Thursday, November 15, 2012

Offense vs Defense Cribbage Strategy

Last week we brought up the idea of choosing your cribbage strategy based on board position (that is, based on your score and your opponent’s score). This week we’ll help you decide when you should play offense and when you should play defense.

What does playing offense mean? It means scoring points at the risk of giving your opponent points: throwing a pair in your opponent’s crib in order to keep a great hand for yourself, or pairing your opponent’s lead at the risk of letting them score three-of-a-kind.

What does it mean to play defense? It’s the exact opposite: keeping a bad hand so that you can throw terrible cards in your opponent’s crib, or ignoring the opportunity to pair or score a run during pegging so that you can keep your opponent’s score down.

How to choose your strategy 

We’ll get to the gory details below, but here’s an executive summary to get you started. When your score is close to your opponent’s (and it frequently is), you can use the table below to help decide your strategy. Follow these steps:

  1. Count the points in your hand.
  2. Make a guess at the points in your crib (if you’re the dealer).
  3. Add in a point or two for pegging.
  4. Look up this end-of-hand score in the table below to decide how to play.

If your score is close to or tied with your opponent’s, and your end-of-hand score is between...
Dealer should play...
Pone should play...
61 to 75
Defense
Offense
76 to 86
Offense
Defense
87 to 101
Defense
Offense
102 to 112
Offense
Defense
113 to 120
Defense
Offense

For instance, if you’re playing as dealer and your and your opponent’s end-of-hand scores are between 102 and 113 then you should play offense. If you’re playing as pone and your and your opponent’s end-of-hand scores are between 76 and 87 then you should play defense.

If you’re more of a visual person you might prefer to look up your end-of-hand score on the board below to choose your strategy. Dealer should play offense (and pone should play defense) when her end-of-hand position puts her in a green area, and defense when her end-of-hand position puts her in a red area. Pone should do the opposite -- play offense in the red areas, and defense in the green areas.


We’re going to dig into the gory details now, but if you want some more advice on what exactly “offense” and “defense” mean you can just skip to the end of the article.

Remember, look up your estimated end-of-hand score on the board (and in the table), not your current score.

Dealer’s win probabilities

Where did these strategies come from? Let’s take another look at our chart from last week, which shows the probability the dealer will win based on dealer’s and pone’s scores immediately before the deal:


Remember that the point of offensive play is to score points even at the risk of letting your opponent score. So let’s see what would happen to dealer’s chances of winning if she had played offensively, and each player had pegged an additional two points above and beyond what they would have scored with typical play:


These charts are getting more and more complicated, and this one deserves a little explanation. This chart shows the increase or decrease in the probability that the current dealer would win if both dealer and pone had scored an additional 2 points in a previous deal. Or to put it another way, it tells us whether or not the dealer should have played offensively on her previous hands. The chart is marked green in areas where the dealer is more likely to win when both players scores are increased by two, and red where the dealer is more likely to lose.

Looking at our first (orange/blue) chart, we can see that the dealer has a 59% chance of winning the game when the score is tied at 95 (one line past the one labeled “90”). If the score were tied at 97 (that is, if each player had scored 2 more points), the dealer would instead have a 65% chance of going on to win. (I know it’s hard to tell the difference between a 59% and a 65% on that first chart -- you’ll have to trust me that those are the correct numbers). If dealer had played her previous hand more aggressively she’d be in a better position to win: an extra two points for dealer and for her opponent would actually give dealer an extra 6% chance of winning (65% - 59%). Because of this the chart shows a nice bright green at position 97/97.

The story would be different if the scores were tied at 80 (two lines before the one labeled “90”). With both players tied at hole 80 the dealer has a 62% chance of winning the game. Tied at 82, a red area, the dealer has a 59% chance of winning the game. This time dealer would have been better off if she’d play more defensively in previous hands. She’d have increased her likelihood of winning by 3% if she’d forced her opponent to peg two fewer points, even it had cost her two points.


The red and green blobs in this chart correspond to the white wave running through the first chart. The blobs get fainter and fainter as you move to the bottom left of the chart, but they’re still there -- alternating red and green. An interesting fact is that the areas repeat every 26 points: there’s a green area centered around 95 points, and another 26 points earlier centered around 69 points; and the red area centered around 80 points has an echo 26 points earlier centered around 54 points. This 26 point cycle shows up because players average 26 points every two deals (16 points as dealer, and 10 points as pone), and is the basis for DeLynn Colvert’s Theory of 26.

Use the red and green blobs to your advantage

This red/green chart gives us another method for deciding when to play offense and when to play defense: estimate your and your opponent’s next starting position, find the position on the chart, and base your decision on the chart’s color.

For example, let’s say you’re at hole 85 as pone, with dealer at hole 80. You keep 6-6-7-8 in your hand, and dealer cuts a 10. You've got 10 points in your hand, so you’ll start in hole 95 next turn if you don’t peg. Dealer averages 16 points (between hand, crib, and pegging), so she’ll probably start around hole 96 next turn. Looking at our chart you see that 95/96 points for dealer/pone puts you squarely inside a green section of the chart. You’d like to start further ahead, so if you see the opportunity to peg a few extra points you should take it, even if it means giving your opponent a few points, too.

If the positions were reversed (you as dealer from hole 80, pone starting at 85) you should reverse your strategy and play defensively.

Play for position or play for points?

Note that the strength of the colors increases as you get closer to the end of the game, which means the importance of offensive and defensive play increases as the game goes on. Our chart shows a very faint pink area around 25 or 30 points. That early in the game the effects of correct offensive or defensive play are so small that it’s probably better not to concentrate on offense or defense, and instead to focus on increasing your lead over your opponent (or catching up with her).

Exercise moderate offense or defense between 60 and 86 points
Position really starts to be important after you’re halfway through the game, as shown by the darker green and red blobs. When you and your opponent are between 60 and 86 points you should start practicing moderate offense or defense: try to manipulate your opponent’s position without sacrificing too many points.

Moderate offense means:

  • Pairing your opponent when she leads a 9 or higher
  • Getting involved in a run if you think you’ll end up with as many points as your opponent

Moderate defense means:

  • Not pairing your opponent when she leads a 3 or a 4
  • No getting involved in a run, even if you think you’ll end up with more points than your opponent

Exercise serious offense or defense between 87 and 109 points
Between 87 and 119 points things you've started getting into the endgame, and the importance of positional play has become very important. Proper positional play here can make a big difference in your win rate. Start playing serious offense or defense, depending on where your end-of-hand score puts you on the board.

Serious offense means:

  • Trying to entice your opponent into getting involved in a run. For example, if you’re holding A/2/3/J and opponent leads a 4, play your 3 and hope that she plays a 5 or a 2 (run for three) so that you can make a run of four
  • Lead a ten card from A/J/Q/K, in the hopes that you’ll end up with a 31 for 2 points (even if means you’ll get paired)

Serious defense means:

  • Lead from a card lower than 5, even if you’ve got a pair. For instance, normally you should lead a 7 from 4/6/7/7, hoping for a three-of-a-kind. In defensive mode you should lead the 4, even though you’ll average 0.6 points more when you lead the 7
  • Lead the 6 from 6-7-8-9 (you’ll score more with the 8, but your opponent scores fewer points if you lead the 6)

Exercise extreme offense or defense starting at 110 points
At the very end of the game everything changes. If your opponent is going to be starting as pone next turn within 10 points of winning then you have to do everything in your power to keep them from scoring. This means being willing to peg 0 points and score 0 points in your hand in order to keep them as far away as possible from the final hole. Pone will average 10 points between her hand and crib, so putting her within 10 points of pegging out is a very dangerous situation.

On the other hand, if you’re going to start in this position as pone you shouldn't play particularly offensively. You’ll count first, and you’ll likely go out before your opponent has a chance to count. Giving your opponent more points would just increase the chance that she’ll be close enough to peg out next turn.

Extreme defense means:

  • Don’t play cards that would let your opponent start a run. For instance, if your opponent plays an 8 you shouldn't play the 7 for 15-2, since she could play a 9 or 6 for a run. Even if you have a 9 and a 6 and could make a run of four you still don’t want to do this, since you could give your opponent points
  • Lead low pairs. For example, lead the A from A,A,2,3. You won’t score 15 for 2 if your opponent plays a 10 card, but you cut down on her ability to pair your lead

That’s plenty for this week

That’s the end of a whirlwind tour of a dense topic. Do you have a specific hand you’re interested in analyzing? Would you like some advice on a tricky board position? Let us know and we’ll tell you how and when to play it for offense and defense. 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).

15 comments:

  1. This is very interesting and helpful. I love that it is based on extensive data of real hands.

    Just a question regarding the first cribbage board depicted in this post, which is mean to be a visual depiction of the info in the table immediately above it: Shouldn't the final green zone be 3 holes long than it is? As I read the final green zone, it goes from 102-109 but the table above has the final green zone going from 102-113.

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  2. Hi Craig,

    You're absolutely right. The table is correct, and the final green zone should in fact be three holes longer.

    Thanks for pointing it out.

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  3. Thanks for the reply Aaron.

    I've got another question about using this method for determining whether you should play offensively or defensively.

    Near the start of the post you specify that this method is meant for contexts in which "your score is close to your opponent’s." Your examples (and indeed, logic)lead me to think that the method is meant to apply in cases where my end-of-hand score projection suggests that the end-of-hand scores will be quite close, and is NOT meant to apply in cases where the current scores on the board (before start of play) are close.

    I'm curious about cases where your end-of-hand score projections are NOT close. I suppose someone might recommend a rule like the following: "If you predict that your end-of-hand score will be significantly behind your opponent's then play offensively; if you predict your end-of-hand score will be significantly ahead of your opponent's then play defensively."

    Yet my hunch is that that rule is too simple; given the seesaw nature of cribbage scoring, I would imagine there are times when you could be,say, 4 points or so behind at the end of the hand but still be more likely to win (for instance, that might true near the very end -- say, if you are dealer and you have 105 and your opponent has 109; the odds slightly favor you).

    Perhaps what you should do is make a prediction for the scores after this hand AND the next?

    Let's say for instance that the scores on the board at the start of this hand are 78-78 and I am the pone. After the cut, I find that I have an average hand; it will score an 8. So I estimate that the scores at the end of this hand will be around 88-94. Those are not tied, so I can't just use the table at the start of this blog post to determine whether to play offensively or defensively. So I estimate one hand further ahead. We'll likely start at 88-94 next round when I am dealer and opponent is pone; odds are that I will get around 16 and he'll get around 10, making the likely end-of-hand score for the hand after next 104-104. Using your table, I can thus predict that in the next hand I will be playing offensively as dealer-- so should I start playing offensively now as pone? Or is that reasoning flawed?

    I hope that is not too confusing. In general, I guess my big question is what to do when I predict that the end-of-hand scores will NOT be a point or two within each other? How is it best to determine whether to play offensively or defensively?

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  4. I thought some more last night about my question in my previous comment. (Sorry for the excessive posting lately!) I realized that (as you indeed say) the red/green blob graph can be used to answer questions like the one I posed in my previous comment.

    First, though, a caveat to others about using this red/green blob graph: the graph actually depicts how the current PONE should play, not the current dealer (the reason being that green in essence says that had a played more offensively in the previous hand as PONE then he'd be in a better position now; red in essence says that a player played more defensively in the previous hand as PONE then he'd be in a better position now). That is why if you look at the table at the start of Aaron's post you will see that red/green text in the pone column matches the red/green blobs in his graph.

    So if I have reasoned correctly, then you should use the graph as follows:

    ** make a projection of the likely end-of-hand scores

    ** estimate where this point is on the graph -- e.g. if you are now pone, then at the end of turn you will be the next dealer, so find your score on the x-axis and your opponent's score on the y-axis. (Aaron, is this right?)

    ** then...
    If you are now pone: red means play defense, green means play offense
    If you are now dealer: red means play offense, green means play defense

    Assuming that is right, let's revisit my question in my previous comment. This was a case in which I am now pone and I estimate that at the end of the turn the scores are likely to be opponent 94, me 88. So I should find this point on the red/green graph. An easy way to do this is to find the point (88,88) on the diagonal and then move straight up vertically to increase the my opponent's score to 94 (go up because my opponent will be pone at the start of the next turn). Doing this puts my firmly still in green. So as pone now I should play offense.

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  5. [I've had to split my post in two to get it past the character-length limit -- very sorry!! Here is the rest of it.]

    Since it'd be hard to play always with Aaron's red/green blob graph constantly in front of you -- especially in face to face games, let's ask: are there any rules of thumb you can follow to approximate the guidance of the graph even when you do not project the end-of-turn scores to be roughly tied?

    Maybe. Here is one way to think about things. Look at the graph and imagine the pone projects he will be at 76 at the end of the turn. Moving up from 76, 76 represents a projection of being behind the opponent. The graph says to play defense in this case. (But the pone should play offense if he is projected to be ahead of the opponent at the end of the turn, which is represented by moving down from 76,76).

    Now imagine the pone is at 86. The graph says pone should play defense if the pone projects he will be ahead of the opponent at the end of the turn; but play offense if he projects he will be behind the opponent.

    (Similar reasoning would apply to a dealer at 76 and and a dealer at 86 -- as opposed to a pone -- but with "offense" and "defense" reversed.)

    Generalizing from this, perhaps we can offer the following advice:

    ** Use Aaron's table at the start of this blog post.

    ** Determine if you are in the middle of one of the table's ranges, at the back end of a range, or at the front end of a range. (E.g. the first table range is 61 to 75; so 68 is in the middle, 61-63 is at the back end, 73-75 is at the front end -- understanding the left side of a range as "back" and the right side as "front.")

    ** If you are in the middle of a range, then the table's advice is robust: do what it says even if you are behind or in front of your opponent (even by 10 points, as best I can tell from the graph).

    ** If you are in the back of a range, then the table's advice is good if you are tied or "in back of" your opponent (that is, your score is lower than your opponent's score).

    ** If you are in the front of a range, then the table's advice is good if you are tied or "in front of" your opponent.

    In short: the table is always good advice if you project the end of turn scores to be close; its advice is still good if you project yourself to be in the front of a table range AND in front of your opponent score-wise; its advice is still good if you project yourself to be in the back of a table range AND in back of your opponent score-wise.

    Still, that is a lot to keep in mind! Maybe one is safer only making a conscious effort to play offensively or defensively when one projects that scores will be close at the end of the current hand...

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  6. Hi Craig. Sorry for the late reply. I agree with your analysis, and I think it's a great way to simplify the red/green blob chart. I'm going to start using your approach.

    Thanks for the deep thoughts about cribbage. Please let us know if there are any topics you'd like us to cover.

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  7. Thanks for the positive feedback, Aaron. I'm surprised you didn't find a mistake in my reasoning; I was sure that somewhere I would reverse the dealer / pone or I would move up in the graph when I should have moved down or left/right!

    As for ideas for future posts: Hmmm... I suppose one thing I am curious about is how some of the Top 50 players on Cribbage Pro achieve such high win ratios. E.g. the number 1 player (BigDouche) has a 76% win rate, Jameson33 has a 72% win rate, etc. Those are higher than I would have thought humanly possible in a game like Cribbage with a significant luck element. I know strategic play mitigates luck, but I wouldn't have thought it could make THAT kind of difference!

    So I'd be curious to learn more about the Top 50 players with the very top win rates. For instance, are they playing primarily "best of 3,5,7" matches (which increases the better player's win rate)? What are their win rates against A players? Against D players? Is their advantage mainly in pegging? In the discard? Etc. (Basically, this would be doing for the top 0.0001% the kind of analysis you earlier did for the top 3%, i.e. the A players.)

    Of course, this could only be done with their explicit permission, and that might not be possible, for all I know. But if it is possible, I would find it interesting!

    Oh, and by the way, I gave a shout-out to this blog (and to the Cribbage Pro) app in a couple of Cribbage-related posts of mine on Boardgamegeek:

    Toward the bottom of this one: http://www.boardgamegeek.com/thread/917813/calling-all-mathematicians-a-game-probability-que/new

    Also here: http://www.boardgamegeek.com/thread/917884/how-match-play-helps-skilled-players-win

    Keep up the great work!





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    1. Hi Craig! That's a great question. I've noticed while playing Cribbage Pro that the players with the highest win rates are generally playing muggins games.

      In the http://blog.cribbagepro.net/2012/10/how-should-you-respond-to-pones-lead.html post I said that if you can average on extra point per game you can expect to see a 1.7% increase in your win rate. You'll see an additional 1.7% increase if you can make your opponent average 1 point fewer per game. Every point of muggins is one extra point for you and one fewer point for your opponent, for a big 3.4% total increase in your win rate. If you can average just three points of muggins per game you should see your win rate increase by about 10%.

      The analysis for this blog doesn't include those games, since the articles focus on strategy rather than hand counting.

      Thanks for spreading the word about the blog, and let us know if there are any other topics you'd like us to cover.

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  8. Ah, muggins. Interesting. Yes that might indeed explain it. Thanks!

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  9. Hi Aaron,

    I trust that the Expert level AI on Cribbage Pro plays using this data. I'm curious: What is its win rate? (In general, and against "A rank players")

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    1. Hi Craig, we actually don't track single player games to this same level of detail and so really can not answer your question right now outside of what is already shown in the "Top 50". However, the AI in single player does not use hard data like this in it's strategy, but instead it calculates probabilities dynamically for primarily a "discard table" like strategy. It is not this sophisticated, but more of a "brute force" approach.

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    2. [Too many typos in my first try at posting, so I deleted it and am re-typing it here.]

      Hi Josh, Thanks for the reply.

      Aaron in an earlier reply to me had asked whether there is any topic I could suggest for a future post. Here is one idea.

      You can find other sites that list the frequency (by rank) of cards appearing in dealer's hand and in pone's hand (e.g. here: http://www.cribbage.org/tips/schell1.asp). I assume your data matches the bar graphs there pretty well.

      One thing I'd be curious about is which PAIRS are most likely to appear in pone's hand and which are most likely in dealer's hand. A bar graph of that might be pretty flat -- one tends to keep pairs -- though the dealer is more likely to toss AA or KK in the crib I would think.

      Still, it would be interesting to know these frequencies. If you could know that certain pairs are statistically somewhat less likely to appear in your opponent's hand than other pairs, then when he plays a card from such a pair and you happen to have a matching card in your hand, you could pair him with a somewhat-lower-than-average worry about him retaliating with a triple...

      Just a thought!

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    3. Hi Craig, Aaron is currently traveling, but he tells me he is planning on doing some digging and answering your question in more detail when he returns in early April.

      Thanks,
      Josh

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    4. Great. Looking forward to it. And after further thought, I think that AA or 22 might be unlikely to be tossed into the crib by a dealer, since those can be very good pegging cards... Curious to see what Aaron learns. Thanks.

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