A.G.
Your questions are good ones and I sort of felt like I wasn't explaining it very clearly or comprehensively in that post, and frankly I'm not all that mathemetically inclined or interested in mathematics or its applied theories anyway.
First of all, one even starts to wonder why, if let's say the hole on some course where it's most necessary to allocate a handicap stroke to equalize between players of different handicaps (let's say for discussion an 18 against a scratch), the proverbial #1 handicap hole, is hole #6 on a course. And let's say the next most necessary hole on the course for the same purpose is #8. The USGA Handicap System and manual does not recommend that #8 get the 2nd handicap stroke allocation, they recommend it be a hole on the back nine, and the 3rd handicap hole be on the front nine and the next on the back nine and back and forth that way for all 18 stroke holes.
They recommend that pretty much due to the essential nature of the match play format---eg scored via "holes" not strokes, and I guess because in tournaments players can start off both #1 and #10.
And then there is this whole ESC thing which is really only relevent to match play because in match play golfers don't have to actually finish holes which they certainly do in stroke play. Because of that golfers can estimate what they might have made on a hole via ESC which is not possible in stroke play because you have to putt out and finish all 18 holes in stroke play or you are not considered to have played a round of golf
And then you get into this whole thing of handicap posting and how it's done---via ONE single round GROSS score. That is not the match play format or scoring mechanism, it's only the Stroke Play format and scoring mechanism, so why do they have it for handicap posting purposes when probably 90% of the golf played in the world is with match play?
In my opinion, because it's easier to post that way; it's simpler.
The reason I know that is about twenty years ago me and a few guys from Philadelphia CC, the club I mentioned to you has both now, made a formal recommendation to the USGA (it took us about two years
) that they consider requiring hole by hole posting for handicap purposes simply to solve all the problems of sand-bagging in one fell swoop!
I remember the Director of the USGA's GHIN System back, Dean Knuth, (I think he actually invented the GHIN System and he has henceforth been known as "The Pope of Slope") said to us; "Guys, it has taken me 20 years to get handicap posting compliance up to about 40% asking golfers to post a single score (a GROSS single round score which is the stroke play format and scoring mechanism) and now you are asking me to tell them to post EIGHTEEN scores (hole by hole)?"
Well that stopped us cold but we told him: "Dean, do you realize if you do require golfers to post hole by hole for handicapping purpose in a single act you will virtually solve every handicap problem known to man in one fell swoop?" He said; "Of course I do, and maybe it can happen when scanning gets better or computers and their programs do. That was about twenty years ago!
We even told him that he wouldn't need to try and teach golfers what ESC is or how to calculate it before posting a single round GROSS score (by the way, they kept changing the ESC procedure and formula so much even I couldn't remember what it was in any given year) because it could just be imbedded into the computer system so if a golfer posted a 10 on a par 3 the ESC formula in the system would automatically reduce it to the ESC requirement for the handicap of that particular golfer.
Basically they are still trying to fit a square peg into a round hole with the handicap posting procedure by using a single gross score for handicap posting purposes (the stroke play format) with the match play format and scoring procedure which is by holes (hole by hole) and not by a single 18 hole round GROSS score.
Essentially, and mathematically and realistically with most all golfers if they had two handicaps, a match play handicap and a stroke play handicap, inherently their stroke play handicap is always going to be greater than their match play handicap due to the inherent differences between the two formats and their vastly different scoring mechanism!