Drivers learn city-scale dynamic equilibrium

The paper formalizes a multi-market oligopoly game and proves: (i) existence of a Nash equilibrium (NE), (ii) symmetry of every NE, (iii) an “essentially unique” aggregate characterized by φx(sx) ≤ ν with equality on active markets and the inverse relation s*x = φx^{-1}(ν) with Σx s*x = n, and (iv) global asymptotic stability of the NE under projected gradient adjustment. These appear in the appendix as Propositions 1–7 and associated definitions of φ and Φ, together with a Lyapunov calculation for V(s)=Φ(s*)−Φ(s) (e.g., definitions and concavity of Φ in , ; existence and symmetry in , , ; characterization via φ and inverse mapping in ; Lyapunov stability in , ). The candidate solution reaches the same substantive conclusions but via a different path: it constructs a planner’s potential Φ and shows the symmetric profile at a maximizer s* is a NE by a deviation-vs-potential inequality (existence), derives symmetry via KKT with strict monotonicity (symmetry), recovers the inverse φ−1 characterization and uniqueness of s* (characterization), and presents both aggregate and microscopic Lyapunov arguments for global convergence (stability). The only tension is terminological: the paper claims the game is “not an aggregate game” in Selten’s sense, whereas the candidate informally calls it “aggregative.” This does not affect correctness. One proof detail slightly underdeveloped in the paper is the concavity of each player’s payoff (Prop. 2), which is stated without full derivation; the candidate’s existence proof does not rely on this and therefore fills the gap constructively. Overall, both are correct, with different proofs and emphases (paper: convex-game/KKT; model: potential/Lyapunov).