duaLab

Game value

Contents

1. Definition

Write $U:\mathbf{Games}\to\mathbf{Set}$ for the functor taking a game to its set of positions. A game value is a cocone under $U$: a set $A$ with functions $v_X:U(X)\to A$, one for each game, such that $v_X=v_YU(f)$ for every game morphism $f:X\to Y$.

See Definition 3.17.

2. Properties

Proposition 2.1 (Recursive calculation)★

Let $\mathcal P_{\mathrm{fin}}(A)$ denote the finite subsets of $A$. Every algebra $\alpha:\mathcal P_{\mathrm{fin}}(A)\to A$ determines a game value by recursively applying $\alpha$ to the values of the available options.

See Proposition 3.18.

3. Examples

Outcome, Grundy number, endedness, birthday, misère outcome and remoteness arise this way. See §3.5 and Corollary 3.25.

Definition 3.1 (Terminal game)

The terminal game has the hereditarily finite sets as positions and a move $x\to y$ exactly when $y\in x$. Every game has a unique game morphism to it. A hereditarily finite set is a finite set whose elements are themselves hereditarily finite.

See Proposition 3.31 and Definition 3.32.

5. References