Bidding system designed by computer

[helene_t]

Yeah if the response scheme has adapted to 1♠ showing spades and now suddenly it becomes 1NT that shows spades, then it may not function properly.

 

But what I envision is that the synapses governing responses evolve over a time during which they have been exposed to constantly changing opening schemes. What is sustainable in such an environment is not a response scheme compatible with a particular opening scheme, but rather a generic scheme which can handle unforeseen situations like "LHO makes a Missery 3♣ opening defined as blablabla, p makes a kgr 3♦ overcall defined as blablabla, RHO makes a KenRexford 3♠ freebid defined as blablabla, now you". What is needed to make that decision is

- the information conveyed by the other three players

- which bid is the latest, by whom, has it been doubled/redoubled

It can't afford to assume any particular relation between the last bid made and the information conveyed by the player that made it, because that relationship is subject to change from generation to generation.

 

I am not claiming that this is feasible. Maybe it is. Maybe one day it will be, when we get quantum computers etc. In any case I think it is a better approach than trying to establish a direct link from the auction-so-far to the meaning of calls.

[hotShot]

I think that the focus is to much on "the bid" right now.

 

Perhaps one should deal with abstract actions like "raise", "support", "pass",

"preemptive raise", "limit raise", "strong raise", "game forcing", "slam forcing" .....

[bab9]

I think that the focus is to much on "the bid" right now.

 

Perhaps one should deal with abstract actions like "raise", "support", "pass",

"preemptive raise", "limit raise", "strong raise", "game forcing", "slam forcing" .....

Should we be looking at "interference"?

 

Would we be simplifying the initial development if we looked at something like the sequence (1♥) - P - (2♥) - ? and have the computer/system determine the best bid? In which the 1♥ and 2♥ bids can be well defined, say using SAYC.

 

I recently read that the idea of "support" is based on the law of total tricks. It would be interesting to see if a computer would base support on a similar method, or have another determinant to indicate support.

 

With the work done on the neural networks so far, are you in a position to indicate if a response to an opening bid can should support? Either in a natural or artifical sense.

[Dirk Kuijt]

It was pointed out that, when I said my experiment was a failure, that really wasn’t true. We learn from failures. Indeed, true, which is actually why I posted it.

 

I rated the results as a failure in the sense that, as far as I could tell, it was bidding at random. Final contracts didn’t bear any relation that I could see to what I was trying to teach, nor to the way it bid similar hands. This was still true even after trials with as many as 1,000,000 hands.

 

My previous approach suffered (I realized in retrospect) from too much linearity, while bridge bidding systems are not linear. As such, even if I set the weights on input, I could not define standard bids. Specifically, five key inputs were HCP, number of spades, hearts, diamonds, and clubs. Consider a standard 1NT (let’s say 15-17, balanced). As the number of HCP goes up, the weight for the 1NT opener line should increase, but, when you get to 18 HCP, the value has to drop to zero. This is pretty much inexpressible in my previous approach.

 

New approach: I now have 9 inputs for ranges of HCP, and 7 ranges for the length of each suit. That makes 37 inputs, which are essentially binary. This at least allows me to express a Standard American 1NT opener.

 

I have also bitten the bullet and now have a set of weights for each possible bidding sequence. Gulp! Yes, that does take a lot of memory. At the moment I’m only dealing with 2 rounds of bidding (4 bids). I limit the opener to calls up to 2NT, the first response to 3NT, and the next two bids to 5D but it is still a lot of memory. At least I can now express a standard Stayman sequence or Jacoby Transfer.

 

Results are still random, much the same as the previous approach. Why, I don’t know.

[Dirk Kuijt]

Next approach (yes, I'm still at it):

 

First, I'm going to just try to get a system that gets to the right suit, rather than penalizing both for the wrong suit and the wrong level.

 

Second, I'm going to try having the program bid a hand until it gets it right, then modifying the hand *slightly* (at most one card in each hand, perhaps just one card in the 26), and having the program bid again. The idea is to make the program discover that similar hands should be bid in similar ways.

 

We'll see how this turns out. More to come.

[Ant590]

I am not claiming that this is feasible. Maybe it is. Maybe one day it will be, when we get quantum computers etc.

Blimey, just realised that GiB on a quantum computer will be able to simulate every possible hand in a fraction of a second.

[Dirk Kuijt]

I decided on tackling an easier problem. Rather than trying to get to the right contract, I would try to have the program just get to the right strain (suit or NT). The weights for HCP were all set to zero, and the weight changes were all based on whether the program got to the right suit (or at least an acceptable suit) or not. Specifically:

1. If the contract is in a suit, the suit must be 7 cards combined or more, with penalty weight changes increasing linearly the shorter the suit, and reward weight changes increasing with the length of the suit.

2. If the hands have 9 or more spades, don’t play in hearts unless there at least 8, don’t play in a minor unless there are at least 9, and don’t play NT.

3. If the hands have 9 or more hearts, don’t play in spades unless there at least 8, don’t play in a minor unless there are at least 9, and don’t play NT.

4. If the contract is NT, the shortest suit in the combined hands must have at least 4 cards.

The program was permitted to make 3 calls at most, with dealer’s initial pass allowed, but counting as one of the three calls. That is, opening bid, response, opener’s rebid is the longest possible sequence.

The initial weights were random, as were the hands, so the results listed below might well not be repeated if the program were rerun (as I plan to do). During the training phase, each hand was bid until the program got it right or it tried 100 times. Also, during the training phase, successive hands differed only by the exchange of one card with one other in the deck. Often, such a change will make no difference in the hands, and it will never make a large difference. During the examination phase, however, hands were (pseudo) randomly dealt fresh for each pair of hands, and the weights were not changed. The opening call may be Pass, or up to 2NT. The response may be up to 3NT, and the rebid may be up to 5D. There were 900,000 hands in the training phase.

The results are obviously not optimal in any sense; for one thing the program never opens 2S, or, at least, never did in the 50,000 deals of the examination phase. I have tried to eyeball the results, and can summarize the opening bids of the ‘system’ as follows:

Pass: long diamonds, may also have long clubs

1♣: 5 diamonds, 6 or more clubs

1♦: usually 4 spades and 6 clubs, sometimes 4 diamonds and 7 clubs

1♥: exactly 2-1-5-5

1♠: a catchall for normally balanced hands, may be 4-4-4-1

1NT: long hearts

2♣: long spades

2♦: long clubs, may have 4 diamonds

2♥: exactly 4-2-0-7

2♠: unused, as far as I can tell

2NT: 5 hearts and a 5 card minor

I recall an earlier poster had noticed a tendency to bid over the promised suit, rather than bidding the promised suit (as natural systems do, by definition) or bid under the promised suit, as in transfer situations. That happens in this set of results as well, with the 1♦, 1♥, 1NT, 2♣, and 2♦ bids, and arguably other cases as well. Why, I have no idea.

More to come.

[NickRW]

Pass: long diamonds, may also have long clubs

1♣: 5 diamonds, 6 or more clubs

1♦: usually 4 spades and 6 clubs, sometimes 4 diamonds and 7 clubs

1♥: exactly 2-1-5-5

1♠: a catchall for normally balanced hands, may be 4-4-4-1

1NT: long hearts

2♣: long spades

2♦: long clubs, may have 4 diamonds

2♥: exactly 4-2-0-7

2♠: unused, as far as I can tell

2NT: 5 hearts and a 5 card minor

I recall an earlier poster had noticed a tendency to bid over the promised suit, rather than bidding the promised suit (as natural systems do, by definition) or bid under the promised suit, as in transfer situations.  That happens in this set of results as well, with the 1♦, 1♥, 1NT, 2♣, and 2♦ bids, and arguably other cases as well.  Why, I have no idea.

More to come.

As I understand it there is no penalty in your training methods for getting too high - consequently it is going to tend to assign some random opening to show some useful shape. I don't think it is particularly selecting any sort of transfer.

 

Emphasising this is that your opening pass and 1♣ - which should be fairly frequent in any system are not particularly frequent at all.

 

Nick

 

Later P.S. You might try adding a very limited hcp input - say the ranges that Helene's work highlighted: 0-6, 7-9, 10-12 and 13+ - which might allow some meaningful way of training it to not get too high - without a big increase in the complexity of your input layer (or indeed the middle layer).

[Dirk Kuijt]

Nick

 

Yes, my current training takes no notice of level or HCP at all. Obviously unrealistic, but my previous attempts to include them didn't lead to any convergence. I don't mean convergence on anything useful; I simply didn't get convergence at all. So, I decided to try a simpler problem that perhaps I could get some answer.

 

I would definitely like to bias the bidding to lower calls; I've pondered this in the past, but have been unable to find a way to do that. That investigation will also continue.

 

Incidentally, the problem that I set for the computer is not all that easy, given that you are restricted to three bids.

[bab9]

Dirk,

 

Interesting results. Are you able to determine how the responding sequence might proceed after the opening bid?

[Dirk Kuijt]

Bab9

Yes, though it takes a bit of searching. Here are the current results. Note that these don’t relate to the previous run; the program comes up with a new ‘system’ each time:

Pass – 1♣: 5+ spades

Pass – 1♦: catchall, I couldn’t find a pattern

Pass – 1♥: 4+ hearts

Others seemed to not occur

1♣-1♦: catchall, I couldn’t find a pattern

1♣-1♥: 6+ spades

1♣-1♠: 5+ hearts

1♣-1NT: normally, both majors

1♣-2♣: seems not to occur

1♣-2♦: normally 3=6=2=2

1♦-1♥: normally balanced

1♦-1♠: 3 spades and a long red suit

1♦-1NT: seems not to occur

1♦-2♣: balanced (only one example)

Others seemed not to occur

1♥-1♠: balanced or long spades or long clubs

1♥-1NT: long spades or both minors

1♥-2♣: seems not to occur

1♥-2♦: long diamonds

[bab9]

Bab9

Yes, though it takes a bit of searching.  Here are the current results.  Note that these don’t relate to the previous run; the program comes up with a new ‘system’ each time:

Pass – 1♣: 5+ spades

Pass – 1♦: catchall, I couldn’t find a pattern

Pass – 1♥: 4+ hearts

Others seemed to not occur

1♣-1♦: catchall, I couldn’t find a pattern

1♣-1♥: 6+ spades

1♣-1♠: 5+ hearts

1♣-1NT: normally, both majors

1♣-2♣: seems not to occur

1♣-2♦: normally 3=6=2=2

1♦-1♥: normally balanced

1♦-1♠: 3 spades and a long red suit

1♦-1NT: seems not to occur

1♦-2♣: balanced (only one example)

Others seemed not to occur

1♥-1♠: balanced or long spades or long clubs

1♥-1NT: long spades or both minors

1♥-2♣: seems not to occur

1♥-2♦: long diamonds

Dirk,

 

As the results do not relate to the first table of results, the meaning of the opening bid is unclear, and hence how the bids combine is unclear. However, it is very interesting that certain sequences do not occur, eg 1♣-2♣. This could open up to having additional conventions added to the system that do not result in the loss of a 'natural' bid.

 

I noticed there were no 1NT or 2 level openings in this run. Was the system generated with the same hands as the previous system you posted?

 

In the bidding sequences, how often did the 3rd bid choose the correct suit/NT contact? Did the simulation choose the longest combined suit, or did it somehow give preference to the majors?

 

It is interesting that the simulation was able to find a transfer method in the majors over a 1♣ opening.

 

Given that the system changes each time it is run, I would be interested in seeing the results of each run that you have time to extract the information.

 

Barry.

[Dirk Kuijt]

Further progress, or at least understanding of what is going on: (I'm still just trying to get to the right suit; getting to the right level is another large step.)

 

1. The program "never" (at least in several runs) uses all the opening calls available to it, and rarely more than 5. Finding a way to rebalance the weights such that it does use all the calls when doing training is my main concern at the moment.

 

2. The opening pass is invariably the most common call, and invariably converges on a generally balanced hand. Balanced here is not as restrictive as you would normally think, it may have a singleton. However, both short suits (0-1) and long suits (7+) are clearly weighted against, having negative weights when I allow them. Having dealer's pass be the most common seems right.

 

3. Higher bids are more specific than lower bids (which seems right). Often, opening bids from 1S on up have very specific meanings, like only 4=1=1=7.

 

4. Intermediate opening bids (1C to 1H) tend to fall into two categories: either two suiters, or bids showing an exact holding (typically 4 cards) in a specific major suit. The two suiters are sometimes two specific suits, sometimes an anchor suit and some other, though the other suit is normally one of two, not one of three. E.g. Hearts plus a minor, but not hearts and another. Devoting a bid to a one suiter seems to be rare. When it does "violate" its system, it often fails to get to a good spot; well, at least responder isn't cheating :-)

 

5. Allowing or not allowing negative weights doesn't seem to make a difference.

[hotShot]

If you only look at the opening bid and not on the bidding as one object, the neural net will not be able to find some implications.

 

Example 1:

Historical Acol has strong 2 bids, to avoid the problem that partner and opps pass my opening bid.

Just looking at the opening bid, there is no reason to open strong hands at the 2 level.

 

Example 2:

Preempts are not good for your side, but they can be really bad for your opponents.

If opps where forced to be silent, you could design your system completely different.

[bab9]

Further progress, or at least understanding of what is going on:  (I'm still just trying to get to the right suit; getting to the right level is another large step.)

 

1.  The program "never" (at least in several runs) uses all the opening calls available to it, and rarely more than 5.  Finding a way to rebalance the weights such that it does use all the calls when doing training is my main concern at the moment.

 

2.  The opening pass is invariably the most common call, and invariably converges on a generally balanced hand.  Balanced here is not as restrictive as you would normally think, it may have a singleton.  However, both short suits (0-1) and long suits (7+) are clearly weighted against, having negative weights when I allow them.  Having dealer's pass be the most common seems right.

 

3.  Higher bids are more specific than lower bids (which seems right).  Often, opening bids from 1S on up have very specific meanings, like only 4=1=1=7.

 

4.  Intermediate opening bids (1C to 1H) tend to fall into two categories: either two suiters, or bids showing an exact holding (typically 4 cards) in a specific major suit.  The two suiters are sometimes two specific suits, sometimes an anchor suit and some other, though the other suit is normally one of two, not one of three.  E.g.  Hearts plus a minor, but not hearts and another.  Devoting a bid to a one suiter seems to be rare.  When it does "violate" its system, it often fails to get to a good spot; well, at least responder isn't cheating :-)

 

5.  Allowing or not allowing negative weights doesn't seem to make a difference.

Dirk,

 

You mentioned that it rarely uses more than 5 opening bids. Are these opening bids simulation dependant?

 

If you have two 7 card fits, is the simulation likely to find the higher suit for the contract or NT?

 

Given HotShot's examples, it is likely that the range of opening bids may change when HCP are taken into account. As a result, you may not need to rebalance the weights at this stage.

 

Looking again at the previous results we seem to have to bidding sequences that give the same result. 1♦ - 1♥ and 1♦ - 2♣ (realise there is only 1 example) both show responder is balanced. It might be worth looking at the weights on the responses to ensure no two sequences give the same information. Alternatively, there may be something subtle going on here that may be interesting to look at.

[Dirk Kuijt]

hotShot

There is no question that both taking into account the level of the contract will make a difference, and taking into account those pesky opponents will also make a difference. I admit that, at the moment, I’m doing neither. My justification for this is that I’m working on a problem that I hope to solve. I tried to take into account level (that is, HCP) earlier, and got nowhere. If I can get it to solve the simpler problem, then I’m definitely going to try the next one as well.

 

Bab9

 

The bids it uses are simulation dependent, to some extent. It always uses dealer’s opening pass, 1C, and 1D. After that, usage drops off quickly, so that an opening 2NT (the highest I allow for dealer) is the rarest, occurring maybe one run in 10. However, it is not linear, and there are often gaps, such as no 1S bids, but 1NT, or no 1NT, but 2C. In any case, bids at 1S and above are rare and tend to show very specific hands, such as exact distribution.

 

As to which fit it finds, my impression is that there is no difference between 7 card fits, though I don’t have any statistics to prove that. The weight adjustment does distinguish between two 8 card or longer fits (one major and one minor), so the program does favor the major suit fit in that case.

 

As for the weights, clearly two sequences should not show the same hand. In fact, there must be some difference between them, though that difference may not jump out looking at a few hands, or at least jump out to me. If the weights were really exactly the same for two bids, then one or the other would always be chosen (either first or last, depending on the details of the program). Of course, not only should the bids mean something different, they should mean something substantially different. However, how to quantify “substantially” is not obvious to me, nor is it obvious how to adjust the weights to make things better if the weights didn’t differ “substantially”.

[Free]

I haven't read every post in this thread, but I think it's an interesting idea. Sorry if some of this is already suggested or answered...

 

Dirk, do you have a limit how high your program can bid? I mean, if it can bid up to 7NT, then obviously your system will look random every time you run a simulation. This is because there's no real difference between opening 1♠ or 1♣ with a 6 card ♥ suit, partner can bid 2/1 or jump to get his suits in.

 

Another question: why do you look for 7 card fits, and not 8 card fits like we do in the real world? You always have a 7 card fit, so at this point NT calls will be used to show suits.

 

I think it would be interesting to see what happens when your system can only bid up to 2NT and opener & responder each get 2 calls to find an 8 card fit (or NT in case no 8 card fit exists). Since the possibilities are quite limited and there's not always a fit, this might be less random.

Or even make it more specific and look for 9 card fits, it should be easier to develop a decent system for this.

[Dirk Kuijt]

Free

 

I limit opening bids to 2NT, the first response to 3NT, and subsequent bids to 5D. I do this to reduce the size of the tables; the memory used is enormous as it is, and constructive bidding doesn’t normally start above 2NT anyway.

 

I’m not sure what I said that implied that I’m looking for 7 card fits; I’m not. The scoring system for the program has the highest weight for a major suit fit of 8 cards or more. I have special, extreme penalties for finishing in a 6 card or shorter “fit”, but that doesn’t mean that I’m looking for 7 card fits.

 

Limiting all calls to a maximum of 2NT is an interesting idea, and I may try that. I’ve got to check how easy that will be in the program. (Yes, it should be simple, but these things don’t always turn out that way.)

[bab9]

I haven't read every post in this thread, but I think it's an interesting idea.  Sorry if some of this is already suggested or answered...

 

Dirk, do you have a limit how high your program can bid?  I mean, if it can bid up to 7NT, then obviously your system will look random every time you run a simulation.  This is because there's no real difference between opening 1♠ or 1♣ with a 6 card ♥ suit, partner can bid 2/1 or jump to get his suits in.

 

Another question: why do you look for 7 card fits, and not 8 card fits like we do in the real world?  You always have a 7 card fit, so at this point NT calls will be used to show suits.

 

I think it would be interesting to see what happens when your system can only bid up to 2NT and opener & responder each get 2 calls to find an 8 card fit (or NT in case no 8 card fit exists).  Since the possibilities are quite limited and there's not always a fit, this might be less random.

Or even make it more specific and look for 9 card fits, it should be easier to develop a decent system for this.

Free,

 

Quick recap, Dirk indicated that he was looking at the bidding when only 3 bids were used to find the right suit contract, ignoring HCP for the time being. There were two other systems briefly discussed, one being a binary decision tree and the other referred to as the silent spade system (is there any update on this?).

 

I was asking about the 7 card fits (did not mean to imply that Dirk was specifically looking at this situation) to see how the program handled those decisions where there is a choice between 2 suits and NT (a followup to some of my previous inquiries).

 

 

Dirk,

 

I was not aware that the program was weighted for 8+ card fits (weighting for major is fair enough). Two things:

1. If you were to look at hands with just 8 card fits, do you see the simulation preferring 4/4, 5/3, 6/2 fits?

2. Do you have a weighting for NT? If so, how is it implemented?

[Dirk Kuijt]

Bab9

 

I'm back, after a detour through the ICU via the ER; don't try this at home.

 

I don't attempt to distinguish between kinds of fits in terms of length. 4-4 is the same as 5-3 is the same as 6-2, and they are all worse than 5-4, or anything with 9 cards, which are worse than anything with 10 cards, which etc. I do favor major suit fits over minor suit fits.

 

In terms of NT, I deduct for having too much shortness in any suit. 3-2 is considered the shortest "normal" holding. A deduction is made for 3-1, and a bigger deduction for 2-2. 3-0 is still worse, and 2-1 worse yet. OTOH, I do add for length in suits in NT, though not as much as if one played in the suit itself. Thus 2=3=4=4 opposite 2=4=3=4 gets a deduction for the 2-2 spade holding, and no bonus, while 2=2=2=7 opposite 2=3=4=4 gets a deduction for the 2-2 spade holding but a bonus for the 7=4 club holding.

 

This adjustment for deciding whether NT is right or not "feels" right. Of course, the exact adjustments are highly debateable. In real life, the race between the stoppers in the short suits and declarer's long suits would be critical; that level of analysis is far beyond anything I'm trying at the moment.

 

I'm working on including values, but right now, not so good. (So far, the program has bid every hand the same :huh: ) More to come.

[tysen2k]

The Silent Spade is still mostly silent.

 

I haven't put too much time into it, but I'm trying some tweaks to try and make it use some of the higher opening bids. My current iteration now bids up through 2♣, but I'm slowing getting it to bid higher.

 

One thing that I am seeing is that even though I always start off with random initial conditions, it almost always gravitates towards pass = spades, 1♣ = balanced.

 

The current iteration is approximately:

 

Pass - 0-37, all hands 4+ spades unless qualifies for 1♣, 1♦, or 2♣. Also pass with any 14+ balanced.

1♣ - 0-13 balanced with exactly 2 or 4 spades

1♦ - 0-37, all unbalanced hands with 5+ diamonds unless spades are equal/longer. Prefer 1NT over this if qualified.

1♥ - 0-13 balanced with exactly 3 spades.

1♠ - A strange one. Again about 0-13, 1-2 spades, 2-4 of both red suits, 5-6 clubs.

1NT - 0-37, 7+ clubs or 11 cards in the minors.

2♣ - 0-37, 6+ hearts. If has exactly 4 spades, bid 2♣ with 0-10HCP.

 

Tysen

[helene_t]

I have been thinking about whether genetic algorithms are adequate for this exercise.

 

In neural networks there are complex interaction effects which I think justifies some kind of Monte Carlo approach. But I see a problem with mating. Gene transfer only make sense if a gene can play the same role in the offspring as it does in the parent. This is why we don't mate with animals of distant species. Now if (say) the first layer of a two-layer NN was fixed within a subpopulation then the roles of the synapse weights in the second layer would be the same for each organism so one could mate within that subpopulation.

 

As for the approach I have in mind in which the parameters are weights assigned to competing heuristics that guide the construction of decision trees, I think the interactions are sufficiently simple that a deterministic approach might work. I thought of modeling the expected matchpoints achieved by a system as a linear function of the parameters plus interaction terms in which an L1 penalty could be put on the interactions. Then one could arrange a second tourney where each participant was the estimated optimum in the first tourney for a particular value of the L1 penalty. Basically an elastic net.

[Zelandakh]

Rather than allowing the program a certain number of bids before it must choose a final contract I think it would be preferable to pick a level. A logical initial point would be 2C as this is often the boundary between weak and constructive openings in many systems. To simplify matters it's also not a bad idea to only feed dealer good (opening or better) hands and to ban a initial pass. Without writing it out to check I believe these conditions would allow 63 possible bidding sequences up to the point of decision. Is that too still many to handle?

 

The point of this is both to get a starting off point that is based purely around constructive bidding (can add preempts later), that is vaguely legal (no forcing pass systems), and that is more efficient when it attaches more meanings to a 1C opening than a 2C opening (since it has more bidding space). When the models are up and running then some of these restrictions can be relaxed but when a human designs a bidding system they usually start with the constructive, so why not make the same assumption for the program? Finding a way to add in the useful space principle has to be a core of any workable bidding system, and this way is more natural than simply forcing the system to assign x number of meaning to each bid. And finally, removing an initial pass option almost halves the potential auctions. If the complexity is still too high then add further restrictions until the system is simple enough to be workable but complex enough to produce workable results. I don't believe the results shown so far for 2-bid systems allowing a strong pass are really useful at all for mutating from.

[helene_t]

Just got another idea:

 

We can measure the "similarity" of two "hands" X and Y by how often our PAR is the same if opener has X as if opener has Y, assuming responder will have the same "hand" Z in both cases (where Z runs through the set of possible responder hands). Here a "hand" could be something like distribution+HCPs rather than a specific hand, and the Z's would be weighted by conditional probability.

 

Then openers "hands" are put into a tree by means of hierarchical clustering. Opening bids are mapped to nodes in the tree by principles similar to those I used for the induction trees.

 

As a first shot (to make things easy), we could assume uncontested auctions and let responder bid the first step as a relay, asking opener to branch further through the tree, and a relay break would just be to play.

 

A funny variant could be to let the similarity (partly) reflect similar safety level rather than same PAR. This might allow the procedure to invent multi-bids. For example, a weak 2♥ and a weak 2♠ are similar in that often the safety levels will be 2♥/2♠ or 4♥/4♠ so the procedure might invent the multi 2♦.

[tysen2k]

Just got another idea:

 

We can measure the "similarity" of two "hands" X and Y by how often our PAR is the same if opener has X as if opener has Y, assuming responder will have the same "hand" Z in both cases (where Z runs through the set of possible responder hands). Here a "hand" could be something like distribution+HCPs rather than a specific hand, and the Z's would be weighted by conditional probability.

That's an interesting idea. It seems like a lot of work and would be a good similarity measure, but it still doesn't tell you if one set of opening bids is "better" than another. Any ideas on that front?

 

I'm still working on Silent Spade and it's making some good progress. Once I got it over the hump of never opening up with higher bids, it's now taking off on its own. It's slowly refining itself each day and hasn't peaked yet. Here is the current iteration (the spades aren't so silent anymore):

 

Pass –0-9 HCP, 3-5 spades, 1-5 in other suits OR 10+HCP, 4-5 spades, 1-4 clubs, 1-5 in reds but no 4441 shapes.

1♣ – 0-12 balanced with 2 spades or 10+ balanced with 3 spades

1♦ –any 5-4 or 5-5 hand that isn't passed, or 10+ HCP any 4441

1♥ – any hand with 5+ hearts that can’t open 1♦/2♣

1♠ – any hand with 6+ clubs that can't open 2N/3♣

1N – any with 6+ spades, 0-4 clubs

2♣ – any with 6+ hearts, 4-5 clubs

Higher bids are all distributional 1- or 2- suiters with no HCP dependence

 

The computer doesn't like the 1♣ definition that much, so it seems that the room for improvement will come from there.

 

Tysen