Buying from a Group

The paper studies how a buyer can optimally purchase an indivisible, commonly owned good (e.g., land or a business) from multiple sellers when each seller privately knows his cost and has a veto right (voluntary participation). The buyer must pay a uniform price per share; individual transfers cannot be differentiated ex post. Motivated by applications such as land acquisition and redevelopment, the research question is to characterize the profit-maximizing mechanism under Bayesian incentive compatibility and interim individual rationality when payments must be proportional to ownership shares. The study aims to identify how private information and the uniform-per-share pricing restriction shape the optimal allocation and pricing, and to determine which sellers exert more influence in the mechanism.

The paper connects to public goods and collective trading mechanisms with private information. Classic public goods mechanisms (d'Aspremont and Gérard-Varet, 1979) allow arbitrary transfers; Rob (1989) and Mailath and Postlewaite (1990) show strong inefficiency in profit-maximizing or incentive-feasible mechanisms with many agents. Cramton, Gibbons, and Klemperer (1987) study efficient dissolution under symmetric endowments; Ekmekci, Kos, and Vohra (2016) consider selling fractions of a firm to a buyer. Güth and Hellwig (1986) characterize profit-maximizing public good provision with IC and IR; the present paper’s buyer problem is equivalent up to a sign, with the added proportional-transfers constraint. Virtual values/costs are central in optimal mechanism design (e.g., Myerson, 1981), with extensions to multidimensional types (Cai, Daskalakis, Weinberg, 2013). Here, virtual costs are aggregated with endogenous weights due to proportional transfers. Related voting literature without transfers studies efficiency and weighted-majority rules (Rae, 1969; Schmitz and Tröger, 2012; Krishna and Morgan, 2015; Azrieli and Kim, 2014), where weights often arise from IC constraints; in contrast, weights here are a feature of optimality, not all IC rules. Bargaining-based collective-decision models include Bergstrom (1978), Che (2002), Grossman and Hart (1980), and Oliveros and Iaryczower (2022), often featuring holdout; here holdout is captured via IR constraints. The paper also relates to uniform- versus discriminatory-price auctions (Friedman, 1960; Malvey and Archibald, 1998; Ausubel et al., 2014), noting uniform pricing can reduce collusion incentives.

Model: N≥2 sellers each own share σ_i (sum to 1) of an indivisible good. The buyer’s benefit b is common knowledge. Seller i’s per-unit cost is θ_i drawn independently from distribution F_i on [θ_i, θ̄_i] with continuous, strictly positive density f_i and strictly increasing virtual cost φ_i(θ_i)=θ_i+F_i(θ_i)/f_i(θ_i). Ex post transfers must be proportional to shares: a collective transfer m(θ) is paid and each seller receives σ_i m. Payoffs (after rescaling by σ_i): buyer bx(θ)−m(θ); seller i m(θ)−θ_i x(θ). A direct mechanism consists of allocation rule x:Θ→[0,1] and transfer rule m:Θ→R subject to Bayesian IC and interim IR for each seller. Implementability and reduction: Define interim allocation X_i(θ_i)=E[x(θ_i,θ_{−i})] and interim transfer M_i(θ_i)=E[m(θ_i,θ_{−i})]. Lemma 1(i) shows that despite the proportional-transfer constraint, an allocation rule is implementable (IC and IR for some m) iff it is interim monotone (each X_i weakly decreasing). The standard payment identity holds: M_i(θ_i)=U_i+X_i(θ_i)θ_i+∫0^{θ_i}X_i(t)dt for some constant U_i≥0. The proportional-transfer constraint implies interim transfers must coincide in expectation across agents. Lemma 1(ii) yields the buyer’s maximal profit for a given implementable x: Π(x)=min{i∈N} E[x(θ)(b−φ_i(θ_i))]. Thus, the buyer’s problem reduces to (BP): maximize over interim-monotone x the objective min_i E[x(θ)(b−φ_i)]. Solution via a zero-sum game and convex program: Relax interim monotonicity to obtain a zero-sum game where the Maximizer chooses x and the Minimizer chooses a mixed strategy ω∈Δ^N over which agent’s IR binds. For a given ω, the Maximizer’s best response is deterministic: x_ω(θ)=1{b≥ω·φ(θ)}, i.e., trade iff b exceeds the ω-weighted sum of virtual costs. The minimax value is min_{ω∈Δ^N} E[(b−ω·φ)^+], a convex program in ω. Theorem 1: The essentially unique optimal allocation rule is the weighted allocation x_ω* with ω*∈argmin_{ω∈Δ^N} E[(b−ω·φ)^+]. Equivalently, support(ω*)⊆argmax_{i} E[φ_i | ω*·φ<b]; any agent with positive weight has a binding IR constraint in expectation. The allocation is deterministic and randomization is not beneficial. Weight comparisons: Theorem 2 uses reversed hazard-rate order to rank weights: if φ_i stochastically dominates φ_j in reversed hazard-rate order (possibly up to a shift), then ω_i≥ω_j (strictly if the dominance is strict and ω_j>0). Intuitively, agents ex ante more reluctant to trade (higher per-unit value/cost distributions) receive higher weight and hence greater influence. Example and indirect implementation: Example 1 (two sellers, power distributions F_i(θ)=θ^{α_i} on [0,1]) yields closed-form weights ω*_i=α_i/(α_1+α_2). Virtual costs are linear φ_i(θ)=(1+α_i)θ. Thus, trade occurs iff b≥(α_1+α_2)^{-1}[(1+α_1)θ_1+(1+α_2)θ_2]. An indirect “name-your-price” mechanism implements the optimum: sellers bid s_i; trade occurs iff τ_1 s_1+τ_2 s_2≤b with τ_i=(1+α_i)/[(1+α_1+α_2)α_i]; the buyer pays s_1+s_2+κ b upon trade, where κ=[(1+α_1)(1+α_2)−1]/[(1+α_1)(1+α_2)]. In equilibrium, type θ_i bids ((1+α_i)/α_i)θ_i, inducing the optimal allocation and transfers. Posted-price analysis: Define collective posted-price mechanisms as those with m=px. Proposition 1 shows the optimal within this class is a unanimous posted price (trade if and only if all accept at price p). Proposition 2 shows posted prices are strictly suboptimal when at least two sellers can veto (under mild conditions), since optimal weighted mechanisms require type-dependent pricing even conditional on trade. Group vs single-agent benchmark: Comparing to a benchmark where a single seller owns the whole asset with value v=σ·θ, the single-agent optimal mechanism trades iff b≥v+G(v)/g(v). In the group setting, trade occurs iff b≥ω·φ. Distortions relative to efficiency (b≥σ·θ) differ: the group setting exhibits (i) rotation (weights ω independent of shares σ), (ii) curvature (from inverse hazard terms F_i/f_i), and (iii) downward distortion (virtual cost addition). Interactions can lead to over-trading (trade when inefficient), which does not occur in the single-agent benchmark. Efficiency comparisons depend on b: with i.i.d. θ_i, group mechanisms are more efficient when b is high and less efficient when b is low. Extensions: The paper discusses dominant-strategy implementability (DIC), showing proportional transfers severely limit DIC mechanisms (they reduce to posted-price-like forms and cannot attain the buyer-optimal allocation). It also studies ex-post IR, provides conditions under which buyer-optimal mechanisms can satisfy it, characterizes the entire Pareto frontier (weights on actual and virtual costs), analyzes pre-market share trading (uniform-per-share pricing reduces incentives to trade shares under i.i.d. per-unit costs), and generalizes to approval by subsets of sellers.

• Optimal mechanism structure (Theorem 1): Trade occurs if and only if the buyer’s benefit b exceeds a weighted sum of sellers’ virtual costs, ω·φ(θ). The weights ω are endogenous and uniquely determined by minimizing the convex objective E[(b−ω·φ)^+]. The allocation is deterministic; randomization gives no gain.
• Binding IR and weights: Agents with positive weights have binding IR constraints in expectation. Weights reflect influence: sellers ex ante less inclined to trade (higher per-unit value/cost) receive higher weights.
• Weight ranking (Theorem 2): If seller i’s virtual cost distribution dominates seller j’s in the reversed hazard-rate order (possibly with a positive shift), then ω_i≥ω_j (strict if the dominance is strict and ω_j>0). Thus, more productive landholders (higher per-unit values) get larger weights; if productivity is inversely related to plot size, smaller landholders receive larger weights, and vice versa.
• Posted-price mechanisms: The best collective posted-price mechanism is unanimous (Proposition 1), but any collective posted price is strictly suboptimal when at least two sellers can veto (Proposition 2). Optimal mechanisms require type-responsive pricing conditioned on trade.
• Group vs single-agent: Relative to efficient trade (b≥σ·θ), group-optimal allocations involve rotational and curvature distortions in addition to the usual downward distortion. Interactions can produce over-trading (trade even when inefficient), unlike the single-agent benchmark. With i.i.d. costs, group mechanisms are more efficient when b is large and less efficient when b is small (formal statements in the appendix).
• Example 1 (two power distributions): Closed-form weights ω*_i=α_i/(α_1+α_2); trade iff b≥[(1+α_1)θ_1+(1+α_2)θ_2]/(α_1+α_2). An indirect bidding mechanism with linear equilibrium strategies implements the optimum.
• Pareto frontier (Online Appendix): Pareto-optimal mechanisms trade iff b exceeds a weighted sum of actual and virtual costs; weights on virtual costs are endogenous (influence), while weights on actual costs are Pareto weights. All Pareto-optimal mechanisms entail complex pricing.
• Collusion/robustness: Uniform per-share pricing reduces incentives for sellers (with i.i.d. per-unit costs) to trade shares pre-mechanism; discriminatory pricing can induce such incentives.

The study addresses how to design an optimal buying mechanism from a group of veto-wielding sellers under a fairness/institutional constraint of uniform per-share payments. By characterizing implementable allocations and recasting the buyer’s problem as a convex minimization/zero-sum game, it shows that optimal trade rules depend on endogenous weights on virtual costs. This resolves the allocation problem under proportional transfers and identifies who exerts influence: agents with higher ex-ante per-unit values receive higher weights. The results imply that simple posted prices fail once more than one seller is involved; optimal mechanisms must let the price respond to reported types. In applications such as land acquisition, the mechanism endogenously tilts influence toward more productive landholders (however productivity is related to plot sizes), offering insight into perceived fairness and efficiency. Relative to a single-agent setting, the group environment changes the distortion geometry (rotation and curvature), explaining when over-trading can occur and when the group mechanism can outperform or underperform in efficiency depending on the buyer’s benefit level. The Pareto-frontier characterization further shows how adjusting Pareto weights on seller welfare changes the trade rule while preserving tractability, which is relevant when the buyer is a public entity with broader welfare objectives.

The paper provides an explicit and essentially unique characterization of the buyer-optimal mechanism for purchasing an indivisible, commonly owned good under proportional transfers and private seller costs. Optimal trade occurs when b exceeds a weighted sum of virtual costs, with endogenous weights reflecting sellers’ ex-ante propensity to trade; the mechanism strictly outperforms posted-price schemes. It ranks seller influence via the reversed hazard-rate order and highlights key differences from single-agent mechanism design, including potential over-trading and efficiency comparisons that hinge on the magnitude of b. Extensions characterize the full Pareto frontier, clarify limits of dominant-strategy mechanisms, provide conditions for ex-post IR, and show how uniform per-share pricing mitigates pre-market share-trading incentives. Potential future research includes dynamic or multi-round acquisition, correlated types, networked ownership, heterogeneous fairness constraints, and implementation under legal or political constraints specific to land acquisition.

• Proportional (uniform per-share) transfers are imposed ex post; while motivated by institutional fairness, this constraint rules out discriminatory transfers that might raise buyer profit or change collusion incentives.
• Types are independent with continuous, strictly positive densities; the analysis assumes increasing virtual costs (regularity). Correlation or irregular distributions are not fully addressed.
• Buyer’s benefit b is common knowledge and fixed; uncertainty about b or private buyer values is not modeled.
• The main optimality results are under Bayesian IC and interim IR; dominant-strategy and ex-post IR variants are discussed but restrict implementability and may fail to achieve the buyer-optimal allocation.
• Results are static and one-shot; dynamic considerations (e.g., sequential bargaining, investment, or learning) are not analyzed.
• Efficiency comparisons with the single-agent benchmark rely on parametric or asymptotic arguments (details in the appendix) and may depend on distributional assumptions.