Proposition 5 — neither firm implements PP when decisions precede positioning

Let consumers be uniform on $[0,1]$, let $t>0$ be the quadratic misfit parameter, and let $F\ge0$ be the fixed PP implementation cost. Firms first choose PP or uniform pricing, then choose any real positions, then compete in prices. The uniform-price firm moves first when exactly one firm uses PP. The source's uniform, both-PP and two reflected asymmetric positioning outcomes are equilibria, and optimal uniform-leader prices exist in both asymmetric outcomes.

For every such pair of optimal leader prices, uniform pricing is strictly dominant for each firm. Consequently, the unique pure Nash equilibrium of the implementation game is

Thus neither firm implements personalized pricing at any nonnegative implementation cost, including $F=0$.

Formalization Note Stage-1 payoffs are computed from consumer-level pricing profits, not entered as the numbers in Table 1. Position deviations range over all real locations. At off-path coincident locations the asymmetric position predicate selects an optimal pricing continuation; the paper does not specify off-path tie selection. The mirrored B-implements-PP positioning case follows the paper's symmetry convention. Reservation value is omitted under full market coverage; uniform-price ties are split equally.