Research proof index

prop4d12 derives the equality μmin μ TotIntvl = μmax μ TotIntvl from the stronger equality μmax μ TotIntvl = μ TotIntvl, provided a pointwise dichotomy that rules out “intermediate” points simultaneously satisfying both comparisons.

prop4d14 is the dual analogue of prop4d12: starting from μmin μ TotIntvl = μ TotIntvl and a suitable dichotomy, it deduces μmax μ TotIntvl = μmin μ TotIntvl.

prop4d16₂ is the main bridge: under SlopeLike μ and both chain conditions, Nash equilibrium is equivalent to the equality μmin μ TotIntvl = μmax μ TotIntvl. The proof packages the slope-like axiom into weak slope-like data on restrictions, and then combines prop4d11₁ and prop4d11₂ with the earlier characterisations.

prop4d18₁ shows that semistability implies the best-response inequality μBstar μ ≤ μAstar μ in a linearly ordered setting.

prop4d3₁ is the dual form of Proposition 4.1: under hypotheses h₁ and h₂ phrased for strict anti-chains and bottom-anchored alternatives, the best-response value μBstar μ coincides with the global supremum μmax μ TotIntvl. The proof reduces to prop4d1₁ on the order dual, and then translates the result back via the duality lemmas.

Proposition 4.8 (implementation form): μQuotient r d is slope-like. Assumptions: - Additivity of d and r along composable intervals: for x<y<z, we have d(x,z) = d(x,y) + d(y,z) and r(x,z) = r(x,y) + r(y,z). - Positivity condition: if r(x,y) = 0 then d(x,y) > 0. Conclusion: - The quotient construction μQuotient r d satisfies the slope-lik...