Disjunctive:
G1 + G2 =
{ { L + G2 : L \in L1} U { G1 + L: L \in L2} |
{R + G2: R \in R1} U {G1 + R: R \in R2} }
{R + G2: R \in R1} U {G1 + R: R \in R2} }
Informally, a player may choose one of the subgames, and make one of the moves available to them normally on that summand.
Lots of the discussion of last month concerned "where to apply misereness" to a game. I was trying to apply it to the game objects themselves, while it is usually applied to the play of a game instead. Paul Ottaway stepped in and said this was okay while evaluating games using the short disjunctive sum. When taking the short disjunctive sum, a player does not need to choose a subgame on which they have a move, but if none legal move exists, they immediately lose.
This is not as immediately obvious to define in the above set-notation, so let's give it a try:
(Warning: Attempted Ascii Brackets ahead! Update: Ascii Brackets removed!)
Short Disjunctive: (Update: fixed spacing issue. Did I get it right, though?)
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}
(Did I do that right?)
Alternatively, one may want to consider the sum where each player chooses to make a move on each of the available game boards. This is called the conjunctive sum of games:
Conjunctive:
G1 + G2 = { { l1 + l2 | r1 + r2} : l1 \in L1, l2 \in L2, r1 \in R1, r2 \in R2}
This may seem simpler, but after getting used to the disjunctive sum, it likely seems much more complicated from a gamester's point of view. Perhaps we will cover these sums more in the future!
Are there sums I missed? Please let me know!
