A puzzle for you

[SossMan]

I was involved in a hand recently that reminded me of a brain teaser from an Economics class in college. I will tell you the puzzle now and post the answer and the hand information a little later....

Suppose there are 5 envelopes sitting on a table. Each of the envelopes contain a dollar amount.
envelope #1 - $5
envelope #2 - $10
envelope #3 - $20
envelope #4 - $40
envelope #5 - $80
None of the envelopes are marked in any way, and there is no way of knowing how much is in each envelope without looking inside.
Now, suppose there are two (rational) people, David and Mason, who are given one envelope each randomly. Each person knows how much is in their respective envelopes, but not how much is in the other person's nor how much is left on the table. Here's the deal: if agreed upon by both parties, they can trade envelopes. If either David or Mason don't agree to the trade, then no trade is made.
If David looks in his envelope and sees $10 (envelope #2), should he offer a trade to Mason?
What is the expected value of this trade?
Why? /forums/images/icons/confused.gif
Results to follow....(and relevance to poker, hopefully) /forums/images/icons/tongue.gif

-SossMan
P.S. You must assume that both David and Mason are trying to maximize their dollars. (i.e. they prefer more money than less)

[pudley4]

No he shouldn't offer to trade.

If Mason has the $80, he obviously won't agree.
If Mason has the $40, he only gains if David offers to give up the $80, which won't happen. So Mason won't give away the $40.
If Mason has the $20, he knows David won't give away the $80. He also knows David won't give away the $40, because David knows Mason won't give away the $80, so David will never give away the $40.
Mason can't have the $10 because David has the $10.

So Mason will only give away the $5 and David will lose $5.

[pudley4]

The EV is [(3*$0) + (1 * -$5)]/4 = -$1.25

[Punker]

Further proof of the value of position. Whoever has to offer first in this game is -EV.

[SossMan]

Pudley got the answer right on the nose (makes me wonder if he'd seen this before /forums/images/icons/tongue.gif ) The reason that many people (obviously not 2+2er's) think this is counter-intuitive is that they don't differeciate between trading their envelope with a random envelope and accepting a trade with a rational thinking person. The EV of trading with a random envelope (i.e. opponent accepts all trades) is positive:
4 other envelopes all have equal chance of being selected -- .25(-$5)+.25(+$10)+.25(+$30)+.25(+$80)=-1.25+2.50+7.50+20=$29.25=EV
Of course, as Pudley pointed out, your opponent will only accept the trade if he has the $5 envelope. So, in essence the EV is 0 since a trade will never happen.
So, how the hell does this have anything to do with poker?
(Besides position, which I hadn't thought of before but is a valid point)
It's when you have a hand that is so good, that it is virtually impossible to be called by a worse one on the end.
I had a A /forums/images/icons/spade.gif K /forums/images/icons/spade.gif on the button and raised after 6 limpers preflop. All called
The flop was J /forums/images/icons/spade.gif 8 /forums/images/icons/spade.gif 9 /forums/images/icons/spade.gif
MP bet,2 callers, I smooth call, BB calls.
Turn Q /forums/images/icons/spade.gif
MP checks, caller 1 bets, caller2 folds, I raise, MP check-raises. Caller 1 folds. Now what! The only card that beats me is the 10 /forums/images/icons/spade.gif , and heads-up I can't fear the one card in the deck that could beat me, could I??
I raise, he calls.
River = blank (offsuit, doesn't pair board red 3 i think)
He checks, I bet, he check raises again. Now, I don't know this guy, but he doesn't seem like a big bluffer, and I've had a pretty tight image. I raised preflop, have shown all the indications of having at least A /forums/images/icons/spade.gif . What (besides a stone cold bluff) would he check-raise me with twice besides the 10 /forums/images/icons/spade.gif ? The 7 /forums/images/icons/spade.gif ?
I meekly call.
He shows the 10 /forums/images/icons/spade.gif .
This is a clear case, I think, of checking behind him on the river because if he doesn't have the 10, he's got to be scared of it or the K or the A. I think betting was pointless, as I am pretty sure he wouldn't call with a set or a smaller flush.
Comments appreciated.
/forums/images/icons/smirk.gif

[Mangatang]

I find it amazing that your economics professor used the names David and Mason. LOL.

[SossMan]

Quite a coincidence, huh? LOL! /forums/images/icons/tongue.gif

[iblucky4u2]

This guy has the $5 envelope and you have the $80 and are offering to trade! Your statement "and heads-up I can't fear the one card in the deck that could beat me, could I??" shows you are willing to ignore all the evidence in front of you. What else could he have? He might have had the 10-7s and had you on the flop!

What the heck did caller one have other than another $80 envelope?

[happyjaypee]

and heads-up I can't fear the one card in the deck that could beat me, could I??

Yes you can, this is'nt exactly heads up. You were in a multiway pot and got down to 2 plrs after the turn betting. If he toke and give all that action, he's in there whit something. Your statement would make more sense if both of you had see the flop heads up.

I can't imagine laying it down but your right about it being a chk behind case.


-Happy /forums/images/icons/blush.gif

[ElSapo]

I like the puzzle quite a bit - things like this, simple that they may be, tend to fascinate me. But if you extend this to, say, 10 envelopes, does the same logic hold true? No one ever trades b/c they can't benefit? And does it work in a real life setting this way?

[SossMan]

It doesn't matter how many envelopes there are: 5,10,100...the same logic applies. Also, it doesn't matter how much is in each envelope either, as long as there is an increasing amount in each one (i.e. 5,10,20,40,a million). As far as real life, that's tricky...if I knew you were logical thinking and you knew I was logical thinking and we both knew that the other one knew, then yes it would work in real life. Now, do all these conditions hold in every situation? Probably not. An easy proof of this is that most people that hear the riddle for the first time say that they would trade the $5,$10,$20 envelopes, keep the $40 and $80 envelopes. If this was the case, then one could clearly profit from trading away the $10 envelope to this person. However, it wouldn't take very many trials for the person to realize that he aint makin' any money. So, I guess you could say that it works in real life in the long run.

-SossMan /forums/images/icons/grin.gif

[APokerGuy]

Paraphrasing Sklansky (or Malmuth or someone), there are *exactly* 2 reasons to bet on the river ... to get a better hand to fold (won't happen as the only better hand in this case is the nuts) or if you can get a worse hand to call (which you conceded was unlikely). So as this situation met neither criteria, no good could come from a bet. Be careful with your words though, betting was not "pointless" (i.e. a waste of time), it was a mistake (i.e. -EV).

APokerGuy!

P.S. I liked your puzzle and the analogy you used to link it to poker.