## Wednesday, December 16, 2009

### Short Disjunctive Game Sums

A few weeks ago, I attempted to define Paul Ottaway's Short Disjunctive sum in terms of game summands instead of the method of play. I am interested in doing this because other game sums are written this way and also because there is a big difference between game definitions and methods used for play.

For example, combinatorial games are defined just in terms of plays available to each player, but then Normal and Misere are two play methods used in competition. Thus, the basic operations do not sum two misere games together, but rather sum two games together and then to compete on that composite state, a play method is chosen. Winning and losing is not defined via the games, but by the way they're played.

The short disjunctive sum definition seems to refer to a play method, except that it doesn't describe who wins or loses:

The short disjunctive sum of games is played by first choosing a component then making a legal move in that component if such a move exists. The game ends when a player chooses a component in which he has no legal move.

The other week, I tried to express this in set notation the following way: (G1 = {L1 | R1} and G2 = {L2 | R2})

G1 + G2 = { Left | Right } where
if either L1 = null or L2 = null, Left = { L + G2 : L \in L1} U { G1 + L: L \in L2} U { | 0}
otherwise, Left = { L + G2 : L \in L1} U { G1 + L: L \in L2}
and
if either R1= null or R2=null, Right = {R + G2: R \in R1} U {G1 + R: R \in R2} U {0 | }
otherwise, Right = {R + G2: R \in R1} U {G1 + R: R \in R2}

The idea behind this is that in the case where one of the children on the appropriate side is null, then there's an extra "bad child" that player could choose to move to (for example, { | 0} for the Left player). In this case, the right player can choose to move to 0 = {|} next turn, and then the game has ended, just before the Left player's turn.

Why did I use this in my definition? With these sum rules in place, in both Misere and Normal play, the correct player wins. In normal play, the Left player, upon making the bad move, would give the Right the last chance to play and win. In misere play, Left would choose that game and, if I understand short disjunctive sums, they would not complete a move and would win. With the above model, that choice would leave Right with a final complete move, which would lose them the game.

This was only my basic logic; I don't know how well it holds up under other game operations. It is often the case that preserving the value of strategies is not the same as actual game equivalence! Also, I should apologize for not having studied the notation Paul has actually introduced for these game sums!