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Subject: Check versus bet: some poker theory (long)
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The first version of this post was a collaborative effort that a friend and I wrote under the "Mark Glover" pseudonym. It was called "Check vs. bet: theory (was HPFAP error)" and appeared on 11 Nov. 1999 in the 2+2 Gambling Forum.

Much of this second version is taken wholesale from that essay, with the permission of my friend.

If you are unclear about the concept of expected value, you first might want to check www.google.com for the four RGP "Understanding EV" essays that I posted on the 7th, 9th, 13th, and 17th of December, 2001.

In this post, I discuss some important factors you should consider when pondering a check (or call) versus a bet (or raise). This analysis certainly isn't original, and the theories that support it should be familiar to those readers who understand the fundamental concepts of poker. But this discussion could prove helpful to newbie theorists.

Let's begin by correcting a misconception that is surprisingly common among poker players. There is a notion that if the pot odds justify a call, then they also justify a bet. For example, a top 2+2 forum hand analyst (according to Sklansky) once noted:

     [I]n the example you give, the pot contains 28 small 
     bets. . . .  So, you're getting (for now) 28-1 if 
     you bet, when you're around an 8-1 dog to improve on 
     the turn.  Even if someone raises you, you're still 
     getting 15-1 odds, which is a mighty fine overlay.[1]

You primarily use pot odds to determine whether you should play or fold a hand. If you are thinking about betting (or raising) just for value, then pot odds are irrelevant. (Note: you can bet or raise for reasons other than for value, and pot odds sometimes play a role in these situations. To keep things simple, however, this post will focus on value bets and raises.)

An example should help illustrate this concept. Suppose you hold AcTc in middle position, there was lots of pre-flop action, but you are heads up at the turn and see a board of Kc6h2c/8s with nine big bets in the pot. To make the math easier, assume you know your opponent holds AhAd.

Your opponent bets the turn. Should you fold? There are 44 unknown cards that could come on the river. (Normally, there would be 46, but you know your opponent's hand in this example.) Some 35 river cards hurt you, while 9 help you. The pot (after your opponent's turn bet) is offering you 10-to-1 pot odds when the odds against you winning are 35-to-9 (or about 3.9-to-1). Clearly, you should not fold with this kind of profitable overlay.

But should you just call or should you value raise? To help answer this question, let's alter the example somewhat.

Suppose your flop call put you all-in. You are heads up, so both of you roll over your hands. The turn brings that 8s. Before the dealer puts up the river card, however, your opponent offers you a proposition. "I know you are all-in and there cannot be any more betting action on this hand," he says. "But can I interest you in a little side bet? I'll bet you $20 at even odds that I will win this hand." Would you accept his offer?

Of course not. When you are getting only 1-to-1 odds in a situation when the odds against your winning are 3.9-to-1, then you can expect to lose (on average) $11.82 on this proposition bet.[2] The size of the pot is irrelevant. Regardless of whether the pot contains $50,000 or $5, your expectation for the proposition bet is still a loss of $11.82. Thus, you would decline the bet.

(Note: You certainly should not propose the proposition bet yourself. But that's essentially what you are doing if you opt to "value raise" rather than just call your opponent in our earlier scenario.)

If your opponent puts up $40 against your $20, then you are getting 2-to-1 odds, but the proposition bet still costs you an average of $7.73.[3] If the pocket aces bets $60 versus your $20, expect it to cost you $3.64.[4]

It is profitable to bet only when the money odds you are receiving exceed the odds against your winning. If you can win $80 by risking $20, for example, your money odds (4.0-to-1) exceed the odds against your winning (3.9-to-1), and you can expect to make a $0.45 profit by accepting the proposition bet.[5]

These kinds of proposition bets don't happen very often at the poker table. But something similar does occur regularly. You find yourself competing for the pot against multiple opponents and must decide whether to check or value bet. Or another player bets, several opponents call, and you must decide whether to call or value raise.

In a modified version of our scenario, if you expect an average of, say, five opponents to call your raise, then you will be getting 5.0-to-1 money odds in a situation when the odds against your winning are only 3.9-to-1. Thus, you should expect to make a profit by raising, regardless of whether the pot already contained $50,000 or just $5.

On the other hand, if you expect an average of three opponents to call your raise, then you will be getting 3.0-to-1 money odds. And you can expect your raise to cost you money. This is true regardless of the pot size.

Let's look at the original scenario from a different angle.

Your excellent draw means you can expect to win (on average) 1.84 of those 9 big bets that entered the pot before the turn, so you shouldn't fold just because its going to cost you part of one big bet.[6] But for every bet you must add to the pot on the turn, you are kissing goodbye to 0.59 bets (on average).[7] Clearly, a raise on the turn is not a value raise in this situation. And if you were first to act, you would check instead of value bet (and hope your opponent checks as well).

To complicate matters somewhat, there is the issue of "implied" odds. If your stack, at the turn, was exactly one big bet, then implied odds would not enter the picture. Your opponent would bet, and you would call if the pot odds were at least 3.9-to-1. If the pot was smaller, you generally would fold. (One exception might be when you think a suck-out by you could cause your opponent to tilt.)

But suppose you have plenty of chips, you saw your opponent's pocket rockets and plan to fold if your flush draw fails, your opponent didn't see your cards, and you are certain your opponent will check-and-call your bet if a flush card appears on the river.

Now, you don't need a full 3.9-to-1 pot odds to call a turn bet. Because you are going to win an extra bet from your opponent if a third club comes on the river (and are not going to lose an extra bet if a non-club arrives), you only need 2.9-to-1 pot odds to call the turn bet in this situation.[8]

You should adjust the minimum necessary pot odds if your opponent sometimes might check-and-fold to your river bet or if they sometimes might contribute more than a single bet on the river. Also note that if there are multiple opponents who might call your river bet, your implied odds increase (which decreases the pot odds you need to justify calling on the turn).

The concept of "effective" odds further complicates matters. Suppose three of you still were involved at the turn. The first player bets, and you have to decide whether or not to fold. Now you generally need bigger pot odds than when only two of you are fighting it out, because the player behind you could raise. Extra bets that you might have to call on additional betting rounds also can increase the pot odds you need to justify a call.

Players with a little poker theory under their belts use these concepts of pot odds, implied odds, and effective odds to decide whether they should fold. But once they have decided to play, what factors do they use to determine whether to check/call or bet/raise for value? The key considerations are: (1) your estimated chances of winning the pot, and (2) the number of players you expect to call your bet/raise. Again, the size of the pot is irrelevant when you are thinking about value betting/raising.

That's my (simplified) take on the major theoretical considerations that determine whether you should check (or call) or whether you should bet (or raise) for value.

[1] GD, 8 Nov. 1999, "Re: Another HPFAP 'Loose Games' error," 2+2 Gambling Forum.

[2] ( ( 35/44 ) * -$20 ) + ( ( 9/44 ) * $20 ) ~= -$11.82.

[3] ( ( 35/44 ) * -$20 ) + ( ( 9/44 ) * $40 ) ~= -$7.73.

[4] ( ( 35/44 ) * -$20 ) + ( ( 9/44 ) * $60 ) ~= -$3.64.

[5] ( ( 35/44 ) * -$20 ) + ( ( 9/44 ) * $80 ) ~= $0.45.

[6] ( 9/44 ) * 9 ~= 1.84.

[7] ( ( 35/44 ) * -1 ) + ( ( 9/44 ) * 1 ) ~= 0.59.

[8] ( ( 35/44 ) * -1 ) + ( ( 9/44 ) * ( x + 1 ) ) = 0

( -35 / 44 ) + ( ( 9x + 9 ) / 44 ) = 0

-35 + 9x + 9 = 0

9x = 26

x = 26 / 9

x ~= 2.9