Collusive equilibria with power costs

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

In a collusive equilibrium one player builds a lead in the penultimate round while the other rests. With effort costs c(e) = eθ, the map shows, for each exponent θ from 1 up to a bound you choose and each head start Δ of player A, which player can be the one who leads. Change the values to see regions shrink or vanish.

Solid curves are participation thresholds (IR); dashed curves are no-catch-up thresholds (IC). Stippled bands lie outside the contested range −wA ≤ Δ ≤ wB, where the lead is decisive and both rest. The dotted line marks θ*.
  • Only A leads B rests; A plays (wB − Δ)/2
  • Either player leads two pure-strategy equilibria
  • Only B leads A rests; B plays (wA + Δ)/2
  • No pure-strategy equilibrium neither player can lead while the other rests

Green never reaches past θ* = ln 2 / ln(4/3) ≈ 2.409, whatever the values.

Where the regions come from

Write \(w_i = c^{-1}(v_i)\) for player \(i\)'s reach in one round and \(w^i_2 = c^{-1}(v_i/2)\) for the effort per round at which two rounds cost \(v_i\). The four thresholds are

$$\begin{aligned} \mathit{IR}_A &= w_B - 2w^A_2, \\ \mathit{IC}_B &= 4w^B_2 - 2w_A - w_B, \\ \mathit{IR}_B &= 2w^B_2 - w_A, \\ \mathit{IC}_A &= 2w_B + w_A - 4w^A_2. \end{aligned}$$

On the contested range \(-w_A \le \Delta \le w_B\), the profile \(\bigl(\tfrac{w_B-\Delta}{2},\,0\bigr)\) with A active is an equilibrium exactly when \(\Delta \ge \max(\mathit{IR}_A, \mathit{IC}_B)\): A gains from building the lead and B prefers resting to catching up. The profile \(\bigl(0,\,\tfrac{w_A+\Delta}{2}\bigr)\) with B active is an equilibrium exactly when \(\Delta \le \min(\mathit{IR}_B, \mathit{IC}_A)\). Blue, green and red are where the first, both, or the second hold; white is where neither does, so there is no pure-strategy equilibrium.

With \(c(e) = e^\theta\) one has \(w^i_2 = 2^{-1/\theta} w_i\), so

$$\mathit{IC}_A - \mathit{IC}_B = (w_A + w_B)\bigl(3 - 4\cdot 2^{-1/\theta}\bigr),$$

which changes sign at \(\theta^\ast = \ln 2/\ln(4/3) \approx 2.409\) whatever the values. Beyond \(\theta^\ast\) the two conditions cannot hold together, so green is empty: with costs that convex, at most one player can lead collusively.