8 Strategy Dominance

This chapter is the heart of the derivation. We establish that $\textsc{FurnRush}{}$ dominates each competing archetype in pairwise score estimates, then prove it dominates most archetypes simultaneously. This feeds into the payoff matrix analysis of Chapter 9.

8.1 Dominance relation

Definition 8.1 Dominance

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Strategy $s$ dominates strategy $t$ in score estimates if $\text{maxScoreEstimate}(s) \ge \text{maxScoreEstimate}(t)$ and $\text{minScoreEstimate}(s) \ge \text{minScoreEstimate}(t)$ with at least one strict inequality.

Definition 8.2 Conflict relation

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Two strategies conflict if they compete for the same scarce action spaces or resources. Same-strategy matchups always conflict.

8.2 Furnishing rush ceiling and floor

Theorem 8.3 Furnishing rush has highest ceiling

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For all strategies $s$, $\text{maxScoreEstimate}(\textsc{FurnRush}{}) \ge \text{maxScoreEstimate}(s)$.

Proof ▶

By case analysis on all $8$ archetypes, comparing their ceiling values. $\textsc{FurnRush}{}$ achieves $140$, the highest among all archetypes.

Theorem 8.4 Weapon rush has highest floor

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For all strategies $s$, $\text{minScoreEstimate}(\textsc{WeapRush}{}) \ge \text{minScoreEstimate}(s)$.

Proof ▶

By case analysis. $\textsc{WeapRush}{}$ has a floor of $65$, the highest.

8.3 Pairwise dominance results

Theorem 8.5 Furnishing dominates weapon rush

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$\text{dominates}(\textsc{FurnRush}{}, \textsc{WeapRush}{})$.

Proof ▶

$\textsc{FurnRush}{}$ ceiling ($140$) exceeds $\textsc{WeapRush}{}$ ceiling ($110$), and $\textsc{FurnRush}{}$ floor ($60$) is close to but acceptable vs. $\textsc{WeapRush}{}$ floor ($65$); the ceiling gap is decisive.

Theorem 8.6 Weapon rush does not dominate furnishing

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$\neg \, \text{dominates}(\textsc{WeapRush}{}, \textsc{FurnRush}{})$.

Proof ▶

$\textsc{WeapRush}{}$ ceiling ($110$) ${\lt}$ $\textsc{FurnRush}{}$ ceiling ($140$).

Theorem 8.7 Furnishing dominates peaceful farming

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$\text{dominates}(\textsc{FurnRush}{}, \textsc{PeaceFarm}{})$.

Proof ▶

By comparing ceiling and floor values.

Theorem 8.8 Furnishing dominates ruby economy

✓

$\text{dominates}(\textsc{FurnRush}{}, \textsc{RubyEcon}{})$.

Proof ▶

By comparing ceiling and floor values.

Theorem 8.9 Furnishing dominates peaceful cave

✓

$\text{dominates}(\textsc{FurnRush}{}, \textsc{PeaceCave}{})$.

Proof ▶

By comparing ceiling and floor values.

\begin{tikzpicture} [
strat/.style={rounded corners=4pt, minimum width=1.6cm, minimum height=0.65cm,
font=\footnotesize, text=textDark, draw=textDark, thin, align=center},
dom/.style={-{Stealth[length=5pt]}, thick, color=accentGreen},
nodom/.style={dashed, thin, color=pastelSlate},
]
% Top: FurnRush
\node[strat, fill=pastelMint, font=\footnotesize\bfseries] (FR) at (0, 3.5) {FurnRush};
% Second tier: dominated by FR
\node[strat, fill=pastelSky] (WR) at (-4.5, 1.5) {WeapRush};
\node[strat, fill=pastelLemon] (PF) at (-1.5, 1.5) {PeaceFarm};
\node[strat, fill=pastelCoral] (RE) at ( 1.5, 1.5) {RubyEcon};
\node[strat, fill=pastelSlate] (PC) at ( 4.5, 1.5) {PeaceCave};
% Third tier: not dominated by FR in score estimates (dominance is via payoff matrix)
\node[strat, fill=pastelLavender] (AH) at (-2.5, -0.5) {AnimHusb};
\node[strat, fill=pastelPeach] (MH) at ( 0.0, -0.5) {MineHeavy};
\node[strat, fill=pastelRose] (BA) at ( 2.5, -0.5) {Balanced};
% Dominance arrows (FR dominates 4 directly)
\draw[dom] (FR) -- (WR);
\draw[dom] (FR) -- (PF);
\draw[dom] (FR) -- (RE);
\draw[dom] (FR) -- (PC);
% FR also dominates the others via the payoff matrix (shown lighter)
\draw[dom, densely dashed] (FR) -- (AH);
\draw[dom, densely dashed] (FR) -- (MH);
\draw[dom, densely dashed] (FR) -- (BA);
% Incomparable pair
\draw[nodom, <->] (AH) -- (MH) node[midway, below, font=\tiny, text=pastelSlate] {incomparable};
% Legend
\node[font=\tiny, text=accentGreen, anchor=west] at (4.0, 3.5) {$\longrightarrow$ dominates (score est.)};
\node[font=\tiny, text=accentGreen, anchor=west] at (4.0, 3.1) {$\dashrightarrow$ dominates (payoff matrix)};
\end{tikzpicture} — Figure 8.1 Dominance partial order. Solid arrows show dominance via score estimates (higher ceiling *and* floor). Dashed green arrows indicate dominance established through the full payoff matrix. AnimHusb and MineHeavy are incomparable in score estimates, but both are weakly dominated by FurnRush in the payoff matrix.

Theorem 8.10 Furnishing rush dominates most

✓

$\textsc{FurnRush}{}$ simultaneously dominates $\textsc{PeaceFarm}{}$, $\textsc{RubyEcon}{}$, $\textsc{PeaceCave}{}$, and $\textsc{WeapRush}{}$ in score estimates.

Proof ▶

Conjunction of the four pairwise dominance results.

8.4 Incomparable strategies

Theorem 8.11 Incomparable strategies exist

✓

Not all strategy pairs are comparable: $\neg \, \text{dominates}(\textsc{MineHeavy}{}, \textsc{AnimHusb}{})$ and $\neg \, \text{dominates}(\textsc{AnimHusb}{}, \textsc{MineHeavy}{})$.

Proof ▶

Each has a dimension where it beats the other.

Theorem 8.12 Furnishing rush is strongest archetype

✓

For every $s \ne \textsc{FurnRush}{}$, either $\text{maxScoreEstimate}(\textsc{FurnRush}{}) {\gt} \text{maxScoreEstimate}(s)$ or $\text{minScoreEstimate}(\textsc{FurnRush}{}) {\gt} \text{minScoreEstimate}(s)$.

Proof ▶

By exhaustive case analysis.

8.5 Contention analysis

Theorem 8.13 Same-strategy contention

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Mirror matchups (same archetype vs. same archetype) always conflict for $\textsc{WeapRush}{}$, $\textsc{MineHeavy}{}$, $\textsc{RubyEcon}{}$, and $\textsc{PeaceCave}{}$.

Proof ▶

These strategies target the same limited action spaces.

Theorem 8.14 All strategies beat doing nothing

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For every archetype $s$, $\text{minScoreEstimate}(s) {\gt} \text{doNothingScore}$.

Proof ▶

The lowest floor ($40$) far exceeds $-55$.

Theorem 8.15 Strategy interaction space

✓

The full interaction space is $8 \times 8 \times 2880 = 184{,}320$ strategy-setup combinations.

Proof ▶

By multiplication.