11 Sensitivity Analysis
The point-estimate payoff matrix proves weak dominance conditional on 64 exact values. This chapter strengthens the result: we replace each scalar entry with a closed interval bounding the true payoff, and prove that \(\text{FurnRush}{}\) remains weakly dominant for every matrix consistent with these bounds.
11.1 Interval payoff matrix
A payoff interval \([l, h]\) with \(l \le h\) bounds the true payoff for a particular (row, column) matchup. The point estimate from payoffMatrix satisfies \(l \le M_{i,j} \le h\).
The interval payoff matrix assigns a \(\text{PayoffInterval}\) to each cell. Error bounds are asymmetric and per-cell:
Mirror matchups (diagonal): \(\varepsilon = 2\).
Cross-archetype matchups: \(\varepsilon = 5\).
Weapon rush vs. furnishing rush: \(\varepsilon = 3\).
For all \(i, j\), the point estimate \(M_{i,j}\) lies within the interval \([\text{lo}(i,j),\, \text{hi}(i,j)]\).
11.2 Robust weak dominance
For every column \(j\) and every non-furnishing alternative row \(i \ne 0\),
Equivalently: for any true payoff matrix within the intervals, \(\text{FurnRush}{}\) weakly dominates every alternative.
The robust margin in column \(j\) is \(\text{lo}(0,j) - \max _{i {\gt} 0}\, \text{hi}(i,j)\). A non-negative margin implies robust weak dominance in that column.
The tightest robust margin is \(0\), occurring in column \(0\) (mirror matchup): \(\text{FurnRush}{}\) interval \([83, 87]\) meets \(\text{WeapRush}{}\) interval \([77, 83]\) at the boundary \(83 \ge 83\).
Every column has a non-negative robust margin.
All columns except the mirror (\(j \ne 0\)) have a robust margin of at least \(20\) points, making them resilient to substantially larger estimation errors.
11.3 Robust Nash equilibrium and welfare
\((\text{FurnRush}{}, \text{FurnRush}{})\) is a Nash equilibrium for every payoff matrix consistent with the interval bounds.
The Nash welfare lies in \([166, 174]\). The point estimate \(170\) is contained in this interval.
The social optimum candidate (\(\text{FurnRush}{}\) vs. \(\text{AnimHusb}{}\)) has welfare in \([200, 220]\).
Even in the best case for Nash welfare (\(174\)) and worst case for the social optimum (\(200\)), the social optimum exceeds Nash welfare. The price of anarchy is robust: selfish play always costs welfare.
11.4 Error bound analysis
Every point estimate differs from its interval boundary by at most \(\varepsilon _{\max } = 5\) points.
If all error bounds were widened by \(1\), robust dominance would fail in column \(0\) (furnishing rush lower bound \(82\) vs. weapon rush upper bound \(84\)). This precisely quantifies the fragility of the mirror-column result.