Optimal Play in Caverna: A Formal Proof of Weak Dominance

5 Animal Husbandry

Animals contribute to scoring (Chapter 2.1) and food (Chapter 3.1). Housing animals requires infrastructure: pastures, stables, and dogs.

5.1 Capacity system

A small pasture holds \(2\) animals; a large pasture holds \(4\). Each stable doubles the capacity of its pasture. A large pasture with two stables holds \(16\) animals.

Definition 5.2 Dog-sheep capacity
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Dogs act as sheepdogs: \(n\) dogs allow housing \(n+1\) sheep in living spaces (no pasture needed).

Theorem 5.3 Stable doubles capacity

A stable doubles small pasture capacity (\(2 \to 4\)) and large pasture capacity (\(4 \to 8\)).

Proof

By computation.

Theorem 5.4 Two stables quadruple large pasture

A large pasture with two stables holds \(4 \times 4 = 16\) animals, the maximum single-space animal capacity.

Proof

\(\text{largePastureTwoStableCapacity} = 16 = 4 \times \text{largePastureCapacity}\).

Theorem 5.5 Dog-sheep linear scaling

\(\text{dogSheepCapacity}(1) = 2\), \(\text{dogSheepCapacity}(3) = 4\), and the function is strictly monotone.

Proof

Direct from \(f(n) = n + 1\).

Theorem 5.6 Dogs worse than stabled large pasture

For \(n {\lt} 15\), \(\text{dogSheepCapacity}(n) {\lt} 16\). Dogs are inferior to a fully stabled large pasture for mass animal housing.

Proof

\(n + 1 {\lt} 16\) for \(n {\lt} 15\).

Theorem 5.7 Dog breakeven

\(\text{dogSheepCapacity}(1) = \text{smallPastureCapacity} = 2\). One dog matches a small pasture for sheep housing.

Proof

Both equal \(2\).