Balancing starts

[...]It is a triviality that the usual method of playing games where there is an obvious first-move advantage, is unfair to the second player. "First" is clearly an average of half a move ahead of "Second", e.g. at random times. A typical attempted rectification of this, in abstract-game-playing circles, is the 1 2 2 2 2 ... move transformer, whereby after the first move, each player plays two moves consecutively. Though inappropriate for most games if used directly, it may have its uses if some further restrictions are added.

It has the Cesaro-propery of "evening-out" the starting advantage, (though
for VERY short games a further integration to 1 3 4 4 4 4 may be suitable),
and it is nice to see the sum of 1-2+2-... coming to 0 by almost every method.

Now, another move transformer often used is the "Progressive" transformer, whereby the moves are taken in series of 1,2,3,4,5 etc. It makes for fun games, if hardly very serious ones; and e.g. Progressive Chess already has quite a long history. But it often struck me that even so, there was a very slight advantage to First. (e.g. His number of moves ahead is successively 1 -1 2 -2 3 -3... so that First is always first to get to a new number of moves ahead, rather than Second.) And so it now appears this advantage is real! There is an advantage to 1/4 of a move to First!

So one way I have considered for some while of rectifying the Progressive transformer was to make it an "Odd-Progressive" transformer. This has move series of 1,3,5,7... , which gives the number of moves First is ahead each time as being 1 -2 3 -4... , which is clearly fairer than the above.

ps: A later idea about progressive games is the "slowing-down" mutator 443322111... which is excellent for slow starting games!