Auctions versus Negotiations: Hoffmann & Vladimirov (2025)

What this is. The paper’s core propositions, the model it builds on (a seller choosing between depth and breadth of bidder competition when payments can be contingent), and the theoretical mechanism it contributes: enough to know what it found and how, without reading all 45 pages. To replicate or extend it, read the full source at the original.

TL;DR

The paper develops a theory of auctions versus negotiations that allows for general (state-contingent) payment structures. A seller choosing between optimal negotiations with a small group of bidders and an ascending-bid auction with one more bidder can strictly prefer negotiations - even against the benchmark result of Bulow and Klemperer (1996) that auctions dominate. The key driver is not the reserve price but bargaining power over the payment structure: when the asset is complementary to bidder productivity (synergies increase in types), negotiating for contingent payments (equity, royalties, performance bonuses) extracts more rent than cash competition. The paper builds on the rent-extraction efficiency trade-off studied by Inderst and Vladimirov (2019), extends the full-surplus extraction result of Liu and Bernhardt (2021) to general securities, and uses the security-bid auction framework of DeMarzo, Kremer, and Skrzypacz (2005). The motivating empirical fact that negotiations with few bidders are as common as auctions without lower premia comes from Boone and Mulherin (2007). The auction revenue benchmark draws on Myerson (1981)‘s revenue equivalence theorem. Negotiations dominate if the type distribution is sufficiently dispersed, absolute valuations are high, and the complementarity condition holds.

Core results

#	Result	Locator	Magnitude
R1	If the seller can extract the full surplus in bilateral negotiations, her expected revenue is strictly higher than from an auction with two cash-bidding competitors	Proposition 1(i), p. 1780	With uniform $\theta$, bilateral negotiations yield $\Pi^{fb}_{neg} = (\alpha_Y - \alpha_N + \frac{1}{2}(\beta_Y - \beta_N))\Delta x$ versus auction $\Pi_{comp} = (\alpha_Y - \alpha_N + \frac{1}{3}(\beta_Y - \beta_N))\Delta x$; negotiations up to 50% higher in Example 1
R2	Full-surplus extraction requires the seller to negotiate for contingent payments: the first-best contract has $\Delta w^{fb} > 0$ and is feasible iff valuations are increasing in productivity (complements case, $\beta_Y/\beta_N \in (1, \alpha_Y/\alpha_N]$)	Proposition 1(ii), pp. 1779-1780	The first-best contract is $\{w^{fb}, \Delta w^{fb}\}$ with $\Delta w^{fb} = (1-\beta_N/\beta_Y)\Delta x > 0$ and $w^{fb} = \alpha_Y(\beta_N/\beta_Y - \alpha_N/\alpha_Y)\Delta x$
R3	In the substitutes case ($v'(\theta) < 0$), an auction with one more bidder always yields higher expected revenue than optimal negotiations	Proposition 2, p. 1786	In substitutes, seller optimally demands pure cash in negotiations (Lemma 2) and bidders choose cash in auctions (Lemma 3), so payment structure is irrelevant; Bulow-Klemperer result applies directly
R4	In the complements case ($v'(\theta) > 0$) when full surplus extraction is infeasible, efficient bilateral negotiations dominate competition if and only if the Gini coefficient $G$ of the productivity-type distribution satisfies $G \geq \frac{\alpha_Y/\alpha_N - 1}{\beta_Y/\beta_N - 1}$	Proposition 3, equation (20), p. 1788	Condition is always satisfied if $\alpha_Y/\alpha_N \geq 1/G$; always holds for any $\beta_Y/\beta_N > \alpha_Y/\alpha_N$ if $\alpha_Y/\alpha_N \geq 1/G$
R5	Setting the payment structure takes precedence over setting a reserve price in the seller’s pecking order: bargaining power over the payment structure has equilibrium value on its own, but bargaining power over the reserve price is valuable only when combined with payment-structure power	Proposition 5, p. 1791	Formally, $\Pi^r_{neg}$ is always smaller than $\Pi_{comp}$, while $\Pi^{s+r}_{neg}$ can be larger than $\Pi_{comp}$, and $\Pi^s_{neg}$ can also be larger than $\Pi_{comp}$
R6	The optimal selling mechanism with $n \geq 2$ bidders is a two-stage mechanism: Stage 1 is a standard English auction to identify the highest valuation bidder; Stage 2 is a take-it-or-leave-it offer with the seller’s preferred payment structure to the last remaining bidder	Proposition 6, p. 1793	The Stage 2 offer is: complements case with full surplus extractable, demands $\omega^{FB}$; otherwise demands $w = 0$, $\Delta w = (1 - p_{N}(\tilde\theta)/p_Y(\tilde\theta))\Delta x$ at the optimal reserve

Overall (paper’s conclusion). Negotiations over payments are valuable in many corporate finance settings. The paper resolves the theoretical puzzle that negotiations are widely observed even when their revenue advantage over auctions is difficult to explain with reserve prices alone. The key value driver in negotiations is bargaining power over the mix of cash and contingent pay, not the reserve price. Negotiations are more likely to dominate when the asset creates higher synergies at more productive types, valuations are dispersed, and the type-independent component of valuations is high relative to the total upside.

Theory / model

The paper studies a single seller (she) selling an indivisible asset to $n$ risk-neutral bidders (they/he) indexed $i = 1, \ldots, n$. The asset can be a takeover target, patent, or employee’s human capital. All parties are risk-neutral and there is no discounting.

Project cash flows and bidder types. Each bidder $i$ has a productivity (quality) type $\theta_i$ drawn independently from distribution $F$ on $[0, 1]$. The project either fails (cash flow $x \geq 0$) or succeeds (cash flow $x + \Delta x > x$). The probability of success depends on bidder type and whether the bidder acquires the asset ($a_i = Y$) or not ($a_i = N$), with the linear specification (equation (2), p. 1776):

$$p_{a}(\theta) = \alpha_a + \beta_a \theta, \quad a \in \{Y, N\}, \quad \theta \in [0,1], \tag{2}$$

where $\alpha_a, \beta_a > 0$ and $\alpha_a + \beta_a \leq 1$. A bidder’s expected cash flow under allocation $a$ is $X_a(\theta) = x + p_a(\theta)\Delta x$, strictly increasing in $\theta$. A bidder’s valuation is his willingness to pay for the asset (equation (1), p. 1776):

$$v(\theta) := X_Y(\theta) - X_N(\theta) = (p_Y(\theta) - p_N(\theta))\Delta x. \tag{1}$$

The complements case arises if $\beta_Y/\beta_N > 1$ so that $v'(\theta) > 0$ (more productive types have higher willingness to pay). The substitutes case arises if $\beta_Y/\beta_N < 1$ so that $v'(\theta) < 0$.

Payment contracts. Payments can be in general securities. If bidder $i$ acquires the asset, he pays the seller $w_i$ in the low-cash-flow state and $w_i + \Delta w_i$ in the high-cash-flow state. Here $w_i$ is the cash payment, $\Delta w_i \geq 0$ is the contingent payment, and the payment structure is captured by the ratio $\omega_i = \Delta w_i/(w_i + \Delta w_i)$. Examples of contingent payments include royalties, stock options, and performance bonuses.

The seller’s expected payment from a contract $\omega = \{w, \Delta w\}$ when the buyer’s type is $\theta$ is (p. 1779):

$$\pi(\theta, \omega) := w + p_Y(\theta)\Delta w.$$

Full surplus extraction requires $\pi(\theta, \omega) = v(\theta)$ for all $\theta$, which from equation (3) (p. 1779) gives:

$$w + p_Y(\theta)\Delta w = (p_Y(\theta) - p_N(\theta))\Delta x \quad \text{for all } \theta. \tag{3}$$

The first-best contract solving (3) is (equations (4)-(5), p. 1779):

$$\Delta w^{fb} = \left(1 - \frac{\beta_N}{\beta_Y}\right)\Delta x, \tag{4}$$ $$w^{fb} = \alpha_Y\left(\frac{\beta_N}{\beta_Y} - \frac{\alpha_N}{\alpha_Y}\right)\Delta x. \tag{5}$$

This contract is feasible ($w^{fb}, \Delta w^{fb} \geq 0$) if and only if $\beta_Y/\beta_N \in (1, \alpha_Y/\alpha_N]$, that is, in the complements case but not too steeply.

Game structure. At $t = 0$ the seller decides between (i) negotiations: choosing the optimal mechanism for the $n$ bidders already present, including setting the payment structure; or (ii) competition: attracting one more bidder so that $n + 1$ bidders compete in a standard ascending-bid (English) auction where bidders choose their own payment structure. Cash flows are realized at $t = 1$ and the winning bidder pays according to the agreed contract (pp. 1777-1778).

Method

The paper’s solution method is mechanism design with state-contingent payments and bilateral-contract analysis under asymmetric information, building on principal-agent contracting and elements of bayesian-persuasion (the seller’s mechanism design with general securities).

Optimal negotiations (seller designs the mechanism). The seller maximizes expected revenue over a menu of contracts $W$, with the set of accepting types $\Theta_W \subseteq [0,1]$. The seller’s problem (equation (6), p. 1781) is:

$$\max_W \int_{\Theta_W} \pi(\theta, \omega_\theta)\,dF(\theta) + \int_{[0,1]\setminus\Theta_W} \underline{w}\,dF(\theta), \tag{6}$$

subject to feasibility ($w_\theta, \Delta w_\theta \geq 0$), individual rationality, and incentive compatibility. Participation requires (equations (7)-(8), p. 1781):

$$v(\theta) - \pi(\theta, \omega_\theta) = \max_{\omega \in W} v(\theta) - \pi(\theta, \omega) \geq 0 \quad \text{for all } \theta \in \Theta_W, \tag{7}$$ $$\max_{\omega \in W} v(\theta) - \pi(\theta, \omega) < 0 \quad \text{for all } \theta \notin \Theta_W. \tag{8}$$

The cutoff type $\tilde\theta(\omega)$ indifferent between acquiring and not is (equation (9), p. 1782):

$$\tilde\theta(\omega) := \frac{w + \alpha_N \Delta x - \alpha_Y(\Delta x - \Delta w)}{\beta_Y(\Delta x - \Delta w) - \beta_N \Delta x}. \tag{9}$$

The seller’s information rent for type $\theta > \tilde\theta$ under the contingent-only contract ($w = 0$) is (equation (10), p. 1782):

$$v(\theta) - \pi(\theta, \omega) = (\theta - \tilde\theta)\beta_Y \left(\underbrace{\left(1 - \frac{\beta_N}{\beta_Y}\right)\Delta x}_{= \Delta w^{fb}} - \Delta w\right). \tag{10}$$

Competition (bidders choose payment structure). The English auction establishes a reservation price that each active bidder must match. Remaining active bidders choose whether to compete in cash or other securities, subject to the seller’s acceptance constraint (equation (12), p. 1785):

$$\int_0^1 \pi(\theta, \omega)\,d\widetilde{F}(\theta|\omega) \geq \underline{w}. \tag{12}$$

The equilibrium outcome is given by Lemma 3: with $n \geq 2$ bidders, they optimally offer pure cash payments ($w > 0$, $\Delta w = 0$), and the winner pays the second-highest valuation (p. 1785-1786). The seller’s expected revenue is the expected valuation of the bidder with the second-highest valuation (equation (16), p. 1787):

$$\Pi_{comp} = \int_0^1 v(\theta)\, 2(1 - F(\theta))\,dF(\theta). \tag{16}$$

Revenue difference decomposition. Taking the difference between $\Pi_{neg}(0)$ (efficient negotiations without a reserve price) and $\Pi_{comp}$ (equations (17)-(19), p. 1787):

$$\Pi_{neg}(0) - \Pi_{comp} = \int_0^1 v'(\theta)F(\theta)(1-F(\theta))\,d\theta - \int_0^1 \varrho(\theta, 0)\,dF(\theta), \tag{19}$$

where the first term is positive in the complements case ($v'(\theta) > 0$) and equals $\Pi^{fb}_{neg} - \Pi_{comp}$, while the second term is the bidder’s expected information rent in negotiations. Negotiations dominate iff the rent is sufficiently small, which is governed by condition (20).

Empirical specifications

This is a pure-theory paper. There are no regressions, datasets, or empirical specifications. The paper’s propositions are established by analytical proofs in the Appendix (pp. 1801-1813). The paper does derive comparative statics and testable implications for M&A, patent licensing, and employee compensation.

Key theoretical comparative statics (Proposition 4, p. 1790). The revenue advantage of efficient bilateral negotiations $\Pi_{neg}(0) - \Pi_{comp}$ is higher if:

(i) The bidders’ productivity type distribution becomes more dispersed in the sense of a mean-preserving spread (higher Gini coefficient $G$), because more dispersed valuations lower the auction revenue $\Pi_{comp}$ (the second-highest valuation falls in expectation) while leaving $\Pi_{neg}(0)$ unchanged.

(ii) The type-independent component of bidders’ valuations, captured by $\alpha_Y/\alpha_N$, is higher, because higher $\alpha_Y/\alpha_N$ allows the seller to demand a larger contingent payment $\Delta w$ acceptable to all types, moving closer to the full-rent extraction contract $\Delta w^{fb}$.

Necessary and sufficient condition for negotiations to dominate (equation (20), p. 1788):

$$G := \frac{\int_0^1 F(\theta)(1-F(\theta))\,d\theta}{\int_0^1 \theta\,dF(\theta)} \geq \frac{\beta_Y/\beta_N\,/\,\alpha_Y/\alpha_N - 1}{\beta_Y/\beta_N - 1}, \tag{20}$$

where $G \in (0,1)$ is the Gini coefficient of the productivity-type distribution $F$. This condition is always satisfied if $\alpha_Y/\alpha_N \geq 1/G$.

Multi-bidder extension (Proposition 7, p. 1794-1795). The condition for efficient negotiations with $n$ bidders to dominate competition with $n+1$ bidders in the complements case (equation (21), p. 1795):

$$\frac{\int_0^1 F(y)^n (1-F(y))\,dy}{\int_0^1 \left(\int_y^1 \frac{\alpha_Y(q-y)}{\alpha_Y + \beta_Y y} dF(q)\right)(n-1)F(y)^{n-2}\,dF(y)} \geq \frac{\beta_Y/\beta_N\,/\,\alpha_Y/\alpha_N - 1}{\beta_Y/\beta_N - 1}. \tag{21}$$

This condition holds for any $\beta_Y/\beta_N > \alpha_Y/\alpha_N$ if $\alpha_Y/\alpha_N \geq 1/G_n$, where $G_n$ is explicitly defined in the Appendix.

Datasets used

Dataset	Role in paper	Wiki page
None (pure theory)	All results are derived analytically	N/A

No empirical data were used. The paper’s claims are theoretical propositions derived from the formal model. Testable implications for M&A, patent licensing, and compensation are discussed in Section VII (pp. 1796-1800).

When to read the full paper

Use the original if you are: studying optimal mechanism design with general security payments; designing M&A sale processes and evaluating when negotiations versus auctions maximize revenue; modeling patent licensing or employee compensation negotiation; extending the framework to seller private information, common values, or risk aversion (Internet Appendix); or checking the full formal proofs (Appendix, pp. 1801-1813).

Attribution and rights

Source: peer-reviewed, The Journal of Finance 80(3), June 2025. This distillation was extracted by an LLM on 2026-06-06 and is not human-verified or independently reproduced. The CC BY-NC 4.0 licence permits reproduction for non-commercial purposes; the verbatim PDF is not hosted in this batch.

Citation: Hoffmann, Florian, and Vladimir Vladimirov. “Auctions versus Negotiations: The Role of the Payment Structure.” The Journal of Finance 80, no. 3 (June 2025): 1769-1813. DOI: 10.1111/jofi.13446. © 2025 The Author(s). Licensed under CC BY-NC 4.0. This page is an adaptation by the Institute for Automated Research: core results extracted and re-expressed; changes were made.