9 Payoff Matrix and Nash Equilibrium

This chapter contains the main results. We construct an $8 \times 8$ payoff matrix from the strategy archetypes, prove that $\textsc{FurnRush}{}$ is the unique weakly dominant strategy, that $(\textsc{FurnRush}{},\textsc{FurnRush}{})$ is the unique pure Nash equilibrium, and quantify the price of anarchy.

9.1 Payoff matrix

Definition 9.1 Payoff matrix

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The payoff matrix $M : \text{Fin}\, 8 \to \text{Fin}\, 8 \to \mathbb {N}$ assigns $M_{i,j}$ as the expected score of archetype $i$ when the opponent plays archetype $j$. Entries are derived from score estimates adjusted for contention and synergy effects. The matrix is indexed via $\text{toFin}$ and $\text{ofFin}$ conversions between $\text{StrategyArchetype}$ and $\text{Fin}\, 8$.

\begin{tikzpicture} [
cell/.style={minimum width=1.15cm, minimum height=0.75cm, inner sep=0pt,
font=\small\bfseries, text=textDark},
hdr/.style={cell, font=\small\itshape, fill=pastelSlate, text=textDark},
]
% Column headers (opponent)
\node[hdr] at (0, 0.75) {\rotatebox{55}{\scriptsize Opponent}};
\foreach \j/\name in {1/FR, 2/WR, 3/AH, 4/MH, 5/Bal, 6/PF, 7/RE, 8/PC} {
\node[hdr] at (\j*1.15, 0.75) {\name};
}
% Row headers + cells
% Row 0: FurnRush 85 130 135 130 125 135 135 130
\node[hdr] at (0, 0) {FR};
\node[cell, fill=heatMid] at (1*1.15, 0) {85};
\node[cell, fill=heatHigh] at (2*1.15, 0) {130};
\node[cell, fill=heatHigh] at (3*1.15, 0) {135};
\node[cell, fill=heatHigh] at (4*1.15, 0) {130};
\node[cell, fill=heatHigh] at (5*1.15, 0) {125};
\node[cell, fill=heatHigh] at (6*1.15, 0) {135};
\node[cell, fill=heatHigh] at (7*1.15, 0) {135};
\node[cell, fill=heatHigh] at (8*1.15, 0) {130};
% Row 1: WeapRush 80 75 100 85 95 105 100 85
\node[hdr] at (0, -0.75) {WR};
\node[cell, fill=heatMid] at (1*1.15, -0.75) {80};
\node[cell, fill=heatMidLow] at (2*1.15, -0.75) {75};
\node[cell, fill=heatMidHigh] at (3*1.15, -0.75) {100};
\node[cell, fill=heatMid] at (4*1.15, -0.75) {85};
\node[cell, fill=heatMidHigh] at (5*1.15, -0.75) {95};
\node[cell, fill=heatMidHigh] at (6*1.15, -0.75) {105};
\node[cell, fill=heatMidHigh] at (7*1.15, -0.75) {100};
\node[cell, fill=heatMid] at (8*1.15, -0.75) {85};
% Row 2: AnimHusb 75 100 70 100 95 105 100 100
\node[hdr] at (0, -1.5) {AH};
\node[cell, fill=heatMidLow] at (1*1.15, -1.5) {75};
\node[cell, fill=heatMidHigh] at (2*1.15, -1.5) {100};
\node[cell, fill=heatMidLow] at (3*1.15, -1.5) {70};
\node[cell, fill=heatMidHigh] at (4*1.15, -1.5) {100};
\node[cell, fill=heatMidHigh] at (5*1.15, -1.5) {95};
\node[cell, fill=heatMidHigh] at (6*1.15, -1.5) {105};
\node[cell, fill=heatMidHigh] at (7*1.15, -1.5) {100};
\node[cell, fill=heatMidHigh] at (8*1.15, -1.5) {100};
% Row 3: MineHeavy 75 85 95 65 90 100 95 85
\node[hdr] at (0, -2.25) {MH};
\node[cell, fill=heatMidLow] at (1*1.15, -2.25) {75};
\node[cell, fill=heatMid] at (2*1.15, -2.25) {85};
\node[cell, fill=heatMidHigh] at (3*1.15, -2.25) {95};
\node[cell, fill=heatLow] at (4*1.15, -2.25) {65};
\node[cell, fill=heatMid] at (5*1.15, -2.25) {90};
\node[cell, fill=heatMidHigh] at (6*1.15, -2.25) {100};
\node[cell, fill=heatMidHigh] at (7*1.15, -2.25) {95};
\node[cell, fill=heatMid] at (8*1.15, -2.25) {85};
% Row 4: Balanced 70 85 85 85 70 90 85 85
\node[hdr] at (0, -3.0) {Bal};
\node[cell, fill=heatMidLow] at (1*1.15, -3.0) {70};
\node[cell, fill=heatMid] at (2*1.15, -3.0) {85};
\node[cell, fill=heatMid] at (3*1.15, -3.0) {85};
\node[cell, fill=heatMid] at (4*1.15, -3.0) {85};
\node[cell, fill=heatMidLow] at (5*1.15, -3.0) {70};
\node[cell, fill=heatMid] at (6*1.15, -3.0) {90};
\node[cell, fill=heatMid] at (7*1.15, -3.0) {85};
\node[cell, fill=heatMid] at (8*1.15, -3.0) {85};
% Row 5: PeaceFarm 65 75 75 80 75 60 80 75
\node[hdr] at (0, -3.75) {PF};
\node[cell, fill=heatLow] at (1*1.15, -3.75) {65};
\node[cell, fill=heatMidLow] at (2*1.15, -3.75) {75};
\node[cell, fill=heatMidLow] at (3*1.15, -3.75) {75};
\node[cell, fill=heatMid] at (4*1.15, -3.75) {80};
\node[cell, fill=heatMidLow] at (5*1.15, -3.75) {75};
\node[cell, fill=heatLow] at (6*1.15, -3.75) {60};
\node[cell, fill=heatMid] at (7*1.15, -3.75) {80};
\node[cell, fill=heatMidLow] at (8*1.15, -3.75) {75};
% Row 6: RubyEcon 65 75 75 75 70 80 55 75
\node[hdr] at (0, -4.5) {RE};
\node[cell, fill=heatLow] at (1*1.15, -4.5) {65};
\node[cell, fill=heatMidLow] at (2*1.15, -4.5) {75};
\node[cell, fill=heatMidLow] at (3*1.15, -4.5) {75};
\node[cell, fill=heatMidLow] at (4*1.15, -4.5) {75};
\node[cell, fill=heatMidLow] at (5*1.15, -4.5) {70};
\node[cell, fill=heatMid] at (6*1.15, -4.5) {80};
\node[cell, fill=heatLow] at (7*1.15, -4.5) {55};
\node[cell, fill=heatMidLow] at (8*1.15, -4.5) {75};
% Row 7: PeaceCave 70 80 80 85 80 85 80 60
\node[hdr] at (0, -5.25) {PC};
\node[cell, fill=heatMidLow] at (1*1.15, -5.25) {70};
\node[cell, fill=heatMid] at (2*1.15, -5.25) {80};
\node[cell, fill=heatMid] at (3*1.15, -5.25) {80};
\node[cell, fill=heatMid] at (4*1.15, -5.25) {85};
\node[cell, fill=heatMid] at (5*1.15, -5.25) {80};
\node[cell, fill=heatMid] at (6*1.15, -5.25) {85};
\node[cell, fill=heatMid] at (7*1.15, -5.25) {80};
\node[cell, fill=heatLow] at (8*1.15, -5.25) {60};
% Highlight: diagonal cells get a thicker border
\foreach \i in {0,...,7} {
\draw[accentBlue, thick] (\i*1.15+1.15-0.575, -\i*0.75-0.375) rectangle ++(1.15, 0.75);
}
% Highlight: FurnRush row bracket
\draw[accentGreen, ultra thick, rounded corners=2pt]
(1*1.15-0.575, 0.375) rectangle (8*1.15+0.575, -0.375);
% Nash cell marker
\node[anchor=south, font=\tiny\itshape, text=accentGreen] at (1*1.15, 0.4) {Nash};
% Labels
\node[font=\small, text=textDark, rotate=90, anchor=south] at (-0.7, -2.25) {Row player};
\end{tikzpicture} — Figure 9.1 The $8 \times 8$ payoff matrix. Green border marks the FurnRush row (weakly dominant). Blue borders highlight the diagonal (mirror matchups, always depressed). Cell colour intensity reflects payoff magnitude: light green ($55$) to coral ($135$).

Definition 9.2 Row sum

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$\sigma (s) = \sum _{j=0}^{7} M_{\text{toFin}(s),j}$, the total payoff of strategy $s$ summed across all opponent choices.

Definition 9.3 Best response

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$\text{bestResponse}(s)$ is the strategy $t$ that maximizes $M_{\text{toFin}(t),\text{toFin}(s)}$, i.e. the best reply to an opponent playing $s$.

Definition 9.4 Nash equilibrium

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A pair $(a, b)$ is a Nash equilibrium if neither player can unilaterally improve: $a = \text{bestResponse}(b)$ and $b = \text{bestResponse}(a)$.

Definition 9.5 Social welfare

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$\text{socialWelfare}(a,b) = M_{\text{toFin}(a),\text{toFin}(b)} + M_{\text{toFin}(b),\text{toFin}(a)}$. The Nash welfare is $\text{socialWelfare}(\textsc{FurnRush}{},\textsc{FurnRush}{})$. The social optimum is the maximum social welfare over all pairs.

9.2 Matrix properties

Theorem 9.6 All payoffs positive

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$\forall \, i\, j,\; M_{i,j} {\gt} 0$.

Proof ▶

By checking all $64$ entries.

Theorem 9.7 Minimum payoff is 55

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$\forall \, i\, j,\; M_{i,j} \ge 55$, and $M_{6,6} = 55$. The worst outcome is a $\textsc{PeaceCave}{}$ mirror match.

Proof ▶

By exhaustive check of all $64$ entries.

Theorem 9.8 Maximum payoff is 135

✓

$\forall \, i\, j,\; M_{i,j} \le 135$, and $M_{0,2} = 135$. The best outcome is $\textsc{FurnRush}{}$ against $\textsc{PeaceFarm}{}$.

Proof ▶

By exhaustive check.

Theorem 9.9 Diagonal is always depressed

✓

For every strategy $i$, there exists an opponent $j \ne i$ such that $M_{i,j} {\gt} M_{i,i}$. Mirror matchups are never optimal.

Proof ▶

By exhibiting a witness $j$ for each $i$.

9.3 Weak dominance

Theorem 9.10 Furnishing rush is weakly dominant

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For all opponents $j$ and all alternative strategies $s$,

\[ M_{0,j} \ge M_{\text{toFin}(s),j}. \]

That is, $\textsc{FurnRush}{}$ (row $0$) weakly dominates every other row.

Proof ▶

By checking all $8 \times 8 = 64$ pairwise comparisons in the matrix. For each column $j$ and each alternative row $s$, the $\textsc{FurnRush}{}$ entry is $\ge $ the alternative entry.

Theorem 9.11 Strict improvement exists

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For every $s \ne \textsc{FurnRush}{}$, there exists an opponent $j$ where $M_{0,j} {\gt} M_{\text{toFin}(s),j}$. The dominance is therefore weak (not just tied everywhere).

Proof ▶

By exhibiting an explicit witness column for each of the $7$ non-$\textsc{FurnRush}{}$ strategies.

Theorem 9.12 Best response is always furnishing rush

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$\forall \, s,\; \text{bestResponse}(s) = \textsc{FurnRush}{}$. No matter what the opponent does, $\textsc{FurnRush}{}$ is the best reply.

Proof ▶

Since $\textsc{FurnRush}{}$ weakly dominates all alternatives, it maximizes the payoff in every column.

9.4 Nash equilibrium

Theorem 9.13 Furnishing mirror is Nash

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$\text{isNashEquilibrium}(\textsc{FurnRush}{}, \textsc{FurnRush}{})$.

Proof ▶

Both players are playing their best response ($\textsc{FurnRush}{}$) to the other’s strategy ($\textsc{FurnRush}{}$).

Theorem 9.14 Unique Nash equilibrium

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$(\textsc{FurnRush}{}, \textsc{FurnRush}{})$ is the unique pure Nash equilibrium. If $(a, b)$ is any Nash equilibrium, then $a = b = \textsc{FurnRush}{}$.

Proof ▶

In any Nash equilibrium $(a,b)$, we need $a = \text{bestResponse}(b) = \textsc{FurnRush}{}$ and $b = \text{bestResponse}(a) = \textsc{FurnRush}{}$.

9.5 Row sum analysis

Theorem 9.15 Row sum ordering

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The row sums satisfy the strict ordering:

\[ \sigma (\textsc{FurnRush}{}) = 1005 {\gt} \sigma (\textsc{AnimHusb}{}) {\gt} \cdots {\gt} \sigma (\textsc{RubyEcon}{}) = 570. \]

$\textsc{FurnRush}{}$ has the highest aggregate payoff; $\textsc{RubyEcon}{}$ the lowest.

Proof ▶

By computing all $8$ row sums and comparing.

Theorem 9.16 Furnishing rush highest row sum

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$\forall \, s,\; \sigma (\textsc{FurnRush}{}) \ge \sigma (s)$.

Proof ▶

Direct from the row sum ordering.

Theorem 9.17 Total game value

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The sum of all row sums is $5595$. The average payoff per cell is $5595 / 64 = 87$.

Proof ▶

By computation.

9.6 Price of anarchy

Theorem 9.18 Nash welfare

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$\text{nashWelfare} = 170$ (both players score $85$ in the mirror).

Proof ▶

$M_{0,0} + M_{0,0} = 85 + 85 = 170$.

Theorem 9.19 Social optimum

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$\text{socialOptimumValue} = 210$, achieved by pairing strategies that avoid contention.

Proof ▶

By checking all $64$ social welfare values.

Theorem 9.20 Price of anarchy

✓

$\text{socialOptimumValue} {\gt} \text{nashWelfare}$: i.e., $210 {\gt} 170$. The price of anarchy ratio is $21/17 \approx 1.24$. Rational play sacrifices $\sim 19\% $ of social welfare.

Proof ▶

$210 {\gt} 170$ and $210 \times 17 = 3570 = 170 \times 21$.

$\begin{tikzpicture} [ bar/.style={rounded corners=3pt, minimum width=1.6cm, anchor=south west, font=\small\bfseries, text=textDark, inner sep=0pt}, ] % Axes \draw[thick, ->, color=textDark] (0,0) -- (5,0); \draw[thick, ->, color=textDark] (0,0) -- (0,4.8); \node[font=\footnotesize, text=textDark, anchor=east] at (-0.1, 4.5) {welfare}; % Nash bar \fill[pastelSky, rounded corners=3pt] (0.8, 0) rectangle (2.4, 3.4); \node[font=\small\bfseries, text=textDark] at (1.6, 1.7) {170}; \node[font=\footnotesize, text=textDark, anchor=north] at (1.6, -0.15) {Nash}; % Social optimum bar \fill[pastelMint, rounded corners=3pt] (2.8, 0) rectangle (4.4, 4.2); \node[font=\small\bfseries, text=textDark] at (3.6, 2.1) {210}; \node[font=\footnotesize, text=textDark, anchor=north] at (3.6, -0.15) {Social opt.}; % Gap annotation \draw[accentAmber, thick, <->] (4.6, 3.4) -- (4.6, 4.2); \node[font=\scriptsize, text=accentAmber, anchor=west] at (4.7, 3.8) {$\Delta = 40$}; % Ratio \node[font=\scriptsize, text=textDark, anchor=west] at (4.7, 3.3) {PoA $= 21/17$}; \node[font=\scriptsize, text=textDark, anchor=west] at (4.7, 2.9) {$\approx 19\%$ loss}; % Gridlines \foreach \y/\v in {1.0/50, 2.0/100, 3.0/150, 4.0/200} { \draw[pastelSlate, thin, dashed] (0.4, \y) -- (4.6, \y); \node[font=\tiny, text=textDark, anchor=east] at (0.35, \y) {\v}; } \end{tikzpicture}$

Figure 9.2 Nash welfare ($170$) vs. social optimum ($210$). Rational play sacrifices $19\% $ of total welfare, yielding a price of anarchy of $21/17$.

9.7 Contention penalties

Theorem 9.21 Furnishing contention penalty

✓

$M_{0,0} = 85$, $M_{0,1} = 130$, $M_{0,2} = 135$. The $\textsc{FurnRush}{}$ mirror ($85$) is the worst $\textsc{FurnRush}{}$ outcome, showing the contention cost of both players racing for the same furnishings.

Proof ▶

By reading matrix entries.

Theorem 9.22 Asymmetry bounds

✓

The maximum payoff asymmetry is $70$: $\forall \, i\, j,\; |M_{i,j} - M_{j,i}| \le 70$. Among non-$\textsc{FurnRush}{}$ pairs, asymmetry is at most $35$.

Proof ▶

By checking all off-diagonal pairs.

9.8 Score bounds from dominance

Theorem 9.23 Dominant strategy safety margin

✓

$\text{practicalFloor} - \text{doNothingScore} = 115$. Even in the worst case, $\textsc{FurnRush}{}$ scores $115$ points above the do-nothing catastrophe.

Proof ▶

$60 - (-55) = 115$.

Theorem 9.24 Dominant strategy variance

✓

$\text{practicalCeiling} - \text{practicalFloor} = 80$. The score range of $\textsc{FurnRush}{}$ spans $80$ points depending on matchup.

Proof ▶

$140 - 60 = 80$.