Interesting formula (Shove EV)

[ZeeJustin]

When you're in the SB, and you have X BB for your stack, your shove EV w/ any given hand is easy to calculate if you know the range of hands your opponent will call with.
X= your stack size (in terms of BB, assume the BB has an equal stack or has you covered. If you have the BB covered, X = the size of the BB's stack)
C = Percent of the time you are called
W = Percent of the time you win when you're called (use poker stove to figure this out)

Your EV is then:
(1-C)1.5 + WC(X+.5) - C(1-W)(X-.5)
translated this is:
1-C = % of the time you are steal the blinds
1.5 = the amount you win when you steal the blinds

WC = the percent of time you are called and win
X+.5 = the amount you win when you are called and win

C(1-W) = the percent of time you are called and lose
X-.5 = the amount you lose when you are called and lose

This formula simplifies to:

SHOVE EV = 1.5 - C + CX (2W-1)
X= stack size
C = % of time you are called
W = % of time you win when you are called

Example: You have 32o in the SB, and your opponent will call you with 22+, 54+, 86+, T7+, J6+ Q2+ (the loose range of hands in MJ's article).

According to poker stove, 32o wins 31% of the time against this range of hands, therefore W=.31

This range of hands accounts for 64% of all hands (13 pocket pairs x 6 combinations)+ (48 non pair hands * 16 combinations) = 78+768 = 846 combinations.
846 / 1326 = .64
Note that 1326 = total # of combinations of hands.
C=.64

Let's say your stack size is 10x BB.
We now have X=10
C=.64
W=.31

SHOVE EV = 1.5 - C + CX (2W-1)
Shove EV w/ 32o and 10x BB = 1.5 - .64 + (.64)(10)(.62-1)
Shove EV w/ 32o and 10x BB = -1.572 x BB

Any ideas on how to make a formula that estimates your EV on the button? The fact that there are 2 players left to act makes it much tougher.

I'm not sure how many applications there are for this formula, but I find this stuff interesting nonetheless.

[parappa]

[ QUOTE ]
Any ideas on how to make a formula that estimates your EV on the button? The fact that there are 2 players left to act makes it much tougher.


[/ QUOTE ]

Of course, the tough part is assigning a probability that they will call, but once you've settled on one (it's much easier mathwise if you come up with the same range for both), then you just take (1-pCall)^#OfPlayers (on the button it's 2) and use it for your fold% in the calculation you mentioned. Of course, 1-that is the chance that they'll call. This assumes that only one will call, of course. Beyond that, it's a bit of a mess.

[gergery]

So wmajik’s article talked about what your correct play should be late in SNG tourneys. Overall, I love the article and think it’s a great foundation for why you should be pretty aggressive.

One area that is a bit fuzzy is how multiple opponents affect things. Since if there are 3 left to act and each will play 15% of hands dealt, then that kinda like being up against a loose player, since three players who will each play only 15% of hands means you’ll get called by a top 15% hand ~40% of the time. ZeeJustin points out the other big adjustment that should happen to wmajik’s article – the fact that some hands have much less Caught Equity than shown.

So it got me thinking that heads up is really where this analysis should be most relevant. And it got me thinking that if this strategy is so successful, and can’t be stopped, then what would the best counter-strategy to it?

Or, what would happen if two very good players played optimally --- What would happen if ZeeJustin played against head to head at the end of an SNG against himself?

We’ll use game theory to answer this. To start with some hands have much lower Caught Equity, hand like 62, 43, etc. are going to be 4:1 dogs, or 2:1 dogs if called, for an average of maybe 2.5 to 1 dogs. So those are clearly EV- to push with. Let’s adjust that to, say get rid of the 30% of hands that are worst heads up.

So, player 1 in the SB pushes with his top 70% of hands, and that is roughly worth 1BB in wmajiks calculation tables. If SB makes 1BB worth in ev value, then Player 2 in the BB must be losing that much. So bb player should put in chips if his expectation is >1BB. If he pushes and is called then he’s risking 9bb to win 11bb (other guys stack plus his 1bb put in). So he needs to win exactly 50% of the time or better to have >1bb expectation. So he should call with any hand that is even money or better vs. the range of hands that’s top 70%. Those hands are any ace, any pair, K6+, Q9+, J9s+. That corresponds to about the top 40% of hands or so.

Of course, his alterego will know this. So now each player will push with his top 70% of hands, and call with his top 40% of hands.

And knowing this, what adjustments will Zee make as the player first-in? You know you’ll be called by the top 40% of hands, and that will happen 40% of the time. So 60% of the time the opponent folds and you win 1bb, so your outcomes the other 40% of the time need to be -1.2 bb or better. Or, your hand selection needs to be ~ 1 to 1.2 underdog or better to lose less than -1bb net. And the hands that are a at least that vs. top 40% of hands are any ace, any pair, K7+, Q8+. That set of hands is about the top 40% of hands.

So now the first player in should only be pushing with the top 40% of hands or so, and the second player calls with the top 40% of hands or so. I’ve run this out another iteration or so, and this seems to be where equilibrium is reached.

Conclusion:
So with both of you at ~10xBB, against players who adjust appropriately (by both pushing all-in themselves, and calling more when pushed against), your optimal strategy is to tighten back up to about the top 40% of hands for both pushing and calling.

Against players who will only call you with the top 20% of hands or so (any ace, any pair, KQ), you should be pushing your top 67% of hands or so (any J or better, good Ts).

Comments?

--Greg

[dethgrind]

I like game theory. But I'd like even more to see someone do this sort of analysis taking into consideration the non-linear value of tournament chips. Eastbay posed an interesting question (http://forumserver.twoplustwo.com/showflat.php?Cat=&Number=1021631&page=&view=&sb=5& o=&vc=1) a while ago, and the discussion looked like it might head in that direction, but it didn't get too far.

What I got from that earlier discussion is that you need a significant overlay to call on the bubble, if it's a question of double up or bust and you have a decent stack. So I suspect that all the calling figures in your post would need to be tightened up considerably, given the raiser's range of hands.

But if the raiser knows the caller is tightening up, he should raise with more hands. Where is the new equilibrium in this scenario, taking into account the real $ value of the tournament chips in a 50/30/20 payout structure?

In this thread (http://forumserver.twoplustwo.com/showflat.php?Cat=&Number=1104446&page=&view=&sb=5& o=&vc=1) a 70/30 strategy is mentioned for head-up play. Push 70% / call 30%. That won't work unless it's just the two of you left though, otherwise you're just bleeding away TEV with every confrontration to the players not in the hand.

[gergery]

[ QUOTE ]

So I suspect that all the calling figures in your post would need to be tightened up considerably, given the raiser's range of hands.

[/ QUOTE ]

Keep in mind that my figures are for a narrow set of circumstances: 1) its head to head, 2) you both have ~10xbb, 3) you have equal stacks, 4) you are both good players optimizing your play against another good player who is optimizing back against you.

With more players you need to tighten up your raising standards because it becomes exponentially more likely one of them will have a hand that can call. And if the callers know the raiser is tightening up, then they must tighten up too as the range of hands they will be facing is better. I think the equilibrium will net out in different places based on function of # of players, looseness of players, position, and blinds to stack ratio, etc.

And the thinking laid out in my previous post suggests that 70/30 isn’t right. You should call with more hands. And knowing you’ll be called with more hands, the raiser should tighten up. To get to 40/40. Of course, the giant caveat is that both players have optimized both their own play, and are up against opponents who have successfully adjusted to that play.

In practice, from what I’ve seen, I think most opponents probably don’t push enough, and most probably don’t call enough. So against the average opponent, you should raise more frequently. And recognizing that raising correctly against an opponent that is too tight is highly profitable, you probably want to call with fewer hands, meaning that 70/30 could well be correct as a default in practice.

--Greg

[El Maximo]

I just wanted to revive this old post. I found this article a while ago and it really changed my outlook on pushing from the SB and Button. There is alot of discussion about what hands to push. I use to read the push posts and wonder how did the veterans get a good grasp on what hands to push from SB and BB. I got pokerstove and took a few days to run numbers against various ranges of hands. I did the ev calcs and it really helped me better understand the EV of pushing from SB and BB. So I revived this post for all the posters who are asking

Do I push this?

Why do I always get blinded down before I make the money?

Why am I always so shortstacked on the bubble?

Hope this article helps others like it helped me.