Why Giving Your Opponent Incorrect Odds Is Not Enough

[supercomputer]

This may be rambling, or old news...but maybe not. This all stems from a post I made last night about whether it is better to go all-in on a flop where you know you have the best hand, or value bet with hopes of your opponent calling with incorrect odds to beat your hand on the next card. My argument was that you want your opponent to call because making him fold is giving up a +cEV situation. But the more I heard people say “Just push. Your hand is too vulnerable.”, the more I believed it. But the math still clearly pointed to a value bet. So I set out to find concrete evidence as to why a push is better than a value bet, and this is what I came up with.

Here is the hand with some hypothetical embellishments to make things simple.

4th level of a $109 SNG. 9-handed. Starting stacks are $2000. Hero starts this hand with $1525. Blinds are 50/100.

Hero has 10 [img]/images/graemlins/spade.gif[/img] 10 [img]/images/graemlins/diamond.gif[/img] in MP and raises to $325. Villain calls from the button and everyone else folds.

Flop ($800) 9 [img]/images/graemlins/spade.gif[/img] 8 [img]/images/graemlins/club.gif[/img] 2 [img]/images/graemlins/heart.gif[/img]

Hero has $1200 chips left behind at this point and villain has us covered.

Suddenly before hero acts, by some magical force, he catches a brief glimpse into villain’s soul. He suddenly knows all his hopes, dreams, ambitions, etc. He also gains a few valuable pieces of information about villain’s hand and plan of action:

1. Villain has paired one, and only one, of his hole cards. (K9, A8, 32, etc.)
3. He will fold if we go all-in.
4. He will call, with hopes of improving on the turn, if we bet $500.
5. He will fold the turn if he doesn’t improve.

Forget about whether villain is playing his hand properly. Villain will be villain and this is his mindset. Let’s also assume, for the sake of simplicity, that if he hits on the turn, he wins (no outdraw for us on the river).

Now...knowing all of this to be fact...what is the correct play? Do we push, or bet $500?

The math:

Villain is 5/45 to improve on the turn, or 1/9. So if we play the hand 9 times, betting $500, we end up with 0 chips once and $2500 the other 8 times, or $2500 x 8 = $20,000.

If we push every time, we end up with $2000 x 9 = $18,000.

So over 9 hands like this, we are gaining $2000 total chips by value-betting as opposed to pushing. This seems like a big advantage. (In a cash game, this would be the end of the argument. $20,000>$18,000.) But I believe in a tournament that the correct play here is to push and give up this +cEV situation. Here is my reasoning:

Say you bought into 9 tournaments for $109 each, or $981 total. Your starting stack is $2000. You are then offered the option of forfeiting one of the tourneys, in exchange for an extra $500 chips from another player’s stack at each of the other 8. This is essentially the exact choice we are faced with in the hand above. You will bust in one tourney out of nine, but there is a net gain in chips:

$2000 x 9 = $18,000

(1 forfeit) + ($2500 x 8) = $20,000

Say you were then offered another +cEV choice: you forfeit one of the remaining 8 tourneys in exchange for another $500 chips. Net gain:

(2 forfeits) + ($3000 x 7) = $21,000

Do you see where I’m going with this?

Next we forfeit another one in exchange for an extra $1000 at the remaining 6 tables.

(3 forfeits) + (6 x $4000) = $24,000

After a few more +cEV exchanges we end up with this:

(7 forfeits) + ($18,000 x 2) = $36,000

We have forfeited 7 of the 9 tournaments and we have 90% of the chips in the other 2.
But we have doubled our total number of chips from 18,000 to 36,000!

Say now we are offered a +$EV deal on the payouts without completing the 2 remaining tournaments. Say, $480 for each (this is clearly a good deal for us, since 1st prize is $500 and even though we have 90% of the chips, we all know that is no guarantee.)

So after all our +EV trades we end up with $480+$480= $960, for a whopping ROI of... -2%

So the next time you give an opponent slightly incorrect odds on a draw and they call you, remember that it is -$EV for you, -$EV for your opponent, and +$EV for everyone else.

[pr0crast]

Great post! I always wondered about the math on this.

[Freudian]

Good post.

[gumpzilla]

Each one of those situations that you keep going through is getting worse and worse odds - in the first case you say your opponent is 1/9 to improve, but in the next case you're implying that they are 1/8 to improve, so on and so forth. Also, 500 is a smaller chunk of our stack each time and so the expected gain in equity is going to be less and less significant. (EDIT: I see you took that into account. But my first point still stands.) So I don't think this analysis makes much sense. (FURTHER EDIT: You are also by no means obligated to go broke on the turn when an A comes.)

If you want to demonstrate your point, why not just look at ICM and contrast it with the chip EV decision here. My suspicion is that you'll find that this is still a good place to take the +cEV move.

[fatmanonguitar]

Great post. The only thing I would add (without the math to back it up), is that this probably only holds true when pushing would not be a huge overbet (i.e., not when stacks are deep relative to the blinds early on). Betting 1500 chips into a t200 pot with TPTK and a drawing opponent only allows your opponent to play mistake-free poker.

[OatmealJoe]

This analysis would be even more pronounced if your opponent had two overcards (i.e. AQ, KQ, etc). Then they would have 6 outs to beat you instead of 5 which means giving them chances to draw would be even less in your favor.

[supercomputer]

[ QUOTE ]
Each one of those situations that you keep going through is getting worse and worse odds - in the first case you say your opponent is 1/9 to improve, but in the next case you're implying that they are 1/8 to improve, so on and so forth. Also, 500 is a smaller chunk of our stack each time and so the expected gain in equity is going to be less and less significant. (EDIT: I see you took that into account. But my first point still stands.) So I don't think this analysis makes much sense. (FURTHER EDIT: You are also by no means obligated to go broke on the turn when an A comes.)

If you want to demonstrate your point, why not just look at ICM and contrast it with the chip EV decision here. My suspicion is that you'll find that this is still a good place to take the +cEV move.

[/ QUOTE ]

Only the first choice is based on the hand in question, thus the 1/9 correlation. I just did that to show how these choices to forfeit matches in exchange for chips in other matches are similar to giving your opponent odds on a draw.

The next choices are just arbitrary +cEV decisions not related to the hand in question. As you see, in each one we end up with more chips, thus each is +cEV. So if you were comparing them to giving an opponent odds on a draw, we would be getting the best of it in each case.

[supercomputer]

[ QUOTE ]
Great post. The only thing I would add (without the math to back it up), is that this probably only holds true when pushing would not be a huge overbet (i.e., not when stacks are deep relative to the blinds early on). Betting 1500 chips into a t200 pot with TPTK and a drawing opponent only allows your opponent to play mistake-free poker.

[/ QUOTE ]

True, this does not apply in that case because we are not risking our tournament life. (If villain hits his hand we are not paying him off.

[gumpzilla]

[ QUOTE ]

The next choices are just arbitrary +cEV decisions not related to the hand in question. As you see, in each one we end up with more chips, thus each is +cEV. So if you were comparing them to giving an opponent odds on a draw, we would be getting the best of it in each case.

[/ QUOTE ]

The conclusion that you reach that making the +cEV play is bad, though, is based on those later plays. (EDIT: And the assumption that we lose our stack whenever they improve, which I still think is a bad assumption for this particular hand. There are certainly cases where it's relevant.) You haven't really done anything to show that this play that you started out describing is bad by itself. And I think you're going to be somewhat hard pressed to do so.

In reality, the situation is closer to what you describe, because you aren't going to be able to get that precise a read, and so there are many cards that are potentially scary - straight cards, etc. But I think you're giving up a pretty considerable amount of value if you're pushing in every situation where there's a remote chance that somebody could catch up.

[craigthedeac]

I agree with your conclusion, nice post.

However, I think it's easier to understand this just by thinking about it logically. In part of your post you assumed one would gain $500 on each table or $1000 for a +cEV situation. It's important to note that these numbers don't appear to have any mathematical backing, they just seem to be numbers chosen by you. That being said, this number will vary in practice. If you offer your opponent terrible odds and he still draws, you are making a +cEV move and likely a +$EV move. However, as you've shown, in a situation where the odds are just slightly in your favor, it is -$EV for you if he calls. That's important to reiterate, that this conclusion is only for situations where you barely give him bad odds.

In the situation where we have 10-10, your math correctly proved that having him draw was +cEV. You are absolutely right that this cEV advantage isn't great engouh. We gained 2,000 chips in 9 trials with this move, which is only 222 chips per move. (2000/9)

Here's where logical thinking comes in. We all know in SNGs it is important to be ITM. In order to get there, we have to play strategically, not just in terms of +cEV. Therefore, I believe a mere 222 chips isn't enough for me to risk busting OOTM. Say you were offered that situation on the bubble, would you take the few chips or rest assured that you won't bust out in 4th? The answer now becomes clear, we don't want to risk our tournament life for situations that are slightly +cEV.