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We're all familiar with the winning average statistic in competitions that pit two players or teams against each other at a time. You simply tally up your number of wins and divide by the total games played to get your winning average. If you win 32 times in 50 games, your winning average is .640. (This is called the won-lost "percentage", which is not mathematically correct terminology, but I don't think anybody minds.)

What if the sport or game involves more than just 2 competitors? It's obvious who the winner is, but how do you handle the 2nd, 3rd, 4th... places? What if you wanted a performance indicator like the above winning average statistic for card games, board games, trivia games, track and field events, etc.? In my web pages, for example, this concern may come up with respect to Scrabble, Monopoly, and croquet.

Here's my suggestion. Use the Average Place Statistic (APS). It is in every way equivalent to the winning average statistic mentioned above, but generalized for multi-player (or multi-team) games. Not only that, but the number of players (teams) may vary from game to game. For instance, sometimes your croquet game may have 6 players, sometimes 3, etc. The APS handles that with no problem.

The Average Place Statistic shows where you expect to finish up no matter how many players (teams) are involved in the competition. Just as in the familiar statistic, an APS of 1.000 corresponds to "always first"; .000 corresponds to "always last"; .500 means you expect to place right in the middle of the field.

It's worth reflecting on that last example. Taking ice hockey as an example, a winning average of .500 can be gotten in several ways. A team may have won exactly half its games and lost the other half; or maybe it tied every single game; or, more likely, it a had a combination of ties and an equal number of wins and losses. No matter how it happened, a .500 statistic indicates that team is smack dab in the middle of the pack, performance-wise.

In this Average Place Statistic, .500 will have the exact same interpretation. Taking croquet as an example, your .500 APS could come from placing first in half your games and placing last in the other half; or from placing 3rd out of 5 in every game; or balancing 2nd and 3rd place finishes in 4 person games; or any overall performance where, on the average, your finishes above the middle-of-the-pack are precisely balanced by finishes below.

The Average Place Statistic is easy to calculate. In a game involving N players, each player is awarded a fraction of a win (Win Fraction) which depends on his finishing place. The Win Fractions, from first to last, can be calculated as follows:

             Place:    First   2nd    3rd    4th  ...  Last 

                               N-2    N-3    N-4  
      Win Fraction:      1     ---    ---    ---  ...    0      
                               N-1    N-1    N-1    

That mathematical generalization might look scary, but here are the actual Win Fraction values for games of different numbers of players:

            Win Fractions for First- to Last-place finishes


        2-person game:  1   0
                        1  .000

        3-person game:  1   1/2   0
                        1  .500  .000

        4-person game:  1   2/3   1/3   0
                        1  .667  .333  .000 

        5-person game:  1   3/4   2/4   1/4   0
                        1  .750  .500  .250  .000

        6-person game:  1   4/5   3/5   2/5   1/5   0
                        1  .800  .600  .400  .200  .000

        Etc.  

Dig the simple pattern? The denominator is always 1 less than the number of players in the game. (I left 2/4 unreduced to keep the pattern clear.)

To calculate your Average Place Statistic after a certain number of games, just add up all your Win Fractions - converted to decimal - for all of your games and divide by the number of games you played. Some of those games may have been 2-person, some may have been 5-person, etc. No matter; just throw them all into the stew.

You probably don't need an example, but suppose you came in 2nd in three games of Confusion Rummy, one of which was a 4-person game, one 5-person game, and one 6-person game. Add up .667 + .750 + .800 = 2.217. Divide that by 3 to get your APS = .739 .

To belabor the point, the calculated APS is in every way equivalent to the winning average statistic we are comfortable with in 2-team sports, such as baseball, where a .739 won-lost "percentage" indicates the equivalent of 739 wins out of 1000 games.

Which ain't bad. In fact, it may be better than the APS of that guy who beat you in the last game (my brother-in-law, Tom, for example) who likes to rub it in that, "Nobody remembers 2nd place!"

I know I'm in a minority position when I say, "I like ties." I've heard people liken a tie to "kissing your sister." I view it from the "cup is half full" point of view - a tie says you are the best. Nobody can take it away from you. At the very end of the last quarter of the Super Bowl game, after a round of playoffs, after a whole season of football, with the score tied 24-24, your team is the best in the world! The other team is too! Imagine that, two best teams! Both can jubilate. They can jubilate together! Neither one has to walk off the field completely shattered after a few minutes of some piddling, makeshift, tie-breaker rule tacked on like a wart to the fundamental body of rules laying out the game's objective. (And how's that for a paradox? - the most devastating feeling in all of sports being reserved for the competitor who proves himself to be the second best among the whole field! I don't have a solution for that.)

As I write this, I realize I haven't given much thought to ties and baseball. Extra-inning games have been around forever, and might even have a certain aura about them. I'm not starting a campaign against extra innings in baseball, but let me raise the question of how necessary they really are, and if baseball might be nicer in certain ways by allowing ties. "And the winner is . . . whoever has the most runs after nine innings." Period. It's simple; it's elegant. And lots of games will have two winners! (Sez me.)

When I took a fresh look at the Scrabble boxtop rules while starting up my Scrabble club, I found a tie-breaker rule that Scrabble implemented in 1976. After the adjustment to the players' scores for leftover tiles, the player with the highest score wins. But, "in the event of a tie, the player with the highest score before tallying the value of unplayed letters is the winner." The way I see it, that's a kick in the snoot of the player who played his guts out to be the one to go out and snag all the leftover points. That's a fundamental part of the game's strategy; why should it all of a sudden not count? I claim that player's performance in the endgame makes him every inch the equal of the player he caught up with. In my club, it goes down as a tie.

Ties are good for the soul.

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Note: For years this page had been calling the APS the "average position statistic". In Jan 2007, I found I liked "place" better.

And if you liked this one, please visit my page of Scrabble thoughts!