Poker Articles

Short-Handed Poker: The Overcard Quandary Part I
by Jason Pohl

Disclaimer: This two-part article includes a lot more mathematics than usual subjects would require. If you are a beginning player, please try your best to follow along. Understanding probabilities and how to calculate the profitability of different outcomes is definitely useful. I have also double-checked all equations, but I cannot promise there are not some minor mistakes since all calculations were completed by hand. If you see an error, please email me at Jason@PokerPages.com, and I will have the math corrected. I hope you enjoy the article.

What Makes a Player Lose?
The main ingredient common to most losing players is a relative looseness. Losing players play too many hands, but more importantly, they will play losing hands too far. Some losing players are loose and aggressive. To some extent, their aggressiveness may help camouflage their vulnerability because they'll buy some pots, but eventually the loose aggressive player will run into a strong hand. When the confrontation occurs, the cost will be high. Loose passive players face a greater dilemma and can really only thrive if their competition bluffs too much. Known affectionately as calling stations, loose passive players will be pummelled by attentive players.

Against any player that is too loose, the winning strategy is uncomplicated: wait for a good hand and bet. Bluffs will be consistently unprofitable if the competition calls (or raises) with weak hands. Profit will be earned on big wins with real hands. A winning shorthanded player is aware that the vast majority of competition will play too loose, even for a shorthanded game. In fact, many players will justify their flimsy calls with bottom pair or Ace-high by assessing, "This is a shorthanded game. My opponent is far more likely to be bluffing." Against such weak opposition, patience and showing down big hands are required to be victorious at the tables.

This fundamental idea is at the heart of one of the toughest predicaments many tight aggressive players face. If a real hand is required to win because our competition is too loose, how does one play overcards? An absolute answer does not exist, because so much depends on the level and style of competition. But it can be helpful to examine some emblematic circumstances every shorthanded player faces.

Overcards in a Heads-Up Pot.
Scenario 1. Button has K Q. Big blind has 9 8.
Flop is 2 6 8. Big blind bets out. 5.5 small bets in the pot.

With this sort of flop, the big blind's bet is very straightforward. Top pair is a strong hand shorthanded, but definitely vulnerable, and a bet puts only 5.5 small bets in the pot. The button faces a difficult decision. The button has 6 outs (in this case, all 6 are clean.) If the button knew what the big blind held, he could calculate odds of 6/45 or 6.5 to 1. At first glance, 6.5 to 1 odds appear insufficient to call, but the button can pretty much count on winning at least one more big bet if a King or Queen falls. In other words, the implied odds are sufficient to justify taking a card off.

If we presumed the button would win one extra big bet when a King or Queen falls on the turn, then the EV for a call would be calculated as follows:

• Queen or King on Turn: (6/45 * 7.5) = +1.0 Small Bets
• No Queen or King: (39/45 * -1) = -0.8666 Small Bets
• Total EV: 1.0 - .8666 = 0.133 Small bets/hand profit

So, the computations concur that a call is profitable. In this ideal scenario, where all 6 outs are clean, the button would make money by continuing to see at least the turn card. But would a raise be superior to a call? It depends. If we assume that the big blind will call a raise on the flop and then check on the turn, the button will be able to take a free card and add even more profit. Let's examine three possible outcomes.

• Outcome 1: Button Does Not Improve: (39/45) * (38/44) = 1482/1980 = 74.85%
  The button does not improve when any of the 39 of 45 cards not a King or Queen fall on the turn and any of the 38 of 44 cards not a King or Queen fall on the river.
• Outcome 2: Both Hands Improve: {(6/45) * (5/44) + (5/45) * (6/44)}= 60/1980 = 3.03%
  Both hands improve when either a King or Queen (6 of 45) falls on the turn, followed by a Nine or Eight (5 of 44) on the river or when a Nine or Eight (5 of 45) falls on the turn, followed by a King or Queen (6 of 44) on the river.
• Outcome 3: Only Button Improves: {(6/45) * (39/44) + (34/45) * (6/44)} = 438/1980 = 22.12%
  -Only the button improves when either a King or Queen (6 of 45) falls on the turn, followed by anything but a Nine or Eight (39 of 44) or when a blank (34 of 45) falls on the turn, followed by a King or Queen (6 of 44) on the river. A blank means any card NOT a King, Queen, Nine, or Eight. For those who are wondering, the blank card in the second half of the equation can not include a King or Queen because the first part of the equation already includes the times that a King or Queen falls on both the turn and the river.

Finally, we guess how much the button would earn or lose in each scenario. In possibility 1, that's easy. The button would lose 2 small bets since it raises the flop. In possibility 2, we can assume the button will lose the 2 small bets plus either 2 or 3 big bets (we'll average and say 2.5.) In possibility 3, the button will gain the 6.5 small bets in the flop plus an average of 1.5 big bets (2 big bets when the King or Queen falls on the turn, 1 big bet when the King or Queen falls on the river.) I know there are a lot of assumptions involved. But we're not trying to come up with a perfect answer, just an approximation.

EV = (.7485 * -2) + (.0303 * -7) + (.2212 * 9.5) = -1.497 -.2121 + 2.1014 = 0.3923 small bets/hand profit

Whew. Hope that math isn't too confusing. The ability to take a free card makes the button's hand three times more profitable than a call alone. But there's a real problem. Most shorthanded players beyond the lowest limits are quite familiar with semibluff or free card raises, and they will not let them work so often. If the big blind bets out again on the turn (even if they only bet when a nine or less falls), then the button's raise on the flop becomes a significant loser. A nine or less would fall 27 of 45 times on the turn. In other words, the button will now only improve (when the big blind doesn't improve) about {(6/45) * (39/44) + (12/45) * (6/44)} = 306/1980 = 15.45% of the time. Instead of .3923 small bets/hand profit, a raise would now result in approximately a .2 small bets/hand loss, even when we factor the times when a Nine or Eight falls on the turn and the big blind's bet prevents the button from improving (and losing additional money.) A call remains superior to a raise unless the button can be very sure to gain a free card.

Scenario 2. Button has K Q. Big blind has 9 8.
Flop is 2 6 8. Button bets. Big blind checkraises. 7.5 small bets in the pot.

Let's compare the EV for a call (since we have already concluded that a raise is less profitable than a call against all but the most passive players.) If we again presumed the button would win one extra big bet when a King or Queen falls on the turn, then the EV for a call would be calculated as follows:

• Queen or King on Turn: (6/45 * 9.5) = +1.2666 Small Bets
• No Queen or King: (39/45 * -1) = -0.8666 Small Bets
  Total EV: 1.266 - .8666 = 0.4 Small bets/hand profit

The calculations are fairly straightforward. With a larger pot, a call with overcards will be more profitable. Although the profit for a call after the big blind's checkraise is higher than after the big blind's bet out, this does not mean the big blind made a mistake by checkraising. The button will earn back .4 small bets profit on average, but that figure is after the 1 small bet already lost on the flop. If the button knew he was about to be checkraised, the best play would have been to check and take the free card. As they say, hindsight is 20/20.

Scenario 3. Button has K Q. Big blind has Q 8
Flop is 2 6 8. Button bets. Big blind checkraises. 7.5 small bets in pot.

Both Scenarios 1 and 2 were near ideal situations for the big blind since all its overcard outs were clean. The reality is that this will often not be the case. If one was to presume that the big blind's checkraise meant it had a real hand, we could limit the possible holdings somewhat. Possible holdings that leave the Queen and King clean include A2, A6, A8, J8, T8, 98, 77, 99, and TT. Other holdings that counterfeit one of the button's outs are K2s, K6s, K8, or Q8. In other words, the button will be thwarted even when a King or Queen falls, because it will make two pair for the big blind. How bad does this counterfeiting hurt? Let's look again at the specific hands in Scenario 3. The button earns an extra big bet when a King falls and loses two big bets when a Queen falls.

• Outcome 1. King on the Turn (3/45 * 9.5) = +.6333 Small Bets
• Outcome 2. Queen on the Turn (2/45 * -5) = -.2222 Small Bets
• Outcome 3. No Queen or King (40/45 * -1) = -.8888 Small Bets
  Total EV: .6333 - .2222 - .8888 = 0.4777 Small bets/hand loss

Although it took a while to get to it, we have reached the first major point of this article. On an average flop, a player with a legitimate hand will have a kicker matching one of the button's overcards some portion of the time. And there is one last possibility. If the big blind holds a big hand such as 22, 66, 88, 86, KK, or AA, the button is drawing dead or nearly dead. By adding these holdings to the mix, calling with overcards against a legitimate hand becomes unprofitable. There are now 22 combinations which reduce the button to 3 outs, 22 combinations which reduce the button to virtually zero outs, and 88 combinations where a King or Queen are clean outs.

The overall EV of calling is calculated by figuring when all 6 overcard outs are clean (2/3 of the time), one of them is counterfeited (1/6th of the time), and when the button is drawing nearly dead (1/6th of the time). I have taken the liberty of doing two additional formulas to plug into our final EV formula. First, I calculated Scenario 3 as if the big blind had only bet out. Second, I calculated Scenario 3 as if the King on the turn led to a 5 small bet loss.

• EV when big blind bets out:
  (2/3 * .133) - (1/6 * .611) - (1/6 * 1.444) = .0887 - .1018 - .241 = .2541 small bets/hand loss
• EV when big blind checkraises:
  (2/3 * .4) - (1/6 * .4777) - (1/6 * 1.444) = .2666 - .0796 - .241 = .054 small bets/hand loss
  
  

And so, we reach the foremost thrust of the overcard quandary. At first glance, drawing to two overcards appears profitable, but it is frequently a losing play. Even if the loss is 1/4th a small bet on average, the impact in the long run is significant. Until a player becomes very good at identifying their opponent's possible holdings, they may not be able to recognize when their overcard outs are safe. Without that skill, the proper play is often a simple fold.

In part II of this article, we will examine overcards in a multi-way pot and overcards against a habitual bluffer. We will also counter a worry of many advanced players: will folding overcards lead to more bluffs?

You can email me at Jason@PokerPages.com. Good luck!