Research proof index

Proposition 3.8 (part 2): decomposition at a stable breakpoint. Under convexity and the comparability/attainment hypothesis, if x ∈ StI μ I and x<y in I, then μA (I.left, y) = μA (x,y). Intuition: once x is chosen as a stable breakpoint, the “best value” up to y is fully determined by the subinterval starting at x.

prop4d1₁ is the core statement behind Proposition 4.1: under the two hypotheses h₁ (a weak “eventual improvement” along strict chains) and h₂ (a weak slope-like alternative towards the top), the best-response value μAstar μ coincides with the global infimum μmin μ TotIntvl.

prop4d11₁ shows that if the global extremal values on TotIntvl coincide, then the best-response inequality μBstar μ ≤ μAstar μ holds.

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.