Discard averages in the real world
By Craig Hessel. Runs under MS-DOS or Windows. Freeware, available for download here
Michael Wortley Nolan:
|It's hard to overstate the importance of
getting familiar with discard averages. Knowing the relative value of
combinations of cards thrown to the crib will greatly improve your game. The
best way to acquire this knowledge is to study
discard tables of the sort published at Cribbage Forum. But one caveat
with these tables is that their values are static, and do not take
into consideration the distribution of your original six-card hand. In a
real game, the presence or absence of helper cards (cards that create
scores in combination with your toss) among the four cards you've kept can
have a statistical impact on the value of your discard. How significant that
impact is has long been a topic of debate among cribbage theorists.
Unfortunately this debate has been mainly fueled by conjecture, since there
was little hard data to go on. Until now. Thanks to Craig Hessel, author of
for Windows 97, we're in a much better position to accurately
measure dynamic discard averages.
But first, let's frame the issue in a more tangible way by examining some excerpts from the classical cribbage literature. Here is DeLynn Colvert in Play Winning Cribbage (p. 117 of the Third Edition):
And here is Joe Wergin, in Win at Cribbage, (p. 104), analyzing how to play 5-5-6-6-7-7:
Colvert is guessing that the 2-2 in your hand sufficiently diminishes the value of A-A in your opponent's crib that in this particular case it is safer than tossing A-4, even though under normal circumstances the opposite would be true. His point is that A-A cannot get into a multiple run without at least one 2, and since you already have two of those locked up in your hand, it's particularly unlikely for one to be cut or dropped into the crib by dealer. Whether this translates to a ½ point shift in the relative value of A-A and A-4 is the question. So is whether Wergin's discarding of 5-7 as pone (instead of the natural 6-7) and 5-5 as dealer (instead of the natural 5-6) is justified based on the original 5-5-6-6-7-7 distribution.
Enter Craig Hessel with DS.EXE, a command line utility that dynamically calculates discard averages. It works by consulting a table of discard frequencies that lists the relative likelihood of your opponent tossing any of the 91 possible discard combinations, from A-A to K-K. (These in turn were derived from Hessel's earlier algorithm for empirically calculating static discard averages). DS adjusts these discard frequencies based on the six cards you were dealt (which your opponent obviously can't be holding), then calculates the value of each combination when added to your own discard and each of the possible starters. The results are averaged together to produce an expected crib value.
DS also calculates your average hand, and provides an expected average: the sum (or difference if you're pone) of average hand and average crib. The program also indicates the amount that the average crib varies from the normal static value for that particular discard (as reported by Hessel's earlier algorithm). This amount — known as delta — measures the statistical impact of the four retained cards on the value of your toss. As it turns out, delta is not as significant a factor in normal play as has generally been thought.
Installing DS is easy. There is no setup program to run. Just download the Zip file from Hessel's Web site, and extract its contents to a folder on your hard drive. To run it, open a command prompt window in the directory where you extracted the Zip file, then type ds followed by a space, followed by the six-card hand you want to analyze. Use a for aces, and t, j, q and k for ten-cards. You can optionally append a space followed by six letters indicating the suits of the six cards. In this way, DS will allow for flushes and precise Nobs probabilities. Press Enter to start the calculation.
Inputting Colvert's example hand of A-A-2-2-3-4 produces the following:
The columns on the left half of the screen are for opponent's crib — those on the right half are for your crib. For each possible toss, DS displays the average hand (HnAve), average crib, delta, and expected average. For your crib, the average crib and expected average are labeled DlAve and DlTot respectively. For opponent's crib, they're rather confusingly labeled PnAve (even though it's dealer's crib) and PnTot. All of the values are dynamically calculated. If you're not versed in how to interpret these numbers, I suggest studying the Cribbage Forum article series How to analyze discards.
The delta values are particularly useful, since they let you quickly scan the effect of card distribution on discards. If you prefer to use a different source for static discard values, you can apply the delta reported here to those values. I prefer to take the static values from my own discard tables, then modify them based on DS's delta numbers. The figures I'll be citing henceforth were derived this way.
What does DS say about A-A-2-2-3-4? Well, there's virtually no delta for either A-A or A-4. A-4 normally gives up fewer points, and it gives up fewer points here too. This is a bit surprising, since dealer is less likely to toss himself, say, 2-3 when two of the 2s are already accounted for. But another common way for A-A to give up a large crib is combination with mid-cards, such as 6-6-8 or 6-7-8. You have no mid-cards in your hand, so this possibility is actually higher than normal. You have one 4 in your hand, but that's not enough to seriously reduce the chances of the A-A combining with 4-x, which would cost you anywhere from six to thirteen points. DS figures that it all evens out in the end. As for A-4, the extra A left behind diminishes the possibility of giving up an A-A-4-x-x crib, but you have no ten-cards to reduce the possibility of A-4-x-x-x. Again, DS figures it's a wash.
The 2-2 in your hand does have a dramatic impact if you toss a 3 though. If you need to play off, A-2 or even A-3 looks very attractive. Unlike A-A and A-4, these tosses are not automatically worth two points going into the crib. And since their main potential lies in combining with other low cards, they do exhibit considerable negative delta here (-0.37 and -0.63 respectively).
Now let's see what DS tells us about 5-5-6-6-7-7. Remember, in these numbers I'm applying DS's delta to my own static discard averages, so the average crib and expected averages will be slightly different than those reported by the program:
As pone, tossing 6-7 both scores the most points for you and gives up the fewest points in the crib. 5-7 is not a serious contender. Note how the discards that include a 6, which is blocked off by two cards of each adjoining rank, benefit the most from the original card distribution.
As dealer, the obvious 5-6 comes out best. 5-5 does get about ¼ point of favorable delta compared to 5-6, 5-7 and 6-7, since it's less likely to be affected by the blocked 6s. But it's still an inferior toss, giving away 1½ points of scoring potential, and keeping a hand that pegs poorly against anything but mid-cards. I'd only consider it if nothing short of cutting a 5 or 8 to 6-6-7-7 (with 5-5 sitting in the crib) would make a difference. A "must-skunk" game, or a game where I'm way behind might qualify for this toss.
In each of the above cases, the author's recommendation to make an unconventional toss based on hand distribution proves inferior to the toss you'd ordinarily select based on static discard averages. That's not a coincidence. There has been a tendency in the literature to overestimate delta, particularly for bunched up hands like A-A-2-2-3-4 and 5-5-6-6-7-7. What Hessel's program shows us is that the effects of delta are more modest and nuanced than we'd thought.
Delta and you
Is it practical to try to factor delta into your over-the-board discard decisions? Delta is a complex phenomenon that can vary tremendously depending on the specific six-card hand you're holding. Nevertheless, at the risk of grossly oversimplifying matters, I'll offer a few general guidelines. Remember, as pone you want negative delta, while as dealer you want positive delta.
The most important point is that you're unlikely to have any significant negative delta without at least two blocked helpers. Blocking a single helper rarely moves the averages much. And if you have no blocked helpers, the delta is usually positive. Have a look at 2-3-6-8-8-8:
If you toss 2-6 or 3-6 as pone, there's very little delta, since you're only blocking one helper (the other half of the 2-3 combo). If you toss 2-3, you're blocking no helpers. All sixteen ten-cards, and all the other low cards, are outstanding. As a result, this discard gives up .17 points more than usual here.
If you're wondering why 2-6 drifts up a tad while 3-6 drifts down, it's because of the subtle effect of the three 8s in your hand. They don't diminish the chances of your opponent being dealt a 7 (which helps 2-6), but they do make it unlikely she'll have an 8 to go with it. This slightly increases the chance that she'll discard a stray 7 to her crib. The 3-6 combo is less affected by this, since it doesn't directly combine with a 7 for a score.
Now look at A-3-7-10-Q-K:
Note how all the candidate tosses return more points than usual. Generally, discards from weak, uncorrelated hands like this will exhibit positive delta, while tosses from highly correlated hands (like 5-5-6-6-7-7) will have negative delta. Keep this in mind if — in the course of using DS — you begin to wonder why you are seeing so much negative delta when, in the end, it must all add up to zero. You're probably just looking at a disproportionate number of correlated hands, which are, after all, the kind you're most likely to want to analyze.
If you do have two or more helpers blocked, this alone can produce point of negative delta. Blocking either of the following is typically good for ¼ point or more:
A discarded 2 through 8 is less valuable if you've reduced its possibility of forming a run by retaining two adjacent cards. This is well demonstrated by 2-3-3-4-5-9, a hand that presents a choice if you catch it as dealer. Do you keep the powerful 2-3-3-4 double run, throwing the normally remunerative 5-9 to your crib? Or do you keep 3-3-4-5, which fetches up to twenty points after the cut and pegs well if pone is holding ten-cards:
I used to like keeping 3-3-4-5 from this hand, until I saw how little the 2-9 was worth in the crib:
Relative to tossing 5-9, the delta on 2-9 is a tidy -.39. It's blocked not only by the 3-3, but also by the 4, which cuts off a possible 15 combination. All told, I don't think the pegging potential of 3-3-4-5 can quite offset the extra ¾ points of scoring potential you'll get keeping 2-3-3-4.
Now consider 2 3 4 7 9 J as pone:
The 2-4 diminishes the value of 3-J in opponent's crib by ¼ point relative to tossing 7-J (or 7-9). This makes the flush the best keep in most circumstances. Note that without the delta, it would be a tossup between the flush and 2-3-4-9.
As we saw with A-A-2-2-3-4, cutting off runs is not as significant when you discard aces. It also tends not to mean much when you throw cards higher than an 8. For example, suppose you're dealt 2-4-10-J-J-K as pone. Since the two Js limit the possibility of a high card run, it looks like tossing 10-K should be even safer than usual. But DS says otherwise:
Here is an example of blocking two 15 helpers as pone. You start with A-A-4-4-10-K:
Do you see how A-K comes out almost as safe as 10-K, which would ordinarily give up 0.4 less? If you're playing off, just keep A-4-4-10, even if you don't need the extra scoring it gives you. By retaining the 10 as an "out" card, you're less likely to get trapped in the pegging, an advantage that's worth the extra point in the crib. If it wasn't for the delta, you'd probably be better off tossing 10-K, hoping to bust your opponent's crib.
Here are the numbers for A-A-2-2-4-9. Note how the retained 2-2 doesn't affect an A-A toss, but it does reduce the value of 4-9, since you're blocking two 15 helpers:
5-5-6-7-J-Q is a slightly different case:
As pone, the 6-7 toss is beguiling since you have two cut-off neighboring cards. But it gets only point better delta than the generally safer J-Q, since in the latter case you've also blocked two helping cards.
As dealer, the 5-5 reduces the 6-7 toss by .42, but the J-Q by only .09. Pone is not too likely to toss you a 5 to begin with, so the blocked 5s don't have as great an impact on ten-card tosses when you're throwing to your own crib.
If you have 2-5-6-7-Q-Q as dealer, both 2-5-6-7 and 2-5-Q-Q look like legitimate plays. If you toss Q-Q, you're blocking a single 15 helper. But if you toss 6-7, you're blocking one 15 helper and one run helper. This is enough to degrade 6-7 by ¼ point, though it turns out Q-Q would be the more productive toss even without considering delta:
One phenomenon I haven't mentioned is breaking up a pair. It turns out this does little to the value of the toss. Suppose you have 9-9-J-Q-K-K as pone:
Note how little delta there is on the 9-K toss, even though you're holding a second 9 and a second K.
If you have two or more blocking cards that cut off both runs and 15s simultaneously, you'll often get quite large swings. This means that tosses involving blocked 7s and 8s are particularly volatile. Take a look at 6-7-7-8-8-K:
Normally 6-K is about 0.1 point less productive in the crib than 8-K. But in this case, it's 8-K that's less productive, due to the effect of the remaining 7-7 (and to a lesser extent, the second 8). So you'll get the highest expected average by tossing 8-K as pone, and 6-K as dealer.
In practical play, you'll rarely see delta larger than this. When bigger fluctuations occur, they're generally for obviously wrong choices, like throwing your opponent 7-7 from 6-6-7-7-8-8 (delta -0.79), or obviously correct ones, like throwing your opponent 4-6 from 4-5-5-5-6-K (delta -1.02). Most serious discard decisions involve comparatively negligible swings, so it's unusual for delta to be a decisive factor in determining the best play. Even among the hands we've analyzed thus far, there are only three or four cases in which delta would legitimately cause us to throw the hand differently.
A case study
In my experience, when the question of delta materializes in a real game, it's usually in the endgame as pone when I'm facing a difficult choice between keeping points and making a defensive discard. If it's a borderline decision, delta might well tip the balance. Consider the following hand, which I drew at the 2002 ACC Tournament of Champions. I'm pone at 101-97*:
Obviously there are two ways to win this game:
Thus the choice is between keeping 2-6-7-7, with potential for sixteen points after the cut, and keeping 2-2-6-7, which can't improve to more than twelve points, but keeps a safer pegging hand and makes a safer toss (A-7 as opposed to A-2). Is it worth taking a modest risk to hold 2-6-7-7, which gives me a chance of cutting myself to within pegging range of home?
Tackling this kind of problem over the board requires doing a cost/benefit analysis. I'll start by looking at the offensive option. 2-6-7-7 gets sixteen points on an 8 cut. That leaves me four holes short. It's tough to peg that much as pone, but with dealer at only 97* it's very likely — though by no means certain — that I'll also get to peg as dealer next hand. Moreover, although my opponent will probably want to play defensively here, his ability to do so is limited by his own marginal position (he's only +1, and could easily fall out of position with lackluster cards). All told, I'll guess that an 8 cut makes me a slight favorite to peg out before my opponent can win. Since cutting an 8 is a 4 in 46 shot (9%), I'll estimate that my front-end (offensive) winning chances are 5% better holding 2-6-7-7 instead of 2-2-6-7.
(Note that I'm not going to worry about cuts that get me twelve points. A 2, 6 or 7 will do this whether I keep 2-6-7-7 or 2-2-6-7, so the odds there are basically the same. Besides, twelve points only gets me to 113, which is too far away to have a significant chance of pegging my way home. All that really matters here is that an 8 cut gives me forward chances with 2-6-6-7, and not with 2-2-6-7.)
Now, what does it cost me to keep 2-6-7-7? As already noted, 2-2-6-7 is a safer pegging hand. It gives me a low-risk, covered 2 lead, which forms a magic eleven with the remaining 2-7. By comparison, holding 2-6-7-7 I'm pretty much forced to lead the 7, since leading the lone 2 traps me on a 5 through 8 reply. I estimate that on average, 2-2-6-7 will give up ½ point less in the pegging than 2-6-7-7. As for the toss, A-7 ordinarily gives up 4.9 in the crib, while A-2 gives up 5.1. Adding this to the pegging risk, it appears that keeping 2-6-7-7 will cost me about ¾ points on defense. How significant is that compared to the 5% extra winning chances I'm getting?
A good rule of thumb when dealer is close to the Fourth Street positional hole (96) is that his chances of going out in three counts increase by 7% for every extra point he scores. This means my back-end winning chances go down 7% for each additional point I give up on defense. In this case, giving up ¾ points more would translate to a 5% (¾ times 7%) decrease in back-end chances. Since this is equal to the front-end chances I gain keeping 2-6-7-7, it appears that the choice is a wash.
But now let's add delta to the mix. I can't expect a significant swing tossing A-2. There's only one blocked helper — the second 2 — and as we've seen, neither splitting a pair, nor blocking an A against runs is particularly effective. On the other hand, if I toss A-7, I'll be retaining two bona fide helpers: the 6 (adjacent to the 7) and the second 7 (valuable here mainly because it blocks a 15 combination). Thus I should expect about a point swing in favor of A-7. And sure enough, running the numbers through DS shows that A-7 gets a delta of -0.16, relative to A-2.
This increases the defensive cost of 2-6-7-7 to about .9 points, which translates to a 6% cost to my back-end winning chances. It's a close call, and a lot of guesswork has gone into it, but I'm not going to relinquish 6% winning chances on defense to gain just 5% on offense. I'll keep 2-2-6-7, which should win a trifle more often in the long run.
The big picture
Learning how to estimate delta over the board is unlikely to have a dramatic impact on your game. It's much more important to learn static discard averages, both because knowing how to estimate delta is worthless if you don't know the underlying static values, and because static values constantly come into play in making good discard decisions, whereas delta rarely does. To put it in perspective, consider that delta infrequently exceeds ½ point in practical play, whereas static discard averages range from 2.8 points (throwing yourself 10-K) to 9.4 points (throwing your opponent 5-5). Unless you're already playing at expert level, your time is probably not best spent worrying too much about delta.
If you do wish to polish your discarding skills by developing a heuristic feel for delta, the best way to do so is to start using DS.EXE to analyze discard decisions. Pay particular attention to six-card hands where you suspect the dynamic averages might vary significantly from the static ones. Enter surprising or interesting findings in your personal cribbage database.
Even if you aren't an expert, I still recommend downloading and using DS.EXE. At the very least, it's a quick and dirty calculator of average hands and expected averages. Unlike Cribbage Hand Evaluator, it doesn't have a graphical interface and doesn't let you choose among several sources of discard averages. But it is faster to use (assuming you can type decently) and displays both pone and dealer results on the same screen. Hessel believes his program offers "the most accurate discard calculation to date that excludes pegging potential". It's hard to quarrel with this assessment, even if his numbers don't reflect the possible biases of human opponents.
The DS.ZIP file includes a couple other small programs, one of them a routine used to generate Hessel's most recent static discard averages. It's all free, and Hessel even throws in the original C source code. I'm in heaven!
Complete Cribbage Discards
|Michael Wortley Nolan has undertaken a project
very similar to Craig Hessel's. His Complete Cribbage Discards is a
catalog of the 18,395 possible six-card hands (with suit disregarded). For
each hand, Nolan lists the "correct" discard (that is, the one producing the
highest expected average) for both pone and dealer, this determination
having been made by a computer program that uses an empirical calculation
algorithm closely related to Hessel's. Like DS, Nolan's program computes
discard averages dynamically, so that the distribution of the original six
cards is reflected in the result. Instead of actually releasing the program,
Nolan has gathered his findings into a 182-page book presented as the
culmination of a programming effort that stretches back to 1978.
Complete Cribbage Discards includes explanatory notes, tables of discard frequencies, a bibliography of cribbage books and 38 pages of original source code. The bulk of it, however, is devoted to a straightforward listing of six-card hands, ordered from A-A-A-A-2-2 to Q-Q-K-K-K-K. Look up any hand, and you'll find a recommended discard to either your or your opponent's crib:
Unfortunately, Nolan only gives you his algorithm's top choice for each hand. Second- and third-place finishers are ignored. Furthermore, no expected averages or other statistical measures are provided. Regrettably, these shortcomings make Complete Cribbage Discards of little use as a serious reference source.
Suppose, for instance, that you want to know how to handle 2-5-7-8-J-K as pone. The book laconically tells you to toss J-K. In fact tossing 2-K, 7-K and J-K all return about the same expected average (J-K gets the most offense, while 7-K gives up the fewest points — 2-K is in between), and any one of them could be the right toss depending on your board position. But since Nolan's algorithm gives J-K a microscopic edge over the alternatives, it's the only one you hear about.
Or consider 4-5-6-J-Q-Q as dealer. The book says to keep 5-J-Q-Q. But 4-5-6-J looks like a much better pegging hand, so should you keep it instead? Without real numbers, it's hard to say. As it turns out, though, there's less than point's difference in hand/crib potential between 5-J-Q-Q and 4-5-6-J — not enough to justify eschewing the superior pegging value of 4-5-6-J.
The good news is that Nolan's results appear to be highly accurate for what they are. I checked a sampling of 378 hands against DS, comparing the results for both dealer and pone. There were only eleven cases of disagreement between the two (a correlation rate of almost 99%), and the average amount of the discrepancies was only 0.05 points. That's not surprising, given the similarities between Nolan's and Hessel's approaches to dynamically calculating discard averages. I also regression-tested Nolan's discard frequency data — generated as an intermediate step in his calculations — and found them to be of high quality, closely matching the findings of Hessel, Schempp, Rasmussen and myself.
Ultimately, the question is: given that you can download and run DS.EXE for free, why should you spend money on a book that provides limited information and doesn't handle flushes? Complete Cribbage Discards may be of use to programmers and analysts who wish to examine Nolan's discard frequency statistics, obtain a second opinion on certain hands, or study the algorithm used to generate the results. It has some obvious value to players without IBM-compatible computers who cannot run programs like DS or Cribbage Hand Evaluator, and it will interest people who simply like to collect cribbage books. But it's probably fair to say that if you're not sure why you need this book, then you don't need this book.
Despite that, I would like to praise both Nolan and Hessel for illuminating a subject that most of us have just been guessing about. Thanks to their hard work, we cribbage players are a lot less ignorant than we used to be.
- April 2002 (links updated 2016)
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