Thursday, November 1, 2012

Cribbage Strategy and Board Position

Today we’re introducing a topic that, more than any other, separates casual players from expert players: board position. Expert players take their own score (board position) and that of their opponent into account when deciding how to play. Based on board position an expert player will play offensively (scoring more points themselves at the risk of letting their opponent score), or defensively (limiting their opponents points, even if that means scoring fewer points themselves).

Why play defense?

Why would someone do this? Let’s take an extreme example. The score is tied at 109, your opponent is dealing, you keep 3-3-10-10 in your hand and the cut is a 2:

You lead your 3, and your opponent plays a 3 for two points. What’s your play? Playing your remaining 3 would give you six points, but let’s think about that play a little.

  • You need 12 points to win
  • You’ve got 12 points in your hand
  • Dealer needs 12 points to win
  • You count first
  • So you win as long as dealer pegs fewer than 12 points

The only way dealer is going to win is if he has the fourth 3 and pegs 12 points from four-of-a-kind.  Play the 10 and guarantee yourself the win. Dealer probably doesn’t have the fourth 3, but by not playing your second 3 you’ve gone from “probably going to win” to “definitely going to win”.

Why play offense?

Imagine you have the same hand, but the score is tied at 106.

The equation changes significantly:

  • You need 15 points to win
  • You’ve only got 12 points in your hand
  • Dealer needs 15 points to win
  • Dealer averages 16 points per deal (between pegging, hand, and crib -- see the first post in this series for more details)
  • Dealer will very likely win unless you peg at least 3 points

Executive summary: play your second 3 for six points and you’ll win unless dealer holds that fourth 3 (in which case dealer will score 2 for his pair, 12 for his pairs royal, and score another 1 for a go).

The decisions to play offense at hole 109 and defense at hole 106 both come from straightforward logic, and in both cases the strategy is dictated by board position. A 3 point difference in position completely changes the way you should play your hand.

Winning probabilities

The logic was easy in those last examples because you know you have a 100% chance of winning if you can make it to 121 points. There are opportunities to use this kind of logic earlier in the game. To understand these opportunities better let’s look at another chart built from our millions of hands of Cribbage Pro multiplayer game data. The key to improving your game in this chart: it’s the white wave running along the diagonal line.

The chart shows the probability the dealer will win, based on dealer and pone’s scores immediately before the deal. The line down the middle shows the cases where dealer and pone have the same score, and has ticks marked off every 5, 10, and 30 points. Bright orange means we’ve never seen dealer win from that position, bright blue means we’ve never seen dealer lose, and bright white means that dealer and pone each won 50% of the time. Light orange means pone is favored to win, and light blue means dealer is favored to win. The chart is grey when we don’t have any games in our sample that were at that combination of points.

Of course the chart gets more blue as you move to the right (since the dealer’s more likely to win if his score increases and pone’s doesn’t). And of course it gets more orange as you move towards the top of the chart. But notice that the color changes as you move diagonally up and to the right, because of that white wave. The wave is weak in the bottom of the left, but it repeats itself in 26 points cycles, getting stronger and stronger until it reaches the top right of the chart. The waves are what make board position so important.

Looking closely you can see that the chart is very orange when the score is tied at 115, and very blue when the score is tied at 105.

When the scores are tied at 115 dealer is at a big disadvantage. Pone counts first and only needs 6 points to go out, so pone will usually win if dealer doesn’t peg 6 points. When the scores are tied at 105 the situation is reversed. Pone has to score 16 points to go out, but we’ve seen that pone averages only 10 points between her hand and pegging. We’ve also seen that 16 points is average for a dealer, across hand, crib, and pegging, so starting from hole 105 dealer will usually go out after counting her crib, or will be close enough to peg out the next hand.

Dealer has a huge advantage when the scores are tied at 105, and a huge disadvantage at hole 115. Neither player has a point lead in either of the two positions, but the likely winner has changed. Why? Because the two positions are on different sides of that wave.

How do you ride the waves?

Throughout the game you’re given opportunities to take points at the risk of giving more points to your opponent. Good players will use these opportunities to ride the waves, sacrificing points when necessary to keep their opponents from moving ahead, and giving points to their opponents when necessary to score a few extra points themselves.

Think about your position on the wave whenever you have the opportunity to score points while potentially giving your opponent points. Would you be in a better position if you moved up and to the right in the chart? If so, take the points. If not, turn them down and keep your opponent from scoring, too.

Let’s take a common example. You (as pone) are in hole 96, with dealer in hole 90. You’re holding 5-J-Q-K, for 9 points, so you’ll start from hole 105 as dealer next turn. On average, dealer scores 16 points per deal (between pegging, hand, and crib), so with a typical hand he’ll be in hole 106 at the start of the next turn. 105/106 is a nice blue spot on our chart, so you should peg defensively to keep your position.

You start the play with your Queen and dealer plays a 5 for two points. Should you pair the 5? You’ll be in a good position to win next turn, so you don’t really need the points. Pairing the 5 gives dealer the opportunity to make three-of-a-kind for 6 points, which would knock you out of your sweet spot on the wave. The right play is to turn down the points, and play your Jack or King instead.

If you were starting 10 points back (you at 86, dealer at 80), pairing the 5 would probably be the right play. You’re a little out of position, and you’d really like to move forward at the risk of moving your opponent forward, too.

We’ll go into detail next week, with more precise instructions on exactly when to play offense and when to play defense.


While you’re waiting for next week’s article we suggest reading DeLynn Colvert’s excellent book Play Winning Cribbage. His groundbreaking Theory of 26 gives advice on board position starting from the very first hand of the game, and is well worth studying. Some of his thoughts about the Theory of 26 are also available at the American Cribbage Congress website.

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).

Thursday, October 25, 2012

Cribbage Hand Statistics - You Ask, We Answer

We’ve received a couple of statistics questions recently from Cribbage Pro players. We love getting these types of questions, and to show our appreciation we’re devoting this week’s entry to their answers.

First dealer’s advantage

Cribbage Pro player Zerrick wondered what the odds were that the player with first deal will win the game.  Across our entire sample we saw that the first dealer won the game 55.5% of the time.  Broken down by the players’ skill levels (using the skill levels from the first post in this series), we get:

Probability first dealer wins game
      First dealer’s skill level

First pone’s skill level


It’s interesting to see that when the two players’ skill levels are equal the first dealer wins 55% or 56% of the time.  If you work through the math you see that no matter what skill levels are involved, a player is around 11% more likely to win if she deals first than if she deals second. [Editor's Note: This should be incentive as well for those "D" level players who are not seeing an advantage when facing the higher skilled "A" and "B" players and winning only 45% of the time even as first dealer. Start taking advantage of this and win more cribbage games by improving your cribbage strategy.]

Hand statistics

Last weekend a Cribbage Pro player named Kip beat me two out of three games, then stumped me with a cribbage statistics question: “How often do players hold double runs in their hands?” Good games, Kip. Here’s the answer to your question and a few more, with lots of colorful charts:

Type of hand
% Dealer
% Pone
% Dealer's Crib
Single pair
Two pair
Pairs royale
Full house
Double pairs royale
Single run
Double run
Triple run
Double double run

As usual, all the numbers in the blog post come from our sample of several million cribbage hands. There’s some overlap in these numbers – a hand could hold a flush and a run, or a pair and a run. Interesting to see that double double runs (for example, 4-4-5-5-6) are more common than triple runs (like 4-4-4-5-6).

What about “15”s, you say? Here’s a little chart for you:

# fifteens in hand

The bars for 5, 7, and 8 fifteens in a hand are too small to be visible, but it’s still possible to count 5, 7, or 8 fifteens in one hand. About one hand in 800 has 5 fifteens, one in 600 has 7 fifteens, and one in 15,000 has 8 fifteens.

Note that only 20% of dealer or pone hands have 0 fifteens, which means that most hands have at least one fifteen in them. Not only that, most hands actually have at least two 15s in them.

Points per hand

Put those all together, and how many points do you get? You get this!

# points in hand
Points per hand look fairly similar for dealer and pone.  The most common number of points in a hand is 8. Dealer’s hand holds 8 points 16.9% of the time, and pone’s holds 8 points 18.2% of the time.

Hands with an even number of points are more common than hands with an odd number of points. Looking at the first table explains why – most hands have pairs and 15s (which score even number of points), while only 23% of hands have single or triple runs (which can score an odd number of points).

It’s impossible to get a 19, 25, 26, or 27 point hand, so those entries don’t show up in the chart. 22, 23, 28, and 29 are possible, but show up so rarely that their bars aren’t visible. About one in 15,000 hands on Cribbage Pro scores 28 points, and about one in 220,000 scores 29 points.

The typical crib doesn’t have as many points as the typical hand, so this next chart looks a lot different. You can score more than 16 points in a crib, but we left those bars off because those scores happen so rarely that you wouldn’t be able to see them. Only about 1 in 1,000 cribs scores more than 16 points.

# points in crib

Pegging probabilities

The last chart in today’s post shows pegging probabilities:

# points pegged
Dealer always pegs at least one point, so dealer gets 0% in the “0 points pegged” column (we’ve excluded hands where pone pegs out before counting). The most points we’ve seen pegged in one hand is 29, and in the millions of hands analyzed for this blog, this has only happened once. In that hand dealer and pone each held two aces and two 2s. Ouch.

That’s all for now

Thanks for the questions, Kip and Zerrick. We know more of you have questions, and we’d love to answer them. Just leave us a comment below.

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).