Heads Up Game Theory exercise

[Jiganti]

[ QUOTE ]
jay, if the sb folds his equity is -.5bb, no?

[/ QUOTE ]

Yes, this sounds correct.

[jay_shark]

Actually if he folds then his EV=0 because once he posts the sb , it doesn't belong to him anymore .

For the second question , lets assume that both players pick random numbers from the interval [0,1] and that there is no sb or bb . Since hero acts first , he will bet if he thinks his number is the highest or fold otherwise . Villain can only call the bet and nothing else . Both players post $1 in antes and can only bet $1 or fold .

What numbers should hero bet with ?
What hands should villain call with ?

[mykey1961]

[ QUOTE ]
Ok here is the answer to the first problem . I'll let others think about the second .

1) The sb can either raise an additional 2.5 bb's which means the BB is getting 2:1 on his call.

[/ QUOTE ]

Minor nit, but you said he could "raise 3x BB", not "raise to 3x BB".

[jay_shark]

Hey Mykey , good to see you .

[ChicagoRy]

Isn't that the same thing mykey?

Raise 3x BB would be 3bb, no?

Raise to 3x BB would be raising to 3bb, no?

[TNixon]

[ QUOTE ]
Ok here is the answer to the first problem . I'll let others think about the second .

[/ QUOTE ]

Couple of things.

First of all, doesn't this really become a third-level thinking problem? The big blind's calling range will depend on what range he believes the small blind is betting with, and the small blind can make a more optimal play than betting any hand if he knows the big blind's calling range.

For example, if you *know* that the big blind is going to call anything above a 34, and fold everything else, then you can certainly make a better move than betting 100% of the time.

Second, obviously I *really* need to pick up Mathematics of Poker. I don't really understand why you constructed your EV equation the way you did.

However, what I am perfectly capable of doing, and what I did do, is write a simulator that plays this game out over a large number of hands (1 million).

Assuming player 1 is always the small blind, following the rules you laid out (player 1 always bets, and player 2 calls 34+), the simulator shows that the game is -$EV for the small blind. Which would lead me to believe that betting 100% of the time is not optimal at all.

If you suspect there's a bug in the actual code generating these results, I'd be more than happy to post the code, but it's *very* simple and straightforward.

Here are 3 sample runs, each over 1 million hands, with the bet/call rules indicated:

Start run: p1 bets 0 or greater, p2 calls 34 or greater
Final result: p = -340860.000000, -0.340860/hand
Final result: p2 = 340860.000000, 0.340860/hand

Start run: p1 bets 34 or greater, p2 calls 34 or greater
Final result: p = 55661.000000, 0.055661/hand
Final result: p2 = -55661.000000, -0.055661/hand

Start run: p1 bets 67 or greater, p2 calls 34 or greater
Final result: p = 108485.000000, 0.108485/hand
Final result: p2 = -108485.000000, -0.108485/hand

So, betting every hand is a losing game for the small blind. Which means that either there's something funky

A few extra runs showed that betting somewhere around 53 or 54+ is actually the optimal spot against somebody who will call 34+.

Now I realize I'm not any more familiar with EV formulas and calculations than I was with variance calculations a week or so ago, but here's how I would want to structure it:

1/3 of the time, BB folds: +1
2/3 of the time, BB calls
1/3 of calls, BB wins because your number is less than 34: -3
1/2 of 2/3 calls, BB wins because his number is higher than yours: -3
1/2 of 2/3 calls, you win, because your number is higher: +3

(1/3 * 1) + (2/3 * 1/3 * -3) + (2/3 * 1/2 * -3) + (2/3 * 1/2 * 3)

The last two terms cancel, leaving
.333 - .666 = -0.333

Which matches the "real" results from my simulator.

Now I realize I'm probably missing something about EV calculations, but this seems like another case where an "accurate" EV calcualation (assuming yours was correct given the circumstances) leads to an incorrect conclusion. If I'm playing this game, and I'm told that betting any 2 cards is +EV no matter what your opponent does, then I'm going to be really confused about how I lost 333k betting units over a million hands.

So. Where's the discrepancy? Where does that 333k go?

[TNixon]

[ QUOTE ]
Raise 3x BB would be 3bb, no?

[/ QUOTE ]
No. Raising 3x BB would add 3BB to what's already in the pot (the small blind), up to a total of 3.5xBB.

Raising *to* 3x is a raise of 2.5bb, for a total of 3bb.

[jay_shark]

Tnixon , I'm certain now that you are my favorite poster of all time .

Nothing is better than to sit down and read one of your posts .

I'm serious too .

[jay_shark]

Hey Tnixon , this is not a nl game .

I made a simplified game where you select ONE number from 1-100 and your opponent selects ONE number from the rest of the stock .

[TNixon]

[ QUOTE ]
I made a simplified game where you select ONE number from 1-100 and your opponent selects ONE number from the rest of the stock .

[/ QUOTE ]

Which is exactly what my simulator did. If player 2's card was 34 or greater, he called. If player 1's card was higher, then he wins.

That's also why I'm so confident there aren't any bugs in the simulator. It's brain-dead simple. Contrary to what some people around here would have you believe, I am actually pretty good at reading comprehension, and understood the game perfectly. [img]/images/graemlins/smile.gif[/img]

Actually, there was one slight bug. My deck is numbered from 0 to 99 rather than 1 to 100. It doesn't affect the percentages by very much at all, though.