Comb March Solver

A Theory of Dynamic Contests with Deadlines and Effort Smoothing Joffrion & Parreiras, 2026

Enter a cost function c(·) with c(0)=0 as a Julia expression (variable e only) plus the players' valuations v_A and v_B and player A's lead Δ. The solver returns every σ(m, n) equilibrium candidate for both regimes (u_A>0 and u_B>0) across nine canonical sizes, each labelled with a verdict. Click a case to inspect its atoms and payoff graph — including the failed comb for no-equilibrium cases.
Engine: forward comb march (robust where Newton false-negatives near affine costs).

Cost function c(e)

Allowed: numeric literals, variable e, operators + − * / ^, and exp, log, log1p, cosh, sinh, tanh, sqrt, abs. Must satisfy c(0) = 0, be non-decreasing (c′ ≥ 0) and convex (c″ ≥ 0) on [0, 2].

Presets: e² e³ cosh(e)−1 exp(e)−1 explin e^2.5

v_A

v_B

Asymmetric cases take a while — please be patient. With v_A = v_B and Δ = 0 the solve returns in a couple of seconds. Once the valuations differ or Δ ≠ 0 the march has far more work to do, and Compute commonly takes around half a minute on this server. That is the solver thinking, not a stuck page. Re-submitting the same inputs afterwards is instant, but reloading mid-solve does not cancel anything — it just starts a second solve alongside the first.

Verdict legend

solution the march assembled a comb with all masses positive (in K), interior branch, and no profitable upward deviation.
no convergence the comb march found no q₁ dial root — there is no candidate profile to report (the only case with no comb to draw).
negative mass the comb closed but the closure system returned a non-positive atom mass, so no valid σ(m,n) of this size exists (typically the support is over-built — one atom too many). The atoms still interleave: the march brackets them into K = {(x, y): y₁ ≤ x₁ ≤ y₂ ≤ x₂ ≤ …} by construction, so this is never an ordering violation. The failed comb is still drawn.
deviable atoms lie in K, but the best-response check finds a profitable upward deviation, or the slack player's participation (IR) payoff is negative. When IR fails (u_A < 0) the comb is still drawn: the blue U_A curve sinks below the u_B = 0 floor — that dive is the visual signature of the failure.
wrong regime the candidate's atoms land on a free-win branch rather than the all-interior σ(m, n) regime.

σ(m, n) is indexed by (slack-IR atoms, binding-IR atoms). The u_B>0 column uses the parameter-swap symmetry σ(m, n, v_A, v_B, Δ | u_B>0) ≡ σ(m, n, v_B, v_A, −Δ | u_A>0).

Cases

Verdict legend