There is just as much chance that LHO holds singleton jack as singleton king so it is a guess whether to start with the ace or to run the Q. Because I don't know what to do, I will make the safety play for my contract.
Yes but leading the Q loses to stiff K both sides as well as KJx on the left, small to the 10 (or A if E shows out) only loses to stiff K/J offside.
Yes but leading the Q loses to stiff K both sides as well as KJx on the left, small to the 10 (or A if E shows out) only loses to stiff K/J offside.
I wouldn't say the Q loses to stiff K on either side - it simply doesn't gain the overtrick. You are right that the leading the ace gains against stiff king in either hand while the Q lead only wins with singleton jack in LHO.
I still make the safety play at all forms of bridge unless table feel dictates otherwise.
"Injustice anywhere is a threat to justice everywhere." Black Lives Matter. / "I need ammunition, not a ride." Zelensky
I wouldn't say the Q loses to stiff K on either side - it simply doesn't gain the overtrick. You are right that the leading the ace gains against stiff king in either hand while the Q lead only wins with singleton jack in LHO.
I still make the safety play at all forms of bridge unless table feel dictates otherwise.
At MPs not making the overtrick other people are making is losing.
I think you make the safety play. You only make 5 if you can nail the stiff K by playing the A. Leading towards the A, covering if RHO follows or playing A if RHO shows out, lets you handle both 3-0 breaks. If RHO plays 9, you only lose when LHO has stiff K and playing the A allows you to make . It makes no difference in the other two cases where 9 was stiff or from K9.
But in big and heterogeneous field, I'd always chose to make the game (see below)...
Even in heterogeneous fields there exist flattish contract boards. One has to be able to judge what boards are going to be flat or not based on looking at the hand and your knowledge of the skill of the field. On this hand I'd think the percentage in 4s to be quite high regardless of field. The time to play safe IMO is in low high card point good slams in a weak field, not 24 hcp games with 10 cd fit and shape outside.
Quote
In every deal, a percentage advantage (if exists) of chasing an overtrick is fixed, always the same. Points advantage of making a game (or slam) tends to grow in bigger and more heterogeneous fields...
This isn't correct. The percentage advantage of the overtrick grows proportional to the number of people in the same contract. As more people play the same, overtricks value goes up, the advantage of just being plus goes down. Here, the edge of the overtrick is rather small over the safety play, so you need > 80 percent in game. But if the edge were larger, e.g. a safety play to guard against a 4-1 break, rather than just playing for 3-2, the breakeven point would be much, much lower. If you are in habit of playing safe when in normal contracts your mp score will usually suffer.
Interests:Cricket,Photography,Paediatrics and Community Medicine.
Posted 2018-November-18, 02:29
Sir. as it boils down it all depends upon 1) the type of event.2)the maths of a 3/0 break 3 ) how many of that will west who has already shown both AK of clubs likely to hold a singleton King of spades..4)How many times will East hold a singleton King,when playing the A will win.5)How may times will East hold KJx when leading a small one from dummy and making the safety play wins is likely.. Sir.as I go by the book plus the table feel, if playing a MP event I shall cash the Ace.However in a IMP or team event I shall take the safety play at the cost of one IMP. A friend of mine, missing KQx did not take the safety play when playing in a small slam and lost a handful of IMPS and gained quite a few jeers.
At IMPs, you take the safety play for one loser in trumps. What do you do at Match-point Pairs (mixed field)?
I'm not sure I would play safe even at IMPs! Well, I actually think it is safe to lay down the ace, because a 0-3 trump split must be almost Fukushima-rare given the auction and the vulnerability. (The a priori probability of a 0-3 trump split most definitely doesn't apply!)
I'm not sure I would play safe even at IMPs! Well, I actually think it is safe to lay down the ace, because a 0-3 trump split must be almost Fukushima-rare given the auction and the vulnerability. (The a priori probability of a 0-3 trump split most definitely doesn't apply!)
Not really, how attractive do you think it is to make a 2 suited bid with something like void, xxxxx, xxxxx, AKx or overcall with the 6-4 version
At MPs not making the overtrick other people are making is losing.
Yes, but that assumes everyone else is in the same contract and they do not take the safety play. My experience finds few hands of that type. But, they do exist and field quality has much to do with it.
At the same time, my choice probably shows my bias against MPs.
"Injustice anywhere is a threat to justice everywhere." Black Lives Matter. / "I need ammunition, not a ride." Zelensky
The number of people taking safety play is irrelevant. Each one taking a safety play just changes your dropping a stiff k into a win a mp instead of tying, rather than tying instead of losing a mp. Or if void, vice versa. The math works out the same. The relevant number is the number of alt scores between -100 and 620, rather than those in game.
In every deal, a percentage advantage (if exists) of chasing an overtrick is fixed, always the same. Points advantage of making a game (or slam) tends to grow in bigger and more heterogeneous fields...
This isn't correct.The percentage advantage of the overtrick grows proportional to the number of people in the same contract. As more people play the same, overtricks value goes up, the advantage of just being plus goes down. Here, the edge of the overtrick is rather small over the safety play, so you need > 80 percent in game. But if the edge were larger, e.g. a safety play to guard against a 4-1 break, rather than just playing for 3-2, the breakeven point would be much, much lower. If you are in habit of playing safe when in normal contracts your mp score will usually suffer.
What isn't correct? I reread my statement and it's short, clear and entirely correct.
And yet you somehow managed to misunderstand me. And I don't understand you (talking about the red part).
When I say "percentage advantage of chasing an overtrick", I am talking about the frequency. The probability that LHO has a singleton king is greater than the probability that he has a void. The difference is about 2.63 percentile points. That's the advantage of chasing an overtrick that remains the same, regardless of the size or other attributes of the field.
If you have in mind some other "percentage advantage of chasing an overtrick", unknown to me, please specify.
I was talking about percentage advantage and points disadvantage of chasing the overtrick. You claim that my statement isn't correct and repeat my phrase, but with the word "chasing" removed. Now the question is - whether you're talking about the same thing (and the important word is missing, for any reasons)
OR (more probably)
you're trying to prove the incorrectness of my statement by talking about something else...
The latter approach doesn't look entirely healthy, but I'll not be petty, I'll try to understand your statement.
The percentage advantage of the overtrick grows proportional to the number of people in the same contract.
If you wanted to say:
The POINTS DISadvantage of (chasing) the overtrick SHRINKS proportional to the number of people in the same contract.
That's plausible for discussion. If that's what you meant, we can continue.
If you wanted to say:
The percentage advantage ... grows ... in the sense of probabilities
That's obvious nonsense, the field has no influence on probabilities of card distributions.
If you wanted to say:
Some OTHER percentage advantage ... grows ...
You should first introduce the very existence of the thing you're talking about and THEN claim that "the thing" grows...
If you wanted to say something entirely different, please specify.
As more people play the same, overtricks value goes up, the advantage of just being plus goes down.
Before I try to interpret this sentence, let me notice one small, but absolute flow:
We weren't talking about "just being plus", we're talking about making a GOOD contract. In this particular case, 4s is the very best contract available to NS; 620 can't be bad and can be a great result (most of the time, it will be a VERY good result).
Making 170 or 140 would be "just being plus"...
Did you want to say - when more people play the same, the overtrick will get you a better result? If so:
- The whole setup is wrong (I don't elaborate, because it's far from certain that I'm correctly guessing your claim)
- Specifically, if you watch that parameter alone (the number of pairs playing the same contract), overtricks value (if we're talking about the result) doesn't go UP, it goes DOWN. You're more than qualified to check that...
* * *
I could (and perhaps should) just say "I don't understand you", instead of writing 900 hundred miles long post, but I didn't want to be perceived as arrogant. I hope you'll be less obscure next time...
Apologies to everybody, for low efficiency.
Matchpoint scores can be expressed as either raw point totals, or as percentages of maximum available.
Your matchpoint expectation of one action vs another, depends on BOTH frequency of gain (this part fixed, for a particular hand, I agree) AND the number of points swung by the action (NOT fixed, field dependent, one won't know until scores posted, but you can try to estimate based on hand and experience) Yes the frequency of gain of the uptrick in this case is small edge vs the losing case, but this is multiplied by the number of people in the same contract. So total gain (or non-loss, if the field is all playing for overtrick) is more, the more in game.
Let's say 90 percent of field plays 4s, the rest are plus some amount < 620. Half the field decides to play safe, half goes for the uptrick. The king is stiff offside. So playing for the overtrick scores 77.5 percent. Playing safe only scores 32.5 percent, which is bad. You lose 45 percent relative to me. Half times the proportion in game.
Now another day, same board and players, except now void on left. Now playing safe scores 77.5 percent. I score 22.5 percent, losing 55 percent relative to you. Or half the proportion in game plus the percent not in game and below 620.
So you gain more when you are right, but since my gain is more frequent, when this proportion of people are in game, I win in the long run, very slightly. 13.16% x 45 vs 10.53% x 55. But +ev is +ev, and you take every edge.
Now suppose the field is crazy unrealistically bad, and half the field misses game for some reason. Now an overtrick scores 87.5 percent when it works, while safety play still scores 62.5 percent, losing only 25 percent. Again the margin is half of those in game. So as you can see, if more people are in game, you lose more when wrong about the uptrick than when fewer people are in game. 45 vs 25.
And when safety play was necessary, now playing safe scores 87.5 while going down is only 12.5 percent, losing 75 percent. Under these conditions, safety play wins easily.
So you can see that the amount one gains or loses depends on the percentage of field in same contract. If nearly everyone is in game, making overtricks is essential when possible, if you don't make one the field is making you can get a very poor score. 620 could be a 10 percent score, if 90 percent are scoring 650!
So one has to be able to separate boards where one third or half will screw it up, vs ones where you expect nearly all to be in same contract. This board looks mostly flat to me, and original poster seemed to indicate it was so in practice.
Each board will have a breakeven point for percent of field in game vs other relevant contracts, depending on the edge of the overtrick play vs the safety play. If gap is small, you need very flat board to go for the uptrick, 80+ percent as here. If the gap is huge, one needs much less.
Classic example is Axx vs kqxxx and a now entryless dummy, need 4 tricks to make. If this is a normal 2nt p 3nt and you are going to take safety play of ducking when lho follows to 2nd round, at matchpoints, because of your philosophy of making game is always good in heterogeneous field, to me you are nuts.
For this hand, you are less nuts because the edge is quite tiny, but you are still wrong mathematically if 81+ percent in game, of the scores in range (the 800s mentioned by Tramticket don't matter, as those are lost mp regardless, are excluded from the calculation). And you will get poor board when field is making 650.
Oops mea culpa. I was sloppy earlier trying to do algebra on a phone without access to pen and paper, and the breakeven for this board is actually 88.9 percent of field in game which of course is why the edge for the 90 percent field examples is so low. So perhaps safety play is right on this board. How many were in game, Tramticket?
But the general gist of my prior statements re Povratnik remain correct. For any given edge of overtrick vs safety play there exists some threshold, if proportion of pairs above that in same contract, the safety play should be ignored. Below that threshold, play safe. When edge is very small you need very high uniformity, like 90 percent here. When the edge is large, like playing for 3-2, not playing safe against 4-1 break, you only need very low uniformity to go for the overtrick, like 35 percent of field in same game.
@Stephen Tu
Another misunderstanding . When I said "I don't understand you", I had in mind the two red sentences only. I do understand the rest of your theory. If I wanted to express in one sentence all that I learned about your views, I'd say: Mathematical expectation is Alpha&Omega
That's my perspective too, so all these misunderstandings are only temporary...
Stephen Tu, on 2018-November-20, 01:34, said:
Matchpoint scores can be expressed as either raw point totals, or as percentages of maximum available.
Aaaaaah, now I understand why did you use the word "percentage". Though I still don't like the term "percentage advantage" the way you used it, it doesn't really matter. All that I wanted to ask - is answered by this one sentence.
The rest of the post explains your general approach. It wasn't necessary, because I've been understanding it (and generally approved) from the very beginning.
However, your long essay wasn't completely vain. It helped me to clearly see what's wrong in your set up.
Please, don't understand me wrongly - you made an impression of a sound thinker and I mostly agree with you. I'm pretty sure, you'll choose the same line as me in vast majority of such cases. When we choose different lines (due to different tastes), they'll usually have the similar mathematical expectation.
I don't discuss with you so intensive because I think YOU need my advice. I'm doing this because of many users who'll read your posts without understanding and harden themselves in wrong beliefs.
To shorten this post, I'll cross over the rest very briefly:
Paragraph No 2
Stephen Tu, on 2018-November-20, 01:34, said:
Your matchpoint expectation of one action vs another, depends on BOTH frequency of gain (this part fixed, for a particular hand, I agree) AND the number of points swung by the action (NOT fixed, field dependent, one won't know until scores posted, but you can try to estimate based on hand and experience) Yes the frequency of gain of the uptrick in this case is small edge vs the losing case, but this is multiplied by the number of people in the same contract. So total gain (or non-loss, if the field is all playing for overtrick) is more, the more in game.
I don't know whether to call the red part wrong, incorrect, or just wrongly set - but I certainly can't accept it as correct. It will be explained...
Paragraphs No 3-7
They brought nothing new.
Paragraph No 8
Stephen Tu, on 2018-November-20, 01:34, said:
So you can see that the amount one gains or loses depends on the percentage of field in same contract. If nearly everyone is in game, making overtricks is essential when possible, if you don't make one the field is making you can get a very poor score. 620 could be a 10 percent score, if 90 percent are scoring 650!
That's said less sloppily than in the previous post, but IMO, it's still below the passing bar. It will be explained...
Paragraphs from No 9 to the end
They cover your general approach, that I already perfectly understood.
* * *
So I owe you an explanation for painting some parts of your post in red. I'll do it in separate post. Just before that, I'd like to prevent remaining of any loose ends...
Outside of red parts (in this and previous post), I don't see any source of potential disagreement between you and me. But I have to check that with you.
Do you see anything else that needs a further discussion? If not, I'll write an essay of my own in the very next post.
Stephen Tu's argument seems correct: At match-points, your decision whether to take the safety-play largely depends on your estimate of how many people will play in 5 or more ♠s rather than in part-scores.
If you judge that significantly more people will play in part-scores, then you should take the safety-play.
If you judge that significantly more people will play in 5+ ♠, then you should cash ♠A.
If you judge the numbers about equal, then you should cash ♠A.
Sorry, before I finished this post, a RL issue had forced me to abandon the works for the time being. It turned out to be good, though, since the longer time usually makes a shorter and nicer post . This post has to be big and ugly, but if there wasn't a sufficiently long pause, it would be MONSTROUS!
Stephen Tu, on 2018-November-24, 03:11, said:
Whatever, tell me exactly where you think I'm wrong. Other than my initial blunder calculating the breakeven point which I corrected.
God, if I was practical enough to just ask you that, a lot of yours and my time and effort would be saved...
OK, I'll give you what you're asking for, but I would really like to shorten yours and my further posts, so I'll try to crystallize the things on global level first. It's by far more important than petty mistakes.
About the process, for Stephen Tu exclusively. Of course, others also can read, but can and will be heavily bored...
Spoiler
My first objection is a certain layer of vagueness, indetermination... English is not my native language, so I don't know what word is the most appropriate. Let's see...
You're trying to prove that trying to drop the West's king is clearly a better line than safety play, without EVER actually saying it. And you know what? Not only that you failed to convince me that your line is better; you didn't even convince me that it's your real opinion! I seriously tend to think that the main reason for your efforts is sheer inertia. How this chain of misunderstandings began?
You and Cyberyeti sent me about the same message. In my words: There are certain boards where you may risk the game having only a slight frequency advantage; such line can even bring some +EV. This board is one of them.
I answered to both, without expressing any disagreement. But you somehow misunderstood me and mistakenly said (error No 0) that my final statement wasn't correct. In the next post you've stiffly withdrawn this qualification, but continued to discuss as if I'm somewhere wrong. Not only you didn't say where do you thought I'm wrong, you haven't even pointed in the direction of my apparent wrongness. You're explaining general things - probabilities, mathematical expectation, mechanism of pair tournaments... I never asked, but I am asking you now - what made you to assume that I didn't know all that stuff you're writing about?
All in all, you've written a lot, but didn't offer one single useful conclusion.
I'm not saying that vagueness (or whatever is the right word) is a mistake, but have a mild objection. I'd really like to clarify - what are we actually talking about.
I also see a certain layer of one-sidedness, that's my second objection. In the desire to make this post as short as possible, I'd try to skip the elaboration of this one, maybe just a comment or two in the rest of the post.
Lets go to real mistakes...
The first one is only hypothetical. I am not claiming the very next statement as a fact - it's just my humble opinion:
Your first mistake
You are trying to prove the hypothesis that you aren't deeply convinced of.
If I'm right, this job was never very promising. It's no wonder that you faced difficulties...
Now lets go to to open part of the post.
Your second mistake (a MAJOR one)
You aren't splitting the questions "MAY I" and "SHOULD I".
You MAY choose a certain line of play if it's reasonably good; if it has similar mathematical expectation as other lines. You SHOULD choose a certain line of play if it's clearly the best; if it has the greatest mathematical expectation or is clearly the best for some other needs that you consider important (it's the wisest, the most practical... or whatever).
In this case you correctly estimated that, due to extraordinary high percentage of pairs who'll bid the same contract, you can afford to risk the game. There is no need to repeat your arguments, you did a proper job. Cashing the ace has about the same expectation as safety play, so you can play it.
But SHOULD you? That's a new question, yet you're using the same old means. That means served you well with the old question (may I, could I), but are powerless with the new one (should I). Your writings mainly consist of general considerations and (far from correct) focusing of (wrong) parameters. That gave us satisfactory (over)proof that risking the game is legitimate choice, but I see nothing that could even moderately corroborate the new thesis - that cashing the ace is actually a better line.
Using the common language, I'll just say:
You're looking for answers at wrong place.
THAT's your main mistake. All specific errors are just natural consequences of this one. So I'll not try to determine whether any of your deep convictions are wrong (they probably aren't). I'll concentrate on positive story that I have to tell - how these things really operate...
* * *
But first I need to introduce some terms. They aren't recognized in theory, I just coined them for this occasion.
OUR group - The group of NSs who played the same contract as we did Local TOP, ZERO - The best/worst result in OUR group 800 group - The group of NSs who made better result than our best possible (4s+3) 300 group - The group of NSs who made better result than our worst possible (4s-1), but worse than 4s= BOTTOM group - The group of NSs who made worse result than our worst possible (4s-1) INNER group - The group that consists of OUR group and 300 group
We can't reach 800 group under any circumstances. With any normal line of play, bottom group is also out of reach. So 300 is obviously the most important. More about that, later.
* * *
In your two large posts, there are four parts that I painted red, because (very) wrong conclusions could be implied from them. I'll cross over them, not necessarily in chronological order.
Once again - whether you meant it wrongly, or were you just sloppy - isn't nearly as important, as positive story that I'm going to tell...
Stephen Tu, on 2018-November-18, 01:19, said:
As more people play the same, overtricks value goes up, the advantage of just being plus goes down.
I already partly commented this. Whatever my previous comment is missing, will be made up by my further story...
Spoiler
Before I try to interpret this sentence, let me notice one small, but absolute flow:
We weren't talking about "just being plus", we're talking about making a GOOD contract. In this particular case, 4s is the very best contract available to NS; 620 can't be bad and can be a great result (most of the time, it will be a VERY good result).
Making 170 or 140 would be "just being plus"...
Did you want to say - when more people play the same, the overtrick will get you a better result? If so:
- The whole setup is wrong (I don't elaborate, because it's far from certain that I'm correctly guessing your claim)
- Specifically, if you watch that parameter alone (the number of pairs playing the same contract), overtricks value (if we're talking about the result) doesn't go UP, it goes DOWN. You're more than qualified to check that...
Stephen Tu, on 2018-November-20, 01:34, said:
Yes the frequency of gain of the uptrick in this case is small edge vs the losing case, but this is multiplied by the number of people in the same contract. So total gain (or non-loss, if the field is all playing for overtrick) is more, the more in game.
It would be actually correct, if you could keep it totally isolated, "frozen in time and space". But you can't.
You have a subtle feeling about the matter, but you neglected the rest of the picture. Continue to read, it will be clear...
Stephen Tu, on 2018-November-18, 01:19, said:
The percentage advantage of the overtrick grows proportional to the number of people in the same contract.
No, it doesn't. It can grow or shrink and neither is proportional to the number of people in the same contract....
Let's assume that every NS pair played the same contract. As it's already said - you have frequency advantage, I have points advantage. In this case, my advantage is clean ZERO. So your frequency advantage, no matter how small, literally guarantees you +EV.
The severest case is a team match. Your f-ad works in its fullest, giving you the maximal 2.63 percentile points of +EV. Small, but relevant.
Now, lets add the third table, making it a pair tournament. Now you can't beat me for 100%, because of sharing results. If you have clean TOP, I don't have ZERO, but 25%. If I have clean ZERO, you don't have TOP, but 75%. So whenever I'm wrong, you beat me for 75%. Your +EV is now 2.63%*75%=1.97%.
Let's add the fourth table. Now you beat me for two thirds (formula is n/2(n-1), n is the number of tables where the board is played on). Your +EV is now 2.63%*2/3=1.75%.
And so on... Your original example (ten tables) gives you +EV of 1.46%; 16 tables ~1.4%.
So we can see that your advantage haven't grown, it shrank. And not proportionally, but fast in the beginning, then slower and slower. In the range typical for club tournaments (8-16 tables per board) it shrinks only a promil - from ~1.5% to ~1.4%.
So what's the truth then? Does the number of people in the same contract work in my favor?
No, it just pushes your EV toward the LIMES. I chose this particular case, because it's the only one simple enough to allow us comfortable monitoring; more complicated cases also have a limes, not necessarily the same. In this case limes is 1/76 = 1.316% (Exactly one tenth of original probability! I do believe it's only coincidence, but I'm not 210% sure. I need to check similar problem, with different probabilities...).
So whenever your EV is above 1/76, it will fall in smaller and smaller steps down to 1/76, never reaching it. Whenever it's below 1/76, it will raise in smaller and smaller steps up to 1/76, never reaching it. So it works for me or for you, depending whether your current mathematical expectation is above or below the limes.
Specific truth
Exact number that serves as limes varies depending on parameters, but when number of pairs that plays the same contract grows - it invariably pushes your mathematical expectation toward the limes - whether it's good or bad for you.
General truth
Useful answers are elsewhere.
You corrected yourself in the next post. Or at least I thought so...
Stephen Tu, on 2018-November-20, 01:34, said:
So you can see that the amount one gains or loses depends on the percentage of field in same contract.
I reread your post (this time more carefully) and realized that this silly statement is offered as some kind of point after several entirely sound and correct paragraphs (in spirit, I didn't check the numbers).
Until then, I presumed that it was correction of the blunder from previous post.
So what did you try - to make a correction, to make a new statement, or both?
If I take your statement literally, it's technically correct, but meaningless. My answer is: Yes, everybody's gains or loses depend on percentage of the field in every contract that was played in the board. So what?
If I take it as correction of previous statement, I already told - it's better, but still not good enough. Let me continue my story and all will be revealed...
In your original example, the board has been played on ten tables. Everybody bid 4s, but one pair faced successful sacrifice. Our group has impressive 90% of the field, but your EV is far from impressive - only +0.146%, TEN (!) times smaller than when it has 100% of the field. Lets drop the percentage once again, to see what's going to happen...
Case A
If we remove a pair that was in 300 group and add two pairs to bottom group and two pairs to 800 group, our group will have 9 of 13 or 69.23%. Your EV is now +0.98% - much BIGGER than with 90%...
Case B
This time we'll just add another three pairs to 300 group. Our group again has 9 of 13 or 69.23% of the field. But now, your EV is -2.52%...
So with nearly 70% of the field, your EV could be +0.98%, but could also be -2.52%. What's wrong?
The METHOD is deeply wrong. Save the few complete but limited examples that you gave us, you're trying to achieve something by analyzing our group in isolation. That's bound to failure. In this particular case, you're referring to size/percentage of our group (in the field)as if it's somehow possible to change it, without also changing the percentage of other groups. No wonder results are so erratic...
Specific truth
When the percentage of our group changes, the outcome depends on what happened to OTHER groups.
General truth
Useful answers are elsewhere.
* * *
As I already hinted - the key group is 300 group. The influence of all other parameters put together is practically negligible, in comparison with influence of the ROW NUMBER of pairs in 300 group.
Yes, our mutual result depends on percentage of our group. But in INNER field, not in entire field. And even that is only theoretically. In practice, the only thing that really matters is - whether 300 group has 0, 1, 2 or MANY members. (MANY = 3, 4, 5...)
We could say that row number of pairs in 300 group decides the ORDER of MAGNITUDE. Sizes of other groups in the field (including ours) decide only where we are in the same class of magnitude. In normal live tournaments, there are only four classes of magnitude - 0, 1, 2 and MANY. Lets say that p0, p1, p2 and p4 are their respective probabilities. If you're writing a book, you'd have to find the credible way to estimate and compare these probabilities.
In practice, you can ignore p1 and p4. On bigger tournaments, you can also ignore p2. But you can never, ever forget about P0. Class 0 is bread and butter of your "collecting small profits" strategy.
Lets return to the beginning. You just estimated that cashing the ace is legitimate line of play. Fine. You're done with phase 1 and can immediately play the move if you like it. We do play bridge primarily for enjoyment, after all.
But if you have greater ambitions (want to judge whether this line is actually better), you're already in the phase 2. Now you should entirely change the perspective and take a view from the opposite side. Forget all obsolete arguments and immediately focus on the most essential question:
What's the probability THAT
NOBODY stopped in wrong contract AND
NOBODY faced a successful sacrifice?
Estimate p0 and if you're impressed - go for it! If you aren't - maybe you should just make the game your partner bid...
How could you convince the audience that your decision is right? No idea, I don't think you can. But any sensible try has to be heavily leaned on your estimation of p0. Any other aproach is just mumbo jumbo...
* * *
If you remain biased in the favor of cashing the ace, I'm not worried. I know you're able to take care of yourself. I just feel sorry for many others who'll risk a good game not only when you do it, but also in many other cases when they clearly shouldn't...
Disclaimers:
Spoiler
Disclaimer 1
I planned to explain things in much greater detail, but sheer size of the task has overwhelmed me. This post (in severely shortened version!) is still extremely long and I am very tired...
Disclaimer 2
Maybe I made some error(s) in the math, but wholeness is sound. If someone finds an error, I'll correct it and all my conclusions will remain valid...
Disclaimer 3
All my discussion applies on cases when we bid a slam or game. And not just a game, but a really good game.
When I play e.g. 2 in minor, I'm at least as aggressive as Stephen Tu. I'll frequently try suspicious chances, like some unpromising finesse, just to somehow force an overtrick. When I go down, I accept -50 (or even -100) with philosophical calmness, because I know they have 110. So partial scores are entirely different story...
Help
Winstonm, on 2018-November-17, 20:19, said:
