How to analyze discards, part 2
of this series discussed discarding as dealer, demonstrating how to
calculate and interpret such statistical measures as average hand,
expected average and minimal (Chambers) average. You may wish to
review that article before continuing, since I'll be using the same
terminology here to describe the discard as pone.
First, however, let's review the exercises from last time.
Review of exercises
You should have gotten the following results:
Clearly, the best play is to toss J-Q. These touching cards are worth about .8 more in your crib than the near cards 7-9. Although it's always tempting to toss yourself mid-cards, 7-9 is a surprisingly weak performer in your crib, returning ½ point less than 6-8, and one point less than 6-7.
Furthermore, A-2-7-9 is a better pegging hand. The 2-9 magic eleven is effective if pone has ten-cards, and the 7-9 will let you score on any mid-card lead, should your board position demand it.
Wergin is one of the most respected authorities in cribbage, and most of his advice is sound, even prescient. But writing without the benefit of the discarding stats available today, he had to rely mainly on experience and judgment. Wergin probably figured that near mid-cards would help your crib more than touching high cards. But thanks to Hessel, Bowman and Rasmussen, we now know that's not the case.
It just goes to show you can't believe everything you read — except, of course, what you read here at Cribbage Forum! Evaluate what the experts say objectively, using the analytical tools I'm giving you. Then make up your own mind.
Here are the numbers:
5-9-10-J gets slightly better expected averages than 6-8-9-10, but what really tips the balance is that it also gets the better Chambers average. When choosing between two good tosses, give extra consideration to the one that keeps the most potential in your hand, away from your opponent's influence.
I hope you quickly narrowed this down to a choice between tossing 2-3 and tossing 6-8. I expect most players would toss 6-8 without thinking too much, and this is what Chambers advises too (p. 17 of Cribbage: A New Concept). Is he right?
As you can see, the expected averages are very close, but they do favor tossing 6-8, which is the play I recommend you make most of the time. But if you need to peg defensively (say, if the score is tied 84*-84), you're better off keeping 5-6-8-Q. The 5-6 magic eleven protects you against ten-cards (you'll want to play the 6 on a ten-card lead, not the 5), and the 8 gives you a good reply to a 3 or 4 lead, either of which would be awkward to defend against holding 2-3-5-Q.
This hand is the same as the last one, except you have 7-9 instead of 6-8. As it turns out, this is a critical difference. Here your best play is to toss 2-3:
George Rasmussen, in his cribbage lectures, likes to compare 2-3-5-6-8-Q to 2-3-5-7-9-Q. Since 6-8 improves with a mid-card or an A, while 7-9 improves only with a mid-card, the former toss is distinctly superior (about ½ point) to the latter. This explains why 5-7-9-Q is worth keeping, while the better-looking 5-6-8-Q is not. For me, these two hands are a good demonstration of the value of applying sound mathematics to cribbage.
Discarding as pone
Now let's turn our attention to discarding as pone (non-dealer). Suppose your opponent deals you the following:
What's the best way to discard from this hand? Should you keep the four-card run (4-5-6-7), the most points (4-5-6-9 or 4-5-6-J) or the flush (4 5 7 9)? How can we apply the analytical tools introduced last month to this decision?
As you'll see, adapting the calculations to the discard as pone is straightforward. Interpreting the results, however, is often more complicated.
Calculating the averages
Since I covered the mathematics in detail last time, I'll move more quickly through them here. You compute the average hand as pone exactly the same way you do as dealer. Here are the calculations for 4 5 7 9:
Now look up the average crib from the tables accompanying the article Discarding to opponent's crib. In this case, we want the figures for 6-J. Obviously, lower numbers are better here:
To obtain the expected average as pone, subtract the average crib from the average hand (as you'll recall, you add these two figures when calculating the expected average as dealer):
The result gives you the average net value of your hand minus dealer's crib, after the cut. If you want to allow for crib flushes, subtract .04 from this amount and add .04 to the average crib whenever you toss two cards of the same suit. Personally, I don't bother with this.
If you have a bad hand as pone, it is possible for your expected average to be negative. For instance, the A-3-7-9-10-K hand we examined in Discarding to opponent's crib will produce a negative expected average no matter how you play it. This means that after the cut, dealer's crib will probably be worth more than your hand:
Now let's go back to our example hand (4 5 6 7 9 J), and calculate the Chambers average for the flush keep. The computations for this average work almost exactly the same as they did as dealer. Start by calculating the average hand, then calculate the average value of your discard to the crib (in this case 6 J):
To obtain the Chambers average, subtract the crib value from the average hand (you add these figures together as dealer):
In the Fifth Edition of Cribbage: A New Concept, Chambers, in his explanation of minimal averages, offers the example of discarding 3 J from A 2 3 6 9 J. If you have the book, you probably noticed a error in his calculations: he mistakenly adds two points of crib value (on a J cut) that should be subtracted from a subtotal on p. 155. As a result, he arrives at a minimal average of 3.63. The correct figure is 3.37 using Chambers' methodology. Using my methodology, which includes a calculation for His Nobs, the figure is 3.13.
Interpreting the results
I'll skip over the calculations for the other possible tosses from 4 5 6 7 9 J, but you might wish to work them out for yourself. Your results should match those in the following table:
As you can see, three credible plays present themselves:
Any of these plays could be the right one, depending on board position. This perfectly illustrates the challenge you face as pone, where you must weigh the conflicting goals of maximizing your hand score, minimizing your opponent's crib and holding your own in the pegging. The art of delicately balancing these offensive and defensive priorities is one of the most difficult cribbage skills to master. It requires a solid knowledge of discarding averages and a keen understanding of board strategy.
Since the discard as pone is particularly sensitive to board position, we need to consider how to apply these averages to different situations.
If you're playing desperation offense as pone, it's probably because:
In the first case you should hold for maximum count. From our example hand, the following holdings get 16 points on a favorable cut:
However, 4-5-6-J gets an extra point if you cut a 5, so it's the best keep if you're desperate for points, and willing to make the riskiest toss to get them. No other holding gives you a shot at as many as 17.
In addition to getting maximum count, 4-5-6-J also gets the highest average hand. This is not true all the time. With 3-5-5-6-6-Q, I'd normally keep 5-5-6-Q. This is worth six points going in, and gets the highest average hand and expected average. But the most it can be worth after the cut is 16 points — not enough to make a difference if the score was, say, 70-84*. In that event, I'd keep 5-5-6-6 instead, hoping for a 24 hand, the only thing that would give me a shot at favorable position on Fourth Street.
The second case mentioned above is a common endgame scenario where you need a specific number of points to reach 121 (or 91 if you're trying to avert a skunk). If dealer is close to 121 (or 91) himself, then your only realistic chance is to cross the line now, with this hand, before dealer has a chance to count his hand and crib. The correct toss is normally that which gives you the best chance at cutting yourself enough points to do that.
Look back at the numbers for our example hand. It seems that 4-5-6-7 is the ugly duckling of the lot, since it doesn't excel in any of the statistical categories. It's hard to justify keeping it, unless you're sitting at exactly 109 points, in which case it wins the game for you on twelve different cuts (any 4 through 7) whereas 4-5-6-9 and 4-5-6-J win on only nine cuts (any 4, 5 or 6), and the flush wins on only three cuts (a 6). Here the average hand is irrelevant, as it often is in the endgame. This concept of holding for specific count is covered extensively by Dan Barlow in his fine little book Miracles on Fourth Street, which is devoted entirely to cribbage endgames.
Playing on means trying to maximize your own scoring, even at the risk of being outscored by your opponent. Players do this most often when they're behind, but it is also appropriate when you're slightly ahead but have poor or marginal position. In such cases, play on if you feel that an offensive "boost" will get you to the next positional hole ahead of your opponent. If this seems unlikely, it may be better to play off, hoping to slow down your opponent enough that he falls short himself.
Playing on, the most important statistic is the average hand. It's a highly accurate measure of the offensive potential of a four-card hand. With our example hand, this means keeping 4-5-6-J. Granted, 4-5-6-9 is also good, since it gets an extra two points on a 2 cut. But this is outweighed by the potential extra point for having the right J. Note that if you had a 10, Q or K instead of a J, you'd be better off keeping the 9 instead.
Holding for average hand often requires making a dangerous toss, and in this case your 7-9 could well give up a large crib. But that is the risk you must take if you are committed to playing on. One situation where this would be appropriate is at 86-75*, where you must ensure that you reach the next positional hole (96) on this deal. If you fall short here, you probably won't reach 121 on your first count as pone two deals hence. Dealer is far enough along that he would probably then win the game with his three counts. Fortunately, he's not so far along to have a realistic shot of going out next deal, even if he gets a large crib here. That makes the risk in tossing 7-9 relatively small. Pull out the stops now, so that when you get your three counts on Fourth Street, you'll be close enough to win with them.
Cautious offense is DeLynn Colvert's term for "normal" play, in which you are essentially trying to outscore your opponent. The actual amount you or your opponent score is secondary. Cautious offense is usually played when you have good board position and want to preserve it.
Let's say you're dealt a reasonably good hand at any of the following scores (asterisks denote the player dealing):
Your position is good, but not great. You'd like to score at least an average number of points this deal — preferably more, to give yourself some insurance. But you can't get carried away. Dealer is only a few points behind the positional hole himself, so an exceptionally good crib could erase his positional deficit, enabling him to beat you to the next positional hole.
In cases like this, make the discard that maximizes your expected average. Since you can't control dealer's hand, you should at least maximize the spread between your hand and his crib. This gives you the best chance of improving your relative position. And that will maximize your winning chances in the long run.
In the case of our example hand, you should keep 4-5-6-9. This gets a bit less than 4-5-6-J, but the resulting 7-J toss is much safer than 7-9.
Since expected averages rely on discarding statistics, which are subject to margins of error of .1 points or more, don't get too hung up trying to decide among alternatives whose expected averages are very close. In some cases it's useful to calculate the Chambers average for each candidate, and use that as a "tiebreaker". But usually, psychology and pegging considerations are worth more than a statistical difference of a few hundredths of a point.
There are many situations in which you should play off as pone. The most obvious is when you have good position yourself, but dealer is poised to shoot past you to the next positional hole (at 47-43* for example). Another is when you're behind, and stand a better chance of winning by slowing down the dealer than by accelerating your own scoring. Playing off is most effective when dealer's position is marginal. At scores like 66-71* or 93-96*, a single sub-par crib may be all it takes to ruin your opponent's position.
If you are playing off, your main priority is to hold down dealer's scoring. Therefore, the most important number is the average crib. Depending on the board position, you may need to drastically reduce your hand's scoring potential, or even break it up completely, in order to make a safe discard.
Our example hand offers two fairly safe tosses: the 6-J and the 7-J. Throwing 7-J gives up about 4.7 points in the crib (a tad below the median), but still keeps a powerful offensive hand, returning the best expected average of any toss. This is a good play at 17-20*, where defense should be emphasized, but not at the cost of breaking up your hand. Your plan here should be to lead the 4 and try to curtail dealer's pegging. Your hand value alone will get you well past the first positional hole (18), so you'll be the favorite if you can slow down your opponent — something you have three streets to accomplish.
It's a different story at 97-97*. Now your opponent stands to go out on his three counts, even with average cards. You must slow him down immediately or lose. In this case you might consider sacrificing 2.67 points off your average hand to make the safest possible toss: the 6-J. Since you're at 97 points yourself, and since your 4 5 7 9 will be worth at least six after the cut, you'll still be in good position. But it's your opponent's position that is likely to determine the outcome of the game, and 6-J is a trifle safer than 7-J. Additionally, 4-5-7-9 is a slightly more flexible pegging hand than 4-5-6-9, and the 4-7 combination gives you a covered lead.
As pone, you often have to choose among several possible tosses with various degrees of return and inversely related degrees of safety. What makes the decision more difficult is that making a safer toss usually means sacrificing a disproportionate amount of offense. We saw that just above, when we took several points off our average hand to save about .1 points in the crib. Recently I was dealt A 5 6 8 J Q at 74-98* (not an enviable position!). I could have kept 5-6-J-Q for the best average hand, but that would require making the most dangerous toss (A-8), something I was loathe to do given the likelihood of a skunk if I didn't get a favorable cut. The most defensive option was to toss 6-Q, but this would gut my hand, relinquishing what little chance of victory I still had. So I tossed 8-Q, an intermediate choice that gave me a shot at as many as ten points (on a 4 cut), while giving up only .1 more in the crib than 6-Q. (I ended up cutting a 6 and losing, but I did get across the skunk line.)
As you can see from the following table, the progressively safer tosses from A 5 6 8 J Q reduce the average hand by an amount in excess of the corresponding reduction in the average crib. Specifically, every .1 point I shave off the crib here costs me almost ½ point in hand value:
If you were wondering why aggressive players seem to have an edge in cribbage, there's the reason! Nevertheless you must learn to recognize those situations that call for defense, and — if you're like me — learn to temper your natural aggressiveness when they occur. The former requires experience and mastery of board strategy. The latter requires discipline.
Occasionally you'll be so far ahead of dealer that the best discard is the one that minimizes the chance of giving up a very large crib. This is not necessarily the toss with the lowest average crib. For example, holding A-3-5-6-7-8 at 97-76*, your best toss is A-3. True, A-8 looks a lot safer, and gives up about ¼ point less on average. But with a large lead, and with Fourth Street position already sewn up, about the only thing you need to be concerned about is giving up a barnburner crib. And according to Rasmussen, dealer's chance of getting an 8+ point crib off your A-3 is 13.1%, compared to 20.9% for A-8.
See the article Discarding to opponent's crib for another example of this strategy, the cribbage equivalent of a prevent defense in football.
If you are playing desperation defense, you must be willing to totally gut your hand to make the safest possible toss. In fact, desperation defense often entails increasing the odds of giving up a big score in order to minimize the odds of giving up any score. This is the case when you make an all-or-nothing toss like 10-Q or Q-K, which can easily give dealer a big crib (if he tosses ten-cards or a 5), but can also easily give him a bust crib (if he tosses anything else). Obviously the best time to play desperation defense is late in the game, when dealer has good or marginal position, and is set to win on his three counts if you don't slow him down dramatically.
Suppose you're dealt the following hand at 92-100*:
You could keep 5-J-Q-K and try to win on offense. But the most favorable possible cut (5) will only get you to 110, leaving you eleven points short of the game hole — too many to expect to make up in the pegging. A better strategy, although the odds are still against you, is to play desperation defense. That means tossing Q-K, the discard that gives you the best chance (42.4% per Ras) to hold dealer's crib to two points or less. Tossing 2-K will give up fewer points on average, but it holds the crib to two or less only 36.7% of the time. Q-K is much more likely to give up a 8+ point crib (24.1% vs. 17.9% for 2-K), but that's a risk you have to take. Considering that dealer is already four points past the positional hole, even an average crib will probably get him close enough to count out as pone next deal.
In contrast to the "prevent defense" described earlier, desperation defense is akin to an all-out blitz, where a football team risks giving up a touchdown to take a shot at sacking the quarterback.
Needless to say, in evaluating any discarding decision, you must consider the pegging potential of the hand, something that is not reflected in these calculations. Consider the following:
You could keep 4-4-5-6 or 4-5-6-6, worth twelve either way. Here's what the numbers say:
It appears that 4-4-5-6 is the best keep. It returns a slightly higher average hand, and gets the best expected average, mainly because tossing 6-10 is safer than tossing 4-10. Dan Barlow recommends holding 4-4-5-6 in an ACC Tip Library article. But in most circumstances I would keep 4-5-6-6 instead. Why? Because it's a very effective pegging hand, guaranteed to score at least five points if dealer is holding nothing but 5s and/or ten-cards:
Neither of these plays works without the second 6, which gets the count over 21, taking dealer's ten-cards off the table.
Pegging potential applies to defense too. If you're dealt 4-4-6-6-10-K with the score 107-92*, don't toss 10-K. Toss 6-K instead! True, 10-K gives up the fewest crib points, but it leaves you with 4-4-6-6, a poorly spaced hand that could give up multiple pegs if dealer has similar cards. Better to keep 4-4-6-10, a hand that is less likely to get you killed in the pegging. You'll still win on a 5 cut.
Here's a quick summary of discarding priorities as pone. Other things (pegging potential, psychology, etc.) being equal, you should do the following:
Using the Chambers average
The Chambers average is generally less useful when evaluating discards as pone. Since it excludes dealer's toss from the computation, and since dealer will usually be trying to salt his crib, it tends to significantly underestimate crib potential. Nevertheless it's worth calculating when you have to choose among several close alternatives. It also comes in handy when evaluating clumpy hands where the risk level of certain discards is substantially mitigated by the remaining cards.
A good example of this is 6-7-7-8-8-K, from which you'll probably be looking to toss either 6-K or 8-K. Which of these is really the best discard?
The expected averages give the edge to 6-K, both because 7-7-8-8 returns a trifle more than 6-7-7-8 and because the published discarding tables we use in our calculations favor 6-K over 8-K. But these tables have fixed values — they do not reflect the impact of your remaining cards on the risk level of any particular toss. In this case the 7-7-8 combination that remains in your hand leaves only four single cards available to form a scoring combination with the 8 toss (there are seven cards available that combine with the 6). The expected averages are impervious to this, but the Chambers average is not, so it correctly gives the edge to 8-K, which is undoubtedly the safer toss from this specific hand.
Another reason to keep 6-7-7-8 is that it's a slightly better pegging hand than 7-7-8-8 (the 6 gives you a tad better rank distribution). But if it's the last hand of the night at your Grass Roots club, and the only money you have a chance to win is the 24 pot, then keep 7-7-8-8, which will get you a few bucks on seven cuts, compared to two for 6-7-7-8.
Now it's time to practice your analytical skills. In each of these exercises, you are pone:
The answers will appear in part 3, when I'll wrap up this treatment of discarding and offer some computer tools for use in your own study.
- May 2000 (updated February 2002)
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