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!

Rules

Every good rule is a Pandora box where is hard to see all consequences. If it is easy, the rule may be useless. [T.Sagme, Meditations]

Active and Passive Games

Games can be seen as "active" or "passive"; perhaps "violent" and "sluggish" would be better terms. In active games the pressure is on to get out and make things happen, before your opponent does. In passive, it tends to be better to wait or at least movement is so slow that one cannot get too active.
 
Chess is a bit of both, though I feel the passive tendency dominates a little. Go is finely balanced. Most chess variants tend to be very active, (e.g. Handgrenade and Progressive); indeed most progressives are very active. Connection games tend to be sluggish.
 
(Handgrenade Chess: No direct take allowed. In a move, all opposite pieces at King distance are captured. The one who captures the King wins)

Fractional Hex

[From Edward Jackman conversations] Speaking of Hex, here's another idea to address it's first-player advantage that may not have been explored much. I found it in the Mudcrack Y and Poly Y book, but it applies to Hex as well. The first player draws a line connecting the midpoints of the opposite sides of one cell, dividing it half, creating two 5 sided cells. She plays her opening move to one of the two halves. The fewer sides a cell has, the weaker a move there is. You might even require that the line divides the cell into a 4 and a 6 sided cell and the move goes in the smaller cell -- that would be a very weak move, even in the center of the board.

Standard move, filling entire cell:
        +---+    
       /     \
  +---+       +---+
 /     \     /     \ 
+       +---+       +
 \     /xxxxx\     /
  +---+xxxxxxx+---+
 /     \xxxxx/     \ 
+       +---+       +
 \     /     \     /
  +---+       +---+
       \     /
        +---+

Half move:
        +---+    
       /     \
  +---+       +---+
 /     \     /     \ 
+       +-+-+       +
 \     /  |xx\     /
  +---+   |xxx+---+
 /     \  |xx/     \ 
+       +-+-+       +
 \     /     \     /
  +---+       +---+
       \     /
        +---+

One third move:
        +---+    
       /     \
  +---+       +---+
 /     \     /     \ 
+       +---+       +
 \     /    ,+     /
  +---+    /xx+---+
 /     \  |xx/     \ 
+       +-+-+       +
 \     /     \     /
  +---+       +---+
       \     /
        +---+

Loss

Every loss should always teach a new lesson. [T.Sagme, Meditations]