Thursday, October 18, 2012

Discarding


In our last post we said that scoring an extra point per game will add an extra 1.7% on top of your win rate. Today we’re going to talk about how to discard to score that extra point. Discarding is such a ridiculously exciting topic that we’re not even going to give an overview. Let’s just jump right into the data!

Dealer discard table

This table shows the average number of points you can expect in your crib, based on the two cards you put in your crib.


A2345678910JQK
A5.24.44.65.25.23.73.73.73.33.33.53.33.3
24.45.86.94.65.23.93.93.73.73.63.83.63.6
34.66.95.95.05.93.83.83.93.73.63.93.73.7
45.24.65.05.56.33.93.73.93.63.43.73.53.5
55.25.25.96.38.56.45.85.35.16.36.76.46.3
63.73.93.83.96.45.64.94.64.93.03.23.02.9
73.73.93.83.75.84.95.86.44.03.13.33.13.1
83.73.73.93.95.34.66.45.34.53.73.33.13.0
93.33.73.73.65.14.94.04.54.94.13.72.82.8
103.33.63.63.46.33.03.13.74.14.64.33.32.7
J3.53.83.93.76.73.23.33.33.74.35.14.53.8
Q3.33.63.73.56.43.03.13.12.83.34.54.53.4
K3.33.63.73.56.32.93.13.02.82.73.83.44.4


You can use this table when you’re having a tough time figuring out what to toss in your crib. For example, you’re holding 2-3-5-6-J-Q and it’s your deal. You probably want to give yourself either the 2-3 or the 5-6, but it’s not obvious which. You’ll end up with 4 points in your hand either way, so what should you put in your crib? Discard table to the rescue: if you toss the 2-3 in you’ll average 6.9 points in your crib; if you toss the 5-6 you’ll average 6.4 points. Keeping the 5-6 in your hand and tossing the 2-3 gives you a 0.5 point advantage in your crib. (5-6-J-Q is also a better pegging hand, but that’s a topic for a future article).

Michael Schell put together some good articles describing how to choose your discards using a discard table. I recommend using his technique, but with our discard tables. (We believe ours is more accurate, since it’s based on a very large sample of hands from actual human play, rather than a smaller sample or computer simulations).

Here’s a visual display showing how those discards stack up with each other. Pairs show up in blue, and cards totalling 15 show in red.

As expected, 5-5 is far and away the best discard. Also unsurprising: 10-K, 9-K, and 9-Q show up at the bottom of the rankings.

You’re probably not going to split up a pair or a 15

Hypothetical scenario: you deal yourself 3-3-6-7-J-J, and you’re not sure what you should put in your crib. Looking at the discard table you see that 3-3 will get you 5.9 points (on average), 6-7 will get you 4.9, and J-J will get you 5.1. It looks like 3-3 is the best discard, but the discard table is a little misleading here. You’ll get the “pair for 2” for the 3-3 or the J-J whether or not they’re in your crib, as long as you don’t split them, so you probably don’t want to consider those points when evaluating the discard. The entry in the table for the Jacks has even more built in points -- between the two Jacks you’ve got about a 50% chance of scoring “1 for his nobs”, for an extra 0.5 points or so on average.

If we ignore those extra points, let’s call them the “intrinsic points,” we see that 3-3 and J-J are actually poor discards, averaging 3.9 and 2.6 points respectively. The 6-7 is probably the right discard, beating out 3-3 by a full point.

Here are updated versions of the charts, with pairs in blue and cards totalling 15 in red. The left chart includes the intrinsic points in each average, while the right chart exclude those points. Note that Jacks have an intrinsic value of around 0.25 points, and we’ve accounted for that on the right chart, too.


Removing the intrinsic points changes everything. 5-5 is no longer the top of the heap. 2-3 beats it by almost half a point. And K-10 is no longer at the bottom, since it does better than pairs of 10s, Jacks, Queens or Kings!

Except for 5-5, the pairs look like pretty bad discards. It’s more important to throw yourself cards that add to five or could turn into a run.

Pone discard table

Here’s the corresponding table for pone, showing average number of points in dealer’s crib based on pone’s discard:


A2345678910JQK
A5.44.54.75.35.54.44.34.44.13.94.23.93.9
24.55.76.74.85.54.64.54.44.34.14.44.14.1
34.76.76.05.46.04.44.54.54.34.24.54.24.1
45.34.85.45.76.54.74.34.44.34.04.34.04.0
55.55.56.06.57.46.66.15.65.56.46.76.46.4
64.44.64.44.76.66.25.85.45.53.94.13.83.8
74.34.54.54.36.15.86.26.74.83.94.23.93.9
84.44.44.54.45.65.46.75.85.34.54.13.93.8
94.14.34.34.35.55.54.85.35.54.84.43.73.7
103.94.14.24.06.43.93.94.54.85.15.04.13.5
J4.24.44.54.36.74.14.24.14.45.05.55.04.4
Q3.94.14.24.06.43.83.93.93.74.15.05.04.0
K3.94.14.14.06.43.83.93.83.73.54.44.04.8


Of course you should subtract the values in this table from your hand value when trying to decide what to toss into your opponent’s crib. As before, Michael Schell’s articles do a good job explaining the logic.

Here’s the visual layout of the average crib scores based on pone discard:


That’s it for this week

Want more details on these discard tables? Let us know and we’ll get it for you. Have a question about any other cribbage related topic? We want to answer it! Just leave us a comment.


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).

3 comments:

  1. Hey Josh (and Aaron),

    I just want to take a moment to complement you on your awesome work and thank you for contributing your time to put together all this information. Not many people have access to heaps of cribbage play data like you do over at Fuller Systems, so I think it's really cool what you've chosen to spend time doing with it. I've always been interested in numbers and statistics as well as cribbage, so reading what you've put together here has been fun. Keep up the hard work and I'll look forward to reading these articles as long as you're putting them out.

    After reading, I realized you'd be the perfect person to ask a long-standing question I've had. What the chances are for the player who wins the initial cut to win the game? If you could break it down based on skill level vs. skill level, that would be great, too.

    Thanks again for the awesome game.

    ReplyDelete
    Replies
    1. Hi Zerrick, sorry I didn't get back to you sooner (just finishing a new Apple release). I'm glad to hear you are enjoying the blog. Aaron has done some really great work and we are grateful for his help.

      I'll have to let Aaron respond more specifically to your question about how things look for the person who wins the initial cut (is dealer first). I know that most give the advantage to the first dealer and I would suspect this would prove out for equally matched opponents in our numbers too. This was the driving force behind the "best of" structure in multiplayer today where you can play a "best of 3" or more games to remove any potential "luck of the draw" factor.

      Thanks again, and I hope you continue to enjoy the game and the blog.

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    2. Hi Zerrick. Great question. I ran the numbers on your question and on average the dealer wins 55.5% of games (averaged across all games without taking skill level into account).

      The 55.5% number is pretty consistent when players are playing another player with the same skill level. The first dealer wins 56.0% of "A" vs. "A" games, 55.2% of "B" vs "B", 55.5% of "C" vs "C", and 55.2% of "D" vs "D".

      Players win around 11% more often when they deal first, no matter who they're playing. For example, in an "A" vs. "D" match-up the "A" wins 66.8% of the time if she has first deal, and 56.0% of the time if she has second deal, which gives an 10.8% advantage for first deal.

      We'll see if we can put more details in a future blog post.

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