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Are bidding systems geared toward "optimizing" contracts?
Most contracts can succeed with some distributions and defenses of the 26 opposing cards, and fail with others. If a contract (except a grand slam) makes against any defense and distribution you've underbid. If one never makes against any distribution and perfect defense, you've overbid. The latter two are clearly "non optimal" contracts.
When I learned bridge in the early 1970s, I was taught to "bid to make." With a standard eight card trump fit and 26 points, one should be in a four level contract that would make if e.g. the trump suit broke 3-2 (a 68% chance) but would often fail if the suit broke 4-1 or 5-0. Other equivalent (68%) chances were a combination of 3-3 suit break followed by a finesse.
More recently, scoring rules have changed, especially for IMPs, so that you will come out ahead if your contracts have a 50% chance of making, and in some cases, even 40%. Have bidding systems adjusted to these changes, requiring say, 24 points instead of 26, so that your contract hinges on 50-50 finesses rather than the 68% chance situations described above?

50% of 200 points is worth more than 75% of 120, but 75% chance to win a game is worth more than 50% chance. But your definition of optimality makes both of them optimal at the same time. This doesn't make sense as a word for this situation, and has no further mention in the point of the question anyway. I suggest removing it entirely as a source of multiple potential confusions.

Alexander Woo's answer well addresses the direct question at issue - but there are numerous statements made (and implied) in the question body which are based on misunderstanding at best. Let's take them one by one:

Most contracts can succeed with some distributions and defenses of the 26 opposing cards, and fail with others.

No, there is no basis for this in fact. For any contract at a level above 1NT there is in general far less than a 50% chance that our side has a chance to make it. Further, double dummy cooperative defenses have no bearing on any sane analysis of Bridge. Even very weak players are out to attempt success, and typically will get at least a couple of good results over a 26 board evening.

If a contract (except a grand slam) makes against any defense and distribution you've underbid.

No. There is for example no point to bidding contracts of:

Three of any suit;
Four of a minor; and
Five of a major

except in a (actual or potential) competitive auction to obtain the best tactically achievable score. Sound bidding guidance, for good reason, advises against bidding these levels for constructive reasons other than to invite the next higher "bonus level": game or slam as the case may be.

If one never makes against any distribution and perfect defense, you've overbid. The latter two are clearly "non optimal" contracts.

No; the object if the game is to maximize one's score. As a tactical means of approaching that goal one will occasionally, in the auction, judge that losing a small amount going down is better than risking a much larger loss allowing the opponents to set the contract. This is a optimal score rather than an optimal contract by your definition.

When I learned bridge in the early 1970s, I was taught to "bid to make."

This is sensible advice to give a novice beginner. If you passed through that stage without being introduced to preemptive openings and reopening (ie balancing) calls then your instruction was sorely lacking.

With a standard eight card trump fit and 26 points, one should be in
  a four level contract that would make if e.g. the trump suit broke
  3-2 (a 68% chance) but would often fail if the suit broke 4-1 or
  5-0. Other equivalent (68%) chances were a combination of 3-3 suit
  break followed by a finesse.

Novices never arrive with intuition - as that is only developed in any sound sense with hundreds (for the very fortunate) or even thousands (for the less fortunate) hours of play and study. For some it never arrives at all. This guidance is to enable, in the interim, a modicum of success that will enable the acquiring of sound intuition. You are conflating an improvement in Point Count Evaluation, that occurred in North America contemporaneous with the arrival of "scientific" systems into North America in the late 1950's (see below), with those systems themselves.

More recently, scoring rules have changed, especially for IMPs, so that you will come out ahead if your contracts have a 50% chance of making, and in some cases, even 40%.

The only scoring change I am aware of since the early 1970's, IMPs or otherwise, is the change for fourth and subsequent non vulnerable doubled undertricks from 200 each to 300 each. This was instituted to discourage over-competitive non vulnerable contracts at both IMPs and other tournament formats; and certainly not for the reason you ascribe.

Have bidding systems adjusted to these changes, requiring say, 24 points instead of 26, so that your contract hinges on 50-50 finesses rather than the 68% chance situations described above?

No. What actually happened is that the Goren Point Count Evaluation was refined by making a clearer distinction between High Card Points and (Total) Points. This resulted in an reduction in the evaluation of the average hand by one Total Point, and thus in the total points by a partnership by two Total Points.
The change involves the distinction in how one evaluates the distinction in playing strength between hands of 4333, 4432, and 5332 distribution, the most common holdings:

Goren Point Count:
A 4333 hand is assessed at 0 points of distributional strength (counted at both NT and Suit contracts). A 4432 hand is assessed at 1 points of distributional strength (counted at both NT and Suit contracts). A 5332 hand is assessed at 1 points of distributional strength (counted at both NT and Suit contracts).
Modern Point Count:
A 4333 hand is assessed at -1 points of distributional strength (counted at both NT and Suit contracts). A 4432 hand is assessed at 0 points of distributional strength (counted at a NT contract) and 1 points of distributional strength (counted at a suit contract). A 5332 hand is assessed at 1 points of distributional strength (counted in either NT or its doubleton suit) and at 2 points of distributional strength (otherwise).
Impact of above:
This evaluation change better recognized the weak playing value of a 4333 distribution relative to the more common 4432 and 5332 distributions - and when matched across both partnership holdings allowed the specification of 24 points (instead of 26) as the recommended minimum combined partnership holding for a sound 3 NT game contract. Likewise again it allowed a reduction in the recommended minimum for an eight-card-fit Major game contract from 26 to 25 points (as presumably not both partners were holding 4333 hands where a 3NT contract might be better).

Examples:

The hand AJxx QTx Kx AJxx evaluates to 16 Goren Points and 15 Modern Points for a 1NT opening. The hand AJxx QTx Kxx AJx evaluates to 15 Goren Points and 14 Modern Points for a 1NT opening. Thus under both systems the first hand is suitable for a 1NT opening while the latter is not.
The hand AQx KTx Kx AQxxx evaluates to 19 Goren Points and also to 19 Modern Points, thus in both cases being too strong for a 1NT opening. Note that Goren counts 1 point for the doubleton, Modern one point for the sound 5-card suit.
The hand AJx KJT AQ Qxxxx evaluates to 18 Goren Points and to 17 Modern Points, thus meeting the upper limit requirement in both cases for a 1NT opening. (Modern doesn't count the five-card suit with only medium honour as being worth a point for NT.)

Note how the evaluation process for hands has improved - not the bidding system itself.

Also your implication that these ideas are post 1970's is blatantly incorrect. By the mid 1950's Tobias Stone and Alvin Roth, soon joined by Kaplan, Sheinwold, Kay, Walsh and others, are already formulating these ideas into Eastern and Western variants of a new Scientific approach for North American Standard bidding that would evolve into Kaplan-Sheinwold and Two-Over-One Game Force respectively. Kaplan and Sheinwold publish the material elements of their system (described in How to Play Winning Bridge) in two installments, 1958 and 1962, more than a decade before your claimed beginning of these developments.

I did fail to make the distinction between freely bid contracts and "competition." I meant first that most people (unless pushed) bid contracts they think they can make. I also didn't allow for deliberate overbidding to block opposing contracts in the earlier version. I have made these two changes in the question.

@TomAu: You know better - Don't change questions to invalidate existing answers. Changing questions in response to comments is fine - but to invalidate existing answers destroys the whole purpose of a Q & A.

@TomAu: That too is incorrect. Openings of a Weak Two or Three or Four (preemptive) of any suit as well as all jump raises are not bid to make, but rather with the intent of (1) minimizing loss and (2) by disciplined bidders, providing a precise hand description that occasionally allows an astute partner to find an otherwise unbiddable game or slam.

I was taught to "bid to make." When I learned the game, the notion of "preempt" barely existed (in my circle). It was considered a special case, because the idea was to sacrifice to save a potential slam (only about seven points in hand, "wild" distribution), not a game. No one in my circle would risk "down two" to save a game (of equal vulnerability), but some would go down two or three to save a potential slam.

On the money, as always. A minor blemish is in your item 6. The increased penalty of 300 per undertrick only applies to fourth and subsequent undertricks. Down three doubled, non-vul, is still 100+200+200=500 for the opponents. This is a natural spot for a change IMVHO. The possibly most common application is that if we are non-vul and they are vul, it still pays to sac and go down three vs. game making their way.

(cont'd) According to 1935-87 scoring rules going down eleven non-vul would show a plus in comparison to a vulnerable grand making their way, and that is surely ridiculous.

@JyrkiLahtonen: Good catch; thank you . corrected. Yes, once it was thought of I don't think anyone regarded it as a bad idea.

Bidding systems, particularly ones experts use against experts, are built around optimizing results overall, not contracts.
Half the time, the opponents have better hands than you do, and you are still trying to make their results as bad as possible.  Sometimes this is by sacrificing - getting to a contract that goes down but for which the opponents score less for setting you than they would for making their own contract.  Often, this is simply by disrupting the bidding enough that the opponents cannot exchange enough information to decide on their best contract.
At the slam level, and even frequently for games, experts are frequently thinking about the entire probability distribution of what partner's hand could be, what opponents' hands could be, looking at their hand, and calculating if there is at least a 50% or 35% (depending on form of scoring and vulnerability) for the contract to make.  They are not counting points.

Great answer - but I take issue with your final statement "They are not counting points.". Perhaps "They are not just counting points." instead. I am no expert, but have played with and spent hundreds of hours discussing hands with others who are true experts. All, without exception, translated their total assessment of points and intangibles back into the point framework of their bidding system - except in rare circumstances of very unusual distribution.

@ForgetIwaseverhere: I start with points, and then make adjustments for e.g. the fifth card in the suit, the probable location of adverse honors based on the bidding, etc. But my evaluation is "anchored" in points.

Absolutely - but I think it's more complex than just changes in scoring.  I was taught in the 1990s, and 26 point games would definitely be overly conservative in that time even.  24-25 points was much more reasonable.
I don't think it's just scoring, though; I think it's due to people learning better strategies, particularly through statistical analysis.  More aggressive bidding systems in general have become more in vogue - look at the interference over strong 1NT, for example, that's currently used as opposed to the older methods that required much more unusual hands.  Even the 1NT opening range has adjusted - 15-17 not 16-18, even in fairly standard systems.

One thing people are learning about are "side" factors. For instance, it might be possible to make 3NT with 23-24 and a solid five card suit, or a "gambling" 3NT with 21-22 and a sold six card suit, but you may need 25-26 with 4-4-3-2 distributions in both hands. That's because the "long" cards provide extra tricks. In a trump contract, a 4-4 distribution could be advantageous (trump from either side) but not NT.


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Should I "take out" a double, using a two card suit?
In an earlier question, the consensus was that I should answer a takeout double of one heart with one spade (the cheaper bid) rather than two clubs. (I agree with that consensus.) The exact situation was: "Left hand opponent opened 1 heart. Partner doubled for takeout. Right hand opponent passed." I had something like (S) xxx (H) xx (D) Jxxx (C) Jxxx.
Suppose my major suits were reversed, so I had (S) xx (H) xxx (D) Jxxx (C) Jxxx? Would you still bid "one spade" with only two spades, or is "2 clubs" now the lesser evil?
Suppose one of my minors had five cards instead of four: e.g. (S) xxx (H) xx (D) Jxx (C) Jxxxx (three spades), or (S) xx (H) xx (D) Jxxx (C) Jxxxx (only two spades).
In the original question and answer, "one spade" was the preferred bid versus two clubs when I had one fewer spade than clubs. But the above variations hypothesize that I have at least two fewer spades than clubs. Put another way, where is the "tipping point" of spade-club distributions, where you would bid two clubs rather than one spade?

Not really relevant here, but another consideration possibly worth keeping in mind is the projected effect our bid has on partner's future evaluation. In this precise situation partner is aware that we may have a bust with only 3. If it goes (1C)-double-pass and I'm to bid with 3334 hand with two jacks or less, I will bid 1D. The reasons are A) it keeps bidding low (see Forget's answer for the reason why we want that to happen), and B) it is less like to excite partner unduly should they have a strong hand.

With xxx xx Jxxx Jxxx, the decision between 1S and 2C is very close. Any move that makes spades less attractive (eg holding only a doubleton) or clubs more attractive (eg a 5-card suit) breaks the "tie" in favor of 2C. Any of those three hands would be a clear 2C call for me.

or even xxx xx Jxxx QJxx.  We don't bid 1S with the original garbage hand because it's a good bid, we bid it because it keeps us on the 1 level, letting partner show their suit if they are double-and-bidding, giving us a runout if we need it, but mostly hoping the opponents will let us off the hook.  It's really hard to double at the 1 level (more so now that the double of 1S is likely to be considered takeout); much easier to double a misfit at the 2 level - so they likely are going to rescue us.

I would not be inclined to bid 1S with any of the hands mentioned, nor in most circumstances with a 2-card spade suit. With a minimum hand responding 1S on a three card suit might be better than 2C on four to the jack -- the decision is marginal. But bidding 1S with only 2 spades could easily find you in a 6- or even a 5-card fit, in a very bad place. I would absolutely bid a 5-card club suit over a 2-card spade suit, even 5 small cards. Four to the jack is not as good, particularly in a minor, but is still likely to work better than bidding a 2-card spade suit.
About the only time I would bid a 2-card suit in response to a takeout double would be if the opponents had bid 2 suits and I was 4-4 in those suits, but not strong enough to pass for penalties
Say
!D - P  - 1H - Dbl
P
holding
Jx  xxxx xxxx kxx
Passing 1H doubled could be a disaster, and 1S might be better than 2C on a 3-card suit I am certainly hoping that the opponents bid on and come to grief in a red suit at a higher level.
But holding  (S) xx (H) xxx (D) Jxxx (C) Jxxx? I would bid 2C and hope that opps bid on or partner happens to have a club suit. I don't expect to do well, but anything else is likely to be worse. Sure you can find hands on which 1S is the killing bid. But not many with that bidding and that hand.

Two points not raised by the other answers are:

What's happening on the hand?

Where are we headed?

When partner Doubles 1H with our bust and 3-small Spade holding, expectation is that Partner is either:

Holding 4 Spade cards and short Hearts, investigating our ability to compete, and will either pass or raise 1S depending on strength;

A bit stronger intending a minimum NT bid to show a balanced 18-19 HCP with a Heart stopper; or

A 1-suited hand of his own too strong for a simple overcall.

Our goal is in all cases to exit the auction as soon as possible, without ever giving opponents an opportunity to consider a penalty double that might be worth more than their own contract. To this end, it's imperative to both stay low, and avoid bidding a suit the opponents might hold 7 (or more) of. This is why the 1S response is acceptable on 3-small and a bust: Partner is either bidding on, which we can pass, or has a 4-card Spade himself. If Partner's intent is to show a 18-19 NT hand it is valuable to allow that to happen as 1 NT instead of as 2NT.
However when holding just 2-small the dynamics change. This could be going down 3 or even 4 against a mere part score by Opponents, especially with partner's hand on the table. Bidding a suit they hold discourages them from bidding, when we wish to make their continuation as easy as possible.
Hence I lean very much into bidding 1S on the xxx-xx-Jxxx-Jxxx hand but will never continue with a 1S call holding a 5-card minor of any quality, even 5 small. There is good reason to expect 2 long Club tricks in my hand after a 2C response, even with xxxxx, compared to zero tricks total in any other denomination.

Note that while Partner might have doubled on a three-suiter (short hearts) and only 3 Spade cards, such a call must guarantee better than minimum doubling values. A minimum holding and lacking Spades is better to wait, retaining the option of reopening later.

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Is playing against an opponent's known "misbids" a good way to defend against a contract?
Neither side vulnerable at rubber, South opens one spade. (He claims to subscribe to five card majors, three card minors, and strong 1NT but not "Flannery." He's also probably capable of  "psyching.")
I (West) pass with (s)JT8xx (h) Kxx (d) Kxx (c) Kx. (All x's are 7 or lower.)
North bids three spades. South alerts "Bergen raise." That means that North has exactly four spades, but has less than six points non-vulnerable. He may have zero. (If he had six, he would have raised to two spades.) All pass. East does not have the 12 hcps he needs for a takeout double, nor the good six card suit he needs for an overcall.
South and North clearly aren't 5-4 in spades, and I believe that South is bidding "Bergen" style; that is treating AKQx as better than a weak five card suit (e.g. Qxxxx) because of his high sequence. This would also imply more than minimum values (15+ hcps). Put another way, I have no reason to believe that North doesn't have four spades and that South has five.
My inclination is to lead the J of spades, because South has fewer spades than his bidding implied (and was unlucky to run into a 5-0 split). Does such a lead make sense?

Form of scoring?  2) What would partner have doubled 3S (presumably for takeout) on?  What would they have overcalled over 3S on?

@AlexanderWoo: 1) Rubber. 2) 12+hcps and suitable distribution. 3) Good six card suit or better.

Please edit all clarifications of the question into the question body - comments being ephemeral and subject to arbitrary deletion at any time, and questions being required to be self sufficient and independent of comments.

Also, please provide a more complete outline of Opponents bidding system: Weak or Strong NT has huge inferences on the types of hand South may hold for his opening, and whether I prefer to lead Club K or a small Heart/Diamond as an alternative to the Spade J. What is their Minor opening style, as this again affects the type of hands South may hold for his Spade shortness bid. These inferences are the key to strong Opening Leads.

Are Opponents playing Flannery? Are they they inclined to believe that AKQx is a better trump suit than Qxxxx?

Is South the sort of player to open a first seat 1 Spade psych on something like "x xxxx Qxxx KJxx"? Something is fishy about this auction and I'm trying to figure out what. It seems unusual that both West and South are unable to move over 3S  with your holding.

Assuming no psych bid: South has a balanced hand with 12+ to 14 HCP presumed to include Spade AKQ for his 4-card holding. That leaves very little room for additional values in his hand. With a presumed (23) or (33) holding in the minors the likely reason to have not opened one of those, planning a rebid in NT, is that they are both bare of high cards.
It's Rubber Bridge so I'm leading the Club K expecting to find partner with a good (45)  or (53) holding in the minors with at least 4 Clubs. We will strip South of his side suit exits and make him give me my two Spade tricks. I'm looking for a 2 or 3 trick set here, and at dinner Partner and I will discuss the value of voids in Opener's suit, and why we didn't get 500 or 800 on this hand.
I can only ever lead Spades from my hand safely once; and I need to save that in case I get endplayed later in the hand. So I'm not leading Spades on Trick One.

South's bidding style is something like the following: He would open 1s with (s) AQxxx (h)x (d) Qxxx (c) Axx, that is 12 hcp (maybe 11 because of the singleton h). He would also substitute Ks for xx of s. But he would need 11-12 hcps over and above (not counting) the Ks for his four card 1s bid, that is (s) AKQx (h)x (d)Qxxx (c) Axxx. He actually had Kxxx of d instead of Qxxx.

@TomAu: If South had a 4144 hand with a minor high honour he's both a cheat and a fool. He's playing 4-card majors and claiming to play 5-card majors. He's also creating endless bidding and play problems for his partner.

@TomAu: He's also created giant ethical problems for a partner (who may or may not be in on the scam, but will soon enough infer it if not) who must pretend to be playing 5-card majors while actually playing 4-carad majors. There's simply no tactical reason to be opening 1 Spade here, instead of the better minor, in first seat.

This strikes me as "foolish," to say the least. With three good four card suits to choose from, there is no reason for South to "lock" the partnership into spades. For instance, one minor, one heart might lead to a good NT contract. With four spades and 6+ hcps, partner would respond one spade. And if partner has less than four spades, there does not seem to be any reason to bid a spade contract. There was even less reason to encourage "Bergen" bidding from partner when he only had 3 hcps. Thanks for your help.

The misbid is irrelevant here. The question is simply, given that holding, and either a 4-4 or 5-3 opposition, what’s the right play?
You have ten HCP.  S has say fifteen. N has five. Partner has ten also.
You also have likely four or five tricks in your hand - all three K are odds on to win, and one or two spade tricks also - likely two.  The question is what to lead, to not cost you one of those trucks?
Leading away from any K is probably going to cost you a trick - odds are 60% or so that declarer has that ace. Partner might have a Q, but who knows?
That makes the spade lead seem like the clear winner, even if it weren’t for the bidding. And leading the J is the clear choice (assuming that’s how you lead away from JTxxx in your partnership).
Don’t overthink things!
Now, if you did have a safe side suit lead, do you still lead spades?  I still like the lead - with that 8 it probably won’t cost you a trick, and they may well be ruffing (and you might have a shot to lead another trump before the ruffing).  Even if sunny has 3, it’s still a good idea - avoid short suit ruffs. It’s riskier though because you do have two likely winners as it stands in tromp, and you could be giving one away- but I still like it on balance if there’s not a specific reason to leas something else (and as long as you still feel P has only ten or so points).

So if Declarer holds AKQ7 of Spades as OP imagines;:what will you lead at trick 4 if Declarer reads the situation and leads Club Ace and a small one at tricks 2 and 3? Opening lead is as much about getting off future endplays as anything else. Even when the Spade opening lead is correct (and we don't have near enough info yet about Opponents bidding style and agreements to know), your reasons here are very dubious.

By trick four, my partner would have told me what to lead back, one would hope…

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What are "other" ways you can prevent your opponent from winning in magic the gathering, given the "right" decks?
In a game, if you can prevent your opponent from winning, there are only two possible outcomes: 1) You will win or 2) Your opponent can also force a draw.
This question outlined how you can prevent your opponent from winning in magic the gathering by obtaining "infinite life." What are other ways you can achieve this result, at least in various situations? According to an answer to the question, getting "infinite turns" is not a way for the opponent to avoid defeat, or is it?

I fee like this is too broad... are you counting “play Fireball for 20 damage and win” as a way to prevent your opponent from winning? Or simply having more blockers than they have attackers if their win condition is attacking creatures. Or simply having an Emrakil in your deck against a mill deck. Are you meaning a combo that works against any deck no matter what?

@Gendolkari: I'm going to work backwards from your comments. 1)  I do NOT mean "a combo that works against any deck no matter what. 2) "Having an Emrakil in your against a mill deck" would be a valid answer. 3) Having more blockers than they have attackers is a valid answer.4) I don't fully understand your fireball example. I guess the gist of my questions is what are examples of strategies for preventing defeat, and in which contexts do they occur (other than in the example already given). I made an edit that I hope clarifies, particularly with this comment.

The simple answer is that you prevent your opponent from winning by disrupting their game plan, whatever that is.
If your opponent is trying to win by decreasing your life total to 0, you can prevent them from winning by gaining large amounts of life, or by using Platinum Emperion, which stops your life total from changing, or with other cards.
If your opponent is trying to win by milling your library into your graveyard, you can prevent them from winning by using a card like Emrakul, the Aeons Torn, which shuffles your graveyard into your library when it enters your graveyard, or by using Wheel of Sun and Moon, which puts the cards right back into your library.
Of course, if your opponent is trying to win with milling, gaining lots of life will not benefit you at all, and if your opponent is trying to win with damage, Wheel of Sun and Moon will not have much of an impact.

Taking a step back, you can disrupts strategies instead of the end goal. If your opponent needs to attack you with creatures to win, you can use Blazing Archon to prevent them from winning. If your opponent needs to cast spells to win, you can use a combo such as Teferi, Mage of Zhalfir plus Possibility Storm to prevent them from winning. If your opponent needs to put cards in a graveyard, you can use Rest in Peace to prevent them from winning.
Some popular strategies try to prevent their opponents from winning by using more general disruption that works against many different strategies. Control decks use counterspells and spells that remove permanents to try to stop whatever cards are most important for each particular opponent's deck. And prison decks try to prevent their opponents' strategies from functioning at all using cards like Stony Silence, Chalice of the Void, and Ensnaring Bridge.
There are as many ways to prevent your opponent from winning as there are strategies for winning the game, and those are numerous and varied.

There are also more general ways to prevent your opponent from winning the game. The cards Platinum Angel, Gideon of the Trials, and Angel's Grace all have effects that say "your opponents can’t win the game". Of course, in each case either the effect is temporary or there is a card that can be removed to get rid of it.


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Is there a way to determine how much of a "fighting chance" I had?
As a child, I was playing Monopoly in a local tournament against "Chuck" (the hero of some of my other game questions). Early on, he got a Monopoly of his namesake maroons: St. Charles Place, State Street, and Virginia Ave. After he acquired "stops" on all the other Monopolies, the game should have ended there except for one thing.
Toward the end of the game, a bystander, a girl named Martha, urged Chuck to give me a "fighting chance." He agreed to give me his defense to the Purple monopoly (Baltic and Mediterranean), in exchange for my defense to the orange Monopoly (St. James, Tennessee, New York). The other matters of note were that Chuck had hotels on the maroons, I had three railroads and one utility, and we each had about $1000 of cash.
Through what quantitative analysis (Monte Carlo simulation, perhaps), can I determine how much of a "fighting chance" I had at this point? I would guess that I would win less than one game in a hundred, perhaps less than one in a thousand, but in a million trials, I should have one or more wins.

yes, Monte Carlo will find your winning chance.

made a quick search, several MonteCarlo sim are open source:
https://github.com/wolever/monte-carlo-monopoly
https://www.mathworks.com/matlabcentral/fileexchange/70350-monopoly-monte-

A fat chance/A slim chance, whatever you wish to call it.
Even if you could get the last railroad AND the last utility AND get all hotels on your set for $0, you will still lose, unless a miracle happens, i.e. you never land on his hotels or on the set until his hotels are down.
There is a 7.54% chance (2 decimal places) you will land on his hotels (and pay at least $750) = equivalent to you losing $56.55 (2 decimal places) every turn (assuming you will land on his hotels at EXACTLY 7.54%)
There is a 4.29% chance he will land on your hotels (250 for Medit, 450 for Baltic) ($15.06/turn)
If you have the 4 railroads, 11.38% chance (gives $200) ($22.76/turn)
2 utilities: Average rent 7*10=$70, 5.41% chance = ($3.79/turn)
Total "gain": $41.61/turn
Total "loss": $56.55/turn
So this means that you may lose.
Note again that this is the most general case and I am only working with probabilities, that is, if the dice never or always leads to a space, this answer might be wrong.
Credits http://www.tkcs-collins.com/truman/monopoly/monopoly.shtml for all the probabilities

Welcome to the site. A good first answer. Upvoted, with a chance of "acceptance" in the tuture.


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Is there a way to quantify the effects of luck versus skill in rubber bridge?
Duplicate bridge was created to "eliminate" the luck factor. That is partnerships are compared only against other partnerships playing the same cards.
Rubber bridge is a different animal. Here, "luck of the draw" is clearly a major factor in the result, at least until you get to a "law of large numbers" point where the luck factors cancel out.
Suppose there are two partnerships, A and B, playing ONE rubber. From their play at duplicate, we "know" that partnership A plays one trick better than the other. Clearly partnership A would have better than a 50-50 chance at rubber as well. But how much better?
Is there any computer software, statistical formula, or mathematical algorithm that could estimate the relative chances of partnerships A and B winning the rubber?

As Eric Murray famously said in the case of Crown vs St. Clair Bridge Club:

"The game [rubber bridge] is only one of luck when played as the
  justices of the Ontario Superior Court play it in closed chambers."

That is of course the same Eric Murray of the Canadian partnership Murray & Kehela that was widely ranked the third best in the world during the 1960's and early 1970's.
Ironically, the luck element reduces as the overall quality of the players improves, yet increases in importance because the qualitative differences between the various players is reduced even more.
This question is difficult to address because of the above point. However, consider this advice to the budding "play for money" bridge player:

Choose Chicago against players who are a little bit weaker than you;
  but choose Rubber against players a lot weaker than you.

The reason is that when your skill advantage is small, you want lots of points in play, so that the luck evens out faster. When your skill advantage is larger, you want more opportunity for the opponent to make foolish decisions. A typical rubber-bridge strategy is to sit on a part-score and collect penalties for several hands before converting the game. Then repeat. even if the opponents get "lucky" and win the rubber, they will often have lost much more in penalties than the rubber bonus was worth.

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In bridge, how do you tell when to use holdup versus blocking plays?
You are the declarer at a 3NT contract (1NT, 3NT, no opposing bids). West leads what appears to be a five card suit, and you have Axx in dummy.
Your main concern is that East can get the setting trick by attacking a vulnerable tenace you have in hand; something like Jxx in the led suit, or AQx in a side suit (and maybe the the answer to the question below is found in the difference between one and the other).
At times, you should go up with the ace in dummy (if East has an honor and a small card) to "block" the suit. At other times, you should hold up the ace until the third round to exhaust East of the led suit. How do you tell the difference?
A club kibitzer gave the opinion, "You use a blocking play if you're worried about what will happen in the LED suit (e.g. if your tenace is Jxx, above), and you use a holdup play when you're worried about what will happen in a different suit (e.g, your AQx tenace)." Is there any truth to this statement?
I haven't stated whether West led from "top of a sequence" or "fourth best," but maybe the answer to question lies in the difference between the two.
Because another take of this matter is "avoidance." That is, if the priority is to keep East off lead, then let West win if he led a high card, and cover if West led fourth best, to prevent East from getting the lead.
I have been told that if I have something like Axx in hand opposite xx in dummy, I should hold up, and hope to exhaust East of the suit. But if I have Axx opposite xxx, West led the queen (denying the king), and East plays low, I might take the trick to "block" the king in East. Does this make sense?

It depends on the hand. What are you exactly looking for?

@aryabhata: I'm looking for some examples. You usually have brilliant ones.

Thanks Tom! I am a bit busy these days and don't really have much time unfortunately...

@TomAu: That is an entirely different hand situation, and should be a new question. I will dig up some examples to answer it with if you create the post.

To prevent the lead form RHO through your tenace, you want to win the ace when RHO plays his last card in the suit. If it looks like LHO has 5 cards, with three in dummy and your hand, win the second round with the ace. If you have only two in had with three in dummy, win the third round with the ace.
In regards your Club Kibitzer - yes, the proffered advice is in general sound - however, the cardinal rule is always Count the hand. If you have counted the hand, the answer is often obvious and possibly counter-intuitive from the general rule suggested by heuristic analysis.

There is a little known rule in Bridge called "The rule of 11" that tells you how many cards your right-hand opponent has that are higher than the one led.
If your LHO has not lead an honour card then they are likely to be missing one of them - they would lead one from a sequence or internal sequence. In this case it is likely that their partner has one honour. if you can deduce that the player has HX then you can try and block the suit by going up with your Ace. I would look out for 6-card combinations when you hold the Ace and Ten.
With regard to hold-up plays, the first thing to do is to see whether by holding up you can prevent the defence from being able to communicate with the 'danger hand' - the one whose card holding may cause you to fail in your contract - in the suit led. Even if they may hold an entry in another suit, holding up to prevent (or at least reduce the chance of) this happening will be worthwhile. There may also be occasions when you can hope that the partner has a singleton honour, but the opening leader will have an entry. In which case you might duck even with a finessable honour combination from your right, but safe from attack from the left.
Of course you should also take time to work out whether the opponents (having taken the trick by you holding up) have a possible damaging switch.

The rule of 11 applies if LHO has led the fourth-best, which may or may not be so.

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Are there common themes in "settlement" games?
I'm referring to games like Agricola, Puerto Rico, and Settlers of Catan. But a few themes seem to pop up in each game.
One is the importance of brick/stone/wood for building. Without improvements, your farms might not amount to much.
Another is the need to keep a balance between "income" (as measured in money or food), and victory points. A third is to keep a balance between advancing your own interests, and an eye on blocking threats from others.
Also, there is the element that while some commodities are (a little) more valuable than others under "normal" conditions, if "everyone" wants the same "great" commodity (e.g. stone), that becomes less valuable, while an otherwise ok commodity thqt is "neglected" can become "great."
Are these (and other features) common between the games mentioned (and similar ones)? More to the point, if I learn to play one well (or at least to a certain level of skill), will this give me a head start on learning the others?
For instance, although the mechanics are different, knowing how to play either Monopoly or Easy Money will give you a leg up on learning the other.

Almost too broad a question, but I haven't voted to close. I think my answer is basically that yes, learning to play one well will help you to play the others well, but only insofar as understanding that the little details (what an action is worth now) are only part of a bigger picture (how a less flashy action now will enable me to get more victory points later) is a general skill that's useful in many, many games.  I agree with user1873 that understanding corn in Puerto Rico won't help you be a good grain farmer in Agricola though.

@thesunneversets: Your comment represents the kind of thing I had hoped to learn.

Voting to close because I see the question as being too broad and possibly subjective as well. As written, I think this would be a perfect candidate for discussion in chat; it doesn't seem to represent an actual problem being faced, and I don't have suggestions as to how to address one aspect of the question in a way that would fit well with the site.

@DaveDuPlantis: It is an actual problem, because I have time to learn one game, but not a whole bunch of them. On the other hand, if I can learn two or more games in "batches," for a little more effort than for one, then it would probably be worth my time.

No, those three games play completely differently. Learning how to play well in one game will not translate to another.
Settlers of Catan is a game where you don't trade VP for advancing your economic engine, it is your economic engine. While goods don't have a value per say, certain goods are better than others. This is especially true in the Knights and Cities expansion, where Paper is by far the best commodity. Everyone focusing on paper will not diminish its value, at least until each player has the City improvement that grants a free good when you don't get one.
Puerto Rico doesn't follow the value states that you say exist in Agricola. Corn is one of the highest value plantations, because it doesn't require two colonists to produce. It may or may not be valuable to focus on the same goods depending on if you are selling to the Trading House or shipping. As for balancing income and VP, Puerto Rico doesn't follow any sort of balance. In the beginning, VP is rather useless. In the end game, coins are rather useless. So in Puerto Rico, you focus the beginning game on improving your income, and the late game focused on scoring VP. There is very little balance at all, just a determination of when to switch from income to.VP. This is more like Dominion in my opinion, but realizing this and playing Puerto Rico well will not make you a better Dominion player.

I disagree with your assessment that there is no tradeoff between VP and economy. There are many times where it would be cheaper in terms of VP per resource spent to build one way, but spending more (longer roads/more ships) to build somewhere else advances your economy more in the long run. There is still very often a tradeoff.

There are some situations where you might tradeoff VP for future economy, but in general this isn't the case. Usually, you are building up your economy (cities, or roads and new settlements). The only time you might choose a current VP bonus (like longest road), is if it wins you the game on the spot.

I don't think the theme plays into it that much.  It seems to me that the commonalities are more general than that, in that there are techniques that you can use to analyze just about any strategy game.  These includes things like the economics of points gained per action or the trade-off of scoring points versus preventing someone else from scoring points.  They also include things like understanding probability and how to manipulate it to determine how much is it worth to have twice as much chance of X happening, or what the "expected value" of a random action is.  You can tie the theme around these however you want, and they're still the same concepts.
That being said, you will likely always need some experience with a game before you can apply these concepts to effectively analyze decisions.  You'll be able to improve at a game faster than someone who doesn't understand these techniques, but you shouldn't expect being an expert at one game to instantly make you good at another.


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In bridge, should a defender play differently with most, about half, or few of the partnership's points?
This example is from Victor Mollo's "Test Your Defense, Where the Points are Won."
You are East. The bidding has been 1NT pass 3NT pass. West leads what looks to be the fourth highest spade. You hold:
(s) Axx (H) KQJ9 (d) Kx (c) xxxx.
Dummy shows (s) Kxx (h) Txx (d) Axxx (c) Kxx.
You take the ace of spades. Normally you would return a spade, partner's suit. But Mollo recommends establishing the hearts for a "set."
The key features are: 1) Dummy's T falls under your J, making the 9 good 2) you have the guarded K of diamonds "offside" the ace, hence a sure entry and 3) You have 13 of your partnerhip's HCP.  West has the J of spades at most (Q if South has only 15 HCP).
So is your strength the reason you should deviate from the usual rule of returning partner's suit? Should you return your partner's suit if you didn't have the heart honors (meaning that partner has about 7 HCP), or if you had only the A of spades (partner has  about 10)?
Conversely, if you were "West" (on lead), would you lead differently with the three different hypothetical hands?

Does Mollo give the values of the "x"s? Because if your partner led fourth highest, then the rule of 11 will tell you how many cards higher than the lead are in declarer's hand. Also useful information.

I don't think the number of points in your hand it the relevant feature here at all.
Essentially, you look at your hand, and see that you can easily take down the contract if you establish your 3 hearts winners before your opportunity to regain the lead with the KD is removed.
Returning a spade isn't going to help much as we're pretty certain partner isn't going to win that trick with his Jack.  As such, you have to forget the "usual rule" of returning partner's lead, which isn't going to help in the current situation, and just make the best play for beating the contract instead.  Partner has basically nothing, and you have an excellent defensive hand here: it's not up to him to dictate the whole play of this contract via his lead, just because of the seat he happens to be sitting in.

HCP count is fairly relevant here -- when the opponents bid 1NT-3NT they essentially always have 25+ HCP between the two hands, so you know for sure that you're not setting up partner's suit by returning a spade. Of course, it's also important to see that you have a sure set in your hand, but there are some cases that are not so clear cut and knowing that your partner has at most a Q is important information when planning how to continue the defense.

Yes, defensive strategy can vary considerably depending on where the points are in your partnership.  Your real goal is to look for tricks, of course, but often points are a good guide as to where the tricks are going to be.  There are exceptions, of course, so you must take each hand by itself, but generally if one defender has many more points than the other, that is where the bulk of the tricks will be.
The hand you describe is a typical example.  Partner's spades will be difficult or impossible to set up because partner is weak, whereas your hearts are easy to set up because you are strong.  Generally to set up a long suit against no-trump, you need one of the following to be true:

Running suit
Suit can be set up and you have an outside entry
Suit can be set up, partner has entries, and partner has some length in the suit

The first two cases usually (though not always) require that you have some points.  Occasionally you get the third situation where the weak defender has 5 or 6 of a suit and partner has 3 or 4, enough to set up the suit and then lead to be overtaken by the weak hand.
A consequence of this is that if you are very weak against no-trump, it is rarely worthwhile trying to set up your own suits, even on the opening lead (unless you have reason to believe partner is also strong in your suit).  Generally you are better off looking for partner's suit to attack.  And if you are the strong defender and your partner is very weak, don't continue partner's suit unless you have help for him.  If the two partners are comparable in strength, the one who leads their suit first has taken the first step to setting the suit up, so it is often advantageous to continue it unless you have a clear alternative.
At suit contracts, matters are less clear-cut, although defenders' points can still serve as a helpful guide.  High cards remain a good source of tricks, and the defender with more points has more of them.  However, a weak defender with a lot of trump or just useful distribution can often take multiple tricks.

Basically, if I didn't have 6 HCPs in the heart suit, I'd return the spade suit and hope that partner had some of those 6 HCPs in say, QJ of spades, right?

@Tom I still think you're fixating on points where you should be thinking in terms of tricks.  If you don't have any tricks in your hand, you're hoping to set the contract by partner making some tricks.  You make an educated guess as to where those tricks might be had.  If partner led what appears to be a fourth-highest spade, you have at least some indication that spades might end up being a source of tricks for him.

When West leads the fourth best from his "best" suit, spades, presumably headed by the J (or Q), he is hoping to do one of two things: 1) Establish the suit with your help or 2) lead the suit as a "pivot" for you to take the trick and do something else.
Given partner's scarcity of points, 1) is highly unlikely. Opposite Jxxxx, you'd need at least AQx to establish the suit, and then only if you can force out the opposing king before the third round, because partner has no other entry. This, you simply cannot do, with Kxx of spades visible in dummy.
Partner's lead to your ace did manage to preserve the lead in your hand. This gives you the opportunity to "pivot" into another suit, hearts, where the setting tricks are.


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In bridge, do people go through cycles of under- and -over bidding?
In the board game Go, there are two basic styles, high and low. "High" is all the rage for about ten years, until people have forgotten how to play "low." Then "low" gets "rediscovered," and people win tournaments with itfor about ten years until high becomes popular again.
In bridge, people used to "underbid" in competitive situations. That is, they failed to overcall, make takeout doubles or "balance," even when it was to their advantage to do so (that is when the penalty for going down was less than the opponent's probably game). Then people learned these techniques and did them a lot. Have they ever overdone such competitive techniques (that is, give away more games or gone down more than they should have)?
Have people made "mistakes" in the past that would be considered such by today's standards, even though they may have been acceptable bids at the time they were made?

There is a famous anecdote of Hans Kreijns (playing with Bob Slavenburg) overcalling Benito Garozzo's 4 Spades pre-empt with a 4 Spades overcall, on 4 to two top Honours, sometime in the 1960's. That's an overbid.
Seriously, when I learned to play bridge in the early 1970's, overbidding was everywhere. Precision had just become famous, and everyone was opening ratty 10 and 11 counts because C.C.Wei's Precision team did it.
Since then overbidding has declined in the bridge company I keep, and so has underbidding. Of course, it is not the same company: the quality has increased substantially, and the bidding has generally become more accurate. Less overbidding, and less underbidding, because the bidding is simply better.

A lot of "butt-in" interventions are there not so much in a constructive manner to try to find the best contract but to interfere with the opponents' bidding system, which is usually less equipped to deal with it.
Pick up most bidding systems and it can be pages and pages of bids and responses and rebids etc, and very little on what to do if the opponents come in.
The intervener does of course take a risk but hopes to get away with it, especially as the opponents have not necessarily yet ascertained their holdings and may therefore prefer to continue by showing their partners what they have rather than going for penalties.
The real downside of the intervention is that when the other side does eventually win the hand, they will often be able to place the opponents cards better.
Methods like DONT over a strong NT, (and most of the other bids made over a strong NT) are good examples.
In the early 1990s I recall the top international teams were quite aggressive. The Poles in particular. Opening strong pass systems were not uncommon. If you're able to find archives of old tournaments, the 1991 Bermuda Bowl final between Iceland and Poland might be quite interesting to analyse. Of course you have to bear in mind that the Bermuda Bowl is a "teams" tournament and the final is over a lot of boards, and played against a team who has your system notes and has studied it well.
In a match-point pairs event which is a win or lose situation, where a mishap gets you "only" a bottom (but can potentially be recovered quicker) and against opponents who only see your system card in the last moments, one can often be more aggressive.
In the mid-1990s the "Law of Total Tricks" was popularised by Larry Cohen, and Robson & Segal's "Partnership Bidding" was published, primarily about competitive bidding and I think these two publications improved competitive bidding in general among the middle-skilled bridge players. (Decent club-level players who are not international / Bermuda Bowl standard but are not novices either).


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In bridge, are there different implications of overcalling 1NT in different positions?
Suppose I had a hand like the following: (s) KJxx (h) xxx (d) AJx (c) AKx. If I were the opening bidder, I would, of course, bid 1NT. (Note to commenter below: In this paragraph, I was describing a "generic" balanced 4-3-3-3 16 hcp "no trump" hand with honors in three suits. This was NOT a specific question about the particular hand configuration above.)
If someone opened in front of me with 1 club, diamond, or spade (all of which I have a stopper in), but not 1 heart, I would still bid 1NT as an overcall.
The earliest I could do this is in the second seat, after the original bid. In this case, I know little about the holdings of either the opponent's partner or mine, except that they are constrained by the strength of the first two bidders.
If I overcalled in the third seat, I would know that my partner (first seat) was a passed  hand but I wouldn't know this about my opponent's partner (fourth seat).
If I overcalled in the fourth seat, I would know that both my and the opener's partners had passed hands.
If the "one bid" came from the first seat instead of the third seat, I would be in the "balancing" seat, and possibly overcall 1NT with 12-14 points.
Questions:

Do I need to be more careful in the third seat, since I might be sandwiched between two strong opponents?
If the "one bid" came from the third seat after two passes, can I overcall light in the fourth seat as if I am "balancing," even though technically I'm in the direct seat? Expert Larry Cohen calls this "balancing in the direct seat" and advocates this.
If opener bid a suit in which I have stoppers (that is, other than hearts), can I worry "less" about he heart suit, on the theory that since opener didn't show strength in hearts, my partner and the opener's partner are equally likely to have strength in that suit?

Books can be written on "position" and so would be too broad. Talking about a specific hand is too localized. In any case, I didn't find a too localized close option...! So haven't voted to close.

Learn to evaluate this hand as being worth only 14 points, 14.5 at the very, very, most, and your bridge results will improve.

No 10's, no 9's, no 8's, not even any 7's. The opponents will kill you with these cards leading through any and all quacks held by you and your partner. Then they will lead these cards through your kings to kill those too. Minus 1 point.
Minus one point for 4333 distribution. Goren was the world's best bridge teacher for three decades for good reason. Read his books, and recognize that in Chapter one he always stated that the 3-2-1 distribution points are not ruffing values, but indicative of general hand strength from the shape.

Open aggressively in third seat, for the preemptive value against a strong hand in fourth seat. Open this hand 1S in 3rd seat, and teach partner the Drury convention in that situation.
Open passively in fourth seat, so still  open this hand 1S and again teach partner the Drury convention for this situation.
I don't care that you are playing five card majors.
Open 1spade anyways, and teach partner the Drury convention in this situation. Presumably you are asking here to learn better bridge. Learn to open strong 4-card spade suites in 3rd and 4th seat. This is part of becoming a better bridge player.
Don't open 14 point hands 1NT when playing 15-17HCP as your 1NT range.


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In rubber bridge, when should you "lead your partner's suit rather than your own?"
With equal vulnerability, opponents bid 1 club, 1 spade, 1NT, 3NT.
I'm sitting West, opening from: (s) xxx (h) J8732 (d) T9x (c) xx.
I don't fancy leading my ragged hearts suit with no side entry. So should I lead the other unbid suit, diamonds, and hope that partner has 4-5 of them, headed by enough honors so that my T9 sequence isn't wasted? The opponents probably have 26-27 HCPs between them; my one leaves 12-13 for partner.
Suppose my hand were upgraded to (s) xxx (h) J8732 (d) ATx (c) xx. Now I have 5 HC points, versus 8-9 for partner. Is this a better time to lead "fourth best" from my heart suit?

What style of game: MatchPoints, Rubber, Chicago or IMPS? What is the vulnerability?

In general, you are correct not to lead that ratty heart suit with a bad hand. Instead lead DT hoping to catch a quack in dummy and partner with some cards in the suit. With luck you might get in again with a heart to lead the suit a second time for partner.
That being said however, the opponents bid game aggressively; in MatchPoints it might be better to try the heart 3 with the goal of not giving up an overtrick, rather than defeating the contract.
With the second hand, not playing MatchPoints I definitely lead the heart suit; or if the auction had finished 1NT - 2NT - 3NT I lead the heart because opponents have bid a minimum game. However in MatchPoints, with the opponents having bid to game strongly, I am leading the heart suit only because every other possibility is worse and so the heart lead is least likely to give up an overtrick.
Update:
As pointed out below by Ruds, if the opponents play CBS or NMF over 1NT in this auction, then the failure of responder to do so is a loud denial of a 5-card spade holding. In this case only a spade lead (highest spot usually, but dependent on leading conventions) can be considered at all vulnerabilities and styles.

A spade seems less likely to give up a trick in this auction than a heart. Declarer has 1-3 spades and dummy will likely come down with 4 only (with 5, NMF or a similar convention may well have been bid). Also, partner likely has opening values but didn't act after 1C-1S; I suspect we have more spades than they do.

Actually dummy is more likely than not to come down with 5, because not enough spades are located. Dummy is known to have 4+, we have 3, and declarer has 2.5. 3.5 spades are unaccounted for, and there are only two hands for them to go into: partner's and dummy's. Dummy is very likely to have at least one of the 3.5 spades unaccounted for.

Why is dummy not bidding NMF or checkback to show the fifth spade? After 1C-1S-1NT and holding a 5-card spade suit, 4S could very easily be the best contract from dummy's point of view, and he will likely explore for it.

Because it is not in the conventions listed by OP as being played. One cannot pick and choose which conventions the opponents are playing.

The OP listed no conventions, nor even a point range for opponents' 1NT or whether they open 4-card majors, but OK, let me modify my earlier statement. Assuming opponents play a convention that lets them explore a major-suit fit after 1m-1M-1NT (as many pairs do these days, even in the clubs), consider leading a spade on this auction.


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In bridge, how much support do you need to raise a responder?
Suppose I bid one of a (short) minor. Partner responds one of a major (one-over-one), showing at least four cards in the suit.
What do I need to raise his major to two? Some books say I need four cards, others say three cards to the queen  is sufficient.
Maybe a better way of asking the question is, which systems allow me to raise with Qxx, and which require four cards for my raise? Does it matter if partner's response is in spades or hearts? Using the "four card" raise method, what should I do if partner responds one spade and I have four hearts but only three spades?

I think the title of this question is misleading -- you seem to be asking more about quality of support rather than strength of hand.

@ruds: OK, changed "strength" to "support."

This is a matter of partnership understanding and taste more than systemic; most natural bidding systems may be played either way.
First, you would never make strong raises of partner's major (splinter, jump to 3M or 4M, etc) with only 3-card support. This is exclusively for the auction 1m-1M-2M. 4-3 fits are much less fun to play at the 4-level than the 2-level.
Second, the reason that you're bidding this way is that you expect that 2M will play better than 1NT. Many partnerships will only raise on three with shortness in an unbid suit. Very few partnerships will raise with a 4333 hand. Remember, ruffing values in the short hand are good; if you're forced to ruff in the long hand in a 4-3 fit, you may find yourself losing trump control. On this note, I don't like Qxx as a minimum; if you have shortness, 3 small trumps are probably fine because some of them will be winners.
Third, remember that partner may have enough to invite or bid a game after your limiting bid. If partner doesn't expect you to raise on 3, they may bid inaccurately. Make sure that their second bid differentiates between a 4-card suit and a 5-card suit. A common agreement is that 1m-1M-2M-3M is invitational with 5+ in the major, while 1m-1M-2M-2NT is invitational with 4; similarly 3NT by responder asks opener to make a choice of games. There are conventional treatments that are probably better: this article on bridgewinners describes the spiral raises convention.

You have the question backwards. It is not what system allows you to raise with Qxx instead of xxxx, but rather what system you end-up playing as a consequence of making this decision.
This question and its answer will fundamentally shape the way you bid; as such it becomes a fundamental system question. The notion that you can have a compatible partnership with someone who bids this differently then becomes nonsensical.

4 card support is sufficient in any cases. but what if you have 3?
If you are going to adopt a natural approach to the game, this is a matter of judgement.
the goal of constructive bidding is to find the best contract for your side, keep that in mind and make your choice.
when you only have 3 card in the major, a rule of thumb in your case is to raise partner when:

you can produce extra tricks by ruffing, or
you have concentrated value in the suits bid, which usually means little value in unbid suits and feel uncomfortable to bid notrumps.

Beside,
Do not raise partner when you have 4333 shape, bid NT


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In Puerto Rico, how do you avoid "losing a game after having the most victory points?"
I'm not talking about a situation where, on the last round of the game, you had the most victory points at the end of your turn, but someone playing after you passed you by, based on the victory points following his or her turn.
Suppose you have the most victory points, but aren't close to ending the game by using the captain or mayor. It look like one of your opponents is building, and more to the point, can fill the 12th city square with a building in one or two turns, whenever ready.
Is there any way to "head off" this person from winning by ending the game?
Put another way, should you try to keep piling up more victory points so that this opponent can't earn enough with the last building to have more than you? Or should to try to find a way to end the game as soon as possible so that no one catches up? And should the builder refrain from filling the city block if s/he has fewer victory points than you and can't earn enough to catch up on that turn?

It depends, and it isn't really clear what you are asking here. It looks like you are asking, if you have more VPs (all, buildings and chips) than any other player, is it possible for the change in VPs between you and an opponent over two governor phases to be greater than your current lead. (this is obviously possible, and isn't a good question.) Take for example the simplest case a 1VP lead, a Big Building is worth 4 points (assuming no one else wants to Mayor). That puts the +3 ahead of you, so if your building+shipping this round is worth X, and your opponent can ship X-2 they win.

Your question is not clear. The player with the most victory points at the end of the game will win. It is possible for a player with less victory points to overcome a player with more victory points.

@bengoesboom: I clarified the question by changing "whether?" to "how?"

There is no such thing as "enough victory points to win."  Nor does it matter who initiates the play that triggers the end of the game.  The only issue is, when the game ends, who has the most victory points.
There are three ways to trigger the end of the game.  If you are, in fact, in the lead, then your strategy should turn towards driving the game to its end quickly, in a way that prevents your opponents from scoring more points than you do.  Whether that involves "more building", "more mayoring", or "more shipping" depends on your situation.

So there is no, say, 10 point minimum as in "Settlers of Catan?" Maybe that is the source of my confusion.

Have you played Puerto Rico before?? It's not a race to X points; it's a race to get the most points before the game ends. And as hymie has mentioned, the game end can be triggered in three ways. Sometimes these all happend within the same round.


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Does one need to bid Stayman with a flat distribution?
This example is from a book. South opens 1NT (15-17) with:
(s) AKx (h) JTxx (d) Kxx (c) Axx.
North has (s) Qxx (h)Axxx (d) Axx (c) xxx and goes to 3NT. The author (George Coffin) says that North ought to bid 2 clubs, Stayman, with four hearts. That's technically true.
But I don't fancy playing for ten tricks with a 3-4-3-3 distribution. Would rather take my chances for nine tricks with 3NT. (And I do have a bias toward NT.)
With a flat distribution, may responder jump directly to 3NT over 1NT? Or must s/he call a Stayman two clubs since he has four of a major? Which bidding systems allow a 3NT call in this situation?

The South hand you've given has 14 cards.

Subtract a point for 4333 distribution; that yields a 12-14 NT hand, so open 1C and rebid 1NT to show the  hand best.

@ruds: Fixed, only 3 diamonds.

In the hand given, you can sometimes make 3NT and never make 4H without misdefense. But change South's hand slightly to AKx JTxx Kxxx Ax and 4H has more chances than 3NT:
You have 7 top tricks (3 spades, 1 heart, 2 diamonds, 1 club). At 3NT, you must develop two more tricks, and hearts seems like your best shot at that. You must hope for honor-doubleton on your right (run the jack; if RHO covers, win the Ace and play up to the ten. You will lose to the honor on your left and then your ten and long heart are winners). If the honor holding is unfavorable but the split is 3-2, you still have hope of picking up the fourth round of diamonds (duck a diamond, then play the A and K; hopefully they have broken 3-3). You will have held up the CA on the opening lead. If clubs broke 4-4 then you'll make if you were able to take 3 hearts but go down if you lost two hearts as you still must lose a diamond. If they were 5-3 you're always going down on a club lead (and if hearts didn't behave, the opponents can still defeat you with a club switch). If they're 6-2, you can make in the unlikely case that the 1-3 tricks you must lose before taking 9 tricks are all lost to the doubleton club holder.
In 4H, you have a sure club and heart loser. In order to develop ten tricks, hearts must break 3-2 and you must ruff a club and a diamond, which requires losing a diamond, so you must be able to play hearts for one loser (which means Qx or Kx on your right). However, you make even if clubs don't break 4-4.
Consider some other hands declarer may have:
Ax KJTx KQxx Axx
KJxx KQxx KQx Jx
xxx QJTx KQx AK
All of these have more play in 4H than 3N.
So this hand isn't a great example of a hand that would want to avoid Stayman. That being said, there are times when jumping to 3NT with a 4-card major is indicated. Here's such a hand:
AKx KJxx Q9x JTx
This is a flat, meaty hand. You don't have ruffing values, but your side has a combined 29-31 HCP. Not enough for a slam, but very comfortable for game -- unless opponents can establish a ruff or two before you gain control. Let's give partner (South)
Qx AQ9x KJTx KQx
You'll almost certainly take 11 tricks in NT. On the other hand, in 4H, it may easily go diamond to the A, diamond ruff, club to the A, diamond ruff.

There is also the chance that 2C bid might get doubled for lead, and a making 3NT on a non-club lead would end up going down, when you look for a major fit (and probably don't find it anyway). With 27-31 total points and 4-3-3-3 hand opposite the 1NT bidder, it is usually recommended to blast 3NT, which you expect to make on strength alone.

As ruds has looked closely at Declarer's hand, let's now consider that of responder:  Qxx-Axxx-Axx-xxx.

Start at 10 HCP
4333 distribution is worth -1 pt, leaving it as a 9 pt hand.
None of the honours support each other, or are supported by a 10: -0.5 pt taking it down to 8.5 points.
Ace-heavy: good for suits, bad for notrump; no ruffing values, bad for suits, good for notrump. Call it a wash.

Playing 15-17 NT I would bid 2NT with this hand (matchpoints, and not vulnerable at IMPS/Rubber), as I (have learned to) expect my partners to accept invitations aggressively. ;-) (Somehow, good players always do.) Vulnerable at IMPS I might attempt 3NT, depending on state of the match, and how I expect the opponent holding my cards will bid at the other table; I want to duplicate his/her bidding if possible on this hand, and allow my partner to play it better.
Knowing how to, in conjunction with your partner, slightly vary your bidding for the state of match and conditions of the hand is an important and valuable skill. M experience has been that accepting aggressively and controlling the decisions as the invitor is easier and more natural for most partnerships.
Also, knowing the appropriate odds for bidding game is valuable:

Matchpoints, bid 50% games (translates to aiming at 25 combined points).
IMPS, Vulnerable, bid 40% games (translates to aiming at 24 point games).
IMPS, not vulnerable, bid 50% games (translates to aiming at 25 point games again.

Reverse this pattern for bidding small slam at IMPS: aggressively when not vulnerable and passively when vulnerable.

Disagree with devaluing the hands. What about adding points for the Aces? They are valuable at no-trump too. You can decide exactly how many rounds of a suit are played, and they can be valuable for tempo/communications etc. Frankly, I find it highly annoying when people randomly add or subtract points to try to prove the point they are making. IMPS or MPS, I think responder needs to get to game. These are a pair of normal hands. There is no need to do anything out of the ordinary.

Also, your game % odds seem off for IMPS.

@Aryabhata: Bridge Ace agrees with me: http://www.bridgeace.com/Lessons/Rubber%20Bridge%20vs.%20Matchpoints.pdf

@Aryabhata:Northern Colorado Bridge Association agrees also: "If you're vulnerable, it pays to be a little more optimistic when you're considering a thin game. The IMP odds (your potential gain if the contract makes) favor bidding any game that has even a 40% chance of making." from here: http://www.northerncoloradobridge.com/archives/playerscorner/TeamsVsPairs.htm

@Aryabhata: From Cornell University: The odds are not spelled out precisely, but the basic strategy between Vul and NotVul is verified: http://www.math.cornell.edu/~belk/impmp.htm

These are the first three references I looked up in Google with the search string: "best odds for bidding game at IMPs"

I believe it is 38% for vul and 45% for non-vul game bidding at IMPS. So, I agree about more aggresive at vul etc, all I was saying was, you are being less aggressive :-)

The accuracy to which hands can be evaluated is only about 0.5 points, which is clearly greater than the 2% you are quibbling over. As stated, I aim for 24 point games vulnerable at IMPS, 25 point games in matchpoints and white at IMPS. The key is for both partners to know and agree on how the aim changes, and understand that it is necessary.

I rather not discuss ill-defined subjective matters at length, only state my opinion/disagreement and leave it at that. But, I can point out mistakes in things that have an objective answer, like the odds you quoted, which are wrong. You might call it quibbling, but at least it might reduce the confusion in the minds of some readers who might try to work out the odds themselves. You also seem to have missed the smiley :-)

@Aryabhata: I did miss the smiley; point taken.


text:
In bridge, should East consider "overtaking" his partner's lead if able?
One example is if West leads a Q (top of a sequence) against a NT contract, and East plays K from Kx to unblock. If allowed to hold the trick, East would lead back the x.
My understanding is that bridge play is designed to protect certain vulnerable holdings, such as tenaces, in the declarer's hand from direct attack by the opening leader, West.
But if the tenace suit is led by East, its vulnerability will be fully exposed. Should East be on the lookout for potential vulnerable holdings in South, and overtake West's lead if an examination of his cards and dummy (and a review of the bidding) leads to a suspicion that South may have such a vulnerable holding? Here is an example of what I mean.

Just listen to the auction, both the bids and the silent calls, and play good solid defense without being on the lookout for unusual plays. Looking too long or hard under one particular lamppost will blind you to the goodies under all the others. The circumstances in which it is worthwhile for the defense to waste a high card in this fashion are unusual, which is why bridge columnists love such examples.

@PieterGeerkens: One of the problems I am trying to deal with in bridge is one of what I call "redundant honors." That is being unwilling to sacrifice an honor by "unblocking," or save an entry by overtaking, or ditching a potential throw in honor for good, strategic reasons. I'm not sure about the average player, but I lose more tricks and games by wrongly "conserving" than by sacrificing honors.

You will doom yourself to permanent mediocrity by attempting to memorize rules. Learn to read the cards at the table, and study (books, newspaper columns or other) in order to extend your toolkit. Study squeezes and throw-ins until you can recognize them on defence as well as declarer play. Only this will improve your play beyond an intermediate level.

No; overtaking partners honour should not be on East's list of possible actions as dummy comes down.
That is not to say that it is never right for East to do so; or that East should never take such an action.
Rather, there are far too many more important things for East to be studying as Dummy comes down than a check-list of rare and unusual plays. Any such bold action can only be performed confidently through an extremely good picture of the hand, and how the defensive setting trick is to be developed. Such an understanding of the hand will only be hindered, significantly, by spending (need I say wasting) valuable seconds considering a list of circumstances that only occur every one or two hundred hands.
If the action is correct, and one obtains through more efficient means a sufficient picture of the hand to predict this as required to establish the setting trick, it will then become an obvious action.
In the example hand, a count of Hearts and inspection of the Club tenace over Dummy, makes overtaking Partner's HQ obvious. The Heart AJ are both tricks unless Declarer rushed into No Trump with a singleton HJ; and no other Hearts can promote over East's KJT9. If West has HJ alone remaining, the H8 return by East will make clear that a Club return is desired if West gets in again and started with just 2 Hearts.

I thought that kind of play was a bit unusual (order of magnitude 1 in 10) but I didn't realize that it was "rare" (more like 1 in 100).

@TomAu: In truth, I doubt that even one in a hundred is accurate; that is once in 4 rounds for some East to be advantaged by overtaking West's lead. It seems more likely to be one in 500 or so hands at best.

Yes, East should certainly consider overtaking the lead; thinking is usually better than not thinking. But what conclusion he reaches will depend on the bidding (presumably South has the A if the NT contract was freely bid), the expected layout of the suit, and how many times the defence can expect to get in to lead the suit. Inflexible rules that take no account of what information you have will do more damage to your score than anything except forgetting what is trumps.
Edit: if East believes the Q is from QJ10, it would be sensible to unblock the K; if just the QJ, probably not. But the question as asked is about overtaking in order to tick the box 'lead through a tenace', which is a bad idea, and will remain so even if in this case it (improbably) provides a good result.

I opined that East should CONSIDER overtaking partner's lead, not that he should automatically do so. The "other information" he has is part of that consideration.


text:
Could Axis and Allies be "balanced" by weakening the US just a bit more?
In Axis and Allies (as in the real war), Germany and Japan together have more industrial power, represented by industrial production certificates (IPCs) than Britain and Russia combined. Hence The US holds the balance of power.
Although the game is still tilted in favor of the Allies, it is much closer than in "real life" because US industrial capacity has been reduced by almost two-thirds. In fact, bidding systems have shown that if you give the Axis a one-time boost of only 12 IPCs at the beginning of the game, it would be even, assuming that Russia is not allowed to attack the first turn.
Axis and Allies - Is there a best country?
Could the game be balanced by reducing America's strength just a bit more? Specifically by taking away 1, 2, or all three of the 3 IPCS per turn that the US receives from Brazil? (This would reduce her IPCs per turn to 35, 34, or 33 respectively.  Even allowing for the factors of time and distance, might this not be equivalent to the "one time" boost of only 12 IPCs to the Axis at the beginning of the game?

Which version of A&A are you talking about? There are many, going back to the original Nova Games version, and each one is different enough to require different balancing.

This would be an interesting variation. I'm thinking you would have to remove all 3 IPC from Brazil. The timing of when IPCs are delivered has a huge effect. Early IPCs allow the Germans to build up faster and drive Russia back, put a nice force into Africa quickly, or replace one of their lost fighters that they usually lose 1st turn. Even so, losing 3 IPC/turn from Brazil might not be as much of a factor as 12 IPC early. 3 IPC is only 1 less US infantry. The US threat builds over time, and is mainly in the form of retaking western Africa, threatening Europe, or reinforcing Karelia. All of these sap German resources. By turn 4 when the US had lost 12 IPC from the Brazilian removal, the US forces would already by doing most of the damage they could be doing, and the German lack of 12 IPCs would already be showing in the inability to expand as quickly.
Why not just whack the US first production by 12 IPCs? You could justify it as a late start into the war because of Pearl Harbor and taking time to mobilize its defense industry. (Of course, in reality, the US defense industry was going quite heartily, supplying the Allied forces).


text:
Does it make sense for a defender to lead a short major when opponents fail to find a fit using Stayman?
Both vulnerable, at rubber, South dealt and bid 1NT, North 2 clubs, South 2 hearts, North 3NT.
You, West, have (s) 64 (h) AJ7 (d)J8752 (c) Q84.
North clearly has four spades, but not hearts. Any spade strength on your side is with East, behind North. You like the strength you have behind South's hearts. Apparently, the bulk of North-South strength is in the minors.
The actual West led the 5 of diamonds. It was a "friendly" lead from a weak suit that gave North and South all the time in the world. Reasoning from the previous paragraph, I would have lead a small spade (and hoped for a heart return).
This is a bit unconventional but does this make sense? If declarer thought that I had length and strength in spades, could that cause a misplay? (In the example, East actually had five to the king, but of course, I wouldn't know this until later.)
Put another way, how much weaker could the diamond suit be, before it is not worth leading?

Form of scoring?

@AlexanderWoo: Rubber bridge.

I did a double dummy simulation using the hand and auction you gave.  The results were:

The 5 of diamonds lead has an 8.8% chance to defeat the contract (playing double dummy).
The 6 of spades lead has a 6.6% chance to defeat the contract (playing double dummy).

I used 1,000 hands for the simulation, so statistically both of the figures above should be accurate to within about 1%.  In particular, the higher chance of setting the contract from a diamond lead is statistically significant.
Here were the parameters for the simulation:

South has a balanced hand (no singletons, at most one doubleton) with 15-17 HCP, three or fewer spades, and four or five hearts.
North has 10-15 HCP, exactly four spades, three or fewer hearts, five or fewer of each minor suit, and no voids.

Of course, real players don't play double dummy, so we should take these results with a grain of salt.  In particular, it seems to me that if you lead a spade your partner might not figure out that you have a five-card diamond suit, making him less likely to switch to diamonds on trick two than he would in double dummy.  So if anything we would expect the results from real play to lean even more heavily towards the diamond lead.
Decreasing the quality of the diamond suit doesn't help the spade lead much.  If we change the J8752 to a 65432, the diamond lead is still the favorite by about the same margin.

It would be interesting to know how many times the Spade lead gave away an over trick to declarer compared to the diamond lead. As a beginner and early intermediate I also liked this lead, and learned expensively that not to. As I recall not only did it not set the contract, but it usually gave away an overtrick  (very important at Match Points.

Also of interest is how many hands are in common between being set by a Spade or a Diamond lead - Those hands where all leads set the contract are not of interest in determining whether attacking a short Major in Opening Leader's hand is sound practice.

If opponents have only 24-25 points the auction would likely have involved acceptance of a game invitation. Instead there was a jump to game by responder without a slam try, each opponent having 4 cards in a different major; opponents rate to have 29 points +- 3. You have 8 points in hand, giving partner 3 points +- 3. The likelihood of being able to set up and run a suit in partner's hand is extremely remote.
However if partner has QTx in diamonds and an entry you might be able to win a race against Declarer in that suit. Lead your fourth best diamond as was actually done at the table. This also gives partner a clear signal as to what to lead back to you when he gets in the (probably) one only time that happens.
On this particular hand I see no reason for method of scoring to matter, but as a matter of good form it should be specified.
It is important to note also that partner is going to be squeezed in both Majors. Clubs are splitting 3-3 for Declarer. Combined with the fact that opponents jumped to game rather than accepting an invitation, at Matchpoints I am going to be happy to get four defensive tricks for Average plus against all those who make an overly aggressive, and desperate, attempt to find unavailable tricks in partners hand. Even sitting patiently for my third defensive trick will often be a good Matchpoint result on this hand.
In short, to answer your specific question: No, it is pointless to search for major suit tricks on this auction. Partner has neither the suit strength nor the entry to make searching for his suit profitable. The only chance to set this hand is a miracle of clashing honours for opponents, and even then you will have to have set up cashing tricks in your hand to collect on the good fortune.

Revisiting this, the biggest problem with a Spade lead are that these three scenarios are most likely:

Partner wins the Spade lead and returns it, setting up his Spade winners. You now have no more entries to partner's tricks, and have lost two tempos for setting up Diamonds.
Partner wins the Spade lead but sees no future in the suit, and makes a random guess as to which minor to return guessing wrong 50% of the time. Either one or two tempos for setting up diamonds have been lost.
Partner has Qxxx or Jxxx unsupported in Spades and you have just guaranteed Declarer four Spade tricks and the contract.

Potentially settable Notrump contracts are a race between Declarer and Defense, and the loss of even a single tempo nearly always guarantees losing that race. It is imperative for the Defense to chase long-suit tricks in the hand with entries. Here, it is very unlikely that Partner can have enough points both to make Spades worth setting up and have an entry making them cashable.
Form of Scoring:

In Rubber (as here) of IMPS one always makes an extreme effort to set the contract, as overtricks are very rarely of import. Lead a diamond.
In Matchpoints, one is looking to not get a bottom on the opening lead. Experience has taught me that leading from jack empty fourth is a nearly guaranteed bottom - but leading from jack empty fifth or longer is often a winning play. The decision is now more complex. It is possible that Partner's Spade trick is dead unless you lead the suit now, so that lead might be right. I am personally an aggressive defender, so would usually lead a Diamond, but would accept a Spade lead from partner as reasonable.

Actually, at duplicate match points, assuming a strong NT, I worry the diamond lead gives away an overtrick to declarer's AQT(x).  I've already given up on setting the contract.

Also, Pieter implicitly made an important point, which is that if you had a weaker hand overall, the spade lead becomes more attractive, because partner is likely to have a stronger hand.

One point you seem to be making is that North should have bid two spades  (invitational) with his ten points instead of jumping to game. Would you jump to game as North with an extra jack (11 points instead of 10? Even as it was, the contract was makeable, but the declarer blew it.

@TomAu: Assuming a 15-17 opener, I will usually jump to 3NT with 10 HCP. As always, the particular hand with all spots should be given before drawing firm conclusions. Vulnerable approaching always; non vulnerable not quite so firm.

@TomAu: But to clarify: With a good ten I jump to 3NT on the rebid, with a bad ten I rebid 2NT. There are (almost) no eleven point hands that rate as a bad ten. An exception would be something like J432  J53  AKQ  532 where everything is blocked, no spot cards, 4333 distribution, and no length with the points.

Opening leads are difficult! Either a spade or a diamond could be right. Neither is likely to set the contract often, so it can be difficult to develop a good intuitive feel for which is best.
I'd lead a diamond, since diamond strength with partner seems our most likely road to five tricks. It's also the lead least likely to blow a trick.
For an empirical analysis of NT leads using double-dummy software see:
http://www.amazon.com/Winning-Notrump-Leads-Taf-Anthias/dp/1554947596

My concern is that even if partner has, say the Q or K, we might be missing the T and 9. Change the 8 to a T, so that I'm leading from JT752, and I'd lead a diamond. In the actual example, East had 64 in diamonds.

A diamond could give up a trick, but so could any lead. Much of the time that partner has Q empty we'll find the A and K in the same hand, or dummy with Ax or Kx.

@AdamWildavsky: I would add that leading low from Jxxxx is significantly safer than leading low from Jxxx.

No argument from me on that score, @PieterGeerkens. That's one reason the standard recommendation is to lead one's longest suit versus NT. Like most bridge "rules" it's more of a guideline, and has a number of exceptions.

Not in the example you gave. Because the disparity in lengths between the spade and diamond suits is too great.
After removing your cards from the deck, "average expectations" for your partner is something like (s) Kxxx (h) Qxx (d) Kxx pr QTx (c) Kxx. Note that s/he will actually have less than that because the bidding has further capped partner's hand to about eight HCP. So let us say that one of the kings is removed from the hypothetical hand.
Partner needs to have only Kxx or QTx of diamonds, plus one side honor, that is adjusted "average expectations," to make your hand good. In order to defeat the contract using spades, opposite your xx, partner would need an ideal holding, something like KQxxxx, or KJTxxx, much more than adjusted "average expectations," plus a side honor. A partner that had such a hand might have overcalled two spades. Absent such an overcall, you have no right to think that a spade lead will defeat the contract.
Your question would have been more topical if you had xxx in spades, and Jxxx in diamonds. The third spade would count for something, and more to the point, the missing fifth diamond counts are a lot.

Jim Belk gave perfect answer!
I mean, may be his analysis was not perfect, may be he should enter another conditions (like 14 balanced with 5332), or 9-13 Hcp opposite instead of 10-15, or make it 100000 hands instead of 1000...
But the way he solves the problem is the only correct way.
Let me clear this: everyone who plays bridge more than 1 year met this position at least 2 times :), so we discuss what could we do, if your  choice was unsuccessful. The problem is really common!
But
careful! here is some esoterics
As in life we sometimes think, that we know something (basing on our experience). But Life in wide meaning (you can say G-d or Buddha or Nature or whatever) knows much more than we.
And one(or ten or even 1000) board means nothing. It's like grain of sand opposite Jupiter (or may be 1000 Jupiters).
esoteric ends here
To be honest, computer modeling gives correct % very soon , after 10000 boards matching your conditions, next 100000 boards will not change %.
Don't forget one thing - computer gives you correct %, but it can not give you the correct answer!
Feel the difference?

It's very unclear what you are trying to say here. Less touchy-feely esotericism and more actual bridge would do wonders for this answer, if you in fact have something more to say than a mere comment.

I just wanted to say, that there is no absolute correct answer. Any "i paid expensive for this(or another)" don't give even a clue.

If there was any validity to what you are saying, then Bridge would be a game of pure luck. Yet, as the preeminent Eric Murray noted in Regina v. Regal Bridge Club: "Bridge is only a game of chance as played by the justices of the Ontario Superior Court, in chambers."