[Chamaco]

Trumpace, on Dec 20 2005, 10:55 AM, said:

I think there is a terminology mismatch.

By a 'match' I mean the whole of Bermuda Bowl!

According to you, Italy vs Egypt is a different match as compared to Italy vs Usa in BB2005, but according to me it is part of the same BIG match...

Even so, the long run analysis (just calculating the expected IMPS per board)  which you seem to think should work, can lead to incorrect strategies, as the above example of coin flip game shows.

If you run it over a long timeseries, we get closer to the mathematic odds, that is, the example of the insurance company, who is willing to risk to pay a big amount for a very low frequency risk, but cash in in the rest of the cases.

I think the example of the insurance company is much better than the example of the coin flipped:
the insurance can be one-year only, 2 years or longer, and the risk calculated on a 1 year period can be an analogy to the risk computed of the short-term" match (or set of matches) in bridge.

====

The exmple of the biased coin flip is not well posed, because:

1) at the end of a single set, before we start a new set of coin flips, the % of outcomes are not the same as at the start, because the probability "a priori" has changed because in the previous set(s) the coin exited as tail X times and exited as heads Y times;
so if you have won one set, you are almost sure to lose the 2nd and 3rd set;
you cannot recompute the odds for the following sets with the original probabilities, the odds must be updated as a function of past results.

2) the % of risk is too high compared to the number of coin throws.
If you run the same analysis with every set having, say, 6+ throws, we all know that my side would be a winner.

[hrothgar]

Chamaco, on Dec 20 2005, 02:09 PM, said:

The example of the biased coin flip is not well posed, because:

1) at the end of a single set, before we start a new set of coin flips, the % of outcomes are not the same as at the start, because the probability "a priori" has changed because in the previous set(s) the coin exited as tail X times and exited as heads Y times; so if you have won one set, you are almost sure to lose the 2nd and 3rd set; you cannot recompute the odds for the following sets with the original probabilities, the odds must be updated as a function of past results.

I'm not sure if I understand your point. You seemed to be confusing "bias" with auto-correlation. The coin toss example is typically used to illustrate independent events. The chance that the coin turns up heads or tails on flip N is NOT a function of the results on flip N-1, N-2, N-3, ...

When people talk about a "fair" coin toss, they mean a coin in which heads is expected to come up 50% of the time and tails the other 50%. In contrast, a biased coin has been loaded in some way (weight) such that the percentage chances are not 50/50. However, biased coin tosses as still independent of one another.

Its possible that you're discussing a situation in which you don't know the odds that the coin turns up heads versus tails, but this is a very different type of problem...

[Chamaco]

hrothgar, on Dec 20 2005, 01:27 PM, said:

I'm not sure if I understand your point.  You seemed to be confusing "bias" with auto-correlation.

Sorry, perhaps wrong terminology.

My "bias" term referred to Trumpace premises which were as follows:

"We have a coin which has 75% chances of showing heads and 25% chances of showing tails."

it's obvious that the outcome of the the coin flip is "biased" (if it was a dice throw, the dice would be "loaded"), in terms of an "a priori" probability different from 50%, but feel free to use the most appropriate term, I won't argue about it :-)

Of course, it is right to use the "biased" example, since both in the coin flip AND in the safety play, we have an a priori probability different from random.

What I was criticizing is the fact that, in the analysis of the coin flip bets, one should take into account that, after each set of coin flips, the odds should be updated as a result of the previous flips.
At the second, third, fourth, etc set of coin flip sequences, the odds for head and for tail will be different than 75% and 25% based on the previous outcomes.

This alters considerably the cost benefit analysis, even in Trumpace's example.

[MickyB]

I've only skimmed this thread, so apologies if I'm repeating something that has already been said -

Cross-IMPs is what we have in MBC on BBO. The tactics should be pretty much the same as teams, so I'd go for the overtrick.

Butler IMPs is different - you remove the top two and bottom two scores, take the mean and then IMP up against that. Overtricks tend to be less valuable there.

[42]

mikeh, on Dec 19 2005, 05:47 PM, said:

[snip] the sense [snip] is that the majority of good players take the safety play unless their table feel tells them that the overtrick play will definitely work.
[snip]
Against a tough team, in a short match, I'd risk the overtrick if my antennae were in good shape that day ...

The insurance uses pure data and knows how high the contribution of every client must be to ensure the constant plus for them, even when they must pay from time to time, because they exclude the active brought out ensurance case like committing suicide or so by their clients, or some risks that cannot be controlled or would ruin them with one strike like a war. They try (try because there might be some creative cheaters and the detectives of the insurance company don't find it out) to exclude the human being as a risk that they cannot control, they just work with it as a factor. That is the difference to bridge. One cannot control the partner or the opps (and sometimes oneself ). But one tries to "manipulate" them (I don't mean cheating!!): the partner by delivering the best information under the given circumstances, and opps by creating problems and traps. Think of expert falsecarding: it pays sometimes against good opps who watch the cards whereas it is more or less useless against a palooka who does not pay attention. And one tries to get as many informations as possible, including those things described as table presence.
In the example with coin, may it be loaded or not: one flips the coin and it is assumed in the experiment that the same single person does it under the same conditions. Again the possibility of a manipulation is excluded.

What I think is that pure mathematics and statistics will not make the perfect bridgeplayer, but I don't know.

[Gerben42]

Against a weaker you should take the safety play as you expect to pick up some 11s on other boards. Pay 1 IMP insurance to not give them the chance to get this one right and get 11 back.

Against equal strength teams, train to become better than them but take the safety play. If they don't you have an outside chance of winning.

Against stronger teams you might want to go for the overtrick. If they don't you're already 1 IMP ahead

[hrothgar]

Chamaco, on Dec 20 2005, 04:42 PM, said:

What I was criticizing is the fact that, in the analysis of the coin flip bets, one should take into account that, after each set of coin flips, the odds should be updated as a result of the previous flips. At the second, third, fourth, etc set of coin flip sequences, the odds for head and for tail will be different than 75% and 25% based on the previous outcomes.

Assume that we have a "loaded" coin that comes up heads one out of every 6 throws and tails 5 out of every six throws. This probability is fixed in stone. (If you have a problem believing that a coin can be fixed so precisely, feel free to consider a fair six sided die which will roll a "1" one out of six times and a "not 1" 5 out of ever six times)

Furthermore, lets assume that we flip the coin 8 times in a row. The coin lands tails first each and every time.

Now, lets flip the coin a 9th time.

The odds that the coin will turn up heads is still 1/6.
The odds that the coin will turn up tails is still 5/6.

[Trumpace]

Chamaco, on Dec 20 2005, 06:09 AM, said:

Trumpace, on Dec 20 2005, 10:55 AM, said:

I think there is a terminology mismatch.
Even so, the long run analysis (just calculating the expected IMPS per board)  which you seem to think should work, can lead to incorrect strategies, as the above example of coin flip game shows.

If you run it over a long timeseries, we get closer to the mathematic odds, that is, the example of the insurance company, who is willing to risk to pay a big amount for a very low frequency risk, but cash in in the rest of the cases.

I think the example of the insurance company is much better than the example of the coin flipped:
the insurance can be one-year only, 2 years or longer, and the risk calculated on a 1 year period can be an analogy to the risk computed of the short-term" match (or set of matches) in bridge.

====

The exmple of the biased coin flip is not well posed, because:

1) at the end of a single set, before we start a new set of coin flips, the % of outcomes are not the same as at the start, because the probability "a priori" has changed because in the previous set(s) the coin exited as tail X times and exited as heads Y times;
so if you have won one set, you are almost sure to lose the 2nd and 3rd set;
you cannot recompute the odds for the following sets with the original probabilities, the odds must be updated as a function of past results.

2) the % of risk is too high compared to the number of coin throws.
If you run the same analysis with every set having, say, 6+ throws, we all know that my side would be a winner.

The insurance example isn't right. The money made by the insurance company due to policy holders in that year is _not thrown_ away, unlike the IMPS you win in Bermuda Bowl 2004 are not counted in Bermuda Bowl 2005.


Hrothgar has answered your point 1).

But just to repeat...
Each coin flip is independent of the previous flips. Just because there have been a 1000 tails does not mean the 1001st has higher chances of coming up heads. The chances of heads on 1001st throw are still 75%. No more. No less.

About your point 2)

Consider a round with 7 flips.
If at least 2 heads come up, I make more tokens than you.

Probability of this = 1 - (0.75)^7 - 7*(0.75)^6*(0.25) ~ 0.55

I still win with 55% chances if the rounds are 7 flips each.

But,(I think, havent done the calculations) the long run analysis wins in the following case:

Suppose we pick the length of the round at random say from a very large set of numbers. Then, there is more chance that you will end up making money!


You might argue that in Bridge we have something similar... Each match would have a random number of such overtrick or down boards (which correspond to the round length in the coin flip game). But that is incorrect reasoning, since the total number of boards with the overtrick/down scenario are a fixed percentage of all possible bridge deals. Say that percentage is 2%. In a match of length 256, you should do the analysis for around 6 boards (and probably a few more in the neighbourhood of 6), as we expect 2% of the 256 boards to be overtrick or down boards, and based on the results of that, we should pick our strategy.

Of course, it depends on how the 256 boards are being generated, which by computer these days is pretty close to being truly random.

For 1%, 2%, 3% chances with a 1 IMP gain or 11 IMP loss, it seems that the long run strategy also works in most of the short runs (I have verified for runs upto 23). So the expert is right (play for overtricks to maximise score, if 5% or less chances of losing 11 IMPs while gaining 1 IMP), but by applying the wrong reasoning.

[awm]

A lot of trumpace's examples seem very contrived. Let's see what we can actually show to be true mathematically. A few assumptions will be needed here; I'll write them in bold.

Assumption One: Our goal is to win the match. Of course, this might not be true, especially if it's VP scoring. We may need a big win, or only to avoid a big loss. It might be more important to "look good" in the eyes of our client than to actually win. But let's go with trying to win for now.

There exists a probability distribution over results of the match. In other words, there's a probability that we tie, win by one, win by two, lose by one, and so forth and so on. We could plot this and it would look like a probability density function, although discretized because "win by 1/2" isn't usually a possibility barring some strange director rulings.

Assumption Two: The probability distribution of outcomes is symmetric about zero. This will not be true if the teams are not evenly matched, because you'd expect the distribution to be skewed to give the better side a higher chance of winning. It will also not be true if one side already has a known substantial IMP advantage. Both these factors have been mentioned already, and aren't the main point of this thread. Assuming two equally matched teams at the beginning of a KO match, a symmetric distribution is a reasonable assumption.

Now suppose we take an action (say choosing a safety play or not) which might possibly result in losing an IMP. Our chance of winning would then decrease, since all the results at zero (tie match) now become losses, and the results at +1 (win by one) become ties. We can now determine the cost/benefit, in terms of probability of winning, from making a decision. Let the probability distribution represent outcomes of the remaining boards of the match -- say we are trying to decide what to do on the first board. If we take the safety play, we lose one with probability L and we win 11 with probability W (assuming the other action at the other table). If we don't take the safety play, things are exactly reversed. Let PX be the probability of a result where we "win by exactly X." If the safety play loses an overtrick, our chance of winning decreases by exactly P1 (because win by one becomes a tie). If the safety play is necessary, then our chance of winning increases by P0+P1+P2+...+P10 (since ties and losing by ten or fewer now becomes a win). So the remaining question is, which is larger, P1 times the probability of losing an overtrick, or P0+P1+...+P10 times the probability that the contract would go down without the safety play?

Assumption Three: In general, if X>Y then PX<PY. Also, PX and P(X+1) are quite close to the same. This will not be true for a single board, because certain imp differentials just happen to be unlikely. For example, those differences that correspond to a game swing are a lot more likely than 7-8 IMP swings. However, over a fairly long match between evenly matched teams, it seems more likely that my team will win by a small margin than it is that my team will win by a larger margin. There are also many opportunities to swing one imp in a long match, so it will never be the case that "my team is WAY more likely to win by 8 than to win by 9." Again, none of this applies to a one board match. In fact, it's possible to prove that since each board can swing a bounded number of imps, over a very long match the distribution must look as described. Of course, whether this applies to an 8 or 12 or 24 board match will be very difficult to determine, since very few matches actually pair evenly matched teams at all. Still, it seems like a reasonable assumption that between evenly matched teams a particular small margin is more likely than a particular larger margin, but not much so if the margins are similar.

Under this third and final assumption, we can conclude that P0>P1>P2...>P10 and that P0 and P1 are pretty similar. This leads us to P1 being roughly (1/11) times P0+P1+P2...+P10, or perhaps a little more than that. So if the contract goes down without the safety play one time in twelve, we have (1/12)(P0+P1+...+P10) is about the same as (11/12)(P1) and the safety play is about break-even.

[Al_U_Card]

awm, on Dec 20 2005, 03:31 PM, said:

So if the contract goes down without the safety play one time in twelve, we have (1/12)(P0+P1+...+P10) is about the same as (11/12)(P1) and the safety play is about break-even.

But a lot easier to explain to your team mates when the safety play results in your winning and not making it costs the victory....

[luke warm]

awm, on Dec 20 2005, 03:31 PM, said:

~snip~
Under this third and final assumption, we can conclude that P0>P1>P2...>P10 and that P0 and P1 are pretty similar. This leads us to P1 being roughly (1/11) times P0+P1+P2...+P10, or perhaps a little more than that. So if the contract goes down without the safety play one time in twelve, we have (1/12)(P0+P1+...+P10) is about the same as (11/12)(P1) and the safety play is about break-even.

sorry for snipping your earlier comments, it's to save space.. it was all very intersting.. but what does it boil down to, adam? take this example... it's the first board, you're in 4♠ and you have 9 tricks in the bag...

you have the ♦AQx on the board, and you're leading from your hand... assume you honestly have no clue who has the king... the odds are 50/50 on the finesse for the overtrick... do you take the finesse?

maybe the ones who say to always play for the overtrick are right, i don't have the skills or experience to know first hand... but intuitively it seems to me that to risk a game bonus (what, 7-10 imps?) on a 50/50 shot that gains at most 1 imp seems frivilous to me

'course my intuition isn't right nearly enough, so it could be wrong here

[Al_U_Card]

luke warm, on Dec 20 2005, 03:56 PM, said:

awm, on Dec 20 2005, 03:31 PM, said:

~snip~
Under this third and final assumption, we can conclude that P0>P1>P2...>P10 and that P0 and P1 are pretty similar. This leads us to P1 being roughly (1/11) times P0+P1+P2...+P10, or perhaps a little more than that. So if the contract goes down without the safety play one time in twelve, we have (1/12)(P0+P1+...+P10) is about the same as (11/12)(P1) and the safety play is about break-even.

sorry for snipping your earlier comments, it's to save space.. it was all very intersting.. but what does it boil down to, adam? take this example... it's the first board, you're in 4♠ and you have 9 tricks in the bag...

you have the ♦AQx on the board, and you're leading from your hand... assume you honestly have no clue who has the king... the odds are 50/50 on the finesse for the overtrick... do you take the finesse?

maybe the ones who say to always play for the overtrick are right, i don't have the skills or experience to know first hand... but intuitively it seems to me that to risk a game bonus (what, 7-10 imps?) on a 50/50 shot that gains at most 1 imp seems frivilous to me

'course my intuition isn't right nearly enough, so it could be wrong here

Not quite the same as a safety play at 4% likelihood......take the finesse and bear the brunt of your teams wrath....

[awm]

luke warm, on Dec 20 2005, 03:56 PM, said:

sorry for snipping your earlier comments, it's to save space.. it was all very intersting.. but what does it boil down to, adam? take this example... it's the first board, you're in 4♠ and you have 9 tricks in the bag...

you have the ♦AQx on the board, and you're leading from your hand... assume you honestly have no clue who has the king... the odds are 50/50 on the finesse for the overtrick... do you take the finesse?

maybe the ones who say to always play for the overtrick are right, i don't have the skills or experience to know first hand... but intuitively it seems to me that to risk a game bonus (what, 7-10 imps?) on a 50/50 shot that gains at most 1 imp seems frivilous to me

'course my intuition isn't right nearly enough, so it could be wrong here

Assuming it's an 11 IMP swing if I go down (this will depend on vulnerability of course), then I should be willing to take a 1/12 chance of a set in order to get an 11/12 chance of an overtrick, assuming it is the first board of a match against an equal team.

The finesse example, I am taking a 1/2 chance of a set in order to get a 1/2 chance of an overtrick, so this is obviously lousy odds.

All of this comes to the following:

If you are playing a fairly long match against an evenly matched team, and the state of the match is equal (i.e. board one), then the strategy that maximizes your expected IMPs is extremely close to also maximizing your chances of winning.

[Trumpace]

awm, on Dec 20 2005, 04:03 PM, said:

If you are playing a fairly long match against an evenly matched team, and the state of the match is equal (i.e. board one), then the strategy that maximizes your expected IMPs is extremely close to also maximizing your chances of winning.

Suppose we are concerned solely with the boards where there is an 8% chance of throwing away 11 IMPS but 92% chance of gaining 1 IMP (assuming opp plays for safety).

Also, assume that of all the bridge deals that are possible (which is a finite number, though very large) around 1.1 percent (I am making this up, just to get a point across) are the 8% 1 vs 11 IMP deals.

Consider playing a "fairly long match" of 1000 boards.

How many such 8% deals do you expect to occur among the 1000 boards? 11 right?

Now if there are 11 such 8% 1 vs 11 IMP boards, the chances of gaining IMPS on these boards by playing safe is 61% (as calculated on a previous post).

So if you use your long run strategy (maximising expected IMPS per board) for a 1000 board match, I can beat you (assuming we are equally matched) with 61% chances by playing safe for the 8% 1 vs 11 IMP boards, while using the same strategy as you on the remaining boards!

If the match was say a million trillion boards long, you will probably win if I stuck to playing safe on the 8% boards. (Haven't calculated...)

The reason the 1%, 2%, 3% IMPS add up is that, in shorter runs too, they actually give more than even chances.

The point is that even though the whole match might be long... the number of boards of a particular kind (here 8% boards with 1 gain vs 11 lose IMP) might be small enough to require an approach other than just the long run analysis.

[awm]

You're assuming that these "safety play" boards are the only swing boards in the match. Combined with the (likely) assumption there are only a small number of them, this effectively creates a short match in which each board has only two possible outcomes. This will give you a distribution that does not look (at all) like the one I have postulated, where (for example) win by two is slightly less likely than win by one.

I don't believe these are reasonable assumptions. In fact, I believe that even in a 12 board match between evenly matched teams, the outcome probabilities will be such that win by X is always at least as likely as win by X+1, for any choice of X. Obviously you can construct contrived examples where this is not the case -- for example if you play 11 boards and every board is either win 11 or lose 1, then win by one is much more likely than a tie (actually ties are impossible) and win by 13 is much more likely than win by 12 (actually win by 12 is impossible). But I think realistic matches even of 12-24 boards in length will in fact have the property I am using.

Anyways, my analysis requires the bold face assumptions, and while I think they are very reasonable, you are welcome to try to come up with examples where they don't hold (and try to argue that those examples are reasonable).

[Trumpace]

awm, on Dec 20 2005, 05:41 PM, said:

You're assuming that these "safety play" boards are the only swing boards in the match. 

I am not assuming that at all. I am saying that I dont need to assume that.. I can play with such a strategy which will make those 11 8% boards the only boards that count!

quoting myself...

Quote

So if you use your long run strategy (maximising expected IMPS per board) for a 1000 board match, I can beat you (assuming we are equally matched) with 61% chances by playing safe for the 8% 1 vs 11 IMP boards, while using the same strategy as you on the remaining boards!

Basically for boards other than the 8% boards I will play with the same strategy as you do. For the 8% boards I will play safe, in effect giving me a 61% chance of winning. By using your strategy on all but the 8% boards, in effect, I have made these boards decide the outcome of the match!

[Trumpace]

awm, on Dec 20 2005, 05:41 PM, said:

Anyways, my analysis requires the bold face assumptions, and while I think they are very reasonable, you are welcome to try to come up with examples where they don't hold (and try to argue that those examples are reasonable).

I am not saying that the long run analysis won't work! All I am saying is that there is more to it than just maximising the expected number of IMPS per board.

The reason I need to contrive values is that it is very hard to show real bridge situations where the long run analysis might fail. These are not sufficient grounds to declare that the arguments I am making are bogus...

As a fair approximation, using the long run analysis is probably good enough (though not in all cases, which is my point), as there are other factors which are more important in deciding the outcome of the match.

Don't just blindly use "in the long run it wins hence I use it" approach. Though it might work for most cases, the reasoning is still wrong! That is all. You might turn out to be right, but there could be cases where it is the wrong thing to do! If your opponents just find one such case (which I admit, would probably be pretty hard to do), they can beat you if you blindly stick to the 'in the long run' strategy.

[Joe de Balliol]

Well, in the Portland Bowl last year Oxford B won two matches by 2 IMP

My first-round NICKO match this year was tied after 24 boards and went to a tie-break.

Everyone's drawn some Swiss teams matches.

In my Crockford's Cup match this year my team was playing against Forrester, Mizel, Allfrey, and Tosh. After 8 boards we were 57 IMP down. While we 'only' lost by 70 or so overall, for boards 9 onwards the idea was to lose by as little as possible, hence it was purely take the odds as we had no chance of winning and knew it.

Finally - I think the contract matters. Safety plays are not usually best in part-scores, worth taking in games, should be taken in slam, and should generally be taken in grands unless the chance of an overtrick is VERY high

J

[mikeh]

Joe de Balliol, on Dec 23 2005, 11:43 AM, said:

Safety plays are not usually best in part-scores, worth taking in games, should be taken in slam, and should generally be taken in grands unless the chance of an overtrick is VERY high

I would be very interested to see a 'safety play' that would be appropriate in a grand

[Blofeld]

Clearly he means you shouldn't be playing for a procedural penalty in your favour if it risks the contract.