四次BPR路由中仿射型价格无政府性形状的一个精确反例
An Exact Counterexample to Affine-Like Price-of-Anarchy Shape in Quartic BPR Routing
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中文总结 AI 辅助
本文通过精确计算机辅助反例，证明四次BPR路由中仿射型价格无政府性形状定理不成立，并利用区间证书和有理数算术验证了PoA的非单调性。
中文摘要 AI 辅助
我们给出了一个精确的计算机辅助反例，针对需求依赖的价格无政府性（Price of Anarchy）的仿射活跃网络形状定理的直接公共四次扩展。该实例是一个有向网络，包含五个顶点、六条边和三条起点-终点路径，所有路径都具有正有理数成本$c_e(x)=a_e+b_ex^4$。精确的有理区间证书表明，在需求区间$[17,24]$内，所有三条路径都承载正Wardrop流，同时\\[ \operatorname{PoA}(21)>\operatorname{PoA}(17),\qquad \operatorname{PoA}(21)>\operatorname{PoA}(24). \\]因此，连续性迫使在恒定均衡活跃网络下存在内部局部最大值。Krawczyk包含法隔离了Wardrop和社会最优KKT解，初等边界不等式证明了恒定支撑，区间社会成本证明了两个严格比较。在可微点处，我们还推导了一个串边恒等式，解释了公共四次边如何在不改变任一路径分流的情况下改变PoA的导数。证明和证书对每个区间和符号判定都使用精确有理数算术。
英文摘要
We give an exact computer-assisted counterexample to a direct common-degree quartic extension of the affine active-network shape theorem for demand-dependent Price of Anarchy. The instance is a directed network with five vertices, six edges, and three origin--destination paths, all with positive rational costs $c_e(x)=a_e+b_ex^4$. Exact rational interval certificates show that all three paths carry positive Wardrop flow throughout the demand interval $[17,24]$, while \[ \operatorname{PoA}(21)>\operatorname{PoA}(17),\qquad \operatorname{PoA}(21)>\operatorname{PoA}(24). \] Continuity therefore forces an interior local maximum despite a constant equilibrium active network. Krawczyk inclusions isolate the Wardrop and social-optimum KKT solutions, elementary boundary inequalities certify constant support, and interval social costs certify both strict comparisons. At differentiability points, we also derive a serial-edge identity explaining how a common quartic edge can alter the derivative of PoA without changing either route split. The proof and certificate use exact rational arithmetic for every interval and sign decision.
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