[jdonn]

hrothgar, on Feb 24 2010, 11:57 AM, said:

I recommend searching on the user ID Bebopkid and looking at his travails...

♥

[allfail]

Aha!

Now I know why there is such discrepancy. In fact, dealing randomly and give the best hand to South is quite different from dealing and discard those which South does not have the best hand.

The difference is subtle though, and that's why people (including myself) did not notice this previously. The problem comes in when there are ties: Imagine a layout of point counts. If the HCPs in all hands are different, then the accept rate (of the second method) is 1/4 whereas in the first method it is always 1. Now if there is a tie with two hands at the highest HCP then the accept rate for the second method would be 1/2, and vice versa. That is, the deal-and-throw-away-unwanted method has a higher accept rate for tied deals. But what point counts do the tied boards tend to have? They strongly tend to be weaker hands! For instance, a 10-count highest HCP board would always be accepted since every hand would have a 10-count and a 16-count board would be accepted only with a chance everslightly higher than 1/4. This in total results in a ~0.3 point difference in average, and ~2% chance of having a total of 24 HCPs or more as I have also checked the numbers with the second method.

Now that we sort this out, it is clear to me that BBO uses the deal-and-throw-away method. It should be stated clearly somewhere as there is a real difference.

I thank those who suggested me to look at previous records since they can't be sure about the details in my program and thus my trustability. However, for those who suggests a program like this can go wrong in many ways I think they should either check their own programming ability or stop their ignorance.

[kenrexford]

allfail, on Feb 24 2010, 06:06 PM, said:

Now that we sort this out, it is clear to me that BBO uses the deal-and-throw-away method. It should be stated clearly somewhere as there is a real difference.

I find it rather frightening that anyone would adjust some aspect of their bridge game to account for the slight difference between the apparently assumed method and the deal-and-throw-away method.

[cherdanno]

allfail, on Feb 24 2010, 06:06 PM, said:

However, for those who suggests a program like this can go wrong in many ways I think they should either check their own programming ability or stop their ignorance.

Says the person who
- first had to "stabilize his random generator" until it was working properly, and
- missed the obvious difference between the two possible ways of ensuring that South has at least as many hcp as everyone else.

[allfail]

To my own defense, my number was never off more than a standard deviation so even the not-so-good random generator works ok. In fact, what I changed was just from using the global random number to newing a class member in the RandomLib. I don't really know the difference between the two. But as so much as that it is the only perceptible difficulty in writing such a dealing program.

In addition, I hope I did present all the necessary details of what I did in this post so the "obvious" differences in the result between the two possible ways was not really obvious to all the responders on this post and you may be the only exception. If you think this is obvious why don't you be so kind and let us know instead of just replying a "nope"?

I acknowledged that you verified the 44.3% percentage in previous posts but I just couldn't figure it could be the same person who just responded a "nope" up there. Frankly speaking, why not just bypass the post if you don't want to share your knowledge?

On the other hand, strategy-wise these two method does have some impact. To begin with, we all recognize in a time-limited, total point game, the overall strategy would be to ignore as much partials as possible. Therefore, one might open quite heavily; for example, myself (previously) open only 16+ any shape any vul. and 19+ when nonvel, 3rd or 4th seat. This action can be interpreted as choosing to open with a threshold of game possibility.

However, why doesn't one just open when game is almost certain? The reason behind, of course, is that there is a cost of not bidding. The cost is not quite the time spending passout, instead is that every few boards even if you pass your opps might open. This, although mostly ending up in partials, costs a lot of time (maybe as much as 1.5 mins?) as GiB plays slower in partials. So to maximize the games in a given time period we have to open some number of hands provided that a game is somewhat likely.

Now we compare the two distributions. Note in the current method you would have twice to thrice as many of those tied hands compared to the first method, in which if you don't open your opps might. This reduces the optimum threshold of game possibility for one to open. I am not sure how large an effect this is, but among 20-25 boards I played in a tournament I would encounter, maybe 1/4 to 1/3 of the boards to be in the situation that I want it to pass out but some opp opened. Factoring in that there has to be a tie for us to have a difference, it's maybe like 1-2 boards per round.

I am not sure if this is an observable effect, but I do sense this kind of thing happened more often to me after the end of January... Is it even possible that BBO had changed the dealing policy by then?

[TimG]

I'm confused and I will probably regret asking, but...

It is my understanding that in Best Hand games, south has at least as many HCP as any other hand.

There are probably many ways to deal such hands, but two that come to mind quickly are:

1) Deal randomly, check to see if south has at least as many HCP as any other hand, discard if this condition is not met.

2) Deal randomly, check to see if south has at least as many HCP as any other hand, if not, then rotate* the deal and check again. Repeat as needed.

I don't see why these two methods wouldn't produce the same HCP average for South (and the same HCP average for NS).

Were you using a 3rd way to produce deals?

* Instead of rotating, you could find a hand that did meet the "at least as many HCP as any other hand" and switch that hand with south rather than rotating the whole deal. Should be the same result.

[allfail]

Yes it is indeed very confusing (at least to me) and I did think it is the same for a long time. But in the end
(1) will have S average 14.85 points and
(2) will have S average 15.1 points.

The reason is that in algorithm (1) you are less likely to discard a hand with 10-12 points than discarding a hand with a higher point count. Or to put in other words, the two distribution would be identical if (1) picks up any board distribution as often as (2) does. However, in the boards with the highest HCPs tied (1) is more likely to pick them up.

[Cascade]

This is indeed correct.

I can agree with allfail.

Here are the two probability distributions (calculated not simulated):

HCP  Rotating    Discarding
10	0.001054593	0.003827362
11	0.032779272	0.05152155
12	0.099157473	0.117524213
13	0.154414808	0.161255334
14	0.172892127	0.168641459
15	0.156873806	0.147463208
16	0.126468288	0.116571669
17	0.093131625	0.084991426
18	0.063996213	0.058159037
19	0.041428835	0.03759938
20	0.025741115	0.023355396
21	0.015114687	0.013713674
22	0.008401711	0.007622938
23	0.004476148	0.004061244
24	0.002236134	0.002028862
25	0.001057111	0.000959125
26	0.000466732	0.000423469
27	0.000196266	0.000178074
28	7.43E-05    6.74E-05
29	2.67E-05    2.42E-05
30	8.79E-06    7.98E-06
31	2.45E-06    2.22E-06
32	6.88E-07    6.24E-07
33	1.41E-07    1.28E-07
34	2.82E-08    2.56E-08
35	3.93E-09    3.57E-09
36	3.78E-10    3.43E-10
37	2.52E-11    2.29E-11

[allfail]

Thanks, Cascade.

It is quite interesting to note that if we look at the renormalization of the probabilities, say the 37 count, it shows a ~10% difference, which means the tied highest HCP deals occupied 10% of all the dealt hands! Of course not all of them would be bad for you (roughly 2/3 of them is bad), and this amounts to 1-2 deals every tournament, surprisingly confirming my estimate.

So here's something I'd like to ask: is it possible to change the dealing method of those besthand tournaments to the deal-and-rotate method? I just think that "whenever you get a 15 count or below you would be facing an anomalous high chance of somebody who has the same HCP as yourself" is not a very welcome thought.

Besides there's strategic inferences. I always think that there might be a way to setup GiB and double their contract to gain points. Knowing the distribution this way might slightly favor this approach when one have some 12-15 count perhaps...

[hanp]

Apparently allfail was at least correct that it wasn't obvious to all.

[TimG]

allfail, on Feb 24 2010, 10:19 PM, said:

Or to put in other words, the two distribution would be identical if (1) picks up any board distribution as often as (2) does. However, in the boards with the highest HCPs tied (1) is more likely to pick them up.

I'm still not getting it.

When the highest HCP is tied, (2) always picks it up, because (2) never discards a deal.

Is the problem that (1) discards the ties where south in not involved in the tie? So that in (2) ties are more likely to happen. And, ties are more likely to involve higher HCP totals?

[hanp]

Say the chance of having two equally strong strongest hands is P. If you draw such a hand then the chance that it is kept is 1/2, while the chance that any other hand is kept is 1/4. So after discarding the hands where south does not have (one of the) strongest hands, the chance that there are two equally strong strongest hands is:

(P/2)/ (P/2 + (1-P)/4) = 2P / (1 + P/4).

For small P this is almost 2P.

By rotating every deal so that south has (one of the) strongest hands you just get P.

[Cascade]

Here are the probabilities of ties with the two methods:

Tied   Rotate      Discard
0    0.905157035	0.821256074
1    0.08857877  0.160736425
2    0.005209602	0.014180139
3    0.001054593	0.003827362

Where the tied column tells us how many other hands we are tied with.