Prof Ortner gave a talk on Tuesday about his recent working paper, “Leader-Driven Collusion” (19 August 2026). The paper finds that a patient cartel leader’s best scheme opens with gradually rising prices before settling at the monopoly price. The method is as interesting as the result. In place of the recursive machinery of Abreu, Pearce and Stacchetti (APS, 1990), the field’s standard tool, the paper uses Lagrangian duality.

One constraint, two readings. In repeated Bertrand with zero cost, a firm that undercuts takes the whole market today and earns nothing thereafter. A fringe firm therefore stays in the cartel only if

$$p_t \le \sum_{s\ge t}\delta^{\,s-t}x_s p_s \qquad \forall t.$$

Here $p_t$ is the cartel price in period $t$, with buyers’ reservation price normalized to $1$; $x_s$ is each fringe firm’s market share in period $s$; and $\delta\in(0,1)$ is the fringe firms’ discount factor. The left side is the deviation payoff, the right side the stream of collusive profits.

APS works on $E$, the set of vectors of firms’ total discounted profits attainable in subgame-perfect equilibrium. A vector belongs to $E$ if it splits into today’s profit plus a continuation vector drawn from $E$, with no firm wishing to deviate. $E$ is then the largest fixed point $E=B(E)$, reached by iterating $B$ downward from the feasible set.

The paper’s Lagrangian method reads them all at once. The leader’s problem becomes a program over whole sequences $(x_t,p_t)_{t\ge0}$, with a multiplier $\lambda_t$ on each constraint. The paper guesses the solution, sets the Lagrangian’s coefficients to zero, solves the resulting recursions for the multipliers in closed form, and verifies a saddle point.

APS’s limit. APS returns the entire equilibrium payoff set and needs no guess, but it works on sets: each application of $B$ maps a subset of $\mathbb{R}^n$ to another. In practice the set must be computed numerically. Judd, Yeltekin and Conklin (2003) approximate it with polytopes, and the exact algorithm of Abreu and Sannikov (2014) covers two players with finite action sets. Neither delivers a closed-form price path.

The Lagrangian. The paper’s method instead fixes a Pareto weight, $\alpha\in[0,1]$ on the leader’s discounted profit and $1-\alpha$ on each fringe firm’s, and solves the program over sequences directly. By weak duality, any feasible multipliers give an upper bound on payoffs. Equality requires a saddle point, which the paper obtains by guessing the solution, solving the multipliers’ recursions and verifying. To get there it relaxes the program, dropping the leader’s incentive constraint and imposing $p_tx_t\ge0$ in place of $x_t,p_t\ge0$, then checks that the solution satisfies the original constraints. This is what lets the paper find optimal equilibria “in any direction without the need to derive the entire frontier.”

Connection between the two methods. Both start from the same incentive constraint: the condition above is the APS decomposition unrolled through time, with the deviator’s continuation fixed at zero. Both also locate points on the same frontier, since the optimum for weights $(\alpha,1-\alpha,\dots,1-\alpha)$ is the support function of $E$ in that direction. They differ in what is known in advance. In Bertrand the worst punishment is static Nash, worth exactly zero, so every constraint’s right-hand side is fixed. In general it is $\min_{v\in E}v_i$, a feature of the set itself, which APS computes alongside everything else. With the punishment fixed, the substitution $\pi_t=x_tp_t$ makes the paper’s program an infinite linear program.

sources

  • Ortner, J. (2026). Leader-Driven Collusion. Working paper, Boston University, 19 August 2026.
  • Abreu, D., Pearce, D., Stacchetti, E. (1990). “Toward a Theory of Discounted Repeated Games with Imperfect Monitoring.” Econometrica 58(5).
  • Judd, K., Yeltekin, S., Conklin, J. (2003). “Computing Supergame Equilibria.” Econometrica 71(4).
  • Abreu, D., Sannikov, Y. (2014). “An Algorithm for Two-Player Repeated Games with Perfect Monitoring.” Theoretical Economics 9.
  • Marcet, A., Marimon, R. (2019). “Recursive Contracts.” Econometrica 87(5).