The Monty Hall Trap

[helene_t]

sure. but that won't influence the profit of the casino.

[nate_m]

Edit: OK, I now see how restricted choice plays a role in a spade lead from four cards. Holding 4=4=4=1 he might have led a heart instead of a spade, holding 4=3=4=2 he would lead a spade, except if the diamonds were something like KQJx.. So restricted choice says that the lead of a spade from a four card holding is evidence for a doubleton club. It's not just a matter of counting possible hands, we must also consider that a spade was led instead of a heart. The reason may well be that he has only three hearts.

This is in the "he is known to have four spades" situation, I started off looking only at the "known to have five spades" case.I still believe RC, or Monty Hall, or whatever you wish to call it, does not apply in the fivee card case.

 

 

 

 

Well,his title refers to Monty Hall, which is the classical example of restricted choice. My own history with this is: I was at a restaurant when a former student came up and gave me the Monty Hall problem This was back shortly after the Ask Marilyn article appeared in Parade. He explained it carefully, I thought for a bit, and said "The answer is 2/3, this is just like a problem in bridge where it is referred to as restricted choice"

 

 

A couple more points.

 

I had not gotten as far as Adam. And here is where restricted choice type arguments really have a role. If he led a four card suit, and if we can safely (for the purpose of the problem) infer from this that he has no five card suit, then indeed he is unlikely to have a stiff club.One can reason that he then might have led one of his other four card suits, or more simply (and preferably imo) simply reason that two clubs and 4-4-3 in the other suits can happen in more ways than one club and 4-4-4 in the other suits. This is because two clubs opens up the possibility of 4=4=3=2, or 4=3=4=2. And really we should just count leading from four spades as leading from a four card major.

 

 

It seems to me that one can (again for purposes of analysis) think of the situation as equivalent to the following: INstead of W leading, the game goes as follows. Before a card is led, dummy comes down. Declarer plays a club to the board and a club back, and sees three spot cards. Before he plays from hand, he gets to ask W "Do you hold at least one five card suit?" and W will answer truthfully. Basically, as I read the problem, that's what happens. At crunch time, declare knows the answer to this hypothetical question, and knows nothing else exacept that if declarer has a five card suit, one of those suits is spades.

 

 

Now again Adam observes that the choice between two five card suits is not random. Correct. Unlike the usual restricted choice situation, I think this situation is way too loaded with unlikely hypotheticals to be very useful.

 

 

One further thought occurs to me: With no information at all to go on, we play for the drop. But if knowledge that W has no five card suit makes it more likely that we should play for the drop, then knowledge that he has a five card suit makes it less likely we should play for the drop. The books have to balance here.The initial prbability of Qx in W's hand is the sum of the probability of Qx when holding five times the probability that he has at least one five card suit and the probability he has Qx when not holding five times the probability of no five card suit. If the probability of Qx when he has no five card suit is less than the initial probability of Qx, then the other has to be more.It takes more effort than i have expended to figure out how much more.

I realize I have phrased the above rather badly, but the point is that the books have to balance.

 

Bottom line: Play from the drop if the original lead was from four, finesse if it was from five. Which is pretty much what anyone would do, I think.

 

The inference about the lead from 4 card suits is even stronger than I think you give it credit for. If the opening lead was from 4, it is often correct to finesse opening leader for the queen, even with 9 cards between your hand/dummy. This is certainly true if there are 2 eight card fits for the defense, and I'd guess in the example given on BW hooking through opening leader is a small favorite in the event of a lead from 4.

[barmar]

This is a university level problem in America?!

When I took Probability & Statistics at MIT around 1980, much of the introductory material overlapped with things I'd learned in high school math classes. But I can easily imagine that students who don't have a background that will get them into a school like MIT might not cover it as well in high school.

[ArtK78]

This is a university level problem in America?! :blink: In England, I was given it to solve at a maths seminar somewhere around 13 or 14. I could see it as a first task in a "Basic Computer Science for Mathematicians" course perhaps but as a probability problem it is way too simple for university undergraduates. On the same seminar course was the problem of finding the expected win/loss for the game of craps - that was far more difficult (and interesting).

You have to start somewhere.

 

Courses in statistics and probability were not offered in our curriculum prior to college. I did have advanced calculus, however.

 

By the way, the odds of winning in craps was also one of the earliest problems in the course. If I remember correctly, playing pass results in a 49.47% win probability. And it is not a difficult problem.

[kenberg]

The inference about the lead from 4 card suits is even stronger than I think you give it credit for. If the opening lead was from 4, it is often correct to finesse opening leader for the queen, even with 9 cards between your hand/dummy. This is certainly true if there are 2 eight card fits for the defense, and I'd guess in the example given on BW hooking through opening leader is a small favorite in the event of a lead from 4.

 

Ah yes, I see. And I tentatively agree. in fact, so he said on BW. I was still just too wrapped up in the five card issue to see this.More to think about.

[GreenMan]

I meant it as an opinion of human gambler behavior, not a game theory computation. Many gamblers tend to bet until they are broke, then go home.

 

But if you change the odds to favor the player, fewer gamblers will go broke.

[Stephen Tu]

The odds favoring the casino is certainly one reason they make money, but far from the only. I am convinced that even if they set the odds to slightly favor the player (51-52%), they would still make a profit because of loss lock-in. Most gamblers quit when they are behind (run out of money) but not when they are ahead. Hence losses are in the bank, but wins are negotiable.

 

This is very silly if you think about it more deeply. One has to quit sometime due to fatigue/hunger/etc. Because the games are rigged in favor of the house, most people lose when they play long gambling sessions, hence at quitting time you naturally find most of them are behind. If you in contrast rigged the games in favor of the player, the opposite would be true, after long sessions most people would be *ahead*, and most people would be quitting when they are ahead.

 

Now, it is possible for casino to run games that are slightly EV- against the house, if played perfectly. It's even been done (some video poker machines, blackjack with counting). That's because the vast majority of players *do not* play at all close to perfectly, and many play atrociously badly giving the house an extra 5-10%. A mass population of players playing badly (it's astonishing how people won't even bother to learn basic strategy at blackjack, doing things like never hitting on 16 regardless of upcard, etc.) can provide enough profit to cover a few perfect +EV players. But of course casino doesn't like to lose anything they don't have to, so +EV video poker is now nearly impossible to locate, and if they think you are card counting at 21 they will ask you to leave or play something else.

 

Many gamblers tend to bet until they are broke, then go home.

That's because they are playing -EV games and will nearly always run out of money given long enough session and bringing non-ridiculous sum of money to casino. I am a +EV poker player. If I go into a limit poker game adequately rolled, against a normal set of players (not a bunch of sharks equal/better than me) rotating into the table, I am *never* going broke. Still, I go home, because at some point I am about to keel over from fatigue, or the game breaks because everyone else leaves.

[jjbrr]

If I remember correctly, playing pass results in a 49.47% win probability.

 

If you define winning as, in the immortal words of chim17, "being pretty drunk, betting on 11, yelling 'Yoleven!' and hitting it" sometimes, I'm pretty sure win probability is 100%.

[GreenMan]

By the way, the odds of winning in craps was also one of the earliest problems in the course. If I remember correctly, playing pass results in a 49.47% win probability. And it is not a difficult problem.

 

I don't have the numbers handy, but I heard a while back that Don't Pass is the better bet by a small margin, though it's still -EV because the house keeps a small cut. In fact, unless you're a +EV player in some games as Stephen described, Don't Pass is the best bet in most casinos.

[gnasher]

Well,his title refers to Monty Hall, which is the classical example of restricted choice.

I tend to assume mathematicians are right about most things, but not necessarily about the format of a 1960s game-show.

 

As I understand it, what mathematicians call the "Monty Hall Problem" isn't the same as what Philip Martin calls the "Monty Hall Trap" - in former you're offered a chance to switch doors; in the latter you're offered the chance to swap your chosen door for some cash. I've no idea whether both, one or neither of these scenarios actually occurred on Monty Hall's game show. Do you know? If you don't, the most you can criticise Philip Martin for is using different terminology from you.

 

Martin's main point isn't about restricted choice, which he mentions only in passing. His argument is about how to treat information that merely tells you something you already knew. Monty Hall opens a door and shows us a goat, but we already assumed that one door had a goat behind it, so he's told us nothing at all. LHO tells us that his longest suit is of 5 cards, but we already assumed that he had a 4.5-card suit, so he's only told us about half a card.

[billw55]

But if you change the odds to favor the player, fewer gamblers will go broke.

IMO most gamblers aren't going broke because of the odds, they are going broke because their behavior leads to it.

 

If you in contrast rigged the games in favor of the player, the opposite would be true, after long sessions most people would be *ahead*, and most people would be quitting when they are ahead.

But most of the people who go to casinos don't play long sessions. They stop when they are out of money, which happens pretty fast. So all the paths that start with a loss and end with a win don't happen - they get cut off as losses. And gamblers also do other things to go broke, like substantially increasing their bets when ahead - so that it takes very few losses to hit bottom again. Sort of like the strategy to win by doubling your bet after every loss, but in reverse. Many seem almost hellbent on losing. These types of things are paying the casino handsomely, above and beyond the simple -EV they offer.

[GreenMan]

IMO most gamblers aren't going broke because of the odds, they are going broke because their behavior leads to it.

 

And the main constituent of that behavior is playing casino games with negative expected value. If the odds are tilted in their favor, the results will change.

[billw55]

And the main constituent of that behavior is playing casino games with negative expected value. If the odds are tilted in their favor, the results will change.

I just don't agree. Compulsive gamblers find a way to lose. 51% chance versus 49% isn't going to stop them.

 

OK, some smart bettors and mathematicians could take advantage of +EV player odds, and this is usually enough to stop the casinos offering them. But the rank and file gambler - no.

 

Basically what I am saying comes down to what was already mentioned - these players are using suboptimal strategies.

[Stephen Tu]

Basically what I am saying comes down to what was already mentioned - these players are using suboptimal strategies.

 

No, that's not what your claim was. Your claim was that even if the games were +EV, people's psychology of when to leave when losing would make them lose anyway, which is just wrong. If the games were +EV, a ton more people would find themselves up after however long they like to play, people would leave when they doubled/tripled their money or whatever (of course people in general would just play until casino inevitably forced to shutdown). It's just that the games are rigged to be -EV, so you find a lot more people broke leaving than winners leaving.

 

People lose at gambling because the games are -EV, period. Many players play suboptimal strategies, that just makes their negative EV more negative. Most people leaving as losers, when they have lost all their money is a consequence of the games being -EV. You are claiming that the EV doesn't matter, it is their "stop loss" strategy that is making them lose, their tendency to quit while behind instead of while ahead. This just isn't true. The tendency to be behind is because in a rigged game, you are usually behind!

 

In the long run, when one starts and stops sessions has *no effect*. Gambling is essentially just "one long lifetime session", the timing of your breaks is very much irrelevant. The only things that really matter are:

- how -EV the game you are playing is

- # of bets per hour you are making

- size of those wagers

- how many hours you play given these same conditions

 

Over the long run, no matter what strategy you pursue as to when you stop a session, your losses will trend toward whatever your expected loss per hour is. Doesn't matter if you try to leave when ahead, or play until broke, or always play exactly half an hour or an hour. Doesn't matter!!!

 

Play some poker sometime, against equal opponents, *without rake*. A 0 EV game. The game will last forever! Just because you have a bunch of degen compulsive gamblers together, they don't spontaneously lose money, it'll just randomly circulate around the table. What busts people out at the casino is the -EV, the rake.

 

Compulsive gamblers find a way to lose. 51% chance versus 49% isn't going to stop them.

Sure it will. Compulsive gambler with a +EV edge often becomes *professional gambler*. People have made fortunes at poker or sports betting where +EV is possible. Now, some of these have been known to lose it back, but they don't lose it back in the field they are +EV in, they lose it in -EV wagers and activities. They have other vices, like some poker players who have been known to make millions at the poker table but a tendency to dump a lot of it to casinos playing craps or baccarat or other table games.

[Vampyr]

That is true.

 

And the chances that your door is the wrong one is 2/3.

 

And you also know which of the other two doors is the wrong one.

 

So, if given the opportunity to switch, you should switch, since the chances that the switch is right is 2/3.

 

Yes, quite. But the author states that your chance of being right is still the 1/3 you had originally.

[billw55]

No, that's not what your claim was.

LOL

 

Play some poker sometime, against equal opponents, *without rake*. A 0 EV game. The game will last forever!

No, because people will quit when they lose a certain amount. Although after some number have done so, the rest might play for a very long time.

 

If the stakes are small enough compared to the loss the players can absorb and continue playing, then yes, I would agree.

[GreenMan]

LOL

 

 

No, because people will quit when they lose a certain amount.

 

[citation needed]

[billw55]

[citation needed]

Hello? When they don't have any more cash on them? Or for many, when their chip stack is gone, whether they have more in pocket or not.

[gnasher]

Back to the article:

 

"Since you already knew that at least one of the other two doors held a booby prize, you have learned nothing. You still have the same one chance in three that you started with."

 

This is very obviously wrong, since there are only two doors left. Even if the author doesn't understand that switching changes your odds to 2/3, he ought to be able to see that you can't have a one in three chance if there are only two doors.

I think you should read the article again. In the article, Monty Hall doesn't offer the possibility of switching doors. He offers the options of keeping the original door, or taking some cash. The statement "You still have the same one chance in three that you started with" is in the context of keeping the original door, and it's correct.

[GreenMan]

Hello? When they don't have any more cash on them? Or for many, when their chip stack is gone, whether they have more in pocket or not.

 

You're taking a situation where people almost always run out of money, and shouting that they will behave exactly the same in a different situation where they almost NEVER run out of money. I can't believe it needs spelling out.

[Vampyr]

I think you should read the article again. In the article, Monty Hall doesn't offer the possibility of switching doors. He offers the options of keeping the original door, or taking some cash. The statement "You still have the same one chance in three that you started with" is in the context of keeping the original door, and it's correct.

 

Oh, OK. I guess I misread it.

[cherdano]

Wonderful article, but I agree with Adam that the assumptions ignore the difference between major and minor suits.

[gnasher]

There may be another minor mistake in the article:

 

"for my intended audience [bridge players], I stand by my claim that this problem is easy"

[ArtK78]

I don't have the numbers handy, but I heard a while back that Don't Pass is the better bet by a small margin, though it's still -EV because the house keeps a small cut. In fact, unless you're a +EV player in some games as Stephen described, Don't Pass is the best bet in most casinos.

Well, sure. If pass is 49.47%, then don't pass is 50.53%. However, the cut taken by the house is more than 1.06%, so it winds up being worse than pass.

[GreenMan]

Well, sure. If pass is 49.47%, then don't pass is 50.53%. However, the cut taken by the house is more than 1.06%, so it winds up being worse than pass.

 

That's not how I understand it. Don't Pass pushes on either 2 or 12, depending on the casino, so the house keeps its edge that way. The house's edge is thus a bit better on Pass than Don't Pass.