Iterations and Standard Deviation

This material was originally produced by BuckyBall and can be found here: http://forum.winholdem.net/wbb/viewtopic.php?t=3742

Standard deviation of prwin/prtie/prlos by Number of Iterations (NIT)

Figure 5.21‑59 Standard Deviation by Iterations

Figure 5.21‑60 Probability by prwin

Here is how to use the data in the above graphs:

Decide what error you find acceptable 99% of the time. For example, if you are happy with getting prwin's of 0.846 - 0.852 99% of the time for AA headsup preflop, your error would be (0.849) +- 0.003.

That error is "3 standard deviations". To get 1 standard deviation, divide by 3 (to give std. dev. = 0.001).

Find that standard deviation in the top chart and read off the number of iterations required. In our case, it would be 100,000 required iterations.

The prwin values you obtain will be within 1 standard deviation 67% of the time, within 2 standard deviations 95% of the time, and within 3 standard deviations 99% of the time.

5.11 Weighted prwin

A weighted prwin is one which is calculated by making informed selections of the cards that opponents are likely to hold, and using these cards in the calculation of your win probability. Factors which can be used to make inferences about cards include willingness to pay to see flop, betting behavior when faced by common cards, and historical information about opponent behavior. Essentially it aims to put your opponents on cards as accurately as possible.

OpenHoldem’s “Weighted prwin ” is based on a fairly simple model using pre-flop behavior. For DLL developers, there is also the option to use {Enhanced prwin}.

Figure 5.11‑61 Weighted prwin model

To enable Weighted prwin, various user-defined functions in a formula file must be defined to return the following values, expressed as handrank169 values (see {Calculated Symbols}):

function	explanation
f$mustplay	Opponents holding cards better than this are assumed always to play them. This allows for the situation where opponents slow play the strongest hands in the hope of milking more from the pot with them.
f$topclip	Indicates that hands better than this are less likely to be played, since the betting level is too low.
f$willplay	Hands better than this will be played.
f$wontplay	Hands worse than this are highly unlikely to be played.

The success in using a model of this sort depends on the data available to allow the formula developer to devise appropriate expressions for the f$functions outlined above. Currently in OpenHoldem the most accessible is floppct - the percentage of opponents seeing the flop averaged over a user definable period. This is a reasonably accurate expression of the strength of cards which opponents are prepared to play. An example of its use could be:

f$willplay

handsplayed<10?72:(180*floppct)>(nopponents*2+1)?180*floppct:nopponents*2+1

f$wontplay

f$willplay*1.4

Note: If f$willplay evaluates to zero, prwin weighting will be turned off. f$wontplay should not be less than f$willplay.

Although this is primitive it gives the single greatest advantage of using a weighted prwin rather than an unweighted one; it eliminates the cards from the calculation which opponents are unlikely to invest any money in. Since it depends on statistical information a constant value is given until floppct has had time to establish itself. The '180' is because 90 is the midpoint of handrank169 at which the number of hands above and below are the same (the stronger hands tend to have fewer possible examples).

Ideally the calculation should be based on betting averages at different levels, since it is what opponents are prepared to pay that determines the probable strength of their hand, and should be recorded on an opponent (and even position) basis. The problem with more refined data accumulation is that it takes an increasing number of hands to gain significance, and is sensitive to changes in playing conditions, such as the number of players, which can alter opponent behavior. Nevertheless statistics of this sort have been used successfully in other bots based on a weighted prwin, and may be available from OpenHoldem in the future.

The OpenHoldem weighted prwin uses a handrank169 table (see {Starting Hand }) which is based on published data about hands players actually take to flop, and varies from conventional ones giving win probabilities, particularly in the way it favors suited hands! In the absence of statistical support it is not recommended that you implement f$topclip or f$mustplay unless you have a good picture of the standard of play at a table.

If f$willplay is absent, or returns a zero value, the prwin weighting is turned off. It will also be turned off if your f$P function specifies more than 13 opponents. If your opponents at flop include a big blind who has not been raised, no weighting will be applied to his hand, since no assumptions can be made about it.

You may need to change your f$P function if using weighted prwin. The “P factor” established in your f$P function is in fact a kludge to compensate for the fact that opponents who fold have weaker hands. prwin weighting allows for this, and therefore a prwin bot is best coded using the actual number of opponents playing. A suggestion for an f$P function, if you use prwin pre-flop, could be:

(nopponentsplaying==0)? 0: //zero if no opponents

(br>1)? nopponentsplaying: //just use actual opponents post-flop

(handsplayed<10)? nopponentsplaying: //fallback until floppct becomes reliable

(nopponentscalling+nopponentsraising+nopponentschecking*floppct+0.49)<1? 1: //this should never happen, but.....

nopponentscalling+nopponentsraising+nopponentschecking*floppct+0.49 //make an estimate for opponents who have still to act

This attempts to estimate the number of opponents you are likely to face when you act pre-flop, and is most significant when you are in early position. Whilst the principle is valid, that actual function should be thoroughly tested in practice.

5.12 Enhanced prwin

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