Two Plus Two Newer Archives simple game theory question
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#41
10-20-2007, 12:55 PM
 jogsxyz Senior Member Join Date: Mar 2005 Posts: 1,167
Re: simple game theory question

This RI game is still too complex. Reading his paper.
Gilpin covers too much territory in too little space.
He's too theoretical. Need to see applied game theory.
How did he solve the individual terms in those nodes?
There just isn't enough details on his thinking process.

He really needs a two street game with dynamic equities.
He needs to explain simpler games.
#42
10-20-2007, 01:15 PM
 Alan McIntire Junior Member Join Date: Oct 2007 Posts: 5
Re: simple game theory question

With 2 players, there are 5 cards exposed at the river,
=45 cards remaining. Your opponent could have one of
45*44/2=990 hands. If only 1 hand is the nuts, You can NEVER bet more than 989 times the pot, else you're guaranteed to be making a negative expectancy bet. 989 is an upper limit.

If you have Kxs with a flush, 3 of your suit on board and none is an Ace, there are 7 ways your opponent could have the nut flush- A and any of the 7 remaining cards. In that case, he'll have the nuts 7/990 of the time so you can't bet
more than 990/7 = 141 3/7 of the pot or you're guaranteed to be making a negative expectancy bet.

If you bet 9 times the pot, and your opponent calls you
less than 1/(9+1) of the time, you'll win money by betting anything.
If you bet 989 times the pot and your opponent calls you less than 1(989 +1) of the time you'll win money, but he'll have the nuts at least 1/990 of the time if you don't have something like A of the suit to prevent this.

If you bet over 1/(141 3/7) of the pot with Kxsuited and 3 of your suit on board, your opponent will call you
1(141 3/7) of the time and you'll lose money. - A. McIntire
#43
10-20-2007, 02:22 PM
 Paxinor Member Join Date: Sep 2006 Posts: 87
Re: simple game theory question

@ Alan: nice one! this is one hell of an answer...
#44
10-22-2007, 11:09 PM
 Alan McIntire Junior Member Join Date: Oct 2007 Posts: 5
Re: simple game theory question

Speaking of fraction of the pot to bet, I'm wondering what
a roughly optimum strategy should look like. Say the pot
contains 1 bet, each of two players has a stack of 10, and
the rules allow the first player to bet any amount 1 to 10,
and the second player to call or fold.

If the first player is playing a NASH strategy, and bets 1 unit, the second player will have to call half the time
to prevent being robbed by bluffing on everything. The caller will be losing a large fraction of the time since he's calling with half of all hands, but he'll be calling
a lot with relatively weak hands.

On the other hand, if the first player bets 10 units, he
usually won't be called, yet if he is called, the second player will be calling with a lot higher hands, the top 1/11, and there will be a greater than even chance of losing the bet unless you bet only bet 1/22 of the top hands, or something like this.

I'm guessing the optimum strategy for bet sizing should be something like an even money split:

If you have one of the top 1/22 of all hands, bet
10 units once,
bet 9 units 10/9 times,
bet 8 units 10/8 times,
...
bet 2 units 10/2 = 5 times,
bet 1 unit 10 times.

Of course you'd also have to bluff proportionate to the
pot odds you're offering your opponent.

Any ideas or inputs on the optimum way to vary bet size?
#45
10-23-2007, 12:14 PM
 rufus Senior Member Join Date: Aug 2005 Posts: 425
Re: simple game theory question

[ QUOTE ]
With 2 players, there are 5 cards exposed at the river,
=45 cards remaining. Your opponent could have one of
45*44/2=990 hands. If only 1 hand is the nuts, You can NEVER bet more than 989 times the pot, else you're guaranteed to be making a negative expectancy bet. 989 is an upper limit.

If you have Kxs with a flush, 3 of your suit on board and none is an Ace, there are 7 ways your opponent could have the nut flush- A and any of the 7 remaining cards. In that case, he'll have the nuts 7/990 of the time so you can't bet
more than 990/7 = 141 3/7 of the pot or you're guaranteed to be making a negative expectancy bet.

[/ QUOTE ]

But you can infer things from the opponent's behavior on prior rounds, so that the probability of the opponent having a particular holding could be less than 1 in 990.

For example, let's say you're holding two AsAh, raise, and get called pre-flop. When the turn comes up AdAcKs you check and your opponent raises. The turn and river are 3c and 4c both of which he also raises hard. Now the nuts is the skip-straight flush 2c5c, but a reasonable opponent, might never make that sort of play with that holding so the proposed limit could asymptotically approach infinity.

Making some similar assumptions about the players being familiar with each other, it's also easy to show that the appropriate bluff-raise is equal to the legitimate raise, so it too, can asymptotically approach infinity.

Edit: Altough, I guess with super-deep stacks, all kinds of strange plays will start making sense, so this is really a weak counter-example.
#46
10-23-2007, 12:26 PM
 rufus Senior Member Join Date: Aug 2005 Posts: 425
Re: simple game theory question

[ QUOTE ]

Of course you'd also have to bluff proportionate to the
pot odds you're offering your opponent.

Any ideas or inputs on the optimum way to vary bet size?

[/ QUOTE ]

In NLHE, you've got to consider stack size as well as pot size.
#47
10-24-2007, 10:36 PM
 _D&L_ Senior Member Join Date: May 2006 Posts: 128
Re: simple game theory question

I think Rufus's point is partially valid. He makes a good point that we can't treat all hands equally weighted. That is, we can infer from previous rounds the likelihood that our opponent is on a certain hand.

But that alone can't imply that we would be willing to wager infinite amounts of money on our read, unlesss our read was infinitely accurate. Such a read (or to use game theory lingo, "Baysian updating") would not occur in game theory (or the real world), because as soon as we are willing to put an infinite weight behind a read, it becomes optimal for our opponent to always play long shots, because we give him an infinite payout. In short, the mere enticement of an infinite payout should cause a rational opponent to deviate, and invalidate our read.

A good read occurs when we can deduce that an opponent would have had to pass up more valuable opportunities by playing his hand in this fashion. As soon as we are willing to throw an infinite payout behind that assumption, it turns out our opponent has not passed up any valuble opportunities - instead he's been playing for the highest payout of all - our mistaken read.

Still as i said earlier, I still believe there isn't a theoretical cap where your hole cards eliminate the possibility of your opponent holding the nuts.

In other cases, there will be a theoretical cap to bluffing. As for Admiralfluffs concern:

[ QUOTE ]

Haven't read all the responses, but I don't see how there could be a theoretical cap. If there were, than the last bet made would be exploitable.

[/ QUOTE ]

If we're bluffing the nuts, and our opponent is actually holding the nuts, we can't go on bluffing indefinately. That would make for a strategy with negative infinity EV. So there has to be a theoretical cap when u can't rule out the nuts being held by your opponent.

Now of course we would never stop re-raising with the nuts. And there is a point where we would stop re-raising with a bluff (that doesn't rule out the nuts). When we pass the point that we are no longer representing any bluffs, any rational opponent would fold. Still there is no harm in making this naked value raise (i say naked, because its not supported by bluffs), because our opponent cannot hold a better hand.

If there is no chance that we're holding the nuts, then best hand we can bluff is the second-nuts. If our opponent might be holding the nuts, then we would hit a theoretical cap. Eventually we would call/fold, in place of making a naked re-raise.

----_Dirty&amp;Litigious_----
#48
10-28-2007, 04:50 PM
 Alan McIntire Junior Member Join Date: Oct 2007 Posts: 5
Re: simple game theory question

I didn't make myself clear earlier, but I was trying to get an idea of the optimum- game theoretic way to vary bet size in no limit. Say it's now the river, and your stake balance is, say, 100 times the pot. If you bet 1/2 the pot, your opponent will have to call with 2/3 of the hands he would play, to prevent you from stealing the pot by bluffing with anything. If you bet the pot, your opponent following game theory will have to call you half the time to prevent you stealing with anything.

If you bet 10 times the pot, your opponent will call with the top 1/11 of his hands to keep you from stealing. You'll win more money if he DOES call and loses, but he's much
less apt to be calling with a losing hand, so you're giving up all those 1 pot size calls where his hand ranged between the top 50% and top 9.09%. In addition, if you DON'T have the nuts, he's much more likely to have the nuts if he DOES call. Obviously you can't bet 10 units only with the strongest hands, else you're telegraphing the strength of your hand by bet size. My first guess was that the proportion you bet any amount should be roughly proportional to the strength of the hand you're representing;i.e., if you're
representing a top 1/11 hand, you bet 10 times the pot once for every 10 times you bet the pot. Of course you'll HAVE a top 1/11 hand only 2/11 of the time you'll have a top 50% hand, so you'll actually BET 10 units 1/11*2/11= 2/121 as often as you'll bet 1 unit. Any further inputs on general rules on optimum bet size in proportion to pot size?
#49
10-28-2007, 08:32 PM
 _D&L_ Senior Member Join Date: May 2006 Posts: 128
Re: simple game theory question

Hey Alan, glad to see someone else with some good intuition and thinking.

1- you already understand that your bet is a hedge, you are sometimes raising for value, and sometimes bluffing.

[ QUOTE ]

If you bet 10 times the pot, your opponent will call with the top 1/11 of his hands to keep you from stealing. You'll win more money if he DOES call and loses, but he's much
less apt to be calling with a losing hand, so you're giving up all those 1 pot size calls where his hand ranged between the top 50% and top 9.09%. In addition, if you DON'T have the nuts, he's much more likely to have the nuts if he DOES call.

[/ QUOTE ]

But you seem worried that your opponent won't be calling you if you bet too much - that your missing out on all the little suck bets you could have made.

You should realize that such a concern contradicts the intuition behind your hedge. You've hedged your value raises and bluffs so that your actually indifferent to whether your opponent calls or folds. Don't worry whether he calls or folds - either way he loses on your turn to bet.

Poker rewards you for your aggression; the opponent being raised suffers a defense penalty. For instance, if the pot is infinite, he has to call you, even if you only bluff one time in a million. He can't afford to have you steal an infinite pot. That's the defense penalty. And if you hedge it right - you gain on average, regardless of whatever bias your opponent has towards calling or folding. Either he calls all your value bets, or he loses the infinite pot to your one-in-a-million bluffs. He's a Loser either way.

2- Our optimization strategy is to increase the frequency with which our opponent encounters our hedge. There are only so many hands capable of being value bet, and lots of potential bluff hands.

The more we raise our value bet hands, the more bluff hands we can hide in their strength. For instance, suppose a pot has \$100 in it. If you raise \$1, then you can only sneak in 1 bluff for every 99 value raises. But if you bet \$100 then you can sneak in 1 bluff for every 2 value raises.

Thus, the more we can raise our value/bluff hands, the better off we are w.r.t everything trapped in the middle, or below.

This is why my original example of a bluff hand that ruled out the nuts, and the nuts, could be raised to an infinite level. (Remember that example was the nut flush, and a bluff hand that ruled out your opponent from having the nut flush, but itself was a worthless hand).

3- The limiting factor: Hands the beat your value hands, and other hands with assymetric information.

This is what constrains our bidding. We weigh the advantage of increasing our hedgning frequency versus the loss we suffer to hands that beat even our value bet cards.

4- Diminishing marginal returns.

The bigger our value bets, the more bluff hands we can sneak through. But as bets grow large in relation to the pot, their growth becomes asymptotic. You can never bluff more than 50% of your hands, even if you raise an infinite amount. If you did, your opponent could gain +EV infinity simply by calling you.

6- Because of diminishing marginal returns there will be an equillibrium point where the value of increasing the frequency of your hedge gives way to losses suffered to hands that beat your value cards (i.e. hands not "trapped in the middle" - my term).

7- Finally, even once you've done all this, you then have to look for sandbagging opportunities. This style of betting shows the strength of your hand. So far Each raise/bluff pair is bet in porportion to its strength.

But this gives away too much information. If we always bet our hands strength, then our opponent would always know to go all in, even if he had a hand just one kicker-notch above us.

Thus, you have to go back through each raise/bluff pair, and look at playing some stronger hands with your weaker ones, to protect the weaker ones from being exploited.

The stronger hands expect to receive enough aggression to justify them being used as sandbags (remember all margins have to be equal). This is by far the hardest part to implement in practice. So many margins... Generally speaking, you sandbag some weak hands with strong hands to severely penalize anyone that comes strongly over-the-top of your weak raises.

p.s. i spent a long time writing a program to do all these things. And it even does a few more things, which is kind of hard to explain, unless you got a firm holding on all the above mentioned concepts. But I have faith in u...

----_Dirty&amp;Litigious_----
#50
10-29-2007, 09:57 AM
 curious123 Senior Member Join Date: Dec 2005 Location: not impressed by your perforaments Posts: 585
Re: simple game theory question

Good post.

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