Games theory

1. L, R : sets of games, then G = <L, R> is a game
   
2. DGC: no infinite sequence of games Gⁱ = <Lⁱ, Rⁱ> with G⁽ⁱ⁺¹⁾ ∈ Lⁱ ∪ Rⁱ for all i ∈ N
   

L and R are left and right options of G.

G = {L₁, L₂, … L_n | R₁, R₂ … R_m }

0 = ({} , {}) = {|}
1 = ({0}, {}} = {0|}
-1 = ({} , {0}) = {|0}

n = {n - 1|}
1/2 = {0|1}

Conway induction: ICGN. 0 satisfies automatically.
Proof: infinite sequence of games not satisfying the property, contradiciton.

Conway induction implies DGC

Finitely many positions: short

Same moves: impartial game

Abelian group:

1. 0 = {|}
   
2. G + H = {(G^L + H) ∪ (G + H^L | (G^R + H) ∪ (G + H^R)}
   
3. -G = {-G^R | -G^L}
   

The set of all Conway games forms a partial order with respect to the comparison operations:

1. G=H. If the second player to move in the game G-H can win (G and H are equal).
   
2. G||H. If the first player to move in the game G-H can win (G and H are fuzzy).
   
3. G>H. If Left can win the game G-H whether he plays first or not (G is greater than H).
   
4. G<H. If Right can win the game G-H whether he plays first or not (G is less than H).
   

A basic theorem shows that all games may be put in a canonical form, which allows an easy test for equality. The canon
ical form depends on two types of simplification:

1. Removal of a dominated option: if G={{L_1,L2,…}|G^R} and L₂>=L₁, then G={{L₂,…}|G^R}; and if G={G^L|{R_1,R_
   

2,…}} and R₁>=R₂, then G={G^L|{R₂,…}}.

1. Replacement of reversible moves: if G={{{A^L|{A^(R₁),A^(R₂),…}},G^(L₂),…}|G^R}, and A^(R₁)<=G, then G={{{A^
   

(R₁^L)},G^(L₂),…}|G^R}.

G is said to be in canonical form if it has no dominated options or reversible moves. If G and H are both in canonical
form, they both have the same sets of left and right options and so are equal.