When do you need to think more than 1 decision ahead in limit hold'em?

Question

Say you're in a $10-$20 limit hold'em game holding A♥8♥ against a flop of K♥3♥6♣ so you have a nuts flush draw against an opponent who bet pre-flop, and has just bet again ($10) so you put him on AA, Kx or QQ, or maybe even JJ or TT - at any rate, he's got you beat at the moment, but you could improve to beat him.

Say there's a total of $X in the pot, and it's $10 to call. My intuition is that you need to look two decisions ahead to decide whether to play or not.

Say a non-heart comes up on the turn, and your opponent bets again ($20) so that there's $Y in the pot. Clearly you would only call if Y > 82 (need pot odds greater than your 4.1 : 1 chance to win).

That means that before the turn, you know you will go all the way if X > 52, otherwise you will fold after the turn if you don't get your heart. So we can analyse those cases separately -

1. If X > 52, then you should always call before the turn if X > 47, i.e. you always call in this situation, and call again if the turn doesn't go your way.

2. If X < 52 then you can call as long as X > 42, because you need to be getting at least 4.2 : 1 odds from the pot to win immediately. If you don't get a heart on the turn, you plan to fold to your opponent's next bet.

However, you actually reach the same result if you only think one step ahead, i.e. you consider your pot odds but not your effective odds:

1. You call if X > 42, because you're getting the 4.2 : 1 odds you need.

2. Then on the next round, you do another pot odds calculation, and call if the pot is $82 or more.

The strategy works out exactly the same, even though you're only bothering to think one decision ahead!

My question is - how often can you get away with just thinking one decision ahead, and deferring the rest of your strategy to the next round of betting? It certainly makes the mental load lighter if you know the situations where you don't need to bother calculating the effective odds, and can just work with the pot odds for this round. Conversely, if there are situations where it is beneficial to calculate the effective odds, I want to know what they are!

Answer 1

I decided to have a go at answering this myself. The situation is you against one other player who has a made hand, and you have N outs.

Before the turn, the 1-step EV (ignoring any bets on the river) is

EV1 = N/47 * X + (47-N)/47 * (-10)

The two-step EV, taking river bets into account, is

EV2 = N/47 * X + (47-N)/47 * [ N/46 * (X + 20) + (46-N)/46 * (-30) ]

For a given number of outs, we can plot the EV using the one-step and two-step method against the pot size. For example

8 outs (outside straight draw)

 POT       EV-1       EV-2
  40      -1.49      -5.10
  45      -0.64      -3.52
  50       0.21      -1.95
  55       1.06      -0.38
  60       1.91       1.19

So for a $45 pot, you shouldn't call (ignoring implied odds and bluffs) since you're getting 4.5 : 1 but your odds of winning the hand are 4.75 : 1. For a $50 or $55 pot, you can call, but you should fold if you don't make your hand on the turn, as you won't have the EV to make a second call. For a $60 pot you should call both times, as you have sufficient EV in both cases. But another case is more interesting

15 outs (outside straight flush draw)

 POT       EV-1       EV-2
  15      -2.02      -1.21
  20      -0.43       1.50
  25       1.17       4.20
  30       2.77       6.91

With a $15 pot you shouldn't call as you don't have the EV - you're getting 1.5 : 1 but you need at least 2.1 : 1. For a $25 pot or above you should call both times.

But a $20 pot is interesting. If you just look at the 1-step EV, you conclude that you shouldn't call since you're only getting 2 : 1, not the 2.1 : 1 that you need. But taking the two-step EV into account, you find that you have an EV of

EV = 15/47 * $20 + 32/47 * ( 15/46 * $40 + 31/46 * (-$30) ) = $1.50

since even if you don't make your hand on the turn, your EV from calling on the river is sufficiently good.

So it seems that the answer to the question is that you sometimes need to consider two or more streets, rather than just looking at your pot odds for one street, but it only seems to occur in cases where you have many outs, and the benefit is quite marginal ($2 / hand in this case or 0.1 of a BB) so it might not be worth the additional effort required.