Thanks to Elizabeth Larsson in the CA [Croquet Association] office, I now have the [Leslie] Riggall article [published in the Croquet Gazette, No. 170, p. 1]. Excerpted below is the part that interested me (referred to by [Keith] Wylie [in Expert Croquet Tactics] at p. 84), concerning the advantage or disadvantage of pegging out one ball with paticular clip positions:

A VIEW ON CROQUET TACTICS

While watching some tournament games it seemed to me that most players would benefit from a serious study of tactics.

One of the most common but erroneous ideas is the belief that a player has only to peg out one of the opponent's balls to ensure victory. And when, as so often happens, the result goes the other way and the single ball wins, the first player feels that the result was unfair.

But what has really happened is that the result reflects the fact that the odds were in favour of the single ball. The following is a very common example. Blue goes to four-back, red or yellow misses the lift and black goes round to the peg. Red hits in or takes a bisque and goes to the peg and pegs out black. This is very dangerous, because blue has only a very easy sequence of three hoops to make. Yellow has much greater need of the black ball than blue, because he has to make twelve hoops. In fact I have often seen players peg out an opponent's ball when the other opponent ball has only one hoop to make (one for rover and one for peg is a common situation). A player who pegs out his opponent in such a situation simply deserves to lose.

When I first propounded this theory in England many years ago it was received with scepticism. So I arranged a series of test part-games with Maurice Reckitt, a "grand-master" of croquet. The results surprised him and proved that in the four-back example given above, the odds were heavily in favour of the single ball. Other tests indicated that the odds favoured the single ball if it was for three-back, and that pegging-out the opponent was generally not worthwhile unless the backward ball was only half way round or less.

The corollary of this urge to peg out an opponent ball, is a commonly-observed reluctance to peg out one's own ball. There was an extraordinary example of this in one of the games referred to in the first paragraph, between two top players.

The player of black and blue was in deep trouble, his black had made only two hoops, and blue had a few more to make. Red and yellow were pegging out but the rush did not take the balls to within a safe distance from the peg. Red missed the peg and travelled to between the boundary and the second hoop, while yellow came to rest about three feet from the peg. If yellow had been pegged out the opponent's chance of completing the course would have been remote. But yellow was sent to red, black near the first hoop hit in on blue which was near the fourth, and it was easy to make the third hoop off red and yellow, with a four-ball break to follow. Thus red and yellow snatched defeat from the jaws of victory, by not taking the opportunity to eliminate all possibility of a four-ball break.

lf a player is pegging out an opponent ball it is often important for him to peg out his own ball at the same time. If your backward ball has about a four-hoop lead over the opponent's ball it is essential to peg out your own forward ball as well. Because virtually the only hope your opponent has of catching up is a three-ball break, and he cannot make this with only two balls. A more precise assessment of the lead required might be as follows.

If the opponent has eleven or twelve hoops to make, a safe lead to enable you to peg out your own ball would be five hoops. If nine or ten hoops remain, a safe lead for you would be four hoops. Less than nine hoops means that three hoops is a safe lead. The score should be watched carefully. Suppose that you start with only a four-hoop lead and your opponent has twelve hoops to make. If you make two hoops and your opponent is still four hoops behind, leave your backward ball at its hoop and peg out your forward ball immediately, and thus eliminate the possibility of a three-ball break.

Really interesting stuff. I would like to hear what the top players have to say, but I would disagree with quite a lot of this.

My first observation — I think much of this is too broad brush. An awful lot depends on whether or not it is handicap or advanced, and if the latter whether or not you are giving away a lift.

I don't agree with the argument that you shouldn't peg out oppo unless his backward ball is for 1B or earlier. I would have no hesitation in pegging out an oppo if his backward ball was for 2B or 3B and I think I am on the conservative side here as many would peg out an oppo if his other ball was for 4B.

When talking about when to peg out one of your balls alone, Riggall doesn't mention that you give up your right to a lift if you do that. And given that top players can do a 3-ball break almost as easily as a 4-ball break, pegging out one of your balls may not right. Obviously some of the time it will still be right to peg out one of your balls (especially if playing handicap when there are no lifts to lose), but it is not nearly as clear-cut as he suggests.

I agree it can often be right to peg out both you and your oppo if your backward ball has a substantial lead over oppo's, but I don't think three hoops is enough. E.g. if oppo is for 3B and you are for rover, I think it would often be folly to peg out the other two balls. It depends where the remaining balls are, but if oppo gets in front of 3B then you can't take position at rover until he's through it, and if he then gets in front of 4B before you are in front of rover you are in serious trouble. I'd much sooner peg oppo alone off, and have my two balls in C1 (assuming no lift — else maximum position on the WB) with a rush to rover giving away a 35 yard shot.

Interested to hear other views.

I think it's a perfectly reasonable analysis, but one that has to be adjusted for applicability to the modern game. Whether this is because of the increase in standard of play or decrease in difficulty of conditions is harder to judge. However, the basic premise that pegging out opponent doesn't give you any automatic right to win is still sound, as is the common error of not reconsidering pegging out the second ball either immediately or as the situation evolves.

The article may stand with less alteration for B class play, but for A class, I suggest that some of the references to four balls breaks should probably now be three ball breaks, and references to three ball breaks become two ball breaks.

The Riggall article was written a long time ago (30 years?) and looks reasonably appropriate for the B and C class now. I doubt it was ever intended for the top level, but in any case the game at the top level has moved on.

It is, indeed, "too broad brush" to be a complete answer — no single formula is going to apply to all levels of play, handicap, level, or advanced.

Personally I am unlikely to peg off both rovers unless my other ball is at least for 3-back, even if the opponent is still for 1.

In advanced play, pegging out the opponent when the other opponent ball is for 4-back is sailing rather close to the wind, unless your own back ball is already past 4-back. There will usually be a lift coming, and if you join and the opponent hits, that could be the game. Or if you separate, the opponent gets a free shot at your rover ball, or can simply set up at 4-back, then bide time waiting for the next lift.

I don't agree with this. We suppose oppo is 4b and peg, and you are on a break to the peg with one ball, conceding a lift, but not a contact. If you don't peg out the oppo, you have to join up and he has a lift shot with four balls on the lawn and hitting the lift more or less wins the game. If you do peg out the oppo, then, even if you join up, he is less certain to win if he hits. If you separate, he should have two shots but is much less likely to finish if he hits them and the second will be longer (probably much longer). After those two shots, you would hope to establish a break without giving away another good chance, whatever hoop your backward ball is for.

If you are not conceding a lift, it is even more important to peg out one oppo ball so you can leave a 35 yard suicide shot. If you are conceding a contact, you probably also want to peg one out and separate, to give you a better chance of having another turn.

"If you don't peg out the oppo, you have to join up and he has a lift shot with four balls on the lawn and hitting the lift more or less wins the game."

And missing the lift more or less loses it.

"If you do peg out the oppo, then, even if you join up, he is less certain to win if he hits."

Just as you are less certain to win (on the following turn) if he misses.

"If you are not conceding a lift, it is even more important to peg out one oppo ball so you can leave a 35 yard suicide shot."

Which oppo need not take. Whereas by not pegging out you can cross wire at your hoop and leave a guaranteed easy finish if oppo misses the 35-yarder.

[If you don't peg-out the oppo...] Yes, so your opponent's chance of winning the game is roughly the same as their chance of hitting the lift, say 40%.

[If you do peg-out the oppo...] His chance of winning on that turn if he hits is going to be something like 30% - involving rolling up to 4b and then finishing. So 40% x 30% = 12% chance of finishing that turn.

But you don't have to win on the following turn, because you can make a killer leave instead. His chance of winning on subsequent turns should be low, unless you make a mistake. For his chance of winning overall to get to 40% you probably have to give him a good chance, or a lot of chances.

Presumably you have separated, or oppo wouldn't have to roll up to 4b. Perhaps you have left your balls N of IV and S of II. Oppo has the choice of a free shot at your peg ball, or could simply lay up at 4b. If the latter, what shot do you take now?

I thought you all might find this interesting.

In Paul's game against Ron in the USCA Eights, Paul pegged-out both balls making the score 22-14 (his clip on #4-back and Ron's on #2), and won 26-16.

In Kerry's game against Tim in the MCC Ladder, Kerry pegged-out enemy ball making the score 16-23 (her clip on #5 and Tim's on #5-back), and lost 17-26.

You might also recount two games from the Last Great Grudge Match (Nikora vs. Billings) in each of which Leo pegged out Paul to make the score 21-22 (Leo on 4b and peg, Paul on 4b and out). Paul won after a hit-in in each.

This is a very gray area and something everyone must figure out for themselves based on their AND their opponent's strengths. If you're good at getting wired and rushing to hoop position, pegging out oppo may be attractive — ESPECIALLY if they are not hitting well or cannot run a 3-ball (2-ball?) break very far. This MUST be tempered by how far you have to go — which may have been Kerry's fatal mistake.

The other downside is that the single-ball player need hit-in only once (as I was lucky enough to do in the LGGM). By leaving both oppo balls in the game, you may garner a possible hit-in attempt (often with a lift) for yourself in the event that your opponent does manage to get in.

Kerry, much of your experience against Leo illustrates that he will almost always will peg you out if given the chance. I would suspect, however, that he also had the lead in the vast majority of those occurrences.

I agree that whether or not to peg-out your opponent "is a very gray area".

The fact that "Paul won after a hit-in in each" game in which "Leo pegged out Paul" seems to suggest that I should not have pegged you out. But what if I hadn't? Then when you hit-in in each game, you would have had four (instead of only three) balls on the court, and it would have been even easier for you to win. If you're going to hit in, then I should certainly peg you out.

To maximize your probability of winning the game, you want to maximize the:

1. probability you get to the peg when you have the inning(s), and
2. probability your opponent does not hit in when he does not have the inning(s), and
3. probability your opponent does not get to the peg when he has the inning(s)

That's the product of three robabilities. Maximizing just one, may or may not result in maximizing the product, because the others may also change. That's what makes this a grey area.

Please note that probability (2) is not and independent variable. It depends on how many balls are one the court, because with only one ball your opponent has fewer choices of how to hit-in, you can put one ball farther away than two balls, you can hide from one ball easier than hiding from two, and you can avoid a wiring lift against only one ball. It also depends on how many chances your opponent has to try hitting-in, which is dependent on probability (1).

Not pegging-out opponent increases (1), but decreases (3). Pegging-out opponent decreases (1), but increases (3). In both cases, whether it increases or decreases (2) is not clear.

I only wanted to illustrate the "grayness"; in fact, I later said that I was lucky to have hit in. However, I do not agree with your statement: "If you're going to hit in, then I should certainly peg you out."

You're only thinking one turn ahead with this strategy, trying to minimize the probability I make it to the peg. But you, in fact, said there's more to it than that with the product of probabilities. You're contradicting yourself, amigo. ;-)

From 4b, if I hit in, odds are still pretty good that I will get to the peg with that ball — and that's game over. (see note below)

My feeling is that I would want one last chance to hit after my opponent hits. Clearly the more hoops my opponent has to make the more time he has to organize the easy pegout of striker and partner if I didn't peg it out. As such, my opinion is:

If you're going to hit in, then I should probably not peg you out — the degree of probability adjusts from "certainly not" (if your backward ball is on the peg) decreasing to "never not" (meaning I should always peg you out) if you're on 1. (Subject to my other clip position, however.)

Note: It is true that the odds of getting around are lower with 3 balls on court than with all of them. It would be an interesting thing to quantify the 2-ball ending given a rush to 4b from, say, EB near 4. With Maui hoops for me, I'd say that's better than 50%, but significantly lower for the hoops used at the 8s.) Again, assessment of my opponent's skills would figure largely into my decision to peg him out or not.

One issue that hasn't yet been mentioned is the psychological aspect of being the single ball player. You have the advantage of "resolve" or "clarity from lack of options" — at some point it likely comes down to "hit or die". The lack of doubt increases focus for many players, increasing the probability they will hit in.

Yes, it's a very gray area indeed and one of the more attractive aspects of the game, I might add.

Gray areas are always fun to discuss.

You added the "psychological aspect" to the grayness. I'll add to the grayness the fact that you don't get any more lifts after pegging-out your opponent's ball.

Let's reduce the grayness a little by discussing a specific example: the Last Great Grudge Match between you and me. You're suggesting that it may not have been wise for me to peg you out.

When you reached the peg, you must have thought that leaving four balls on the court was to your advantage, because you did not peg-out striker ball!

When I reached the peg (after you), here's what I thought:

From hoop #4-back, both you and I could get to the peg with either a three- or four-ball break in one turn; and both you and I might take a couple of turns to get to the peg with a two-ball break.

If you did hit-in, then as you say, "I would want one last chance to hit"; so I wanted to limit you to a two-ball break.

To limit you to one-ball breaks, I would never roquet you; that means I would never be responsible for your position, and never have to give you a wiring lift. I would wire both my balls from your ball, with a rush towards my next hoop; let's call that 'hiding'. If the rush was good enough, then I would split the hoop, run the hoop, and hid again. If the rush was not good enough, then I would hid again.

If while trying to hid, I would ever fail to wire partner ball, then I would separate striker ball; and I would be back to the same situation as after pegging you out (I'll have one last chance to hit after you do), except you would not have a lift (so your hit-in is probably longer). With only three hoops left to run, this should not happen very often.

So, I pegged you out.

But you did hit-in. More importantly, you did get the rush on my other ball, and you did build a two-ball break, and you did run it to the peg in one turn. You were unbeatable!

If I had it to do all over again, I'd peg you out again. If you were unbeatable again, you'd win again; but then you'd win no matter what I did. :-)

Nope, I never suggested that it was unwise for you. What I did say was that I did win after a single hit-in, and that is a very real downside to pegging out your opponent, especially with only three hoops to go for his other ball. In two emails now you have incorrectly stated that I said you made a mistake. Since I did not say that and you somehow missed the first correction, I can only wonder if your subconscious mind must think it really was a mistake, causing you to project said sentiment into my comments. ;-)

Clearly, you're not very good at reading minds, because I did not think that [that leaving four balls on the court was to my advantage] (nor can I think of any reason why I "must" have thought it). My thought at the time was that I wanted to have a partner, therefore I did not peg out striker. The further consequence of four balls remaining on the lawn was immaterial to me.

Not sure why you didn't think I could make the three easiest hoops on a single 2-ball break when I routinely would run many more in our past 1-ball games. And those were after three or four of Ramm's beers! But you made a decision based on your assessment, which is all that anyone can be expected to do.

Which is where we differ in how to get that last chance. You felt a 2-ball break would do it, which is fine. I feel that it is harder for me to get to the peg AND position partner where it could be pegged out. Don't forget, I would be separated, and partner would probably be on a boundary (corner 2 if I was making the leave in your situation), and I would still have to run those hoops with a 2-ball break.

Wow, OK, you were right — I now see that it really was unwise for you to peg me out. So sorry to have doubted your subconscious! ;-)

You employ this [hiding and two-balling] line of play a lot more than I do — thanks for the tactics lesson. Since you're seldom on the other end of it, however, you might have missed that if you separate, I'm going to lag for position. You might then join up, but odds are that I'll make that hoop (great odds if my name is Bruno Amby). You don't have a lot of time to be playing the separate-yourself tact when I have so few hoops left to play.

It seems that we'll have to agree to disagree as to what is the "wise" choice, then. But that's why we play this silly game, I suppose!

At that time, I could not see any way to reduce the "downside of your opponent winning after a single hit-in" by leaving the opponent's ball on the court. Later in this message, you describe how burying the opponent's ball might reduce the chance of him winning, and I can now see that. Whether that tips the balance away from pegging opponent out is still unclear in my mind.

When you got to the peg, you wanted to have a partner on your next turn. That's why you did not peg yourself out. When I got to the peg, I wanted you not to have a partner. That's why I pegged you out.

I was sure you could "make the three easiest hoops" on a single four- or three-ball break. My only hope of you not running them (if you hit-in) was to give you only a two- or one-ball break. Your next, and most persuasive argument, isn't about making those hoops, but about how hard it is to peg out two balls after making the last three hoops with one ball.

I see the difficulty in balancing between the difficulty caused by removing an opponent's ball, and putting an opponent's ball in an out-of-the-way place. I don't think I ever gave that enough thought. [On first consideration, I think that if you had a three-ball break (with your partner in corner 2) after running penultimate hoop, you could easily rush (or even just take-off) back to your partner ball, split partner ball out of corner 2 (maybe even half way to the peg), run rover hoop off the pioneer, take-off pioneer to behind partner ball, rush it near the peg, and peg-out both balls.]

In the Last Great Grudge Match, both our backward balls were for hoop #4-back. I thought I would not have to separate more than once or twice, while having to hide only three times. Even if you successfully lagged to position, and then ran a hoop, one or twice, I would still win.

That is why we play the game. And it is why we enjoy discussing strategy so much.

Because of our discussion, I'll give more thought to burying my opponent's peg ball in a corner; and I hope Tim, Ron, and Kerry might also have learned something.