CROQUET:  Opening Tactics

Here is an interesting interchange about the wisdom of Jacques Fournier's second shot at the World Croquet Championships at Sonoma-Cutrer.  If you read nothing else, read Jacques Fournier's reply.

Duncan Reeve's Question
David Maugham's Reply
Jacques Fournier's Reply

Duncan Reeve wrote:

I was thinking about the Sonoma final and was intrigued by the opening tactics.

Robert Fulford placed the first ball in the middle of the lawn, and Jacques Fournier shot and hit with the second ball.

Jacques tactic seems odd to say the least. I don't claim to be an expert in probability but with a quick rough calculation. If:

p = probability of Jacques hitting the 17 yard shot
q = probability of Robert going round third turn following a miss
r = probability of Robert winning if Jacques hits on his first shot.

The probability of Robert getting the first break, B, is then approximately B = (1-p)*q + p*r

I guess q is about 0.95 and r is about .5, which means that for B to be <0.5 then p must be 1!  Surely, from a 'percentage' point of view this shot was little short of suicidal.

Though of course one mustn't ignore the psychological benefit of hitting the 'wrong' shot...

Is this a widely used tactic amongst top players?

David Maugham wrote:
If:  p = probability of Jacques hitting the 17 yard shot
The shot is more likely to be about 12 to 13 yards.
q = probability of Robert going round third turn following a miss
r = probability of Robert winning if Jacques hits on his first shot.
The probability of Robert winning what? The opening, the game or something else? I assume the opening....
The probability of Robert getting the first break, B, is then approximately B = (1-p)*q + p*r
Probability was never my strongest point and has deteriorated significantly from when I was at school, but this seems to be very simplistic and ignores the relative opportunities if Jacques chooses not to shoot.
I guess q is about 0.95 and r is about .5, which means that for B to be <0.5 then p must be 1!
B doesn't necessarily have to be less than 0.5, merely less than the probability of Rob going round if Jacques chooses not to shoot (at a guess this would be closer to 0.65/0.7)
Is this a widely used tactic amongst top players?
Probably not "widely" but it is an acceptable choice.

Here are some probabilities that I have just had a go at (apologies if they are incorrect)
 
j P(R going round if J in II) 0.65
k P(J going round if R misses) 0.90
p P(J hitting 13 yarder) 0.65
q P(R hitting ball in IV from B baulk) 0.05
r P(R going round with balls near II and IV) 0.10
s P(J going round if R misses) 0.65
t P(J going round if R hits) 0.50

P (R going round first) is:
 
if J goes to II if J shoots at R
j + (1-j)(1-k) = 0.685 (1-p) + pqr + p(1-q)(1-s) + pq(1-r)(1-t) = 0.584

Obviously as j increases, p can decrease and still make this play acceptable.

I have ignored the vanishingly small percentages of players failing to go round if they hit in with either a ball in the middle of the lawn, or two balls together, so these are roughly hitting percentages (even though in actual fact, p(1-q)(1-s) wasn't enough for Rob to go round off <g>).

Jacques Fournier wrote:
I don't know much about statistics as a 17 year old, but the ball in the center was in a position where I could line up a shot that if missed would leave me very close to the second corner, which seems to be the ideal position for those who don't want to shoot.  As it turns out I hit, which allowed me to set a leave which got my opponent to take a much longer shot than he would have had if I had hidden, left him very little if he hit, and left me a 12 yarder if he missed.  It seemed like a pretty simple choice to me.

Maybe I should have worked it out on some scratch paper first!