Game Theory Question - Where do you put the number...?

[BarronVangorToth]

A friend and I have been working out the math on this but we both agreed that it would be an interesting 2+2 problem.

The game is Hold 'em.

Let's say it's a 10-20 game (or A-B -- whatever).

Traditional, ie. $5 small blind and $10 big blind.

(or SB = .5A and BB = 1.0A)

Everyone bets preflop like normal.

Once the flop hits, going in turn from the guy UTG on, each play can elect to muck ONE of his cards to get back a portion of his wager pre-flop (the dealer in this game wouldn't gather up the monies -- they would be left in front of every player).

So let's say there's no raising, 4 guys are in, each for $10.

The question: what percentage of your initial bet would you need to get back in order to make this game be one where it was strategically sound to sometimes utilize this function, but not often enough to make it completely foreign.

In other words, if you got back 100% of your initial bet, MOST would always take advantage (especially if they're going to fold).

If you only got back 5%, few (if any) would ever take advantage.

Seeing as how the numbers would have to be divisible in $1 increments, likewise, let's say the percentages need to be 10% intervals.

What would the percentage be to make this game optimal in a strategic sense...?

Thoughts...?


Barron Vangor Toth
www.BarronVangorToth.com

[FyrFytr998]

Smells like another Barron card game in the works. Good luck. [img]/images/graemlins/wink.gif[/img]

[Louie Landale]

Maybe I didn't read this correctly. Sounds like you are saying that folks that don't like the flop will get some percentage of their pre-flop investment back; where they first discard one card figuring to subsequently check-and-fold. Also, those who flop trips with a pair on board and those that have a pair no kicker and those that flop a one-card draw may be tempted to discard even they they figure to continue playing.

If we ignore the relatively rare cases where someone will discard figuring to continue playing, this game (say you get 40% of your money back) is similar to the following: PreFlop bets are $6. There is a after-flop betting round of $4 featuring no-checking-and-no-raising. There is then a normal post-flop betting round of $10.

The bigger the % pay-back the more late position is worth since you get to see what folks do during the no-check-no-raise $4 round: you will often win in last position without risk when everyone folds.

I don't see the point.

- Louie

[The Armchair]

Just to clarify:

Pre-Flop: Normal
Flop: Each person, starting at UTG (why not SB, btw?) can muck one card, leaving them w/a one card hand. If a person does so, they get n% of their pre-flop bet back. After that, play continues as normal.

If that's the case, then there are a few reasons to "reverse buy-back" (neato name, huh?):

1) You've made junk, and intend to fold. Therefore, you'd RBB for n>0.
2) You've made a nut draw but only need one card from your hand (i.e. Ac2s on a three-club board or on a board with QJT rainbow). Your incentive to RBB is rather high, and approaches n>0.
3) You have the third to a pair on the board, but your kicker isn't all that good (i.e. 98x on a board of 992). At that point, you have to assume you're ahead, so n>0 is fine. (If you're behind, you're very behind, but note that A9 has incentive to pitch the A as well.)
4) TPNK. In that case, n is hard to measure, b/c you don't know where you stand. The higher n, the more likely you'd pitch, but that's directly proportional to the size of the kicker.
5) A made four-card hand, which is exceptionally rare -- it'd have to be quads. In that case, one could argue that you'd not pitch, as you have a near-lock on the hand.

Given that #4 is the only violently variable situation (can anyone else provide others?), let's focus on that. Take the following board:

A[img]/images/graemlins/diamond.gif[/img] Q[img]/images/graemlins/club.gif[/img] 6[img]/images/graemlins/heart.gif[/img]

Looking only at Ax hands, AA, AQ, and A6 are not going to toss the undercard, even if n=100. (Remember that they have no reason, giving your hypo, to think they are in anything other than the lead right now.) AK would probably need a very high n -- perhaps even n=100 -- but in that case, note that AK is about as valuable as A-rag.

Any rag 2 through 5 is going to pitch for n>20 or so, and it's not hard to compute. Put them against an A w/a higher kicker. Assume the kicker is going to keep its undercard. They have 3 outs to make their two pair, discounted some for the situation where they are already up against a better two pair. Figure out their EV, and find the break-even point, keeping in mind the possibility that they have the only ace. In any event, n could be as high as 100, but my gut says its closer to 20.

For rags 7 through J, you have a real problem. There simply isn't enough information out there. Obviously, as rag gets higher, there's more incentive to hold on. But you can math it out.

[BarronVangorToth]

ARGH!

Good catch from my post... starting with the SMALL BLIND, NOT utg on the flop.

Barron Vangor Toth
www.BarronVangorToth.com

[BarronVangorToth]

[ QUOTE ]
Maybe I didn't read this correctly. Sounds like you are saying that folks that don't like the flop will get some percentage of their pre-flop investment back; where they first discard one card figuring to subsequently check-and-fold. Also, those who flop trips with a pair on board and those that have a pair no kicker and those that flop a one-card draw may be tempted to discard even they they figure to continue playing.

If we ignore the relatively rare cases where someone will discard figuring to continue playing, this game (say you get 40% of your money back) is similar to the following: PreFlop bets are $6. There is a after-flop betting round of $4 featuring no-checking-and-no-raising. There is then a normal post-flop betting round of $10.

The bigger the % pay-back the more late position is worth since you get to see what folks do during the no-check-no-raise $4 round: you will often win in last position without risk when everyone folds.

I don't see the point.

- Louie

[/ QUOTE ]


That's another factor (if that's what you're saying) about how major it is to be in last position (perhaps moreso than in traditional Hold 'em) and that, likewise, needs to be factored in, especially when you see everyone toss in one card ahead of you which means they don't like the flop (or have a rag to go with their piece of the board).

Shockingly enough, we got jumbled up doing the math on what the percentage should be, not to mention that "value" has to be factored and weighted in, in the sense of making people want to utilize the function sometimes, but not often, and not stilting everything to the total position game.

Further thoughts on what the percentage back should be in this hypothetical game...?


Barron Vangor Toth
www.BarronVangorToth.com

[The Armchair]

The problem is that "piece plus rag" hands gain certainty but not necessarily any more than your late-position opponets.

Example: You have A3. Board is AJ6. (Ignore suits. Thanks!) By tossing your 3, all you gain is the knowledge that your kicker will certainly not play. But your late position opponent now has perfect knowledge. He can be sure that his AQ is good right now.

The liability you incur by RBB is enormous. Therefore, in order to utilize the option, the RBB would have to be very high. Given that drawing hands are often going to be using both cards, that's also the case for them.

I think the outcome is that if I'm going to RBB, it's because I'm intending to fold. I can't see a value that'd make it worthwhile otherwise, save for 100%.

Now, if you did this w/3-card pockets, everything changes.

[BarronVangorToth]

But at 100% -- EVERYONE will stay in on EVERY action pre-flop as they know they'll get 100% back if they so desire... So that certainly can't be the number (or close to the number -- if is even, I don't know, 80%, a lot of marginal holdings (A-X suited, as an example) will never balk at taking a look.

The unfortunate part is that I don't believe it can be truly cracked purely on the math.... which is rather frustrating, as there is a "value" element that needs to be implemented, taking into account a lot of variables and the "feel" people will have by utilizing this system over another.

The horrific thing is that I think the number will end up being something like 50% -- which is easy to compute AND remember AND to divvy up in a formal or informal setting.

Hold 'em with a 50% rebate option of your pre-flop investment by mucking 1 card.

Anyone have an idea that it should be a large degree higher or lower? All feedback is appreciated -- especially since we're going to be running this format in our home games as, otherwise, I'm going to drive my friends crazy (crazier?) with my talk about this Hold 'em Variation.


Barron Vangor Toth
www.BarronVangorToth.com

[BarronVangorToth]

If someone jumps in late and wants to see what this is about briefly, basically, it's this: standard Hold 'em in every way EXCEPT after the flop, BEFORE the betting has begun, every player, in turn, starting from the small blind, has the option of mucking ONE card. If they do, they get a rebate (currently: 50%) of their pre-flop wager. Once every player has either mucked-and-rebated or passed, you start the wagering as normal.

The question is whether the rebate number should be higher or lower than 50%. If so -- why or why not? Remember, also, to make things easy to calculate, it should be divisible by 10 as getting a 42.5% rebate on a $40 pre-flop investment would require people to start getting out the loose change and calculators.


Barron Vangor Toth
www.BarronVangorToth.com

[The Armchair]

I was using 100% as a way to demonstrate why I don't think RBB would work.

In any event, I don't see how a marginal holding such as A2s goes up in value. If it hits it's flop (flush, flush draw, trips, two pair, wheel, gut-shot-plus-backdoor-flush), it most often won't be pitching the undercard. The obvious exception is if it trips the A, but in that case, you've either won the hand or you're losing badly, and the rebate matters little. (In fact, you'd probably keep the deuce b/c you want a chance to fill up in case of a 5-card hand making its draw.)

Other marginal hands go down in value also. 87s? You'll not have the pot odds to make a call. Imagine 5 player limp in (including you) and you are in last position. Flop is A56 rainbow (and none to your suit). Each of the four players before you RBBs. You don't. There are now 3 small bets in the pot. First player bets, next 3 folds, and you're getting 4:1 on a call, and you know all he is ahead of your 8 kicker. Your negative implied odds suggest a fold here. And you're not _really_ getting 4:1 on your call, because had you thought a step ahead, you could have simply RBB'd and folded. You've actually put in 1.5 SB to win 3.5, and are therefore getting under 3:1.

The way to combat this is a lot of pre-flop raising (you get back 50% of your investment, not .5 SB), but that, clearly, kills your implied odds down the road.

A similar analysis applies to small/medium pocket pairs. In this game, if you flop a set, you're even a bigger favorite than in others, as it pays poorly to chase a draw. But let's say you have 22 on the button. If you have six players in (including the blinds), you're getting 6:1 on your call, and a 7.5:1 to set. But again, your implied odds go down, as your full bet has to stay in the pot, while hands that miss are going to RBB and fold. You're getting closer to 3:1 with weak implied odds.

It's true that drawing hands want to see the flop cheaply, but only in multi-way pots. The problem is that, because their hands require both hole cards, they're likely to get the opposite in this game, as they pay twice as much to see a flop.