The Cross-Multiplication Issue
First of all, I have a couple of mistakes to acknowledge here. In yesterday’s "Hey, Bill" answers, I referred to the problem I am now trying to discuss as a "picayune, niggling, trifling, petty, irrelevant little objection." This was grossly inappropriate. First of all, it was grossly inappropriate because it was disrespectful to the contrary opinion, and thus rude to the man who raised the issue and to Tom Tango, who supported his objection. Beyond that, it was grossly inappropriate because, if we are discussing the implications of the method—which we are—then the objection that the gentleman raised was a serious issue that should have been taken seriously, rather than dismissed in this fashion.
The gentleman who raised the question was tjmaccarone, or at least that is the name that he uses here; my apologies to Mr. Maccarone. But the next day, Mr. Maccarone sent another post, trying to expand on the point that he had raised. I read the post, decided not to publish it, and deleted it. This wasn’t improper; I am the editor of this site, I have to make editorial decisions, I have to decide when a discussion is no longer productive and we’re going to move on. But it was a BAD editorial decision. It was a mistake. I deleted his query before I took the time to understand the implications of what he was saying. I’d like to recover the query and post it as a headnote to this article, but. . .it’s gone. Sorry. My bad.
What the gentleman was saying, if I can reconstruct it from memory, from my perhaps imperfect understanding of it, was that the method was. . . well, good enough for government work, but that
1) If you applied it to all of the teams, every game of a schedule, there would be small distortions, and
2) Under some conditions, such as perhaps in the NBA where the standard deviation of winning percentage is larger, these might become meaningful issues.
Well, yes and no. The distortion that Mr. Maccarone suggests DOES exist; it is present in the test that he suggests. I believe it is actually larger than he suggests. But the problem is not in the cross-multiplication approach. The problem is in the test that he wants to use, which is completely irrelevant to the method.
All "closed" leagues have a .500 cumulative winning percentage. The major leagues now, there is inter-league play, but the two leagues together form a closed system which must always center at .500. The "output" strength of every league is a .500 winning percentage.
But does this represent reality? Well, of course it does not. A high school league and a major league both have exactly the same output winning percentage, but is the strength of the leagues the same? In 1953 (or in any season in that era), the National League and the American League both had output winning percentages of .500. Does this prove that the American League was just as strong as the National League in that era?
Well, of course it does not. If the output winning percentages were true representations of the strength of the teams, that WOULD be the case, but it isn’t. The American League in 1960 had a winning percentage of .500. In 1961 they added two expansion teams. In 1961 they still had a winning percentage of .500. Does this prove that the expansion did NOT water down the quality of the league?
Of course it does not.
What I am saying is, the "output winning percentage" of any and every league is a misrepresentation of reality. There is no rule in nature that the composition of every league must contain a .400 team and a .600 team, a .450 team and a .550. There is no rule in nature that a league must be "balanced." It is nonsense. There is no such requirement for the formation of leagues.
The "small inaccuracy" that Mr. Maccarone cites results from imposing on the league a requirement that the league must be balanced. You’ve got a .400 team; well, you’ve got to have a .600 team. No. There is no such rule. This does not happen in reality.
This is what happens in reality. . . let us say this represents 16 teams which are divided into two leagues of eight. The teams have "natural strength levels" which are like this:
Atlanta
|
AL
|
.500
|
Boston
|
AL
|
.563
|
Chicago
|
AL
|
.549
|
Dallas
|
AL
|
.541
|
El Paso
|
AL
|
.465
|
Fresno
|
AL
|
.345
|
Greensboro
|
AL
|
.726
|
Houston
|
AL
|
.562
|
|
|
|
Indianapolis
|
NL
|
.433
|
Jacksnville
|
NL
|
.663
|
Kansas City
|
NL
|
.559
|
Los Angeles
|
NL
|
.515
|
Memphis
|
NL
|
.284
|
New York
|
NL
|
.485
|
Oklahoma City
|
NL
|
.686
|
Phoenix
|
NL
|
.518
|
|
|
|
American League Avg
|
|
.531
|
National League Avg
|
|
.518
|
Or like this:
Atlanta
|
AL
|
.407
|
Boston
|
AL
|
.391
|
Chicago
|
AL
|
.500
|
Dallas
|
AL
|
.695
|
El Paso
|
AL
|
.486
|
Fresno
|
AL
|
.505
|
Greensboro
|
AL
|
.525
|
Houston
|
AL
|
.498
|
|
|
|
Indianapolis
|
NL
|
.496
|
Jacksnville
|
NL
|
.500
|
Kansas City
|
NL
|
.370
|
Los Angeles
|
NL
|
.501
|
Memphis
|
NL
|
.637
|
New York
|
NL
|
.607
|
Oklahoma City
|
NL
|
.709
|
Phoenix
|
NL
|
.448
|
|
|
|
American League Avg
|
|
.501
|
National League Avg
|
|
.534
|
Or like this:
Atlanta
|
AL
|
.520
|
Boston
|
AL
|
.452
|
Chicago
|
AL
|
.489
|
Dallas
|
AL
|
.636
|
El Paso
|
AL
|
.337
|
Fresno
|
AL
|
.720
|
Greensboro
|
AL
|
.506
|
Houston
|
AL
|
.484
|
|
|
|
Indianapolis
|
NL
|
.524
|
Jacksnville
|
NL
|
.583
|
Kansas City
|
NL
|
.337
|
Los Angeles
|
NL
|
.498
|
Memphis
|
NL
|
.324
|
New York
|
NL
|
.423
|
Oklahoma City
|
NL
|
.500
|
Phoenix
|
NL
|
.542
|
|
|
|
American League Avg
|
|
.518
|
National League Avg
|
|
.466
|
Or like this;
Atlanta
|
AL
|
.437
|
Boston
|
AL
|
.472
|
Chicago
|
AL
|
.271
|
Dallas
|
AL
|
.500
|
El Paso
|
AL
|
.265
|
Fresno
|
AL
|
.494
|
Greensboro
|
AL
|
.464
|
Houston
|
AL
|
.750
|
|
|
|
Indianapolis
|
NL
|
.739
|
Jacksnville
|
NL
|
.361
|
Kansas City
|
NL
|
.502
|
Los Angeles
|
NL
|
.516
|
Memphis
|
NL
|
.738
|
New York
|
NL
|
.500
|
Oklahoma City
|
NL
|
.464
|
Phoenix
|
NL
|
.743
|
|
|
|
American League Avg
|
|
.456
|
National League Avg
|
|
.570
|
Whatever the actual strength levels of the teams in each league, the experience of playing through a schedule is going to push every league to a .500 level. But the experience of playing through a league doesn’t have a damned thing to do with this problem. The method is based on the actual strength of the various teams, whatever that is.
Mr. Maccarone has stated directly that the winning percentage of the .600 team against a .400 team plus a .600 team has to be the same as their winning percentage against a .500 team. But there isn’t ANY reason to believe that that is true. The belief that it is true is based on the assumption that the league must be balanced. But the league only APPEARS to be balanced because we force the teams through a schedule which has to result in a .500 record. The league is NOT balanced; the true strength balances at SOME point, but we have no way of knowing what it is. The true strength of each league balances at some unknown point, with no legitimate expectation that any two points will add up to anything.
When you think about it, I think you will realize that that is true. Feel free to convince me otherwise. But we’re not working this out in "Hey, Bill." I don’t have the patience for that, or the right attitude, or something. I’m a very flawed human being.