Bilateral Collusion in Second-Price Sealed-Bid Auctions Under Heterogeneous Risk Aversion

Abstract

Second-price sealed-bid auctions with collusion are well studied, but the role of bidders’ risk aversion remains unclear. We study a two-bidder auction with heterogeneous constant relative risk aversion under the variable-bribe and fixed-bribe mechanisms, characterize the corresponding collusive equilibria, and examine effects on collusion and the seller’s expected revenue. Under the variable-bribe mechanism, we obtain an equilibrium in which lower proposer values separate and higher values pool. Greater proposer risk aversion increases the bribe, the responder’s acceptance cutoff, the probability of successful collusion, and the probability of misallocation, while reducing the seller’s expected revenue. Responder risk aversion does not affect the equilibrium. Under the fixed-bribe mechanism, primitive sufficient conditions establish a regular interior cutoff equilibrium. Within this regular interior equilibrium class, greater responder risk aversion raises the responder’s acceptance cutoff and the probability of misallocation, but lowers the proposer’s offer cutoff and the seller’s expected revenue. Greater proposer risk aversion lowers both the proposer’s offer cutoff and the responder’s acceptance cutoff, while its effects on the probability of successful collusion and the seller’s expected revenue can be positive or negative. Finally, we briefly discuss four extensions: variable-bribe collusive equilibria with positive initial wealth or a reserve price, and fixed-bribe collusive equilibria with general concave utility or an endogenously determined amount.

1. Introduction and Literature Review

1.1. Introduction

Second-price sealed-bid auctions are among the most fundamental mechanisms in auction theory. Vickrey’s classic analysis establishes truthful bidding [1], and its state-by-state dominance argument also preserves truthful bidding under strictly increasing utility. Bolotnyy and Vasserman [2] find infrastructure procurement bids consistent with firms limiting quantity-risk exposure, providing an empirical motivation for risk-averse preferences. When bidders can collude, internal transfer payments may increase their payoffs at the seller’s expense. Decarolis, Goldmanis, and Penta [3] show how common-agency coordination of competing advertisers’ bids for multiple positions can lower revenue, a related concern for bilateral agreements. Existing studies examine arrangements that include bidding rings and direct bribery between bidders [4,5,6,7]. Eső and Schummer [4], in particular, study bilateral collusion in a second-price sealed-bid auction in which one bidder offers the other a fixed or variable bribe to submit a zero bid. Eső and Schummer’s analysis assumes risk neutrality. We extend Eső and Schummer’s [4] analysis of bilateral collusion to risk-averse bidders and further allow the two bidders to differ in their degrees of risk aversion. Because the proposer and responder occupy different strategic roles in bilateral collusion, heterogeneous risk aversion allows us to identify more fully how each bidder’s risk aversion affects the collusive equilibrium. We assume that the two roles have different constant relative risk aversion (CRRA) preferences. The proposer offers a bribe in exchange for the responder’s zero bid; acceptance gives the proposer the object at a zero auction price, while rejection leads to truthful bidding. The two mechanisms differ in the proposer’s choice: under variable bribes, it chooses the amount according to its own valuation, whereas under fixed bribes, the amount is specified in advance and it chooses only whether to offer it.

The main contribution is to identify how heterogeneous risk preferences affect collusion through the proposer’s offer decision and the responder’s acceptance decision, with qualitatively different effects across the two bribery mechanisms. More specifically, the paper makes four contributions. First, it extends the literature on collusion in second-price sealed-bid auctions [4,5,6] by analyzing the effects of bidders’ risk aversion on the collusive equilibrium, allocative efficiency, and the seller’s expected revenue. Second, it extends the two-bidder pre-auction bribery model of Eső and Schummer [4] from risk-neutral to risk-averse bidders, building on Llorente-Saguer and Zultan’s [8] observation that risk aversion changes the bribery schedule. Under variable bribes, although the schedule retains the Eső–Schummer form under a transformed distribution, we further introduce positive initial wealth to examine the potential equivalence arising from utility homogeneity and assess the robustness of our findings on the effects of risk aversion on the collusive equilibrium. Third, by allowing heterogeneous risk aversion, we identify separately how the proposer’s and responder’s risk aversion affects the collusive equilibrium. For fixed bribes, these comparisons hold within the regular interior cutoff equilibrium under the stated primitive sufficient conditions. Fourth, we extend the analysis of collusive equilibria in second-price sealed-bid auctions to reserve prices and endogenously determined fixed bribes.

The remainder of the paper is organized as follows. Section 1.2 reviews the related literature. Section 2 presents the model. Section 3 and Section 4 analyze variable and fixed bribes, respectively. Section 5 examines the seller’s expected revenue under the two mechanisms. Section 6 develops the extensions. Section 7 concludes. Appendix A contains all proofs.

1.2. Literature Review

The closest model is the bilateral pre-auction bribery game of Eső and Schummer [4]. A variable offer reveals lower proposer values, with higher values pooling once acceptance is certain, whereas a fixed offer identifies only an offering pool. Both protocols can generate inefficient allocation, but the different information available at the acceptance stage changes the responder’s decision. This bilateral offer differs from the internal allocation and transfer mechanisms studied by Graham and Marshall [5] for single-object second-price and English rings and McAfee and McMillan [7] for first-price cartels. Meanwhile, without bid control, Marshall and Marx [6] show that a ring can eliminate all internal competition under second-price but not first-price rules. Mailath and Zemsky [9] show that efficient second-price collusion is possible with asymmetric value distributions.

Later bilateral models change the bargaining sequence or available proposals. Rachmilevitch [10] introduces two sequential bribing opportunities and characterizes efficient equilibria under pure strategies and a refinement. Lu, Lu, and Riis [11] allow both an offered bribe and a requested payment, obtaining a robust equilibrium with certain collusion. With random outside competition and difficulties in verifying compliance, Chen and Tauman [12] use a price-dependent side-payment mechanism to address incentives to use shill bidders. Rachmilevitch’s first-price bribery analysis [13] provides an auction-rule comparison, with its corrigendum clarifying sufficient conditions for zero-bribe equilibrium existence. Deng [14] studies pre-auction acquisitions in single-object procurement, another way of reducing competition before bidding. These studies connect the form of an agreement to the gains from collusion and incentives to disclose private information.

The value environment also matters. Troyan [15] studies interdependent values and affiliated signals. Acceptance can lead the proposer to revise its estimate of the object’s value, creating a briber’s curse, particularly with outside competition. In our independent-private-values environment, an offer does not change either bidder’s own valuation. Belief updating instead affects the responder’s distribution of auction surplus after rejection.

Experiments connect these information and transfer channels to behavior. Llorente-Saguer and Zultan [8] study bribery in first- and second-price auctions; failed negotiations can distort subsequent first-price bidding. Agranov and Yariv [16] examine communication and voluntary post-auction transfers, rather than the binding bilateral commitment assumed here. Noussair and Seres [17] find that second-price bidding rings reduce experimental efficiency, connecting collusion to the misallocation examined here. Sujarittanonta and Viriyavipart [18] investigate reserve prices as a deterrent to collusion in first-price auction experiments, finding lower average bribe offers and a lower rate of successful bribes. Hsueh et al. [19] link two-stage Chinese land auctions with secret reserves to patterns consistent with collusion. This motivates attention to reserves, which we examine in a second-price environment. Eső and Schummer’s supplement [20] provides the risk-neutral reserve-price extension of both bribery protocols.

Risk aversion introduces a further distinction between the protocols. Pratt [21] provides the preference comparison underlying our concave-transformation extension. Zhang and Liu [22] study how default affects the revenue consequences of risk aversion with uncertain valuations in second-price auctions. Bos, Gomez-Martinez, and Onderstal [23] examine risk-averse bidding when bidders care about an outside observer’s assessment of their values. These studies connect risk effects to the uncertainty and information entering bidders’ payoffs. Truthful second-price bidding follows from the classical state-by-state dominance argument [1], which strictly increasing utility preserves. Before bidding, however, a revealing variable offer makes the responder’s rejection payoff known, whereas a fixed offer leaves a lottery over proposer values. Because the upper pooled variable offer is also accepted by every responder through a statewise payoff comparison, responder risk aversion affects the fixed-bribe acceptance decision but not the variable-bribe equilibrium. We examine how these information differences generate role-specific risk effects, including the respective extensions to positive wealth, a reserve price, and general concavity.

2. Model

A seller allocates one indivisible object through a second-price sealed-bid auction. There are two bidders. Bidder P is the potential proposer and bidder R is the potential responder. Their values are independent and identically distributed and satisfy

$$V,X\stackrel{\mathrm{iid}}{˜}F\mathrm{on}\left[0, \overline{v}\right],$$

where F(0) = 0, $F\left(\overline{v}\right)=1$, F is continuously differentiable, and f > 0 on $\left(0, \overline{v}\right)$. Each bidder observes only its own value. Their utilities over payoffs y are

$$u_P\left(y\right)=y^a, u_R\left(y\right)=y^d, 0<a, d\le 1.$$

The power utilities are defined over nonnegative monetary surplus from the auction and the agreement. The proposer’s and responder’s coefficients of relative risk aversion are 1 − a and 1 − d, respectively, and belong to [0,1). These parameters describe curvature over gains without initial wealth. Section 6 examines CRRA utility over positive initial wealth and separate general-utility extensions.

A lower exponent represents greater relative risk aversion. If no agreement is reached, both bidders enter the auction and bid truthfully. Truthful bidding remains weakly dominant under every strictly increasing utility, so continuation monetary payoffs are (v − x)₊ for the proposer and (x − v)₊ for the responder, and the seller receives min{v, x}.

The pre-auction stage follows the two bribery mechanisms in Eső and Schummer [4]. Under the variable-bribe mechanism, a proposer of type v chooses a bribe b ∈ [0,v]. Under the fixed-bribe mechanism, a bribe $\beta \in \left(0, \overline{v}\right)$ is specified. A proposer with v ≥ β chooses whether to make the fixed-bribe offer. A proposer with v < β does not make the offer. The responder observes the offer and its own value, then accepts or rejects. Acceptance commits the responder to submit a zero bid. The proposer receives the object and pays the bribe, and the seller receives no auction payment. Rejection leads to the truthful second-price auction. Strategies and beliefs form a perfect Bayesian equilibrium (PBE).

Following the exogenously fixed-transfer formulation of Eső and Schummer [4], under the fixed-bribe protocol, β is a publicly known transfer specified by a pre-existing agreement opportunity before private values are realized. Economically, this can represent a standardized payment stipulated by a prior arrangement between the bidders. The baseline takes that amount as given: neither bidder chooses it within the game, and after observing its valuation, the proposer decides only whether to use the opportunity. Acceptance is binding, committing the proposer to the specified payment and the responder to a zero bid, with no subsequent renegotiation. Thus, the amount is fixed in advance, but payment occurs only if an offer is made and accepted. Section 6.4 instead assigns the public, pre-valuation choice of the amount to the proposer.

For the continuous equilibrium with an upper pooling region studied below, assume that F is log-concave and, for every $z\in \left(0, \overline{v}\right]$, satisfies the elasticity condition

$${\displaystyle \frac{\mathit{zf}\left(z\right)}{F\left(z\right)}}\ge 1-a.$$

Log-concavity makes the equilibrium type map monotone. The elasticity bound guarantees global incentive compatibility (IC) under constant relative risk aversion. It is automatic at a = 1, and power cumulative distribution functions $F\left(z\right)={(z/\overline{v})}^k$ with k ≥ 1 satisfy all these conditions. The fixed-bribe analysis uses the regular interior cutoff equilibria defined in Section 4.

3. Variable Bribe

For a separating offer that reveals proposer value v and specifies bribe b, a responder of value x receives b upon acceptance and the known amount (x − v)₊ upon rejection. The responder therefore accepts exactly when x ≤ v + b. Write this acceptance cutoff as q = v + b and the proposer’s surplus after acceptance as g = v − b. For $q\in \left(0, \overline{v}\right]$, the separating candidate can be parameterized as

$$g\left(q\right)={{\displaystyle \int }}_0^q{\left[{\displaystyle \frac{F\left(t\right)}{F\left(q\right)}}\right]}^{2/a}\mathit{dt}, v\left(q\right)={\displaystyle \frac{q+g\left(q\right)}{2}}, b\left(q\right)={\displaystyle \frac{q-g\left(q\right)}{2}}.$$

(1)

Here, we parameterize v, b, and g by the responder’s acceptance cutoff q.

Remark 1.

The endogenous acceptance cutoff q parameterizes the proposer’s value, bribe, and surplus. Conditional on the proposer’s value v, the relevant objects are the equilibrium bribe b(v), the proposer’s monetary payoff after successful collusion g(v), and the responder’s acceptance cutoff q(v). Appendix A establishes a one-to-one correspondence between v and q on the separating region, so the objects in Equation (1) can equivalently be expressed as functions of v.

Set g(0) = v(0) = b(0) = 0. Log-concavity implies that v is strictly increasing. Let q = v⁻¹ on its range, with q(0) = 0, and set

$$c=v\left(\overline{v}\right), \overline{b}=b\left(\overline{v}\right)=\overline{v}-c.$$

The cutoff c is the proposer value at which the acceptance set reaches the top of the responder’s value support. Our first proposition gives the proposer’s separating offer strategy with an upper pooling region under the variable-bribe mechanism.

Proposition 1.

Under the variable-bribe mechanism in the model of Section 2, a perfect Bayesian equilibrium is given by

$$b\left(v\right)=\left\{\begin{array}{cc}b\left(q\left(v\right)\right),& 0\le v<c,\\ \overline{b},& c\le v\le \overline{v}.\end{array}\right.$$

An offer specifying the bribe b(q) below $\overline{b}$ reveals type v(q) and is accepted iff x ≤ q. The offer specifying the bribe $\overline{b}$ is made by all types v ≥ c and is accepted by every responder type. On the separating part, the equilibrium bribe also solves

$${\displaystyle \frac{\mathit{db}\left(v\right)}{\mathit{dv}}}={\displaystyle \frac{f\left(v+b\left(v\right)\right)\left[v-b\left(v\right)\right]}{\mathit{aF}\left(v+b\left(v\right)\right)-f\left(v+b\left(v\right)\right)\left[v-b\left(v\right)\right]}} \mathrm{for} 0\le v<c, b\left(0\right)=0.$$

(2)

Proof. 

See Appendix A. □

Remark 2.

If F is log-concave and zf(z)/F(z) ≥ 1 on $\left(0,\overline{v}\right]$, then, as a ↓ 0,

$$b_a(v)\to \min \{v,\overline{v}/2\},Q_a(v)\to \min \{2v,\overline{v}\}$$

uniformly on the value support. Moreover, g_a(q) → 0 uniformly for $q\in [0,\overline{v}]$, $c_a\to \overline{v}/2$, and ${\overline{b}}_a\to \overline{v}/2$.

Proof. 

See Appendix A. □

Remark 3.

Log-concavity is sufficient for the continuous construction but is not necessary for the existence of an equilibrium. On [0,1], the non-log-concave distribution F(v) = v exp[0.6(v² − 1)] supports the same equilibrium for 0.2 < a ≤ 1. A second distribution in Appendix A admits a non-collusive PBE even though the candidate separating type map is nonmonotone.

Proof. 

See Appendix A. □

Remark 4. 

Because CRRA utility without initial wealth is homogeneous, the separating schedule has a transformed-distribution representation. For a true proposer value v₀ and an offer indexed by q ≥ v₀, the reduced objective $F(q){[v_0-b(q)]}^a$ has the same maximizers as F(q)^1/a[v₀ − b(q)]. The candidate schedule therefore coincides with the Eső–Schummer schedule under G_a = F^1/a. To compare risk preferences, however, we hold F fixed; a downward deviation with q < v₀ has expected utility

$$F(q){[v_0-b(q)]}^a+{\displaystyle {\int }_q^{v_0}{(v_0-x)}^a}\mathit{dF}(x).$$

Section 6.1 introduces positive initial wealth and establishes the proposer-risk direction beyond homogeneous utility over auction gains.

We next study how the bidders’ risk aversion and the value distribution affect the equilibrium bribe function and the responder’s acceptance cutoff. Define the acceptance frontier

$$Q\left(v\right)=\left\{\begin{array}{cc}q\left(v\right),& v<c,\\ \overline{v},& v\ge c.\end{array}\right.$$

The reverse hazard rate f/F measures the marginal proportional increase in acceptance probability from raising the cutoff. A higher rate makes an expanded acceptance set more valuable relative to the bribe’s cost, inducing a weakly higher equilibrium bribe and acceptance frontier, as Proposition 2 shows.

Proposition 2.

Consider two distributions F₁ and F₂ on $\left[0, \overline{v}\right]$ in Proposition 1. Suppose that, for every $z\in \left(0, \overline{v}\right)$,

$${\displaystyle \frac{f_1\left(z\right)}{F_1\left(z\right)}}\ge {\displaystyle \frac{f_2\left(z\right)}{F_2\left(z\right)}}.$$

Then b₁(v) ≥ b₂(v) and Q₁(v) ≥ Q₂(v) for every proposer value.

Proof. 

See Appendix A. □

We next examine how each bidder’s risk aversion affects the separating offer strategy with an upper pooling region. To simplify notation, objects used in the comparative statics for proposer risk aversion carry a subscript a. For example, we rewrite b(v) as b_a(v). The collusive agreement converts an uncertain monetary payoff without collusion into a certain monetary payoff under successful collusion. The proposer offers a larger bribe to obtain a higher probability of acceptance. A more risk-averse proposer therefore raises the bribe to reduce uncertainty, which also raises the responder’s acceptance cutoff.

Proposition 3.

Let 0 < a₁ < a₂ ≤ 1. For every v > 0,

$$b_{a_1}\left(v\right)>b_{a_2}\left(v\right), Q_{a_1}\left(v\right)\ge Q_{a_2}\left(v\right).$$

Proof. 

See Appendix A. □

Remark 5.

b(v), Q(v) and c are independent of d.

Proof. 

See Appendix A. □

We next examine allocative efficiency under the variable-bribe mechanism and the distribution of monetary payoffs between the bidders. We define the probability of successful collusion, the probability of misallocation, and the expected valuation loss respectively as

$$\begin{array}{c}{C}_V\left(a\right)={{\displaystyle \int }}_0^{\overline{v}}F\left(Q_a\left(v\right)\right)\mathit{dF}\left(v\right),\\ I_V\left(a\right)={{\displaystyle \int }}_0^{\overline{v}}\left[F\left(Q_a\left(v\right)\right)-F\left(v\right)\right]\mathit{dF}\left(v\right),\\ W_V\left(a\right)={{\displaystyle \int }}_0^{\overline{v}}{{\displaystyle \int }}_v^{Q_a\left(v\right)}\left(x-v\right)\mathit{dF}\left(x\right)\mathit{dF}\left(v\right).\end{array}$$

An accepted offer is inefficient exactly when v < x ≤ Q(v). The responder values the object more, but the responder’s zero bid gives the object to the proposer.

Remark 6.

The equilibrium in Proposition 1 is inefficient with positive probability, and

$$I_V\left(a\right)=C_V\left(a\right)-{\displaystyle \frac{1}{2}}.$$

Moreover, I_V(a) and W_V(a) are strictly decreasing with a but independent of d.

Proof. 

See Appendix A. □

Finally, let ${\mathsf{\Pi }}_P^V\left(a\right)$ and ${\mathsf{\Pi }}_R^V\left(a\right)$ denote the bidders’ ex ante expected monetary payoffs. Our seventh remark shows that greater proposer risk aversion raises the bribe, shifting expected monetary payoff from the proposer to the responder even as successful collusion becomes more frequent.

Remark 7.

${\mathsf{\Pi }}_P^V\left(a\right)$ strictly increases with a, while ${\mathsf{\Pi }}_R^V\left(a\right)$ strictly decreases with a.

Proof. 

See Appendix A. □

4. Fixed Bribes

Under the fixed-bribe mechanism, when the proposer offers the bribe β, the responder learns only that the proposer belongs to the offering pool. Consider cutoff strategies in which the proposer offers iff v ≥ B and the responder accepts iff x ≤ A, with $\beta <B<A<\overline{v}$. Bayes’ rule gives the posterior

$$\mathit{Pr}\left(V\le v|V\ge B\right)={\displaystyle \frac{F\left(v\right)-F\left(B\right)}{1-F\left(B\right)}}, B\le v\le \overline{v}.$$

For later use, define

$$N\left(z\right)={{\displaystyle \int }}_0^z{\left(z-x\right)}^a\mathit{dF}\left(x\right),K\left(A,B\right)={{\displaystyle \int }}_B^A{\left(A-v\right)}^d\mathit{dF}\left(v\right).$$

Here, N(z) is the proposer’s expected utility from entering the auction without an offer, and K(A, B)/[1 − F(B)] is the responder’s expected utility from rejection conditional on observing an offer. At the proposer cutoff, acceptance yields monetary surplus B − β, while rejection yields zero because B < A. The responder at its cutoff instead compares the fixed bribe with a lottery over (A − V)₊ under the offer posterior. These two comparisons give the marginal indifference conditions

$$F\left(A\right){(B-\beta )}^a=N\left(B\right), {\beta }^d\left[1-F\left(B\right)\right]=K\left(A, B\right).$$

(3)

Equation (3) therefore determines the proposer’s offer cutoff and the responder’s acceptance cutoff jointly.

These indifference conditions identify marginal types; a single-crossing condition ensures that they also describe the offer decisions of the other proposer types. For β < v ≤ A, define

$$\rho \left(v; \beta \right)={\displaystyle \frac{N\left(v\right)}{{\left(v-\beta \right)}^a}},$$

as the minimum acceptance probability required for the proposer to be willing to pay the fixed bribe. For v ≥ A, define

$$p\left(v; A\right)=v-{\left[E\left[{\left(v-X\right)}^a|X\le A\right]\right]}^{1/a}.$$

This is the maximum fixed bribe that a high-value proposer is willing to pay for the responder’s zero-bid commitment. We call (B, A) regular when Equation (3) holds, ρ(v; β) > F(A) below B and ρ(v; β) ≤ F(A) from B to A, p(v; A) ≥ β above A, and

$${\displaystyle \frac{N^{\prime }\left(B\right)}{N\left(B\right)}}<{\displaystyle \frac{a}{\left(B-\beta \right)}}.$$

The slope inequality makes the proposer incentive cross from no offer to offer at B. The high-type condition is satisfied, for example, when E[X|X ≤ A] ≥ β. Existence can be checked by tracing the two cutoff curves. Let A_P(B) and A_R(B) denote the proposer and responder solutions to Equation (3) for a given B. A closed interval $\left[B_L, B_H\right]\subset \left(\beta , \overline{v}\right)$ is a regular crossing interval when both curves are defined there, the proposer single-crossing conditions and the slope inequality hold throughout the interval, and

$$A_P\left(B_L\right)>A_R\left(B_L\right), A_P\left(B_H\right)<A_R\left(B_H\right).$$

On such an interval, A_P is strictly decreasing and A_R is strictly increasing.

Our fourth proposition shows that the cutoff system underlying the fixed-bribe equilibrium has a unique solution on a regular crossing interval.

Proposition 4.

If a regular crossing interval exists, Equation (3) has a unique regular interior solution (B, A) on that interval. The strategies

$$v\ge B\iff \mathrm{offer}\hspace{0.17em}\beta , x\le A\iff \mathrm{accept}$$

form a perfect Bayesian equilibrium with the posterior stated above.

Proof. 

See Appendix A. □

The next two lemmas give conditions on the primitives that ensure the regular crossing interval required by Proposition 4. The first uses a shape restriction on the value distribution, and the second gives a sharper explicit bound under uniform values.

For $0\le t<\overline{v}/2$, define the certainty equivalent of the highest-value responder’s rejection payoff after truncating proposer values at the primitive level 2t:

$$H_{F,d}(t)={\left[{\displaystyle \frac{{\displaystyle {\int }_{2t}^{\overline{v}}{(\overline{v}-v)}^d}\mathit{dF}(v)}{1-F(2t)}}\right]}^{1/d}.$$

Let β_G(F,d) be the unique solution of H_F,d(t) = t which lies in $(0,\overline{v}/3)$.

Lemma 1.

Suppose f is nondecreasing. If 0 < β < β_G(F,d), a regular crossing interval in Proposition 4 exists. The resulting fixed-bribe equilibrium has

$$\beta <B<2\beta <A<\overline{v},$$

and its positive-offer outcome is essentially unique under the maintained protocol.

Proof. 

See Appendix A. □

The amount bound ensures that the highest responder has a sufficiently attractive rejection payoff, while the distribution shape controls the proposer’s global offering incentives. The bound can be checked directly as

$${\displaystyle {\int }_{2\beta }^{\overline{v}}{(\overline{v}-v)}^d}\mathit{dF}(v)>{\beta }^d[1-F(2\beta )], 2\beta <\overline{v}.$$

Lemma 2.

If values are uniform on $[0,\overline{v}]$ and

$$0<\beta <{\displaystyle \frac{\overline{v}}{1+a+{(1+d)}^{1/d}}},$$

Proposition 4 has a regular interior equilibrium, and the positive-offer outcome is essentially unique. Its cutoffs satisfy

$$\beta <B<(1+a)\beta \le 2\beta <A<\overline{v}.$$

Proof. 

See Appendix A. □

In Lemmas 1 and 2, essential uniqueness means that all perfect Bayesian equilibria in which the fixed bribe is offered with positive probability have the same offer and acceptance cutoffs and induce the same allocation and transfers for almost every valuation pair under the joint value distribution. It does not require unique actions at indifferent cutoff types or unique off-path beliefs, and it does not exclude the zero-offer equilibria discussed in Remark 8.

Remark 8.

Put $A_0=\min \{2\beta ,\overline{v}\}$ and, for v ≥ β, define

$${\Delta }_0(v)=F(A_0){(v-\beta )}^a-{\displaystyle {\int }_0^{\min \{v,A_0\}}{(v-x)}^a}\mathit{dF}(x).$$

A zero-offer PBE exists precisely when Δ₀(v) ≤ 0 for every feasible proposer type. Under the condition in Lemma 1 with a < 1, no such equilibrium exists. Under uniform values with a = 1 in Lemma 2’s region, a non-collusive outcome coexists with the unique positive-offer outcome.

Proof. 

See Appendix A. □

We next examine the effects of risk aversion under fixed bribes. Under the conditions of Lemma 1, the responder equation can be written as the certainty equivalent

$$\beta =(E{\left[\left(A-V{)}_{+}^d\mid V\ge B\right]\right)}^{1/d}.$$

Our fifth proposition shows that greater responder risk aversion lowers the proposer’s offer cutoff and raises the responder’s acceptance cutoff.

Proposition 5.

Suppose both parameter configurations being compared satisfy the primitive sufficient conditions of Lemma 1 or Lemma 2. In the corresponding equilibria of Proposition 4, greater responder risk aversion, represented by a fall in d with all other primitives held fixed, produces

$$B\downarrow , A\uparrow .$$

The offer probability 1 − F(B), conditional acceptance probability F(A), probability of successful collusion

$$C_F=F\left(A\right)\left[1-F\left(B\right)\right],$$

and expected transfer payment between the two bidders βC_F all rise.

Proof. 

See Appendix A. □

Unlike a revealing variable offer, a fixed offer leaves uncertainty about the proposer’s value. Greater responder risk aversion lowers the certainty equivalent of rejection, expanding the acceptance set and making an offer profitable for lower-value proposer types. Their entry shifts the posterior distribution of proposer values downward, making rejection more attractive and partly offsetting the direct risk effect. The regular cutoff intersection implies that the direct effect dominates: the acceptance cutoff A rises while the offer cutoff B falls.

We next examine how proposer risk aversion affects collusion under the fixed-bribe mechanism. At a fixed B, Equation (3) can be written

$$F\left(A\right)=F\left(B\right)E[{\left({\displaystyle \frac{B-X}{B-\beta }}\right)}^a|X\le B].$$

Direct inspection shows that the logarithm of the expectation is convex in a. Because the right-hand side exceeds F(B) at an interior equilibrium, it increases with a. Greater proposer risk aversion therefore makes the fixed-bribe offer more attractive. Our sixth proposition further shows that greater proposer risk aversion lowers both the proposer’s offer cutoff and the responder’s acceptance cutoff.

Proposition 6.

Suppose both parameter configurations being compared satisfy the primitive sufficient conditions of Lemma 1 or Lemma 2. In the corresponding equilibria of Proposition 4, greater proposer risk aversion, represented by a fall in a with all other primitives held fixed, produces

$$B\downarrow , A\downarrow .$$

Proof. 

See Appendix A. □

Greater proposer risk aversion makes the fixed-bribe offer profitable for additional types and lowers the offer cutoff. With the amount fixed, this entry shifts the distribution of proposer values conditional on an offer downward. Rejection consequently becomes more attractive to the responder, which lowers its acceptance cutoff. Successful collusion thus reflects two opposing changes: a larger offering set and a smaller conditional acceptance set. This interaction between proposer risk aversion and the information conveyed by a fixed offer explains why the probability of successful collusion need not increase.

Without an accepted offer, truthful bidding assigns the object to the higher-value bidder. An accepted offer instead assigns it to the proposer, so misallocation occurs exactly on {B ≤ V < X ≤ A}. Thus

$$I_F\left(B, A\right)={\displaystyle \frac{1}{2}}{\left[F\left(A\right)-F\left(B\right)\right]}^2, W_F\left(B, A\right)={{\displaystyle \int }}_B^A{{\displaystyle \int }}_v^A\left(x-v\right)\mathit{dF}\left(x\right)\mathit{dF}\left(v\right).$$

Proposition 7 relates misallocation probability and expected valuation loss to the two cutoff movements induced by responder risk aversion.

Proposition 7.

Under the conditions of Lemma 1 or Lemma 2, in the equilibrium of Proposition 4,

(i) 

I_F > 0, W_F > 0;

(ii) 

d₁ < d₂ ⇒ I_F(B(a, d₁), A(a, d₁)) > I_F(B(a, d₂), A(a, d₂));

(iii) 

d₁ < d₂ ⇒ W_F(B(a, d₁), A(a, d₁)) > W_F(B(a, d₂), A(a, d₂)).

Proof. 

See Appendix A. □

A lower offer cutoff adds lower-value proposers to the agreement set, while a higher acceptance cutoff adds higher-value responders. Both changes expand the region in which the responder values the object more, increasing misallocation probability and expected valuation loss. Greater responder risk aversion moves both boundaries in these directions. Greater proposer risk aversion instead lowers both cutoffs, so the two boundary changes have opposing effects on misallocation.

5. Effects of Collusion on the Seller’s Expected Revenue

Both the variable-bribe and fixed-bribe mechanisms affect bidder payoffs and allocative efficiency. They can also reduce the seller’s expected revenue. Let

$$R_0=E\left[\min \left\{V, X\right\}\right]={{\displaystyle \int }}_0^{\overline{v}}{[1-F\left(t\right)]}^2\mathit{dt}$$

denote expected seller revenue in the truthful no-collusion benchmark. Under the variable-bribe mechanism, a rejected offer implies x > Q_a(v) ≥ v, so the responder wins and pays v. Under the fixed-bribe mechanism, an accepted agreement removes the payment min{v, x} from the seller. The seller’s expected revenue under the two mechanisms is therefore

$$R_V\left(a\right)={{\displaystyle \int }}_0^{\overline{v}}v\left[1-F\left(Q_a\left(v\right)\right)\right]\mathit{dF}\left(v\right), R_F\left(B, A\right)=R_0-{{\displaystyle \int }}_B^{\overline{v}}{{\displaystyle \int }}_0^A\min \left\{v, x\right\}\mathit{dF}\left(x\right)\mathit{dF}\left(v\right).$$

(4)

The first expression integrates the proposer’s value, which the responder pays after rejection. The second subtracts the expected auction payment removed by accepted fixed-bribe offers from the no-collusion benchmark.

An accepted agreement replaces the truthful-auction payment with zero, whereas a rejected or absent offer leaves that payment unchanged. Hence either mechanism reduces expected revenue whenever successful collusion has positive probability.

Remark 9.

Under the variable-bribe mechanism, positive-probability successful collusion implies R_V(a) < R₀. Under the fixed-bribe mechanism, positive-probability successful collusion implies R_F(B, A) < R₀.

Proof. 

See Appendix A. □

Under variable bribes, greater proposer risk aversion expands the acceptance region, replacing a positive auction payment with zero for additional value pairs. Expected revenue therefore falls. Responder risk aversion leaves that region, and hence revenue, unchanged.

Remark 10.

In the collusive equilibrium characterized in Proposition 1, R_V(a) is strictly increasing in a but independent of d.

Proof. 

See Appendix A. □

For the fixed-bribe mechanism, define the payments at the two moving boundaries by

$$m_B\left(A\right)={{\displaystyle \int }}_0^A\min \left\{B, x\right\}\mathit{dF}\left(x\right), m_A\left(B\right)={{\displaystyle \int }}_B^{\overline{v}}\min \left\{v, A\right\}\mathit{dF}\left(v\right).$$

The first is the expected payment lost when a proposer at B joins the offer pool. The second is the expected payment restored when a responder at A leaves the acceptance set. Differentiating Equation (4) gives

$${\displaystyle \frac{d{R}_F}{\mathit{dB}}}=f\left(B\right)m_B\left(A\right)>0, {\displaystyle \frac{d{R}_F}{\mathit{dA}}}=-f\left(A\right)m_A\left(B\right)<0.$$

Because greater responder risk aversion lowers the offer cutoff and raises the acceptance cutoff, both boundary movements remove additional auction payments. This yields the revenue reduction in Remark 11, in contrast to responder-risk invariance under variable bribes.

Remark 11.

Under the conditions of Lemma 1 or Lemma 2, in the equilibrium of Proposition 4, R_F(B, A) is strictly increasing in d.

Proof. 

See Appendix A. □

Finally, we examine how proposer risk aversion affects the seller’s expected revenue under the fixed-bribe mechanism. For proposer risk aversion, define

$${\kappa }_R={\displaystyle \frac{m_B\left(A\right)}{m_A\left(B\right)}}, {\kappa }_C={\displaystyle \frac{F\left(A\right)}{1-F\left(B\right)}}.$$

The first threshold compares the payments lost and restored at the two margins. The second compares the corresponding probability masses. For every interior pair B < A,

$$0<{\kappa }_R<{\kappa }_C.$$

Here, Λ denotes the local quantile-response index along the responder cutoff curve. It is evaluated at the current regular interior equilibrium.

$$\mathsf{\Lambda }:={\displaystyle \frac{\mathit{dF}\left(A_R\left(B\right)\right)}{\mathit{dF}\left(B\right)}}={\displaystyle \frac{f\left(A\right)}{f\left(B\right)}}{A}_R^{\prime }\left(B\right)={\displaystyle \frac{f\left(A\right)\left[{\left(A-B\right)}^d-{\beta }^d\right]}{d{{\displaystyle \int }}_B^A{\left(A-v\right)}^{d-1}\hspace{0.17em}\mathit{dF}\left(v\right)}}>0.$$

Comparing this response index with the two thresholds separates the change in agreement frequency from the change in forgone auction payments. The next proposition gives the resulting local sign decomposition.

Proposition 8.

Under the conditions of Lemma 1 or Lemma 2, in the equilibrium of Proposition 4, an increase in proposer risk aversion produces three strict local regions:

$$\begin{array}{c}\mathsf{\Lambda }<{\kappa }_R: C_F\uparrow , R_F\downarrow ,\\ {\kappa }_R<\mathsf{\Lambda }<{\kappa }_C: C_F\uparrow , R_F\uparrow ,\\ \mathsf{\Lambda }>{\kappa }_C: C_F\downarrow , R_F\uparrow .\end{array}$$

Proof. 

See Appendix A. □

In the outer regions, the probability of successful collusion and the seller’s expected revenue move in opposite directions. The middle region instead allows both to rise because revenue losses depend on the payments removed as well as agreement frequency. A marginal increase in proposer risk aversion lowers B, adding agreements with lower-value proposers, but also lowers A, removing agreements with higher-value responders. At these margins, the added agreements remove less revenue per agreement on average than those that disappear. Thus more agreements can be reached while less total revenue is lost. The following two numerical examples illustrate these regions. These examples establish qualitative attainability rather than necessarily large economic effects. The code used to generate the numerical results and Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 is provided in the Supplementary Materials.

Figure 1. Seller revenue along the first parameter path.

Figure 2. Collusion probability along the first parameter path.

Figure 3. Collusion probability along the second parameter path.

Figure 4. Seller revenue along the second parameter path.

Figure 5. Two-dimensional regions for Example 1 (d = 0.20).

Figure 6. Two-dimensional regions for Example 2 (d = 0.65).

Example 1.

Let values be uniform on [0,1], set d = 0.20 and β = 0.153, and vary a over [0.2,1]. Figure 1 and Figure 2 trace seller revenue and successful collusion, respectively. The revenue response changes sign at a^* = 0.696200, where B = 0.159234 and A = 0.895999. A lower exponent means greater proposer risk aversion. Over the full path, seller revenue ranges from 0.06249942 to 0.06254046, a spread of approximately 0.00004104 (0.06562%). The collusion probability ranges from 0.74684820 to 0.75747832, a spread of 1.06301 percentage points (1.42333%).

Example 2.

Let values be uniform on [0,1], set d = 0.65 and β = 0.094, and vary a over [0.2,1]. The corresponding collusion-probability response changes sign. Figure 3 and Figure 4 report the probability and revenue paths, respectively. Over the full path, the collusion probability ranges from 0.54058575 to 0.54060844, a spread of approximately 0.00226953 percentage points (0.00419827%). Seller revenue ranges from 0.19128210 to 0.19321831, a spread of approximately 0.00193621 (1.01223%).

Figure 5 and Figure 6 display two-dimensional regions under uniform values, which fix d = 0.20 and d = 0.65, respectively, and vary (a,β). The original parameter paths are indicated. Each region classifies a marginal fall in a at fixed d and β. Classification uses Lemma 2’s sufficient bound. Unfilled points lie outside this sufficient domain. The solid and dashed boundaries are Λ = κ_R and Λ = κ_C, where the corresponding first-order effect is zero.

6. Extensions

We first examine variable bribes with positive initial wealth and a reserve price, then study fixed bribes under general concavity and a public amount choice before values are drawn. Some of these extensions test the robustness of the main conclusions obtained in the baseline model by changing the bidders’ utility functions, while the others further examine the effects of bidders’ risk aversion on the collusive equilibrium in more complex environments.

6.1. Variable Bribes with Initial Wealth

First, we consider bidders with positive initial wealth in the baseline model of Section 3. This removes the equivalence to risk neutrality arising from the homogeneity of CRRA utility. Let w_P,w_R > 0 be publicly known wealth held before the auction. Retain the baseline exponents and use utility over final wealth:

$$u_P(y)={(w_P+y)}^a, u_R(y)={(w_R+y)}^d, 0<a,d\le 1.$$

Here, y is the monetary surplus generated by the auction and the agreement. A losing bidder has utility $w_i^{a_i}$, where a_P = a and a_R = d. Relative risk aversion with respect to final wealth remains 1 − a_i. For calculation, define the incremental representation

$$U_{a_i,w_i}(y)={(w_i+y)}^{a_i}-w_i^{a_i}.$$

Subtracting this constant leaves expected-utility comparisons unchanged. The underlying utility is CRRA over final wealth, with absolute risk aversion over auction gains (1 − a_i)/(w_i + y). Propositions 9 and 10 establish the equilibrium and its risk comparisons under positive initial wealth. These results show that introducing positive initial wealth preserves the direction of the effects of bidders’ risk aversion on the collusive equilibrium.

Proposition 9.

Under the variable-bribe distribution conditions in Section 2, positive initial wealth preserves a separating equilibrium with an upper pooling region.

Proof. 

See Appendix A. □

Proposition 10.

Fix w_P,w_R > 0 and 0 < a₁ < a₂ ≤ 1, with both exponents satisfying Proposition 9’s conditions. Write $b_a^w(v)$ and $Q_a^w(v)$ for the equilibrium bribe and acceptance frontier. For every v > 0,

$$b_{a_1}^w(v)>b_{a_2}^w(v),\hspace{1em}{Q}_{a_1}^w(v)\ge Q_{a_2}^w(v).$$

Both objects are independent of the responder exponent d.

Proof. 

See Appendix A. □

6.2. Variable Bribes with a Reserve Price

In this section, we extend the baseline model in Section 3 by introducing a reserve price and examine how the collusive equilibrium characterized in Proposition 1 and the effects of bidders’ risk aversion change. Let V,X be independent and uniform on [0,1], with u_P(y) = y^a and u_R(y) = y^d. A publicly known reserve r is paid to the seller after an accepted positive offer. A proposer of value v > r may offer any b ∈ (0,v − r]; acceptance gives surpluses v − r − b and b. No agreement leads to truthful bidding, with proposer utility

$$N(v)=r{(v-r)}^a+{\displaystyle \frac{{(v-r)}^{1+a}}{1+a}},\hspace{1em}v\ge r.$$

Eső and Schummer [20] discuss reserves under risk neutrality. Here a positive reserve introduces a lower non-offering region. Above it, values separate until the highest offer induces certain acceptance. We use the D1 criterion for refining off-path beliefs, as in the signaling analysis of Eső and Schummer [4], to discipline beliefs about offers omitted by the equilibrium strategy.

The construction requires only the following numerical sufficient bounds, in addition to uniform values:

$${\displaystyle \frac{1}{2}}\le a\le 1,\hspace{1em}0<d\le 1,\hspace{1em}0<r\le {\displaystyle \frac{1}{4}}.$$

For a < 1, put ${\mathcal{l}}_a(t)=t^2/[(1+a)(t+a)]$ and let t_a ∈ (0,1 − a) be the unique positive solution of

$$[1+t+{\mathcal{l}}_a(t)]{\left[1-{\displaystyle \frac{{\mathcal{l}}_a(t)}{t}}\right]}^a=1+{\displaystyle \frac{t}{1+a}}.$$

For a = 1, set t_a = 0. Define the first offering type, its offer, and the associated acceptance cutoff by

$$B_r=r(1+t_a),\hspace{1em}{b}_{-}=r{\mathcal{l}}_a(t_a),\hspace{1em}{q}_{-}=B_r+b_{-},\hspace{1em}{\theta }_a={\displaystyle \frac{a-t_a}{a+t_a}}.$$

For q ∈ [q₋,1], let

$$\begin{array}{c}{g}_r(q)={\displaystyle \frac{a}{a+2}}\left[q-r{\theta }_a{\left({\displaystyle \frac{q_{-}}{q}}\right)}^{2/a}\right],\hfill \\ v_r(q)={\displaystyle \frac{q+r+g_r(q)}{2}},\hspace{1em}{c}_r=v_r(1),\hspace{1em}{\overline{b}}_r=1-c_r.\hfill \end{array}$$

The type map v_r is strictly increasing; write $q_r=v_r^{-1}$ on [B_r,c_r]. The equation for t_a imposes indifference at the first positive offer; the formula for ${\mathcal{l}}_a$ supplies tangency to the no-offer payoff. Both boundaries are endogenous.

Proposition 11.

Under the preceding bounds, the following strategy is part of a PBE satisfying D1:

$$b_r(v)=\left\{\begin{array}{cc}0,& 0\le v<B_r,\\ q_r(v)-v,& B_r\le v<c_r,\\ {\overline{b}}_r,& c_r\le v\le 1.\end{array}\right.$$

A separating positive offer reveals v and is accepted exactly when x ≤ q_r(v). The upper pooled offer is accepted by every responder. We have r ≤ B_r < c_r < 1; if a < 1, then B_r > r and b₋ > 0. As r ↓ 0, the strategy converges pointwise to Proposition 1’s uniform-value strategy,

$$b_0(v)=\min \left\{{\displaystyle \frac{v}{1+a}},{\displaystyle \frac{1}{2+a}}\right\}.$$

Proof. 

See Appendix A. □

Write Q_r(v) = q_r(v) for B_r ≤ v < c_r and Q_r(v) = 1 for v ≥ c_r. The offer probability and the probability of successful positive collusion are

$$O_r=1-B_r,\hspace{1em}{C}_r={{\displaystyle \int }}_{B_r}^1{Q}_r(v)\mathit{dv}.$$

Our twelfth proposition gives the effects of the responder’s risk aversion on the offer probability and the probability of successful positive collusion with a reserve price.

Proposition 12.

Under Proposition 11, holding r and the other bidder’s preference fixed, changing responder risk aversion leaves O_r and C_r unchanged.

Proof. 

See Appendix A. □

6.3. Fixed Bribes Under General Concavity

We next examine whether the fixed-bribe collusive equilibrium retains the form obtained in Section 4 when risk-averse bidders have general concave utility functions. Let U_i(y) = u_i(y) − u_i(0) be continuous on $[0,\overline{v}]$, increasing and concave, and differentiable with positive derivative on $(0,\overline{v}]$. Thus, U_i(0) = 0. The next proposition establishes the fixed-bribe collusive equilibrium under general concave utility functions and provides external conditions for its existence.

Proposition 13.

Suppose f is nondecreasing and

$$0<2\beta <\overline{v},\hspace{1em}{{\displaystyle \int }}_{2\beta }^{\overline{v}}{U}_R(\overline{v}-v)\mathit{dF}(v)>U_R(\beta )[1-F(2\beta )].$$

There is a PBE with

$$b(v)=\left\{\begin{array}{c}0,v<B,\\ \beta ,v\ge B,\end{array}\right.\hspace{1em}\mathrm{accept}\hspace{1em}\beta \hspace{1em}\iff \hspace{1em}x\le A,\hspace{1em}\beta <B<2\beta <A<\overline{v}.$$

The cutoffs are the unique solution in this range of

$$\begin{array}{c}F(A)U_P(B-\beta )={{\displaystyle \int }}_0^B{U}_P(B-x)\mathit{dF}(x),\\ U_R(\beta )[1-F(B)]={{\displaystyle \int }}_B^A{U}_R(A-v)\mathit{dF}(v).\end{array}$$

Proof. 

See Appendix A. □

Next, we discuss the effects of bidders’ risk aversion on the collusive equilibrium in Proposition 13. Greater risk aversion is represented by ${\tilde{U}}_i=\psi \circ U_i$, where ψ is increasing and concave and ψ(0) = 0 [21]. The results show that, even when general concave utility functions represent the two bidders’ risk aversion, the directions of the effects of each bidder’s risk aversion on the two collusive-equilibrium cutoffs agree with those in the baseline model of Section 4.

Proposition 14.

Suppose both preference configurations satisfy Proposition 13.

(i) 

Greater responder risk aversion gives $\tilde{B}\le B,\hspace{1em}\tilde{A}\ge A.$

(ii) 

Greater proposer risk aversion gives $\tilde{B}\le B,\tilde{A}\le A.$

Proof. 

See Appendix A. □

6.4. An Endogenously Determined Fixed Amount

In Section 4, the fixed bribe is exogenous. Now we assume that the proposer publicly chooses β ∈ [0,1] before values are drawn. This adds an ex ante choice of terms to the enforceable bribing contract in Eső and Schummer [4]. Economically, the proposer commits to a value-independent transfer rule: after learning its value, it may offer the announced amount or make no offer, but cannot substitute another amount. Payment is due only upon acceptance. The responder observes the commitment and any subsequent offer. To characterize this choice, suppose values are independent and uniform on [0,1], with 0 < a < 1 and 0 < d ≤ 1.

Define the auction payoff N_a, the responder parameter c_d, and the sure-acceptance amount function s_a by

$$N_a(v)={\displaystyle \frac{v^{a+1}}{1+a}},\hspace{1em}{c}_d={(1+d)}^{-1/d},\hspace{1em}{s}_a(z)=z-{\left[{\displaystyle \frac{z^{a+1}}{1+a}}\right]}^{1/a}.$$

Let z_I ∈ (0,1/2) be the unique solution of s_a(z) = c_d(1 − z) and put β_I = s_a(z_I). This amount depends only on the risk parameters and marks the onset of complete acceptance. Proposition 15 characterizes the fixed-bribe continuation equilibrium and the conditions for its existence.

Proposition 15.

Suppose values are uniform on [0,1], with 0 < a < 1 and 0 < d ≤ 1. For every 0 < β < 1/2, the continuation game has a unique equilibrium outcome with positive-probability offers. Its proposer and responder cutoffs satisfy

$$\beta <B_{\beta }<(1+a)\beta ,\hspace{1em}2\beta <A_{\beta }\le 1,$$

$${\displaystyle \frac{B_{\beta }^{a+1}}{(1+a){(B_{\beta }-\beta )}^a}}=A_{\beta }=\min \left\{1,B_{\beta }+{[(1+d){\beta }^d(1-B_{\beta })]}^{1/(1+d)}\right\}.$$

Moreover, A_β < 1 exactly when β < β_I, and 0 < β_I < 1/3.

Proof. 

See Appendix A. □

For 0 < β ≤ β_I, define the proposer’s ex ante expected utility using the cutoffs in Proposition 15,

$$J_{a,d}(\beta )={\displaystyle \frac{B_{\beta }^{a+2}+{(1-A_{\beta })}^{a+2}}{(1+a)(2+a)}}+{\displaystyle \frac{A_{\beta }}{1+a}}\left[{(1-\beta )}^{a+1}-{(B_{\beta }-\beta )}^{a+1}\right].$$

Set J_a,d(0) = 1/[(1 + a)(2 + a)].

Because the proposer publicly commits to the fixed amount ex ante, it chooses the amount that maximizes its expected utility given the known value distribution and risk preferences. Proposition 16 gives the conditions satisfied by the proposer’s optimal fixed bribe.

Proposition 16.

Under the conditions of Proposition 15, a globally optimal public commitment exists. Every optimal amount β^* satisfies

$$0<{\beta }^{*}\le {\beta }_I<1/3,\hspace{1em}{J}_{a,d}({\beta }^{*})>J_{a,d}(0).$$

The optimal amount maximizes the displayed scalar objective on [0,β_I]. At an interior optimum, the marginal gain from a higher acceptance cutoff equals the marginal cost of the transfer,

$${\displaystyle \frac{A_{\beta }^{\prime }}{1+a}}\left[{(1-\beta )}^{a+1}-{(B_{\beta }-\beta )}^{a+1}-{(1-A_{\beta })}^{a+1}\right]=A_{\beta }\left[{(1-\beta )}^a-{(B_{\beta }-\beta )}^a\right].$$

Proof. 

See Appendix A. □

Let J_a,d(β) denote the proposer’s ex ante continuation utility for every feasible commitment, given by the preceding formula on [0,β_I]. Proposition 16 gives the optimized value

$$J^{*}(a,d)={\max }_{0\le \beta \le {\beta }_I(a,d)}{J}_{a,d}(\beta ).$$

Finally, Proposition 17 examines the effects of the two bidders’ risk aversion on the complete-acceptance threshold in Proposition 15 and on the proposer’s optimized expected utility. Greater responder risk aversion makes a given transfer more attractive relative to rejection, allowing the proposer to secure acceptance at a lower cost.

Proposition 17.

Under the conditions of Proposition 15,

$${\displaystyle \frac{\partial {\beta }_I}{\partial a}}<0,\hspace{1em}{\displaystyle \frac{\partial {\beta }_I}{\partial d}}>0,\hspace{1em}{\displaystyle \frac{\partial z_I}{\partial a}}>0,\hspace{1em}{\displaystyle \frac{\partial z_I}{\partial d}}>0.$$

For fixed a ∈ (0,1) and 0 < d₁ < d₂ ≤ 1,

$$J^{*}(a,d_1)>J^{*}(a,d_2).$$

Proof. 

See Appendix A. □

7. Conclusions

This paper studies how heterogeneous risk aversion affects bilateral collusion in a second-price sealed-bid auction. We introduce a proposer and a responder with different degrees of risk aversion into the pre-auction bribery framework of Eső and Schummer [4] and examine the variable-bribe and fixed-bribe mechanisms separately. The results show that risk aversion has no uniform effect on the collusive equilibrium, allocative efficiency, or the seller’s expected revenue: its effect depends on the bidder’s strategic role in collusion and on the bribery mechanism.

Under the variable-bribe mechanism, the equilibrium bribe has a piecewise structure: it increases strictly with the proposer’s valuation in the lower valuation region and remains constant above the proposer’s separating cutoff. Greater proposer risk aversion raises the equilibrium bribe and weakly raises the responder’s acceptance cutoff, thereby increasing the probability of successful collusion and the probability of misallocation while reducing the seller’s expected revenue. The higher bribe also shifts expected monetary payoff from the proposer to the responder. By contrast, responder risk aversion does not affect the equilibrium. We also show that a higher reverse hazard rate of the valuation distribution weakly raises the equilibrium bribe and the responder’s acceptance cutoff. In the equilibrium studied here, the probability of successful collusion is always at least one half, and the probability of misallocation is exactly one half lower. With positive initial wealth, the proposer-risk direction and responder-risk invariance continue to hold under the stated distribution conditions. With a positive reserve, the D1 equilibrium retains responder-risk invariance.

The fixed-bribe mechanism operates differently. Under the stated primitive sufficient conditions, its collusive equilibrium has regular interior cutoff strategies: only proposers whose valuations reach the offer cutoff make the fixed-bribe offer, and the responder accepts only when its valuation is no higher than the acceptance cutoff. Holding the amount fixed within this equilibrium class, greater responder risk aversion lowers the proposer’s offer cutoff and raises the responder’s acceptance cutoff. Both changes expand the agreement set, increasing successful collusion, expected transfers, misallocation probability, and expected valuation loss while reducing the seller’s expected revenue. Greater proposer risk aversion instead lowers both cutoffs, so more proposer types offer but fewer responder types accept. These opposing effects allow a marginal increase in proposer risk aversion to raise or lower the probability of successful collusion and the seller’s expected revenue. In particular, both may rise together; more frequent collusion therefore need not imply a larger revenue loss. When the fixed amount is publicly chosen before values are drawn, greater proposer risk aversion raises this complete-acceptance threshold, whereas greater responder risk aversion lowers it and increases the proposer’s optimized expected utility.

Taken together, the effects of risk aversion depend on both strategic role and bribery mechanism. Under the variable-bribe mechanism, the proposer can adjust the bribe to its valuation, so its risk attitude is reflected directly in the equilibrium bribe. Under the fixed-bribe mechanism, the amount cannot vary with private valuation, and risk aversion instead operates through the proposer’s offer cutoff and the responder’s acceptance cutoff. A seller assessing collusion therefore needs to consider the form of the agreement, the participants’ bargaining roles, and the value pairs for which competition is removed. The frequency of successful agreements alone does not determine lost revenue or allocative loss. For empirical work, experiments can vary the offer protocol and assign bargaining roles while holding values and wealth fixed. The model predicts that proposer risk aversion changes the amount of a revealing variable offer, whereas responder risk aversion changes offer and acceptance frequencies at a given fixed amount. Field applications would also need information about negotiation opportunities and the selection of transfer protocols.

Several extensions remain open. First, other partially accepting and separating equilibria beyond the reserve-price construction remain to be characterized. Second, with additional bidders outside the agreement, even a revealing bilateral offer leaves uncertainty about the highest competing value, potentially restoring a role for responder risk aversion. Third, a larger ring also requires an internal allocation and transfer rule. Fourth, privately known risk preferences would add a second source of information to the bargaining problem. Studying these environments, together with more general utility curvature and endogenous entry, would clarify how the role-specific effects extend beyond bilateral private-value auctions.