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).


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

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).

Thursday, August 23, 2012

Should You Pair Your Opponent's Lead?

Talk about a classic cribbage question! Should you pair your opponents lead, or is that sly dog planning to sneak in a three-of-a-kind on top of your pair for a big six points? If you stay up late pondering these questions you're in luck: we've got some data to help ease your mind. A word of caution, though. Today's post is nothing but numbers. If you just want the executive summary skip to the bottom of the post.

Before we show the numbers, let's talk about what we're measuring. We looked through our games and found all the hands (hundreds of thousands of them) where the dealer had the opportunity to pair their opponent's lead. We didn't include hands where the dealer could score a 4-of-a-kind in response if the non-dealer scores 3-of-a-kind. In those cases we'd consider the dealer the trapper, not the trappee. We did, however, include hands where the dealer had two cards 8 or higher, since the count would have to go past 30 for them to score their 4-of-a-kind. Perfectly clear? Then let's see some data:

Probability dealer would be trapped if they paired opponent's lead

Lead cardTrap %
(Don't use these numbers, though. We've got more useful stuff coming up soon).

So if you're the dealer, you're sitting on an ace, and some random opponent starts play by laying down an ace, then there's a 49.9% chance that he's got at least one more ace in his hand.

The cut card has some bearing on these odds. If the you've got an ace, your opponent leads an ace, and the cut is an ace, then it's less likely that the your opponent has an ace in reserve. The normal case is that the cut card is a different rank than your opponent's lead. Let's see what the probabilities are in that situation:

Trap probability when lead is different rank than cut

Lead cardTrap %
(Don't use these numbers yet, either)

Turns out that doesn't make that much of a difference. That's mainly due to the fact that the lead card is usually a different rank than the cut, though, so the averages are weighted heavily in that case's favor. We'll talk about the other case (lead is the same rank as the cut) a little later in the article.

What are the A players doing differently?

We talked a lot about the A players in our first post. Let's look at this first table to see what the A players are doing, and how we should respond to their fancy pants play:

Trap probability when playing A level players, lead different rank than cut

Lead cardTrap %
Use these numbers!

The A players are less likely to try to trap with aces or twos (especially aces), but more likely to try to trap with everything else, especially nines and jacks. We see two lessons here: one is that the A players are more likely to lead from a pair, and that's a good sign that we should do that, too. Another lesson is that we should be nervous about pairing a good player's lead.

What if an A player's lead is the same rank as the cut?

Trap probability when playing A level players, lead same rank as cut

Lead cardTrap %
(Note that we've left out statistics for the "5" case. We've got a lot of data, but that particular situation is so rare that we don't have a large enough sample to feel confident reporting on it)

The lead card is a lot less likely to be a trap when its the same rank as the cut. In particular, notice that all the probabilities are lower than 33%.

How should we play against A+++ players?

The keys to taking your game to the A level are study and practice. Getting from A to A+++ is going to take psychology and creativity. Here's a fun story from DeLynn Colvert, four time national champion and author of Play Winning Cribbage, about a game against former national champion Duane Toll:
The game with Duane Toll was a 4th round match in the Grand National in Lincoln City, OR. With the score 2-2 best 3 of 5, Toll was dealing from about 28 out and I was standing about 23 out with first count. Toll has studied my habits consistently leading from a pair. My hand was 3-3-4-k and well short of game. I lead from the single 4!! Toll was also short of winning with first count the next deal and was forced to play offense...and he played a 3 on my 4 lead! I paired his 3 and knowing I consistently lead from a pair immediately played the 3rd 3 for 6!! When I played the 4th 3 for 12 he was stunned, threw his remaining cards in the air and conceded the match. I went to the championship and $5,000! Toll took home $200 for the 4th round! His play of the 3rd 3 was a good play as he would have been 15 or so short otherwise!
Colvert and Toll are possibly the two best players in the world, and it's great to see how they approach a decision as common as "should I pair my opponent's card?"

Executive summary

Should you pair your opponent's lead? This data is an important piece of the answer, but there are other factors to take into account when making a decision of this magnitude. If you need two points to go out then of course you should pair the lead. And if your opponent always, 100% of the time, leads from a pair then you should be even more cautious about pairing their lead.  Your current board position also plays a big part (We’ll talk more about that soon in another article).

But you'll frequently find yourself in a position where you don't have a read on your opponent and you'd like to take a chance at some points as long as you're more likely to come out with more points than the other player. That's where this data comes in handy.

In a large fraction of hands we're happy to take a guaranteed two points for ourselves as long as our opponent will average less than two points in response. That means we should take the pair for two as long as there's less than a 33% probability that our opponent is going to get three-of-a-kind for 6 in response.

When you just want to come out ahead in the exchange, the executive summary is that you should pair your opponent's lead of a 3, 4, or 8 (the three leads with a < 33% trap probability). And if your opponent's lead matches the cut you should feel free to pair it.

Have any insights you'd like to share on this topic? Questions about the data? Suggestions for future posts? We'd love to hear from you. Leave your comments 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).

Thursday, August 16, 2012

Cribbage Pro Multiplayer Analysis - Tips & Strategy

Welcome to the first installment of "Cribbage Analysis - Tips & Strategy"!

Cribbage Pro is the most popular cribbage application for mobile devices and the only one offering cross-platform multiplayer today. We have hosted millions of online multiplayer games since releasing Cribbage Pro Online in 2009. The cribbage community has been very good to us, and we would like to continue to give back whenever we can. So, we will be publishing several articles about cribbage over the next few months, based on actual data from Cribbage Pro multiplayer games to help you develop your cribbage skill and strategy both when playing in Cribbage Pro or in the "real world".  You should subscribe (link on the right) to this blog now to get the very latest tips and analysis as soon as it is available and if you don't have Cribbage Pro yet, go download Cribbage Pro now so you can "play along".

In online multiplayer games we track every game played, including every hand dealt, every card played and every point scored. Armed with our database of millions of hands we can analyze the game in ways that nobody has ever attempted, based on real players playing real games against other real players. Want to know how to play 6-7-8-9? We have seen it played tens of thousands of times, and we know what has worked out best. Should you pair your opponent's lead? We have the data on that too, and we'll be sharing that and much more with you on this blog in the coming weeks.

One of the first questions you will probably have is about privacy in all of this data we are collecting. As our EULA states, all our analysis is done anonymously and in aggregate. We know that thousands of players like to lead a 4 from 4-5-6-6, but we don't track the corresponding names, addresses, or anything else about them and we secure the data using some of the best security technology available. You can rest assured that you can play online without worrying about your privacy. And please don't lead the 4 from 4-5-6-6!

With that out of the way, I would like to acknowledge the hard work of Aaron Harsh in pulling all of this together (more about Aaron in the guest post tag line at the bottom). Aaron has spent a lot of time and effort on pulling all of this together, and we are very grateful for his willingness to do all that work and be a guest author for us on the blog. Everything you read below comes directly from Aaron and his analysis of the data provided unless otherwise noted.

First topic: What makes the good players so darn good?

Some players are better than others. What are they doing differently, and what do we need to do to play as well as them? For our analysis we've ranked the players by the percentage of games they win[1]. We'll name these ranks "A", "B", "C", and "D". Around 3% of the players got an A, 16% got a B, 56% a C, and the bottom 25% got a D. The A players win around 54.3% of the time, on average, B players win 51.5% of the time, Cs win 50% of the time, and the D players on average win 48.1% of their games.

(Editor's Note: View your total win percentage by selecting the statistics button in Cribbage Pro. Refer to the footnote for how this is different than the calculation used here. When considering just "raw" data that includes forfeits and manual counting games, roughly 30% have a 53% or better win rate and 10% are at 60% or better)

Here's what this looks like visually:
The As are in blue, Bs in orange, Cs in yellow, and the Ds are in green. Those As are on a completely different level from the masses in group "C", but they're still not winning anywhere near 100% of the time. It looks like a 56% winning average across a large number of games would be a good goal. So how do we accomplish that? What are the great players doing differently than the casual player? Well, for starters..

Better players score more points

No surprise here: great players score more points than casual players. The surprise is that they're not scoring that many more points:

Average points/hand, by player's rank

Skill levelAverage points as dealerAverage points as non-dealer

The A players, on average, are scoring almost exactly the same number of points as the B players when they deal (taking into account hand, crib, and pegging), and only 0.16 more than the C players. Even compared to the D players they're only scoring 0.34 more points per hand when they deal. Cribbage is a subtle game!
The situation isn't much different for hands played as non-dealer. The A players score essentially the same number of points as the B players, 0.17 more than the C players, and 0.30 points more than the D players.

Let's break those numbers down a little:

Points as dealer, by player's rank

Skill levelHeelsPeggingHandCrib

There's not much skill involved in cutting a Jack, and all players end up with the same "Heels" average. Surprisingly, pegging doesn't show much of a difference, either. All players peg a similar amount each hand. The hand and crib counts are where things really pick up. Once again there's very little difference between the A and B players, but the C and D players have a small but noticable disadvantage. One fun fact is that the A players are apparently thinking more about their cribs than the B players, and lose some pegging points in the process.

The non-dealer numbers are about what you'd expect:

Points as non-dealer, by player's rank

Skill levelPeggingHand

Pegging's still close here, although the D players lose some ground when pegging as non-dealer. And the differences in the hand count show up again for the C and D players.
So now we've got our first step to being a better player: choose your hand and your crib more wisely. We'll talk more about that in future blog entries. So now we know what the good players are doing better: they're scoring more points. What else could there possibly be to cribbage besides scoring more points?

Better players keep their opponents from scoring points

Here's the complement to the first table we saw: average points per hand, based on the opponent's skill level.

Points/hand, by opponent's rank

Skill levelAverage points as dealerAverage points as non-dealer

There's some interesting stuff here. Dealers playing against an A player average only 15.86 points per hand, vs. 16.23 when playing against a D player. That's an advantage of 0.37 points per hand. Non-dealers playing against an A player average 9.88 points per hand, vs. 10.12 against a D player.

Let's break those numbers down and see where the difference is coming from:

Points as dealer, by opponent's rank

Skill levelHeelsPeggingHandCrib

Points as non-dealer, by opponent's rank

Skill levelPeggingHand

Even great players can't keep themselves from cutting a Jack. They don't seem to have much effect on their opponents' hands, either. But great players take away 0.1 or 0.2 points of pegging points from their opponents, and they do about the same to their opponents' cribs, too. This looks like the real difference between the A and B players -- Bs score just as many points as As, but the As are better at keeping their opponents from scoring.

One interesting point in this data -- players' hands contain slightly fewer points when they're playing better opponents. A player can't directly affect the number of points in her opponent's hand, so what's going on here? Is this because the players are keeping different cards in an attempt to peg more? Or is this because the opponents get so far behind they make rash decisions in an attempt to catch up? Maybe we'll investigate this in a future blog post.

Executive summary: in two hands (one as dealer, one as non-dealer), a great player will score 0.34 more points than a D player, and he'll lower his opponent's score by 0.37 points. Altogether, an A player has net advantage of 0.70 points per hand. The A players score more points, and their opponents score fewer points.

That might seem like the end of the story, but there's one more thing:

Better players play differently based on their board position

An understanding of board position is probably the biggest different between a great player and the rest. It's a little deep for our first blog entry, though, so let's save it for future blog posts.

That's it for today

We'll dig deeper into all these topics in future blog posts. Do you have something in particular you'd like us to look at? Leave us a comment and we'll see what we can do.

1. To be precise, we've ranked each player based on a Bayesian posterior mean of the player's win rate, to take into account the number of games played. The more games a player plays, the closer their ranking will be to their win percentage. We've also excluded manual-count games since we want to identify players with good strategy, rather than players with good counting abilities. The formula used is: (25 + # games won) / (50 + # games played) where "games won" and "games played" are the number observed in the sample.

Editor's Note: This is a guest post by Aaron Harsh. 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).