# Games theory

## Table of Contents

## ¶ Conway games gamestheory

- L, R : sets of games, then G = <L, R> is a game
- DGC: no infinite sequence of games G
^{i}= <L^{i}, R^{i}> with G^{(i+1)}∈ L^{i}∪ R^{i}for all i ∈ N

L and R are left and right options of G.

G = {L_{1}, L_{2}, … L_{n} | R_{1}, R_{2} … R_{m} }

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

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

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

## ¶= ({0}, {0}) = {0|0} gamestheory

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:

- 0 = {|}
- G + H = {(G
^{L}+ H) ∪ (G + H^{L}| (G^{R}+ H) ∪ (G + H^{R})} - -G = {-G
^{R}| -G^{L}}

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

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

- Removal of a dominated option: if G={{L
_{1,L}_{2},…}|G^{R}} and L_{2}>=L_{1}, then G={{L_{2},…}|G^{R}}; and if G={G^{L}|{R_{1,R}_

2,…}} and R_{1}>=R_{2}, then G={G^{L}|{R_{2},…}}.

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

(R_{1}^{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.