Math and Bridge

[han]

There was an email in the math department asking for people who could talk about how mathematics comes up in games. I wrote back that perhaps I could talk about bridge, and now it seems I'm hooked. I have 10 days to prepare a talk. I need help.

 

I can think of several topics, there must be more:

 

-Suit combinations.

 

-Hand evaluation.

 

-Restricted Choice.

 

-Possible number of constructive auctions that end with 3NT.

 

The problem is that I can't assume that the audience knows the rules, and I won't have time to give them a good idea of how the game is played. How can I make this talk interesting for math majors who don't know bridge?

[mike777]

Well as an old old Finance major we throw in the human element.

 

Would Math majors enjoy a talk on the limits of math in game theory?

Is Psych too soft and fuzzy a science..Ok .lets say yes..but still....maybe there is a challenge to bring more Math to the human element in bridge?

 

Will Math majors enjoy Karl Popper and his theories or is this just too old school and boring for students today?

 

btw old non math student here.

[Elianna]

For a forced talk in a Math Seminar I took (all first year grad students had to take this), I talked about restricted choice and suit combinations (how if we have an 9 card fit, they must have at least an 8 card fit, etc.) None of my audience had ever played bridge, except for the professor.

 

They all enjoyed it, though.

 

My advice is don't do anything that involves bidding/evaluation/etc. Stick to card play, because even if they don't play bridge, they might be able to relate from other games.

 

Another idea: If it's a longer talk, you might want to talk about evaluating chances of contracts making (ie, if it's on one of two finesses, what percent)?

 

Oh, I also talked about strategy of sacrificing and the theory of Law of Total Tricks (ie, when is something a good sacrifice and other percentages). That didn't go over as well, because I had to explain bridge scoring, and couldn't really do that quickly enough (it was supposed to be a 10 minute talk). If you can do the background material quicker, that is a possibility for you to talk about.

 

I wish that I could think of a way to involve Complex Variables and whatever else it is that you do in this. :)

 

-Elianna

[helene_t]

An introduction to communication theory:

 

Alice and Bob realized that their marriage suffered from inefficient communication, so they consulted a relationship therapist. He gave them the following exercise:

 

They must both, on turn, mention a number between 1 and 100. Constraint: Each number must be higher than the preceding number. Example:

Alice: 1 Bob: 14

Alice: 39 Bob: 40

Alice: 98 Bob: 100

 

The object of the game is to exchange as much information as possible. Alice suggested:

"I start with a number between 1 and 64. That gives you 6 bits. If you have 64 or more option left (i.e. if I mention a number below 36), you can give me 6 bits back. Otherwise, you give me 5 bits. Etc."

 

(You can probably imagine how they improve the scheme which eventually leads to the fibonacci series in relay systems).

 

With two-way information exchange, both can convey 50 bits or ten letters. This is convenient since it allows for conversations like:

Bob: "I_LOVE_YOU"

Alice: "GOOD_NIGHT"

[Gerben42]

Restricted choice: the Monty Hall problem.

 

Suppose you are in a game show and you have to guess where your prize is. It is behind one of three doors.

Say you pick door nr. 1. When you have done so, the game show host opens one of the doors you haven't chosen, leaving you with a choice of two doors. He asks you: "Would you like to change your mind?".

 

If you know that the game show host always opens a second door, you should switch as the chance that the prize was behind your original choice is and stays 1/3, the other door then having odds of 2/3 at this point.

 

Application in bridge. Suppose this is the suit.

 

[hv=n=sak432&w=s?65&e=s?j&s=st987]399|300|[/hv]

You lead the Ace and RHO drops the Jack (or the Queen). This is the same problem as the Monty Hall problem: Because RHO (and the game show host) had a restricted choice which door to open (he had to open a door without the prize) or which card to play (he had to play the honor he had, with two he might have played the other one), the odds are in favor of switching doors and to finesse the Queen.

[Gerben42]

On Fibonacci and relay bidding:

http://www.jimloy.com/algebra/fibo.htm

[helene_t]

- Rating schemes

- Balanced movements

- Games with incomplete information (mixed strategies)

- Matchpoints, butler, Cross-IMPs or Total Points: which scoring will converge fastest towards finding the "true" winner?

[Gerben42]

General: you will have to tell them there are 13 cards in a suit to explain suit combinations and restricted choice. Otherwise use the mathematics as a base and then use bridge examples.

 

For example show that in a deck of cards when you have 8 of the 13 of a suit the chance that the opponents have 3-2 is 68%, and then show the relevant suit combination or this hand (Bridge Master has this one I think)

 

[hv=w=s543hakq6daq5c432&e=sakqh5432d432ca65]266|100|[/hv]

 

You don't have to tell them MUCH about how the game is played to get to the mathematics of things and see that ♦ are a 50-50 chance and ♥ are 68% chance.

[Fluffy]

Restricted choice: the Monty Hall problem.

 

Suppose you are in a game show and you have to guess where your prize is. It is behind one of three doors.

Say you pick door nr. 1. When you have done so, the game show host opens one of the doors you haven't chosen, leaving you with a choice of two doors. He asks you: "Would you like to change your mind?".

 

If you know that the game show host always opens a second door, you should switch as the chance that the prize was behind your original choice is and stays 1/3, the other door then having odds of 2/3 at this point.

 

Application in bridge. Suppose this is the suit.

 

[hv=n=sak432&w=s?65&e=s?j&s=st987]399|300|[/hv]

You lead the Ace and RHO drops the Jack (or the Queen). This is the same problem as the Monty Hall problem: Because RHO (and the game show host) had a restricted choice which door to open (he had to open a door without the prize) or which card to play (he had to play the honor he had, with two he might have played the other one), the odds are in favor of switching doors and to finesse the Queen.

This oen is always popular, it could fill a full hour if you can force teh audition into make guesses, you could even make 20-30 demostrations before they quicly realice staying on the same door doesn't change their chances at all.

[hrothgar]

I don't think that anyone has touched on the most important question...

Who is the audience? How much math / stats do they already have?

 

In particular, do they already know basic probability / statistics? Do they understand the concept of a saddle point? Alternatively, can you "get away" with a discussion based on simple combinatorics?

 

Assuming that I was doing a presentation for a group of college Juniors, I'd focus on one on the following areas..

 

1. The applicated of mixed strategies in bridge. I'll note in passing that any discussion of restricted choice requires an understanding of mixed strategies

 

2. Transitivity and bidding system design with implications for bidding populations

 

3. Information theory and relay bidding (fibonacci sequences

[han]

First of all, thanks all for these quick and useful replies.

 

Somehow I was expecting Richard to come up with mixed strategies. Can you think of easy examples besides restricted choice where it is easy to explain that a mixed strategy is useful? (falsecarding as decclarer?)

 

I don't know exactly what the audience is like, this is the "math club". I indeed expect junior-senior math majors, but maybe some sophomores too. I can probably take as much time as I like, maybe I'll talk for 40 minutes.

[cherdano]

Why are there so many mathematicians among the serious bridge players?

 

In my opinion, it has not so much to do with the fact that we are better at calculating probabilities, or that it is easier for us to understand restricted choice. I think mathematicians are, on average, better at accepting the fact that you may have done the right thing even though it turned out wrong at the end on this specific hand. That it was the right thing to do given the odds on the information you had at the time.

 

Of course, this mindset is not unique to mathematicians (and certainly there are also mathematicians who are result merchants at bridge), and all good bridge players have it. I just think it is a lot more common among mathematicians.

 

Sorry Han, I suppose this won't be so helpful for your talk.

 

Arend

[Al_U_Card]

University students? let's see, how about the relationships, when playing penny a point, for the decision process in bidding games vul vs non vul? Most students play cards for money still, do they not?

[hrothgar]

First of all, thanks all for these quick and useful replies.

 

Somehow I was expecting Richard to come up with mixed strategies. Can you think of easy examples besides restricted choice where it is easy to explain that a mixed strategy is useful? (falsecarding as decclarer?)

 

I don't know exactly what the audience is like, this is the "math club". I indeed expect junior-senior math majors, but maybe some sophomores too. I can probably take as much time as I like, maybe I'll talk for 40 minutes.

I'm not particularly interested in declarer play / defense and haven't studied it extensively. I'm probably the wrong person to be asking about good examples.

 

However, I went and googled "mandatory falsecard" on rec.games.bridge

 

the following thread cropped up which has some useful discussion and some example hands

 

http://groups.google.com/group/rec.games.b...780fbfa119c54ff

 

It is, of course, important to note that by definition, a mandatory falsecard isn't an example of a mixed strategy...

[inquiry]

Rule of 11 (etc)... or rule of 14... how to "see" the missing spots.

 

Combinational math and probabilities

 

Counting HCP

 

Logic (as well as math). Such as Counting distribution (signals used to help), Also the usefulness of the fact that a single suit or a hand will always divided with one odd - three even (4441 is three even, one odd... 5332 is three odds, one enen, and how you can use this info quickly to get the full hand patterns).

 

Restricted choice is always a good one, as noted above.

 

Be sure to give them a link to the ACBL learn to play bridge site and to the BBO (of course) so they can take the game up if you excite them.

[hrothgar]

University students?  let's see, how about the relationships, when playing penny a point, for the decision process in  bidding games vul vs non vul?  Most students play cards for money still, do they not?

Nothing more than basic algebra...

Grade 9 math

 

earlier if you're in an accelerated program.

[cherdano]

However, I went and googled "mandatory falsecard" on rec.games.bridge

 

the following thread cropped up which has some useful discussion and some example hands

 

http://groups.google.com/group/rec.games.b...780fbfa119c54ff

 

It is, of course, important to note that by definition, a mandatory falsecard isn't an example of a mixed strategy...

Look at post no. 14 in that thread. David des Jardins shows that with Q876 in dummy, you hould J9xx as defender, and know that declarer holds AKxx or AKTx, you should play the 9 one third of a time when declarer leads the A.

 

However, I think there should be easier examples of a mixed strategy, where you can assume that defenders know all of declarers cards. I am almost sure that it is absolutely impossible to explain the above example to people whom you still have to remind that defender can only see dummy, not declarer, and who have never seen a mixed strategy before.

 

Arend

[Guest Jlall]

Be sure to tell them at the end that if anyone is interested in bridge you'd be happy to play a club game with them or on BBO :lol:

[Gerben42]

Why are there so many mathematicians among the serious bridge players?

 

Because both math and bridge are fun? :lol:

 

Gerben (M.Sc in Astronomy with Mathematics minor)

[Finch]

Like everyone else, I suggest Restricted choice. It doesn't take long to explain the problem.

 

If they know Bayes' theorem, you can show how this is just a simple example: it's just the terminology that's different "he's more likely to have singleton Q than QJ because with the latter he would have had a choice of cards to play" is not obviously the same thing as the standard algebraic statement of Bayes' theorem using set notation.

 

As an alternative, I agree that the idea of balancing movements, or how many rounds of Swiss do you need to get to the winner, or what's the best format for the Bermuda Bowl... are all mathematically interesting problems that are easy to explain. They aren't really related to _playing_ bridge, however.

 

While there's some neat (and fairly simple) algebra behind some of the mixed strategy questions, I think it requires too much knowledge of bridge to understand.

 

By the way, at one of my interviews for a university place (Imperial College), the interviewer saw I played bridge on my application form, and asked me about PRC.

[hrothgar]

As an alternative, I agree that the idea of balancing movements, or how many rounds of Swiss do you need to get to the winner, or what's the best format for the Bermuda Bowl... are all mathematically interesting problems that are easy to explain. They aren't really related to _playing_ bridge, however.

This is a great suggestion:

 

I find the relationship between statistical sampling and tournament design fascinating.

 

If you decide to go this way, I suggest reading a paper called "Selection in the Presence of Noise: The Design of Playoff Systems" by Adler, Gemmel, Harchol, Karp, and Kenyon.

[cherdano]

Like everyone else, I suggest Restricted choice. It doesn't take long to explain the problem.

 

If they know Bayes' theorem, you can show how this is just a simple example: it's just the terminology that's different "he's more likely to have singleton Q than QJ because with the latter he would have had a choice of cards to play" is not obviously the same thing as the standard algebraic statement of Bayes' theorem using set notation.

And if they don't know Bayes' theorem (which I would rather assume), it is an excellent opportunity to learn one incarnation of it.

 

If I had to pick a single mathematical statement that should be known more widely, there would be no second choice to Bayes' theorem. (E.g. I think it should be used more often for hypothesis testing, since all these 95% confidence tests don't do what people think they do.)

And bridge players are the only group outside of mathematics that I know who have, in a way, formed their own understanding of it (under the name of PRC).

 

Arend

[Al_U_Card]

Possibly you might consider the use of simple hand frequency as a determining factor in bidding methods. The choice of one bid over another should depend on that hand occuring the most often? (This is not always the case, I would imagine.) So, in real life, straight calculations do not always result in a highly correlated response......

[kfgauss]

There's a great article on mixed strategies in cardplay by John Swanson at www.johnninaswanson.com/articles.html called "A Hand for Deep Blue."

 

He analyses the position Axx opposite KJ9xx and determines that a mixed strategy is required for the defenders regarding when they should falsecard the 10. It's a short article, so I won't summarize -- just go read it.

 

The position has a nice history too (see the notes below the article), with a writeup by Edgar Kaplan in the Bridge World after it came up in the Vanderbilt and led to an accusation of cheating (Kaplan's claim was that the accused made the right play, actually).

 

Andy

[tysen2k]

I agree that something along the lines of restricted choice is best. Since almost no one will have bridge experience don't even touch bidding or hand evaluation.

 

Tysen