An interesting mathematics issue?

[Trumpace]

See, this is where my head explodes.

I believe that I have also proven that this tactic is very good against superior tems, but that it is also good against equal teams.  It may be four times better advantage against a superior team than against an equal team, admittedly, but it still succeeds against an equal team.

You have proven nothing substantial.

 

All you have proven is that if you are tied with your so called superior team, with 3 boards to go (if you can do this good so far, you don't need the weird strategy), with 2 close to <50% slam decisions then your strategy works.

 

If you had say 12 boards to go with 6 slam hands close to <50%, your strategy is unsound for BAM. (I am assuming you pair up boards and play each pair independenty with your described strategy.)

[kenrexford]

Why is there a constant switch to BAM when arguing against the principle? As noted, the reasoning involves a large swing, the specific instance of a small slam. A BAM is not a similar situation.

 

Why, also, do people keep talking about the "superior team," when I have screamed that the principle is sound against an EQUAL team. The superior-team situation is desperate, where anti-percentage has its own merits. The equal-team situation is not, and hence the greater mystery.

[Trumpace]

Why is there a constant switch to BAM when arguing against the principle?  As noted, the reasoning involves a large swing, the specific instance of a small slam.  A BAM is not a similar situation.

 

Why, also, do people keep talking about the "superior team," when I have screamed that the principle is sound against an EQUAL team.  The superior-team situation is desperate, where anti-percentage has its own merits.  The equal-team situation is not, and hence the greater mystery.

The fact that you swing X, 0, or -2X is the same whether it is BAM or IMPS. In BAM X is 1, while in IMPS X is whatever... the reasoning stays the same. You have negative expectation of IMPS or BAM points (choose whatever). For "long" runs, this negative point expectation will show itself. In your example of 48% slam, if we encounter 3 such pairs of slams, your strategy becomes unsound.

 

 

Since we are discussing mathematics, I don't care if the team you are playing against is a bunch of LOLS or a team of clones of Hamman.

[kenrexford]

A similar principle, by the way, is present in a gambling theory.

 

Suppose, for instance, that a specific course will net +10 once every twelve tries, but -1 on the other eleven tries. The economics suggests that this course is 11:10 a poor idea.

 

However, if you have a limited number of occurrences, the magnitude of the action is spread across several groupings. If the course was a bridge action, for instance, it might come up once per session. For eleven sessions, you would have an expected loss of one, which is rarely fatal. However, once every eleven sessions, you would have a gain of ten, which may be critical.

 

This might be translated. In teams events, for instance, what happens if you make a routine slam try that has a one-in-twleve chance of finding gold for a 10-IMP gain but that forfeits on IMP worth of overtricks every time the golden hand is not there? On average, you will have a probably-insignificant loss most of the time. Adding all of those up is less rewarding than the once in a while big gain. But, the big gain is meaningful. If twelve matches were expected to be decided by a margin of 3-6 IMPs, then you tactic would never lose anything but would gain once every twelve matches.

 

So, the question would seem to be whether the chances of a one-IMP difference in final score is greater than one-twelfth of the occurrences of a final score that is 1-9 IMP's plus 50% of the occurrences of a tie at the end. The relative total gained IMPs versus total lost IMPs is not as relevant.

[cherdano]

A similar principle, by the way, is present in a gambling theory.

 

Suppose, for instance, that a specific course will net +10 once every twelve tries, but -1 on the other eleven tries. The economics suggests that this course is 11:10 a poor idea.

 

However, if you have a limited number of occurrences, the magnitude of the action is spread across several groupings. If the course was a bridge action, for instance, it might come up once per session. For eleven sessions, you would have an expected loss of one, which is rarely fatal. However, once every eleven sessions, you would have a gain of ten, which may be critical.

 

This might be translated. In teams events, for instance, what happens if you make a routine slam try that has a one-in-twleve chance of finding gold for a 10-IMP gain but that forfeits on IMP worth of overtricks every time the golden hand is not there? On average, you will have a probably-insignificant loss most of the time. Adding all of those up is less rewarding than the once in a while big gain. But, the big gain is meaningful. If twelve matches were expected to be decided by a margin of 3-6 IMPs, then you tactic would never lose anything but would gain once every twelve matches.

 

So, the question would seem to be whether the chances of a one-IMP difference in final score is greater than one-twelfth of the occurrences of a final score that is 1-9 IMP's plus 50% of the occurrences of a tie at the end. The relative total gained IMPs versus total lost IMPs is not as relevant.

If you assume equally strong teams, then a tie or a 1 IMP win is more likely than a 10 or 11 IMP win. So in fact you have it backwards, in a short knockout match you should be more careful about protecting the overtrick rather than looking for the rare slam.

[kenrexford]

What? I'm not saying that you will need to win by 10. I'm saying that the the likelihood of that 10-IMP gain changing the results of a match may be greater than the number of times than all of the one-IMP loss decides the match. You can figure this out by seeing how many matches are lost by about 1-12 IMPs (plus half of the ties) and how many are tied at the end. (Actually, 1/2 of the ties plus 1/2 of the +1's.) I'll bet that many more matches are lost by 1-12 IMPs than are tied, and that the difference is much greater than the likelihood that disclosure costing an IMP overtrick outweighs the gains from the remotes being found.

 

Comparing a 1-imp or tie to specifically a 10-11 imp match is incorrect, therefore. You must compare the 0-1 with the 0-12, because the 13-IMP gain swoops all of these up.

 

I mean, this seems obvious. If this were not so, why would you be so careful to make games and make slams and set games and slams at the cost of overtricks? It seems that one could say that overticks might add up to counter the one lost game, but the match is not long enough for the full odds to be realized, and the overtricks lost in aggressive defense and safety plays are deemed less likely to shift victory than the massive IMP loss/gain from the set.

[Elianna]

Some get it.  When I re-clarify until I'm blue in the face, it is because some do not.

As I have to tell my teenagers (who have a developmental excuse not to grasp this subject): just because someone disagrees with you, doesn't mean that they don't "get" what you're trying to say.

 

Try to argue with specific points people have made. Continuing your arguments in vague generalities not only convinces no one, it just makes people more stubborn in not agreeing with you, because it makes them believe that you don't HAVE counterarguments.

 

As it is, I'm not sure that they're wrong.

[hrothgar]

What? I'm not saying that you will need to win by 10. I'm saying that the the likelihood of that 10-IMP gain changing the results of a match may be greater than the number of times than all of the one-IMP loss decides the match.

Yhere was a fairly extensive discussion about the impact of variance on placing on rec.games.bridge a few months back. Gerben provides some very useful Monte Carlo simulations. The original analysis was framed in terms of skill level, however, its analagous to the point that you're raising.

 

http://groups.google.com/group/rec.games.b...e&rnum=1&hl=en#

[awm]

I think what Ken Rexford is saying can be summarized by the following from game theory:

 

Best response to opponents strategy is not necessarily the same as equilibrium strategy.

 

In other words, if the opponents strategy is somehow fixed and known, you can try to come up with the best strategy to counter that. In a team match format where the goal is "to win" and margin doesn't matter, it is typically best to select some "different" actions from opponents to try to create swings early in the match, and then stop doing this as soon as you have a lead to protect. This is the strategy Ken is proposing.

 

However, in real life the opponents strategy is not necessarily known, and they can modify it on the fly. The opponents may know something about my strategy too. Assuming symmetry and complete knowledge, the best I can do is to select an equilibrium strategy that guarantees the opponents cannot obtain any advantage over me, and that if they also select an equilibrium strategy I will win 50% of the time (if they select some inferior strategy I may well win more than 50% of the time, but nothing I can do will guarantee a win more than 50% of the time against a truly equal adversary who knows my strategy as well as I know his). Trying to select a non-equilibrium best response will work great if opponents strategy is fixed, but leaves open the possibility for them to change their strategy and cause me to lose more than half the time.

 

Of course, there is also a scenario where the game is not truly symmetric; even though my "bridge skill level" may not be superior to the opponents, I may have more knowledge of what "they're going to do" than they have of what "I'm going to do" or maybe I somehow have more control over the swinginess of the match than they do (I play weirder methods, but our skill levels are the same). In these cases I can use my additional knowledge and/or control over the frequency/size of swing boards to obtain an advantage, by varying my strategy. A simple example of this is a barometer match where my table is much slower than the other table, since I have information about "state of the match" that was not available to the opponents when they held my cards.

[cherdano]

What? I'm not saying that you will need to win by 10. I'm saying that the the likelihood of that 10-IMP gain changing the results of a match may be greater than the number of times than all of the one-IMP loss decides the match. You can figure this out by seeing how many matches are lost by about 1-12 IMPs (plus half of the ties) and how many are tied at the end. (Actually, 1/2 of the ties plus 1/2 of the +1's.) I'll bet that many more matches are lost by 1-12 IMPs than are tied, and that the difference is much greater than the likelihood that disclosure costing an IMP overtrick outweighs the gains from the remotes being found.

 

Comparing a 1-imp or tie to specifically a 10-11 imp match is incorrect, therefore. You must compare the 0-1 with the 0-12, because the 13-IMP gain swoops all of these up.

 

I mean, this seems obvious. If this were not so, why would you be so careful to make games and make slams and set games and slams at the cost of overtricks?

My point is, if a tie is more likely than an exact loss 2 IMPs, and more likely than a loss by 3 IMPs, and more likely than a loss of 4 IMPs, ... AND a 1 IMP loss is more likely than a 2 IMP loss, and more likely than a 3 IMP loss, etc, than 50% of (tie or 1 IMP loss) is more likely than 1/12 of 0-11 IMP loss.

 

I think many people would intuitively agree with the former (a tie is more likely than, say, an exact 7 IMP loss) but disagree with the latter. This is clearly wrong, but just one of the many situations where intuitions about probabilities tend to lead many people astray.

[kenrexford]

Some get it.  When I re-clarify until I'm blue in the face, it is because some do not.

As I have to tell my teenagers (who have a developmental excuse not to grasp this subject): just because someone disagrees with you, doesn't mean that they don't "get" what you're trying to say.

 

Try to argue with specific points people have made. Continuing your arguments in vague generalities not only convinces no one, it just makes people more stubborn in not agreeing with you, because it makes them believe that you don't HAVE counterarguments.

 

As it is, I'm not sure that they're wrong.

Yes, very insightful analysis. There one general problem, though, with this analysis. Many disagreements and arguments that very educated adults have, like politicians and academics, are arguments that arise because neither side truly ever understands the other, or is so blinded by preconceptions and agendas that they do not really want to understand the other side. The best way to assess whether the person with whom you are communicating gets your side and to prove that you get theirs, it seems, is to assess the ability to regurgitate the argument of the other back before countering the points contained therein.

 

When the regurgitation includes incorrect articulation of the original point, then you know that the point was not received.

 

By the way, some folks, most notably AWM, are regurgitating better than I originally articulated and are therefore countering (and often bolstering) quite well.

[cherdano]

Of course, there is also a scenario where the game is not truly symmetric; even though my "bridge skill level" may not be superior to the opponents, I may have more knowledge of what "they're going to do" than they have of what "I'm going to do" or maybe I somehow have more control over the swinginess of the match than they do (I play weirder methods, but our skill levels are the same). In these cases I can use my additional knowledge and/or control over the frequency/size of swing boards to obtain an advantage, by varying my strategy. A simple example of this is a barometer match where my table is much slower than the other table, since I have information about "state of the match" that was not available to the opponents when they held my cards.

Of course in your example, this means I can without worries about mixed strategies take the anti-percentage play in the slam on the last hand if I know that the other table couldn't know at the time that they were leading.

But it still doesn't make the necessary assumptions for Ken's strategy valid: if we are tied with two boards to go, and the other table is long finished, then I still don't know that it is me who can try to swing on the last board, or whether it is our table opponents. (Unless we make some strange assumptions about opponents being too stubborn to try swinging etc. - I don't see how weird methods make it easier to control swinging, they rather seem to force us to swing; a standard pair can always choose to make the standard bid of the other table, or do s.th. different.)

 

I think your example about team strategies is the only convincing one in this thread so far.

 

Arend

 

(Btw, in real life my opponent's seem to apply a mixed strategy anyway. They bid some 50% slams, and don't bid others, take anti-percentabe plays on other hands, all highly unpredicatble...)

[Trumpace]

The best way to assess whether the person with whom you are communicating gets your side and to prove that you get theirs, it seems, is to assess the ability to regurgitate the argument of the other back before countering the points contained therein.

 

When the regurgitation includes incorrect articulation of the original point, then you know that the point was not received.

And the best way to assess that the regurgitation was successfully understood, it to have it regurgitated again, and again.. and again.. and so on and so back and forth.

 

Yuck!

 

Sorry, but what you say is pretty ridiculous.

 

The point you tried to make in this thread was simple enough. Regurgitation isn't required. Perhaps if you reread what others have to say...

 

Anyway, I think I will stay away from this thread. Nothing of value seems to left here.

[hrothgar]

A simple example of this is a barometer match where my table is much slower than the other table, since I have information about "state of the match" that was not available to the opponents when they held my cards.

 

Of course in your example, this means I can without worries about mixed strategies take the anti-percentage play in the slam on the last hand if I know that the other table couldn't know at the time that they were leading.

I don't think that this is correct

 

In many cases, an equilibrium strategy in a simultaneous move game is also the equilibirum in a sequential move game. I beleive that this is one of those cases.

 

Lets stick with the standard 52% line versus 48% line in some game or slam contract.

 

Assume the following: The table that plays first is aware that the table that plays second will know the state of the match before they chose their line of play. Furthermore, they know that the table that if the table that plays second is trailing, they will need to plays adopt a different line of play than the table that plays first.

 

The incentive to randomize across the 52% and the 42% line is still there.

[awm]

Here's some actual math.

 

Suppose it is the last board of a barometer team match, with my table being much slower than the other table. I have the opportunity to bid (or not bid) a slam which will make 1/3 of the time.

 

When my opponents at the other table made this decision, they did not know the state of the match. They decided to bid the slam with probability p, or avoid it with probability 1-p.

 

If I am leading by a small amount, then by bidding the slam I will win any time I made the same decision as the other table, or any time the slam makes. My chance of winning is 1/3 + (2/3)p. If I avoid the slam, my chance of winning is 1 - (1/3)p. Examining this, I should bid the slam exactly when (p>2/3). In other words, if I'm winning I should only bid a bad slam if the other table probably bid it.

 

If I am losing by a small amount, then by bidding the slam I will win only when the slam makes and the opponents didn't bid it, for a chance of winning of (1/3)(1-p). By avoiding the slam I will win with probability (2/3)p. Here we can see that if I'm losing I should bid the slam exactly when (p<1/3).

 

Now let's look at the opponents decision. They figure they're equally likely to be winning by a little or losing by a little, since they cannot see an up-to-date barometer result when they play the last board. If they set (p>2/3) then I will bid the slam only if I'm up. If I'm up before the last board, opponents will come back to win with (2/3)(1-p) probability. If I'm down before the last board, I come back to win with probability (2/3)p, so opponents win with probability 1-(2/3)p. The overall chance of opponents winning, since it's equally likely they are up or down with one board to go, is (5/6) - (2/3)p. Thus assuming (p>2/3), opponents should set p=2/3 to maximize their chances, in which case they will win (7/18) of the time.

 

Now suppose they set (p<1/3). I will bid the slam only if I'm losing. In this case I come back to win with probability (1/3)(1-p), so opponents win with probability 2/3 + (1/3)p. If I was winning, then I will not bid the slam, and opponents come back to win with probability (1/3)p. So the overall chance opponents win will be (1/3) + (1/3)p. Thus assuming (p<1/3), opponents should set p=1/3 to maximize their chances, in which case they will win (4/9) of the time.

 

Now suppose they set (1/3<p<2/3). I will never bid the slam. If I was leading with one board to go, opponents come back to win with probability (1/3)p. If they were leading, then I will come back to win with probability (2/3)p and therefore opponents win with probability 1-(2/3)p. Combining these, opponents win with probability (1/2)-(1/6)p. They do best to set p as small as possible, in which case they win with probability (4/9) when p=1/3.

 

So in this case, opponents should bid the slam that makes 1/3 of the time, 1/3 of the time. They will win 4/9 of the time and I will win 5/9 of the time. Note that I did obtain an advantage from my superior knowledge of state-of-the-match.