Collusion with Optimal Information Disclosure

This paper asks how a third-party intermediary (an “algorithm”) that observes market demand or costs superior to competing firms should optimally disclose that information to maximize the firms’ collusive profit in a repeated Bertrand competition setting. The motivation is the rise of algorithmic pricing intermediaries such as RealPage in apartment rentals, A2i Systems in retail gasoline, and Rainmaker in hotel rooms, as well as offline cartel facilitators like AC-Treuhand.

The model extends the canonical Rotemberg–Saloner (1986) repeated Bertrand framework with stochastic demand. The key technical assumption is that firm profit is affine in the unknown state s, so expected profit depends only on the expected state. This holds for binary states, linear demand with unknown intercept (D(p,s) = s − p), and linear demand with unknown per-unit cost. The algorithm observes s and commits to a known disclosure policy mapping s to a public signal. The solution concept is pure-strategy subgame-perfect equilibrium, and the paper solves for the disclosure policy and equilibrium that jointly maximize collusive profit.

The main result (Theorem 1) is that the unique optimal disclosure policy is upper censorship: there is a cutoff ŝ such that demand states s < ŝ are disclosed and result in the corresponding monopoly price p^m(s), while demand states s ≥ ŝ are pooled — only the event {s ≥ ŝ} is disclosed — and result in the monopoly price for the mean concealed state, p^m(s*), where s* = E[s | s ≥ ŝ]. The reduction to a static information design problem (Lemma 1) is the key technical step: optimal collusive profit equals V*, the greatest fixed point of V = max_{G ∈ MPC(F)} E_G[min{π^m(s), δV/((1−δ)(n−1))}]. The “capped monopoly profit” min{π^m(s), π^max} is convex-then-concave in s, and classical results from the static information design literature (Kolotilin 2018; Dworczak and Martini 2019) then imply upper censorship is uniquely optimal.

Two features of the optimal equilibrium are notable. First, prices are rigid (constant at p^m(s*)) whenever s ≥ ŝ — the opposite of Rotemberg–Saloner’s “price wars during booms.” The logic is that pooling high demand states with a lower average state is more profitable than cutting prices, because pooling reduces the current-period deviation gain without sacrificing as much on-path profit. Second, for demand states s ∈ (ŝ, s*), the equilibrium price p^m(s*) exceeds the monopoly price p^m(s) — supra-monopoly pricing occurs for a range of intermediate states. Monopoly pricing is attainable at each such state in isolation, but recommending the higher price p^m(s*) is necessary to make the pooling incentive-compatible at states s > s*.

Comparing to full disclosure, Proposition 1 shows that optimal disclosure leads to strictly higher prices at every demand state, and hence unambiguously lower consumer surplus. Proposition 3 shows that improving the algorithm’s accuracy (a mean-preserving spread of F) reduces expected consumer surplus whenever consumer surplus under monopoly pricing is concave in s — a natural condition. This result is more pessimistic than prior work (Sugaya–Wolitzky 2018; Miklos-Thal–Tucker 2019), which found ambiguous effects because those papers assumed full disclosure.

Comparative statics (Proposition 2): fewer firms or a higher discount factor δ increases collusive profit V* and makes prices more flexible (raises ŝ). Collusion is impossible if and only if δ < (n−1)/n, the same threshold as under full disclosure.

Extensions maintain the core results. With Markov (persistent) demand (Section 4 / Theorem 2), upper censorship remains optimal but the cutoff ŝ(s) depends on last-period demand s: under positive serial correlation, ŝ(s) is decreasing in s, so the algorithm discloses less information following high demand. With differentiated products under a symmetric linear demand system (Section 5 / Theorem 3), the optimal policy censors an intermediate interval [ŝ_L, ŝ_H] and discloses both the lowest and highest demand states, because at high states the absence of an upper bound on equilibrium profit makes disclosure with price-cutting optimal.

Q: What is the core research question and why is it policy-relevant? A: The paper asks how an informed intermediary should optimally disclose demand or cost information to competing firms to maximize their collusive profit. It is directly motivated by antitrust cases against RealPage (sued by the US DOJ in August 2024), A2i Systems/Kalibrate, and Rainmaker, all of which gather market data from competing firms and recommend prices. The theory also applies to offline facilitators like AC-Treuhand, prosecuted by the European Commission for disclosing competitively sensitive information.

Q: What is the affinity assumption and why does it matter? A: The paper assumes that firm profit π(p, s) is affine (linearly increasing) in the demand or cost state s for each price p. This implies that expected profit for any distribution over states equals profit evaluated at the expected state: E[π(p,s)] = π(p, E[s]). As a consequence, any disclosure policy is equivalent, from a profit standpoint, to choosing a distribution G of the firms’ posterior mean beliefs over s, and G must be a mean-preserving contraction of the prior F (by Blackwell 1953). The assumption is satisfied for binary states, linear demand with unknown intercept, and linear demand with unknown cost.

Q: What is the key reduction result (Lemma 1) and what does it achieve? A: Lemma 1 reduces the problem of finding an optimal repeated-game equilibrium to a static information design problem. Optimal collusive profit equals V*, the greatest fixed point of V = max_{G ∈ MPC(F)} E_G[min{π^m(s), δV/((1−δ)(n−1))}], and this is attained by a symmetric, stationary, grim-trigger equilibrium. The reduction works because, under Bertrand competition, static deviation gains are proportional to on-path payoffs, creating a one-to-one correspondence that allows the repeated-game constraint to be folded into a single-period objective.

Q: Why is upper censorship the uniquely optimal disclosure policy? A: The static information design problem has a “capped monopoly profit” objective: min{π^m(s), π^max}, where π^max = δV*/((1−δ)(n−1)) is the maximum per-period profit that satisfies incentive constraints. Because π^m(s) is convex (as the maximum of affine functions) and the cap π^max is constant, the overall objective is convex for s below the cap and constant (then concave) above it — i.e., convex-then-concave in s. Classical results for linear information design (Kolotilin 2018; Dworczak and Martini 2019) imply that the unique optimal policy for a convex-then-concave objective is upper censorship.

Q: What is the supra-monopoly pricing result and why does it arise? A: For demand states s ∈ (ŝ, s*), the equilibrium price is p^m(s*) > p^m(s), meaning firms charge above the monopoly price for the current state. This arises because the pooling policy must recommend a single price for all states s ≥ ŝ, and the recommended price is p^m(s*) where s* = E[s | s ≥ ŝ]. At intermediate states s ∈ (ŝ, s*), this price exceeds the local monopoly price. The algorithm accepts lower profit at these states because it is necessary to maintain the pooled recommendation at higher states where monopoly pricing would otherwise require a price cut.

Q: How does optimal disclosure compare to full disclosure in terms of consumer surplus? A: Proposition 1 shows that collusive prices under optimal disclosure are strictly higher at every demand state compared to full disclosure (Rotemberg–Saloner). In Rotemberg–Saloner, high demand states trigger price cuts (“price wars during booms”) to deter deviation; under optimal disclosure, high states are pooled and prices are instead rigid at p^m(s*). Because prices are higher at all states, consumer surplus is unambiguously lower under optimal disclosure.

Q: What does Proposition 3 say about the effect of algorithmic accuracy on consumer surplus? A: Proposition 3 states that if consumer surplus under monopoly pricing, CS(s), is concave in s, then a mean-preserving spread of F (i.e., improved algorithmic accuracy) reduces expected consumer surplus. This result is more pessimistic than prior work by Sugaya–Wolitzky (2018) and Miklos-Thal–Tucker (2019), which found ambiguous effects. The difference is that those papers assumed full disclosure, so better accuracy tightened incentive constraints and sometimes forced price cuts. Under optimal selective disclosure, a more accurate algorithm always raises average prices because the algorithm withholds information that would have forced price cuts.

Q: What are the comparative statics with respect to the number of firms and the discount factor? A: Proposition 2 establishes that a decrease in the number of firms n or an increase in the discount factor δ increases collusive profit V* and makes collusive prices more flexible (raises ŝ). The intuition for fewer firms making prices more flexible is that with fewer firms, incentive constraints bind for a narrower range of demand states, so less pooling is needed. Collusion is impossible if and only if δ < (n−1)/n, the same threshold as under full disclosure.

Q: How does the model generate empirically testable predictions distinct from other collusion models? A: The model predicts: (1) the equilibrium price distribution has support on an interval [p^m(s_bar), p^m(ŝ)] plus a single mass point at the higher price p^m(s*); (2) prices are pro-cyclical overall but rigidly fixed at p^m(s*) for all but the lowest demand states; (3) the gap p^m(s) − p(s) is non-monotone — zero at low states, negative (supra-monopoly) at intermediate states, and positive at high states; (4) prices are more flexible when firms are more patient or fewer. The rigid high price combined with a flexible interval of lower prices is described as a distinctive collusive marker not present in other models.

Q: How does the model relate to the empirical literature testing Green–Porter versus Rotemberg–Saloner? A: Rotemberg–Saloner predicts counter-cyclical prices (price wars during booms), while Green–Porter predicts pro-cyclical prices. Empirical tests (e.g., Porter 1983, Ellison 1994) have typically found pro-cyclical prices, favoring Green–Porter. The present model generates pro-cyclical prices through a different mechanism — perfect monitoring plus selectively disclosed demand information — showing that pro-cyclical prices are consistent with perfect monitoring when the information intermediary optimally pools high demand states. The paper suggests that distinguishing the theories requires estimating the gap between price and monopoly price over the cycle: under Green–Porter, collusion succeeds better in high demand states; under this model, collusion succeeds better in low demand states.

Q: What narrative evidence from the RealPage case corroborates the model’s predictions? A: The US DOJ complaint against RealPage states that “in down markets… [RealPage] instills pricing discipline in landlords, curbing normal fully independent competitive reactions by substituting them with interdependent decision-making,” and that RealPage advertised that its AI helps clients “avoid the race to the bottom in down markets.” This is consistent with the model’s prediction of flexible monopoly prices at low demand states and a rigid, supra-monopolistic price in normal times. The Kumatori Contractors Cooperative case (studied by Kawai, Nakabayashi, and Ortner 2024) corroborates the censorship result: that organization took drastic steps to limit bidders’ information about costs on the largest projects — exactly the states where deviation is most tempting.

Q: How do results change with persistent (Markov) demand? A: Theorem 2 shows that upper censorship remains uniquely optimal with Markov demand, but the cutoff ŝ(s) now depends on last-period demand s. Under positive serial correlation, ŝ(s) is decreasing in s: the algorithm discloses less information after high demand because firms are more optimistic and thus more tempted to deviate. Under negative serial correlation, ŝ(s) is increasing. The optimal collusive price is no longer always equal to the monopoly price for the disclosed mean demand, and the expected price conditional on last-period demand can be countercyclical (similar to Rotemberg–Saloner), even though the current-period price is always monotone in current demand.

Q: How does the optimal disclosure policy change with differentiated products? A: With a symmetric linear demand system (Section 5, Theorem 3), the optimal policy censors an intermediate interval [ŝ_L, ŝ_H] and discloses both the lowest and the highest demand states. At high demand states s > ŝ_H, the algorithm discloses the state and recommends a price below monopoly (to satisfy incentive constraints), because with differentiated goods there is no upper bound on equilibrium profit and profit is convex in s at high states, making disclosure with price-cutting optimal. Mathematically, the capped monopoly profit is piecewise-convex rather than convex-then-concave, so the optimal policy is intermediate-interval censorship rather than upper censorship. The Appendix A version extends to general demand systems and capacity constraints with the same qualitative logic.

Q: What are the main limitations and directions for future work acknowledged by the authors? A: The paper identifies three main limitations. First, if profit is not affine in s (i.e., expected profit depends on more than the mean state), the information design problem becomes non-linear and upper censorship is typically suboptimal, though it remains approximately optimal when the problem is close to linear. Second, the model assumes the algorithm’s objective is to maximize industry profit; if the intermediary is a profit-maximizing seller of software (as in Harrington 2022), the objective may instead be to maximize the profit differential between adopters and non-adopters. Third, the model assumes all firms use the algorithm; allowing partial adoption would require modeling firms’ incentives to subscribe. The paper notes that incorporating these considerations “could be an interesting direction for future research.”

Upper Censorship (disclosure policy): A disclosure policy in which demand states below a cutoff ŝ are revealed to firms (along with the corresponding monopoly price recommendation), while states above ŝ are pooled — only the event {s ≥ ŝ} is disclosed — with a single monopoly price recommendation p^m(s*) for the mean concealed state s* = E[s | s ≥ ŝ]. This is the uniquely optimal disclosure policy in the baseline model.

Capped Monopoly Profit: The per-period profit objective in the reduced static information design problem: min{π^m(s), π^max}, where π^max = δV*/((1−δ)(n−1)) is the maximum industry profit attainable in a single period without violating incentive constraints. This function is convex-then-concave in s, which drives the optimality of upper censorship.

Supra-Monopoly Pricing: Equilibrium prices that exceed the monopoly price for the realized demand state. In the model, this occurs for states s ∈ (ŝ, s*), where the algorithm’s pooled recommendation p^m(s*) is above the local monopoly price p^m(s). It arises because the pooled recommendation must be incentive-compatible at the highest concealed states.

Price Rigidity: The feature of the optimal equilibrium in which the collusive price is constant at p^m(s*) for all demand states s ≥ ŝ. The algorithm achieves this by withholding information about high demand states, preventing the “price wars during booms” predicted by Rotemberg–Saloner (1986) under full disclosure.

Algorithmic Accuracy: In the paper’s terms, the informativeness of the algorithm’s signal about s, formalized as the precision of the distribution F. Improving accuracy corresponds to a mean-preserving spread of F (Blackwell 1953). A more accurate algorithm always increases collusive profit; under the concavity condition on consumer surplus, it also reduces expected consumer surplus.

Mean-Preserving Contraction (MPC(F)): The set of distributions G of firms’ posterior mean beliefs over s that are consistent with Bayesian updating of the prior F. By Blackwell (1953), a disclosure policy is feasible if and only if it induces a distribution G ∈ MPC(F). This is the feasibility constraint in the static information design problem.

Affinity in the state: The assumption that π(p, s) is affine (linearly increasing) in s for each price p. This implies E[π(p,s)] = π(p, E[s]), so expected profit is determined entirely by the expected state, enabling the reduction of the disclosure problem to choosing a distribution of posterior means.