Cost of a "Free Card"

With one round to go a single opponent has a 10% chance to beat you and there are 5 bets in the pot. If you bet he'll fold. How much does it cost to check and give a "free" card?


Well, 10% of the time he wins 5 bets so it costs only a half a bet. That's no disaster. That's no surprise since if his hand was worth a bet then he'd have the right odds to call.


The only time a free card costs more than a bet is when the player IS getting the right odds to call but won't.


So lets not get all bent out of shape about giving or getting "free" cards. Strategic reasons to check or check-raise can easily earn more than one bet.


- Louie

"Strategic reasons to check or check-raise can easily earn more than one bet."


If you can make someone mistakenly fold a winner then this is obviously true. But in most cases (e.g., heads up) the opponent can call your bet/raise at a cost of one bet. So you can't earn more than one bet by checking or raising without an opponent "mistake".


There is only one important case I know where your action earns more than one bet without relying on mistakes. That is in multiplayer pots where your bet/raise makes multiple draws fold. Each player abandons up to one bet in pot equity. Since all those players were drawing against you, those draws were subsidizing each other, "implicit collusion". So each fold sacrifices .9 bets in pot equity and transfers equity from the other players to you. For example, you hold top pair against 3-4 straight/flush draws with 4-5 bets in the pot. Your equity is small in a no-fold'em game, but a check-raise will win the pot immediately. It is like robbing a crowded bank with only one or two bullets in your gun.

No, no, no. I don't know if I am going to be able to explain this though.


If you bet and he is going to fold because he doesn't have odds, he wins 5 bets 0% of the time. You are giving him equity in the pot by checking. Why give away half of a bet? Half of a bet is a lot, since we really only expect to win about 1 bet an hour. If he incorrectly calls, you lose the 7 bet pot 10% of the time so his equity is .7 bets instead of .5. Your equity is now 6.3/7, however. You have a 90% chance of winning more money than before. You gain 80% of his call on the turn, in addition to your previous EV. If you let him have his 10% chance, this leads to the "mathematical catastrophe".


In the case where your opponent is correctly drawing, he has a sufficient amount of equity in the pot to make his call correct. You can still make money by betting, however, and poker is about making money. If the pot is 8BB before you bet, he is correct to call a bet with a flush draw on the turn. He is paying 1 to win 9 about 20% of the time negating implied odds. A bet on the turn makes him risk money he has an 80% chance of losing. By not making him do this you are technically losing that 80%. This in NOT a negligible amount. You have to make players pay (most of the time) when their ev is negative to make up for the times your ev is negative.


Obviously you can and should vary your strategy to induce bluffs, etc..., but the general rule should be that it is important NOT to give free cards.


-g-

Betting to avoid giving free cards is about the least understood piece of advice in the poker literature. Guys are simply paranoid about this. There are so many situations (particularly when heads-up) that it is correct to check/call rather than bet/raise even when you know that you are ahead. A simple example is when you call with Ax out of the bb against a button raise. The flop comes with an Ace and what most bb's do here is checkraise the flop, bet out on the turn and see the other guy fold. What a wasted opportnity to let the other guy hang himself a little. That is, usually, the better play is to check call flop and turn and bet the river (I realize of course that there are other ways to play the hand and you should do so at times).


I should note that when you have 2 opponents, the free card fear is a lot more real. It simply ain't there in most heads-up situations.

Louie,


And if your opponent will bet behind you 55% of the time, allowing you to checkraise, what's the cost then (assume he correctly folds)?


Regards.

The danger of giving a free card is a function of many things, two of which are the number of opponents and the number of "collective outs" that are against you. Agreed that in a heads-up situation, flopping a top pair of aces means that you have little to worry about when your lone opponent gets a free card. But suppose you have a top pair of eights and six opponents? Now a free card can prove very expensive.

It's a heads up raised (by B)pot and the flop is K-8-2 rainbow. Player A is first with K-3. Player B has A-8. One scenario is player A checkraises the flop and bets the turn and B folds. The other is, player A leads on the flop, B raises, A calls, and A checks the turn and B checks behind.


These are the types of situations where free cards add up because the flop-cost is the same either way. Assuming that B can fold to a river bet after missing, isn't he getting infinitely better odds in scenario two compared to one?


(In the second scenario, A can counter by betting the turn (with or without reraising the flop). For the sake of my question, please assume that B can know that A will become passive when raised on the flop.)


Tommy

I think that is a very good point. I struggle with this concept all the time and others probably do as well.


Pat

Tommy,


Yes. In the particular situation you have presented, you are giving away equity in the pot for no nothing.


As with most poker situations, changing the assumptions changes things alot.


Give player B, the hand AQ. Now he's got three outs to an ace and three to a payoff hand. Also, if the turn gets checked thru, if his calling with ace high on the river frequency goes up, this too is good for player A. We can make reasonable assumptions that make checking the better play. This would involve good judgement on the part of the player. You can also make assumptions that make player A's situtation worse, like having him bet into the river when the 8 pairs, and call a raise.


I think Louie's point was that there are sometimes good tactical reasons for giving free cards (or just checking) that can outweigh their cost. Don't get caught up in the "don't give free cards" notion to not fully consider these options. Know that if he bluffs behind you at a certain rate, then checking becomes the better option. Don't check out of weakness and ignorance, POWER check, with knowledge and EV.


This is similar to the notion of calling the river in big pots. I think we had this discussion before. The argument goes, "you either lose a big bet or a whole pot." But you don't lose a whole pot, you lose or gain whatever the difference is between the correct action and what you actually did. It's just that with very big pots the risk to reward ratio becomes so in favor of calling that its usually not a bad default. The same is true for checking, usually.


R

"I think Louie's point was that there are sometimes good tactical reasons for giving free cards (or just checking) that can outweigh their cost. Don't get caught up in the "don't give free cards" notion to not fully consider these options."


Right. Well put.


It's especially nice to 'give' a free card only to find out that we were getting one.

"The only time a free card costs more than a bet is when the player IS getting the right odds to call but won't."


True for head-up situations, except perhaps in the case that the opponent would raise with an inferior hand if you had bet (particularly when you act first). Good point.


Mike

Lets say a player is getting the right odds to call a single bet but not a double bet, and will in fact call a single bet and NOT call a double bet. By definition, then, his hand is worth more than a bet (but less than two). Successfully check-raising a 3rd player then earns you more than a bet against this player.


You also earn when he is forced to call a single bet twice (he's sandwhiched). If he has only a 20% chance to win then you earn 80% of a bet twice, or 1.6 bets when he calls.


- Louie

Yes, these fractions of bets are important.


I notice that you use the word "catastrophy" for the time when an opponent isn't getting the right odds to call and will fold, but NOT for the time he IS getting the right odds and will call, even though you make more money when you bet against the better draw. And that's my point. Its only a "catastrophy" in the mathematical divide-by-zero sense, not in any EV sense.


Lets say you call on the river with 32 no-pair, you cannot win, and therefore your EV is -1bet. Would you call that a "catastrophy"? I wouldn't.


A "catastrophy" (i.e. a multiple bet bad decision) would be failing to bet on the river with the clear winner when lots of players are going to call, or failing to bluff when its going to work.


- Louie

He's not getting the right odds to call the check-raise even when the pot is 3 bets bigger. So half the time you get a full bet from him for free (since he bets-folds), and half the time he draws slim for free. Sounds like a lucrative check-raise to me.

Yes, its "infinately" better in the mathematical divide-by-zero sense. Never-the-less it costs A less than 1bet.