Game Theory Resolution
 

The object of this game is to select a number from the closed interval [0,1] and to bet if you think your number is the highest . You only play one round so if you fold , the game is over .

a) A generous man decides to give you (hero) and your friend (villain) a free roll to enter this game . Hero posts the sb and villain posts the bb and you can raise to 3bb's or fold . Villain on the other hand can only call . What number should you raise with ??

Solution: Let a be your optimal pushing range ; a>=0
Let x be your opponent's optimal calling range . 1/3<=x<=1

We can write a in terms of x . Notice that (1-x)/(x-a) = 2
x=(2a+1)/3

The probability that your opponent wins given that he calls is 2/3 which is verified since 1/3 + 2/3*1/2 = 2/3

So the probability that you win given that he calls is 1/3 .

EV(x) = (3-3x)/2*[[1.5x + 3.5*1/3*(1-x) -2.5*2/3*(1-x)]

We wish to maximize this function using calculus .

After simplifying you get

EV(x) = 3x-3x^2
EV'(x)= 3 -6x
Set this =0 so we get x=1/2 .

Finally we're done !!

Re: Game Theory Resolution
 

So we plug x =0.5 into the equation relating x and a .
0.5=(2a+1)/3
a=1/4

Hero's EV for employing this strategy is 0.75 which you get from plugging 0.5 into the EV(x)function .

So our shoving range for hero is to push with any number greater than 1/4 . Villain should call with any number greater than 1/2 which ensures that this is the best result for both players.

Re: Game Theory Resolution
 

LOL YOUAREGOODATMATHEMATICSAMENTS

Re: Game Theory Resolution
 

can you explain where you get "Notice that (1-x)/(x-a) = 2" ?

Re: Game Theory Resolution
 

Only one problem.

The optimal play isn't actually anywhere near a=0.25, x=.5.

This is very easy to prove by just calculating a couple EVs.

Figure the EV at a=0.25, and then figure the EV at a=0.5. The EV at a=0.5 will be higher, meaning that it is more +EV to bet 50+ than it is to bet 25+.

Which is a direct contradiction to this statement:

[ QUOTE ]
So our shoving range for hero is to push with any number greater than 1/4 . Villain should call with any number greater than 1/2 which ensures that this is the best result for both players.

[/ QUOTE ]

Re: Game Theory Resolution
 

Your simplification of the EV equation is incorrect.

Here it is step-by-step, so you can point out specific errors if you disagree:

(3-3x)/2 (1.5x + 3.5 * 1/3 * (1-x) - 2.5 * 2/3 * (1-x))
...
(3-3x)/2 (4.5x/3 + 3.5/3 - 3.5x/3 - 5/3 + 5x/3)
...
(3-3x)/2 * (4.5x + 3.5 - 3.5x - 5 + 5x)/3
...
(1-x)/2 * 6x - 1.5
...
(6x - 6x^2 - 1.5 + 1.5x)/2
...
(-6x^2 + 7.5x - 1.5)/2
...
-3x^2 + 3.75x - .75

The derivative is:
-6x + 3.75
so
6x = 3.75
x = 3.75/6 = 0.625

x = 0.625, a = 0.4375

Which, strangely enough, is pretty much exactly what my exhaustive search with the simulator found. (bet with 44+, call with 63+) Finally, math matches reality.

So I guess the small blind didn't matter after all, except for the fact that it lead to EV calculations that appear to be profitable, but really aren't.

Re: Game Theory Resolution
 

Yup , thx for correcting the algebra .

So you understand how the EV equation works , right ?

Specifically the term (3-3x)/2 ?

Now it's easy to solve more elaborate problems using derivatives which is pretty neat .

Re: Game Theory Resolution
 

For a second I thought I had been doing something wrong by not calling bets with 6[img]/images/graemlins/diamond.gif[/img]3[img]/images/graemlins/heart.gif[/img].

Re: Game Theory Resolution
 

[ QUOTE ]
can you explain where you get "Notice that (1-x)/(x-a) = 2" ?

[/ QUOTE ]

since x is the minimum for your opponent's optimal calling range , this means he will be calling 1-x of the time .

Likewise , a is hero's minimum for his optimal betting range. X-a are the numbers that hero would be losing to when villain calls .

So , since villain is getting 2:1 pot odds , he should be calling with any hand with any equity greater than 1/3 . This is equivalent to (1-x):(x-a) = 2:1

Re: Game Theory Resolution
 

[ QUOTE ]
The object of this game is to select a number from the closed interval [0,1] and to bet if you think your number is the highest . You only play one round so if you fold , the game is over .

a) A generous man decides to give you (hero) and your friend (villain) a free roll to enter this game . Hero posts the sb and villain posts the bb and you can raise to 3bb's or fold . Villain on the other hand can only call . What number should you raise with ??


[/ QUOTE ]

You're so obsessed with the fold EV = 0 idea, that you resort to this?

Clearly since it's a freeroll, the BB should never fold.
And if the BB isn't going to fold, neither should the SB.

Re: Game Theory Resolution
 

[ QUOTE ]
Your simplification of the EV equation is incorrect.

Here it is step-by-step, so you can point out specific errors if you disagree:

(3-3x)/2 (1.5x + 3.5 * 1/3 * (1-x) - 2.5 * 2/3 * (1-x))
...
(3-3x)/2 (4.5x/3 + 3.5/3 - 3.5x/3 - 5/3 + 5x/3)
...
(3-3x)/2 * (4.5x + 3.5 - 3.5x - 5 + 5x)/3
...
(1-x)/2 * 6x - 1.5
...
(6x - 6x^2 - 1.5 + 1.5x)/2
...
(-6x^2 + 7.5x - 1.5)/2
...
-3x^2 + 3.75x - .75

The derivative is:
-6x + 3.75
so
6x = 3.75
x = 3.75/6 = 0.625

x = 0.625, a = 0.4375


[/ QUOTE ]

why a = 0.4375

the EV(sb) is the same for a >= 0.0, a <= 0.625
the EV(sb) is worse for a > 0.625

Re: Game Theory Resolution
 

Mykey x=(2a+1)/3 which I already explained to another poster .

Just solve for "a" given that x=0.625

I recommend you read this again because you're not understanding this .

Re: Game Theory Resolution
 

[ QUOTE ]

x = 0.625, a = 0.4375


[/ QUOTE ]

Ok it took a while but the lightbulb finally flickered.

As far as the BB knows, the SB may or may not play optimal, therefore the BB plays an optimal strategy "x" which locks in a win rate >= 5/64BB per hand.

SB also doesn't know if the BB is playing optimal, so SB plays an optimal strategy "a" which locks in a loss rate <= 5/64BB per hand.

Seems to me that if SB determines that the BB is also playing optimal, then SB should open up the range, and look for the BB trying to exploit the new range. Once that happens, SB should tighen up the range to exploit the BB.. and the race is on.

Re: Game Theory Resolution
 

Hey Mykey , let me put you on the spot .

How often should you bluff in this game ?

Re: Game Theory Resolution
 

It appears to me that the SB should:
0.0000 <= SB <= 0.1875 Raise
0.1875 < SB < 0.6250 Fold
0.6250 <= SB <= 1.0000 Raise

Re: Game Theory Resolution
 

Mykey , I already showed that raising with 0.4375+ is optimal for the sb and that bb should call with 0.625+ .

Re: Game Theory Resolution
 

What you did was got an answer from a formula.

That doesn't mean it's the right answer to the question.

Re: Game Theory Resolution
 

My answer is consistent with Tnixon's answer . Ignore the algebra error I made in the first post . It's been corrected and verified using simulation .

Re: Game Theory Resolution
 

what you haven't shown is why (if it is) your strategy is better than what I presented.

In my opinion ,my strategy for the SB is mathematically equivilant in terms of EV. And it allows the BB to Call more hands without losing EV. That creates a higher variance, which is good for the SB since he is in a -EV situation.

Re: Game Theory Resolution
 

Other possible solutions are the following :

raise with [0,0.1875] , raise with [0.625,1] and fold everything else .

BB would always call with [0.625,1]

2) raise with [1,19.75], raise with [0.625,1] and fold everything else .

BB would always call with [0.625,1]

As you can see , there are infinitely many solutions, all of which cannot do better than EV = 0.421875

Re: Game Theory Resolution
 

For the second solution , it should read

raise with [0.01,0.1975] . Hope that's clear now .

Re: Game Theory Resolution
 

[ QUOTE ]

As you can see , there are infinitely many solutions, all of which cannot do better than EV = 0.421875

[/ QUOTE ]

And that is what makes this [0,1] problem different than the 100 #'s problem, less solutions there, and this one doesn't require the SB to raise with the lowest possible value to be optimal.

Re: Game Theory Resolution
 

Interesting enough , this strategy shows that the sb should raise with about 56.25% of all numbers .So if we were to rank our cards in nl, then raising with the top 56.25% of all hands is a pretty darn good strategy .

We should not rank our cards in terms of equity ,hot and cold , but in terms of your average EV gained by those cards .

Re: Game Theory Resolution
 

In this game, you are going to be raising with a certain percentage of numbers that are lower than what the BB will call with.

You're raising everything higher than .625, and .1875 lower than .625, so 1/3 of your hands are lower than .625.

But it *does not matter* where that 1/3 comes from. It could be 0-.1875, it could be .4375-.625, it could be 0-.09375 and .531-.625. It simply does not matter, because anytime you're raising with a number lower than .625, you only win if your opponent folds. The solutions are all identical in this game.

And since you're still playing the same number of hands, (and since he can safely assumes you're not being stupid, playing 0-.5625 and folding .5625+) if the big blind calls with anything lower than .625, then he is no longer playing optimally.

But as I mentioned originally, at that point, this game becomes more about third-level thinking than it is about the numbers. If you can get the BB to play sub-optimally, then you can exploit his adjustments. But playing 0-.1875 instead of .4375-.625 should not cause him to adjust, unless believes you're playing more than 43% of the time. The problem here is that you have to believe he's capable of adjusting to your play, and willing to, because if he simply calls .625 no matter what you do, he will end up ahead in the long run, since the SB has a very clear disadvantage in this game.

If we equate these numbers to percentage hand ranges, this is all very different from poker, where there are certain "bluffing hands" that are going to be better than others, because they have a better chance to win against the top 62.5%, for example, and do better when you do get called. In poker, 56 suited has a better chance to beat AA than just about every other 2 cards. But in this game, 44 loses to 100 every single time.

Re: Game Theory Resolution
 

[ QUOTE ]
But as I mentioned originally, at that point, this game becomes more about third-level thinking than it is about the numbers. If you can get the BB to play sub-optimally, then you can exploit his adjustments. But playing 0-.1875 instead of .4375-.625 should not cause him to adjust, unless believes you're playing more than 43% of the time. The problem here is that you have to believe he's capable of adjusting to your play, and willing to, because if he simply calls .625 no matter what you do, he will end up ahead in the long run, since the SB has a very clear disadvantage in this game.


[/ QUOTE ]

You wouldn't play this game unless you expected to be in the BB as often as you would be in the SB.

So if both of you are playing optimal (and there isn't a rake) the long run wouldn't be so bad.

EV_SB/2 + EV_BB/2 = 0

If in the SB you bet with 7/16 -> 1, and the BB plays non-optimal, you will do better than if you bet with 0 -> 3/16, 5/8 -> 1 against the same player.

Re: Game Theory Resolution
 

The point wasn't that you wouldn't want to play if you had to be the SB all the time, but that because the BB has an inherent advantage, it is profitable to call .625+ from the big blind, no matter what the player in the SB is doing, so there's no need to modify the strategy at all.

Which means that there's some chance that any attempt to get a player to modify their strategy in the BB is going to do absolutely nothing except lose you more money.

Think of it like a game of roshambo (which can involve many levels of thinking), except that in this case, calling rock is always slightly +EV, no matter *what* your opponent is doing.

Re: Game Theory Resolution
 

[ QUOTE ]
The point wasn't that you wouldn't want to play if you had to be the SB all the time, but that because the BB has an inherent advantage, it is profitable to call .625+ from the big blind, no matter what the player in the SB is doing, so there's no need to modify the strategy at all.

Which means that there's some chance that any attempt to get a player to modify their strategy in the BB is going to do absolutely nothing except lose you more money.

Think of it like a game of roshambo (which can involve many levels of thinking), except that in this case, calling rock is always slightly +EV, no matter *what* your opponent is doing.

[/ QUOTE ]

There isn't a need for the SB to change from his optimal strategy either, since whatever he loses as SB he will gain as BB.

Re: Game Theory Resolution
 

jayshark and others,
is it tru that the way to apply these results to holdem is that the sb should push with any hand that has 43% equity vs. a random hand? In that case it looks like SB has to push with something like T4o and better. Then I guess the BB has to call with everything except perhaps the worst (32o - 82o).

Or is it not so straightforward...

Re: Game Theory Resolution
 

Well it's not quite the same situation . In this game , when your number is higher than you win 100% of the time . In hold em , when your card is better(hot and cold) then you don't necessarily win 100% of the time .

This is essentially how the SAGE system was derived but you would need to compare hand ranges for various possible subsets of hands amongst the two players .

Re: Game Theory Resolution
 

[ QUOTE ]
There isn't a need for the SB to change from his optimal strategy either, since whatever he loses as SB he will gain as BB.

[/ QUOTE ]
Yeah. There was some thought buried somewhere about how it *might* be worthwhile to try to get the BB to play sub-optimally and then exploit that, but the more I think about it, the less sense it makes. There's no reason for him to play sub-optimally, and the only thing you can do to try to get him to play a wider range is by playing a wider range yourself. But while you're doing that, even if he doesn't adjust, he's not as well off as he *could* be, but he's still better off than he was before you opened up, and it's going to be fairly obvious when you re-adjust to try to exploit.

So yeah, I guess there's really no third-level thinking involved here at all.

What a silly, silly game. I'm sure glad we have poker instead.

[img]/images/graemlins/smile.gif[/img]