you do have to back off at some point, whether it's after X games, or after you build up to a point where it would be very painful to try to replace your bankroll, either because you don't have the money, or because it would take so much time to rebuild.

[TNixon]

[ QUOTE ]
I'd also like to see the source code. PM me if you're willing to show it

[/ QUOTE ]
Well, it's in C, and I've got it torn up a little bit right now so the winrate is not specified on the command line anymore, but here's the source:

<font class="small">Code:</font><hr /><pre>
#include &lt;conio.h&gt;
#include &lt;stdlib.h&gt;
#include &lt;stdio.h&gt;
#include &lt;time.h&gt;
#include &lt;string.h&gt;

typedef struct
{
float cost;
float prize;
float winrate;
} Level;

#define MAXLEVEL 8
float STEPUPMUL = 20.0f;
float STEPDOWNMUL = 20.0f;

Level levels[] =
{
{2.15f, 4.0f, 0.58f},
{5.25f, 10.0f, 0.58f},
{10.5f, 20.0f, 0.58f},
{21.0f, 40.0f, 0.58f},
{31.5f, 60.0f, 0.58f},
{52.5f, 100.0f, 0.58f},
{84.0f, 160.0f, 0.58f},
{105.0f, 200.0f, 0.58f},
{210.0f, 400.0f, 0.58f},
{520.0f, 1000.0f, 0.58f},
{1030.0f, 2000.0f, 0.58f}
};

int trytolive( int attempts, int games, float money, double &amp;avgcash, int &amp;avgexit, int &amp;latestexit )
{
int fail = 0;
int rake = 20;
double totalcash = 0;
int totalexit = 0;
latestexit = 0;
for( int i = 0; i &lt; attempts; i++ )
{
float cur = money;
int level = 0;
float stepup = 0.0f;
float stepdown = 0.0f;
for( int j = 0; j &lt; MAXLEVEL; j++ )
{
if( levels[j].cost * STEPUPMUL &lt; cur )
level = j;
}
int win = (int)(levels[level].winrate * (double)RAND_MAX);
stepdown = levels[level].cost * STEPDOWNMUL;
if( level &lt; MAXLEVEL )
stepup = levels[level + 1].cost * STEPUPMUL;
for( int j = 0; j &lt; games; j++ )
{
cur -= levels[level].cost;
if( rand() &lt;= win )
cur += levels[level].prize;

if( cur &lt;= levels[0].cost )
{
if( j &gt; latestexit )
latestexit = j;
totalexit += j;
fail++;
break;
}

if( cur &gt; stepup )
{
if( level &lt; MAXLEVEL )
{
level++;
win = (int)(levels[level].winrate * (double)RAND_MAX);
stepdown = levels[level].cost * STEPDOWNMUL;
if( level &lt; MAXLEVEL )
stepup = levels[level + 1].cost * STEPUPMUL;
else
stepup = 100000.0f;
}
}
if( cur &lt; stepdown )
{
if( level &gt; 0 )
{
stepup = levels[level].cost * STEPUPMUL;
level--;
stepdown = levels[level].cost * STEPDOWNMUL;
int win = (int)(levels[level].winrate * (double)RAND_MAX);
}
}
}
totalcash += cur;
}
int wins = attempts - fail;
avgcash = (double)totalcash / (double)wins;
avgexit = (int)((double)totalexit / (double)fail);
return wins;
}

int main( int argc, char *argv[] )
{
if( argc == 4 )
{
if( _stricmp( argv[1], "dump" ) == 0 )
{
STEPUPMUL = (float)atof( argv[2] );
STEPDOWNMUL = (float)atof( argv[3] );
for( int level = 0; level &lt;= MAXLEVEL; level++ )
{
float su = 0.0f;
float sd = 0.0f;
if( level &gt; 0 )
sd = levels[level - 1].cost * STEPDOWNMUL;
if( level &lt; MAXLEVEL )
su = levels[level + 1].cost * STEPUPMUL;
printf( "Level %d: cost %f SU: %f SD: %f\n",
level, levels[level].cost, su, sd );
}
return 0;
}
}
if( argc != 5 )
{
printf( "Syntax: %s &lt;cash&gt; &lt;games&gt; &lt;stepupmul&gt; &lt;stepdownmul&gt;\n", argv[0] );
return 0;
}

srand( (unsigned)time( NULL ) );

float cash = (float)atof( argv[1] );
int games = atoi( argv[2] );
STEPUPMUL = (float)atof( argv[3] );
STEPDOWNMUL = (float)atof( argv[4] );
double avgcash;
int attempts = 10000;
int avgexit, latestexit = 0;

int lives = trytolive( attempts, games, cash, avgcash, avgexit, latestexit );

printf( "Trying %d times over %d games with $%0.2f. SU: %0.2f SD: %0.2f\n", attempts, games, cash, STEPUPMUL, STEPDOWNMUL );
printf( " Survived %d times with %0.2lf avg remaining\n", lives, avgcash );
printf( " %f equity avg exit: %d latest: %d\n", ((float)lives / (float)attempts) * (float)avgcash, avgexit, latestexit );
}

</pre><hr />