Heads Up Game Theory exercise

[TNixon]

[ QUOTE ]
If you're familiar with SAGE then you probably know the answer to the first problem .

[/ QUOTE ]
And btw, SAGE doesn't say push any 2 at 3bbs.

It says push wide, but not any 2.

But SAGE also takes into account that the hand that is ranked at the 66% mark (hand 33 in your game) has some chance of beating a top 10% hand (90 - 100). In your game, 33 *never* beats anything higher than 90.

So SAGE should give looser push recommendations than would be appropriate for your game, where the hand that starts behind can never catch up.

[jay_shark]

Tnixon , my ev calculations are correct .

Did you input that hero posts the sb and villain posts the bb ?

Hero can only raise an additional 2.5 bb's or fold .

[TNixon]

Yes, I took the blinds into account.

And I just realized where the confusion came from: one spot where I said "push with any 2". Sorry about that. Old habits. Much more used to talking about "any 2" than "any 1". [img]/images/graemlins/smile.gif[/img]

The simulator really did use the rules of your game, though, and the results were that the small blind averages a loss of about 1/3 of a bet every hand if he bets 100%, and villan calls with 34+.

Here's the code, should be very simple to verify, even without any direct programming knowledge:

<font class="small">Code:</font><hr /><pre>
printf( "Start run: p1 bets %d or greater, p2 calls %d or greater\n", p1Bet, p2Call );

srand( (unsigned)time( NULL ) );

int hands = 1000000;
float p1 = 0.0f;
float p2 = 0.0f;
for( int i = 0; i &lt; hands; i++ )
{
int c1, c2;

// post blinds
p1 -= 0.5;
p2 -= 1.0;

c1 = c2 = (int)(100.0f * (double)rand() / (double)RAND_MAX);
while( c2 == c1 )
c2 = (int)(100.0f * (double)rand() / (double)RAND_MAX);

if( c1 &lt; p1Bet ) // player 1 folds, p2 wins the blinds
{
p2 += 1.5f;
}
else // p1 bets
{
p1 -= 2.5;
if( c2 &lt; p2Call ) // p2 folds, p1 winds blinds plus gets his bet back
{
p1 += 4.0f;
}
else // p2 calls
{
p2 -= 2.0f;
if( c1 &gt; c2 ) // player 1 wins
{
p1 += 6.0f;
}
else // player 2 wins
{
p2 += 6.0f;
}
}
}
}
printf( "Final result: p1 = %f, %f/hand\n", p1, p1 / (double)hands );
printf( "Final result: p2 = %f, %f/hand\n", p2, p2 / (double)hands );
printf( "\n" );
</pre><hr />

[jay_shark]

ahhh I see the problem .

The simulator assumes that if you fold then you're down 0.5 in EV but this is not true . Once you post the sb , it's not yours anymore , so your EV for folding is 0 and NOT -0.5 .
Notice that -0.5 + 0.1666666 = -0.3333333333

[TNixon]

The simulator is not calculating EV.

The simulator is telling you how much you win or lose over a session of a million hands. The total profit or loss has to include the posted blinds.

Not taking the posted blind into account gives you a calculation that is +EV, but where you will actually lose money over time.

Which means the calculation is misleading at best, and flat out incorrect at worst.

*Edit*

The whole point of trying to figure out the EV of betting 100% against an opponent who will call with any number 34 or higher is to figure out if that is a profitable play, is it not?

Simulating 1 million hands shows pretty clearly that you will lose money in that situation, to the tune of 1/3BB per hand.

Therefore, any sort of math that tells you it is profitable to bet 100% *has* to be wrong, does it not?

[jay_shark]

Let me ask you this .

Lets say I post the sb and I can only call instead of raising and villain can only check . I have card # 26 , what is my EV for this game ?

It should be 1.5*25/99 - 0.5*74/99 = 0.00505....

[TNixon]

Simulator says:

Final result: p1 = -471660.000000, -0.471660/hand
Final result: p2 = 471660.000000, 0.471660/hand

But, again, this is not calculating EV. It's calculating total profit and loss.

And it should be fairly obvious that if you're flipping a weighted coin that is only 26% in your favor, and 74% in your opponents favor, that you're going to lose money in the long run.

Yet your EV calculation shows this to be a profitable situation?

Ignoring the posted blind gives you a very misleading (incorrect, IMO) result.

Which would indicate to me that you cannot ignore the posted blind.

Where math doesn't match reality, the math is wrong.

Try this:

Calculate the EV in your original game of folding 52-, and betting 53+, against somebody who will still call with 34+. This *should* end up being higher EV than betting 100% (if it doesn't, then I'm not sure what to say except that the math does not match the reality here).

But if betting 54+ and folding everything else *is* higher EV, that should throw a serious wrinkle into the "if betting 100% is +EV, then folding can never be the correct play" assumption.

[omgwtfnoway]

jay_shark, in mop the games are solved where sb folding is -.5bb in ev, can you explain why this is not the case here.

tnixon, there is no third level thinking here because we're solving for game theory optimal solutions. a strategy is not game theory optimal unless you could tell your opponent your whole strategy and he could still not exploit you with this information. hence, third level thinking is fruitless.

fwiw i also don't think it's correct to raise with all hands here but can you see how an optimal strategy might still be -ev? for very small stack sizes this game may favor the big blind so if both players play perfectly the bb will be a favorite.

[jay_shark]

Maybe this will clear things up .

If you have to decide before the game begins , whether or not you wish to play , then you would safely decline . You save yourself -0.33333333.

BUT , if you are coerced into playing this game; maybe some guy has a gun to your head , and post the sb , then the correct strategy is to move all in 100% of the time . If you employ this strategy then you will be ahead 0.166666 .

Your overall profit is 0.16666666-0.5 = -0.33333333

Hopefully the confusion is cleared . My assumption was that you were in the game which was outside of your control . I probably should have stipulated this but I thought it was understood .

[omgwtfnoway]

yes that clears it up, thanks