Thursday, August 30, 2012

What card should I lead? (Part 1)

We talked about some basic dealer pegging strategy in our last post. Let's switch it up a little and talk about pone (non-dealer) pegging strategy. The first pegging question for pone is "What card should I lead?" As usual, the answer depends on your objective at the moment (Score at any cost? Keep your opponent from scoring? Go for most net points pegged?). We’ll focus first on optimizing net points pegged.

For our analysis we'll measure net gain for pone as the average (mean) number of points pegged by pone minus the average number of points pegged by the dealer. So if pone pegs 5 and dealer pegs 3 the pone has a net gain of 2, and if pone pegs 10 and dealer pegs 12 the pone has a net gain of -2 (or a net loss of 2).

If there's enough interest we can examine other metrics in a future post (e.g., probability of pone pegging at least one point; probability of keeping dealer's pegging below some threshold).

6-7-8-9

Our data shows that pone keeps 6-7-8-9 in his hand around 0.7% of the time. That doesn't sound like much, but that actually makes it the third most common hand (after 5-J-Q-K and 5-10-J-Q), so it's worth spending some time analyzing this hand.

6-7-8-9 is the hand that got me interested in analyzing this data. It seems like whenever I play the 8 my opponent plays a 7 on top for "15 two"; and when I play my "9 for a run of three" she's got a "6 for a run of four." Is 8 really the right lead? And if it is, is "9 for run of three" really the right play when the dealer plays "7 for 15 two"? Or should I play "7 for a pair" instead?

John E. Chambers (in Cribbage, A New Concept) suggests that we lead the 6 from this hand. Let's stir up some controversy by respectfully disagreeing with Mr. Chambers!

Lead the 8

There's an ugly truth here that needs to be accepted: the dealer is probably going to outscore us. This doesn't mean that 6-7-8-9 is a bad hand, just that dealer has the advantage when pegging. The best we can hope to do is to keep the gap between dealer's pegging and ours as close as possible.  Cribbage Pro players have played this hand tens of thousands times, and on average they see these results:


Pone leadAvg pone points peggedAvg dealer points peggedAvg pone net points
61.843.52-1.69
71.933.68-1.76
82.073.53-1.46
91.873.63-1.75


Leading the 8, on average, gives us a net advantage of 0.23 over the 6 lead (which, to be fair to Mr. Chambers, is the second best lead). The 6 limits dealer’s pegging almost as well, but doesn’t peg as well for pone.

Why does pone score more with the 8 lead? The detailed explanation why is very, well, detailed. If you're interested leave us a comment, and we'll show the gruesome details, with probabilities and results for every possible response to your 8 or 6.

What next?

What should you do when the dealer inevitably plays a 7 on your 8 lead? The answer is (drum roll)... play the 9. Your opponent will have the "6 for a run of four" 42% of the time, but 58% of the time she won't and that's enough to make the 9 the right choice. Average net gain for pone is -2.11 playing the 9 on top of the dealer's 8, -2.80 playing the 6 on the dealer's 8, and -3.17 playing the 7.

Defensive situations: maybe the 6, maybe the 8

Sometimes it’s more important to keep your opponent from scoring than it is for you to score. If your opponent is four points away from pegging out, for instance, you want to maximize the odds that she scores 3 points or fewer. The chart below show probabilities of limiting dealer to any number of points pegged.  It’s interesting to see that the 8 lead is more likely to limit your opponent’s pegging to 5 points or fewer, but the 6 lead increases the chance that your opponent pegs 12 or fewer.



Hail Mary pegging: lead the 8

If you desperately need 10 points to go out, you’re going to need to peg an extra 2 points on top of the eight points already in your hand. The 8 is almost always the best offensive pegging lead. The only exceptions are if you need to peg at least one point (lead the 6), or if you need to peg eight or more points (lead the 7).  The 7 is also your best choice if you need to peg more than ten points -- we've seen pone peg 11 about one time in a thousand when she leads the 7, but we've never seen pone peg 11 with an 8 or 9 lead. The chart below shows the details:


Executive summary

Moral of the story: you should almost always lead the 8 from 6-7-8-9, and feel confident playing that 9 for a run of three. You're probably in for a painful pegging experience, but you're going to get out-pegged no matter what you do.

We'll talk more about pone pegging strategy in the next post. Do you have a hand you're especially interested in? Leave a comment and let us know.


Editor's Note: This is a guest post by Aaron Harsh continuing the series on cribbage strategy and tips. Aaron lives in Portland, Oregon with his wife Stacy and daughter Audrey. He spends his evenings analyzing cribbage strategy for Fuller Systems, and his days analyzing television viewership for Rentrak Corporation's Advanced Media & Information group. You can play him on Cribbage Pro Online as user "aaronhars", or in person at American Cribbage Congress grassroots club #28 (Oregon's Finest).

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