Welfare effects of a concealed information exchange

Müller, Lars; Karos, Dominik Working Paper Welfare effects of a concealed information exchange Center for Mathematical Economics Working Papers, No. 703 Provided in Cooperation with: Center for Mathematical Economics (IMW), Bielefeld University Suggested Citation: Müller, Lars; Karos, Dominik (2025) : Welfare effects of a concealed information exchange, Center for Mathematical Economics Working Papers, No. 703, Bielefeld University, Center for Mathematical Economics (IMW), Bielefeld, https://nbn-resolving.de/urn:nbn:de:0070-pub-30026290 This Version is available at: https://hdl.handle.net/10419/318448 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ 703 April 2025 Welfare Effects of a Concealed Information Exchange Lars M¨uller and Dominik Karos Center for Mathematical Economics (IMW) Bielefeld University Universit¨atsstraße 25 D-33615 Bielefeld ·Germany e-mail: [email protected] uni-bielefeld.de/zwe/imw/research/working-papers ISSN: 0931-6558 Unless otherwise noted, this work is licensed under a Creative Commons Attribution 4.0 International (CC BY) license. Further information: https://creativecommons.org/licenses/by/4.0/deed.en https://creativecommons.org/licenses/by/4.0/legalcode.en Welfare Effects of a Concealed Information Exchange∗ Lars M¨uller†Dominik Karos‡ April 16, 2025 Abstract This paper analyzes the welfare effects of private and unilateral disclosure of sensitive information in a sequential bargaining context. We consider a model where two sellers each propose a take-it-or-leave-it price for a homogeneous good to a single buyer. The buyer accepts or rejects the first seller’s offer before the second seller proposes her price. Crucially, the second seller might learn the first seller’s price and whether it was accepted, allowing her to update her belief about the buyer’s willingness to pay and optimize her pricing strategy. The welfare effects caused by this information exchange are evaluated under general conditions. We show that it benefits the buyer if a rejection is revealed but might harm him if an acceptance is revealed. Additionally, the information exchange improves the societal welfare by reducing inefficiencies and promoting additional trade. This paper strengthens the theoretical framework for assessing the welfare effects of information exchanges by offering new insights and providing tools to assess causality for alleged damages. Keywords: Information Exchange; Collusion; Unawareness JEL Classification: D82; D83; L41 ∗The authors thank Gerrit Bauch, Jurek Preker, Ina Taneva, and Nikhil Vellodi for their helpful comments and suggestions. †Center for Mathematical Economics, Bielefeld University, Postfach 100131, 33501 Bielefeld, Germany. lm[email protected], corresponding author. Lars M¨uller gratefully acknowledges financial support by the German Research Foundation (DFG) [RTG 2865/1 – 492988838]. ‡Center for Mathematical Economics, Bielefeld University, Postfach 100131, 33501 Bielefeld, Germany. dominik.k[email protected] 1 1 Introduction and Motivation The exchange of information among competing firms plays a crucial role in shaping market outcomes. Additional information can contribute to market efficiencies, but can also influence pricing behavior to the consumers’ detriment. Because of the latter, antitrust law is quite restrictive with respect to information exchange, and the last few years have seen several cases where firms have faced substantial fines for exchanging sensitive information. For instance, the European Commission imposed fines of about 32 million euros on two metal packaging companies (European Commission, 2022), and the German Federal Cartel Office fined steel forging companies approximately 35 million euros for exchanging information in violation of antitrust law (German Federal Cartel Office, 2021).1 It seems to be an established principle to assume that private announcements among competing firms can only be motivated by assisting price-collusion and do not enhance efficiency, as argued by the OECD (2012). This paper demonstrates that such a principle must be questioned. We show that even private and unilateral disclosure of information can enhance efficiency and enable welfare-improving trade, which might even benefit third party consumers. Such an exchange among competing firms is nothing extraordinary: for instance, trade associations are platforms where information is easily and often shared.2 With their sheer ubiquity of almost 60,000 in the U.S. alone,3the importance of correctly assessing the positive and negative effects of information exchange becomes obvious. This paper analyzes these effects in a sequential trading game and assesses the potential harm for a consumer unaware of any information exchange. This provides a natural worst-case analysis from their perspective. In this model one buyer sequentially bargains with two different sellers who offer a homogeneous good. Both sellers know the distribution underlying the buyer’s willingness to pay but not its realization when offering a take-it-or-leave-it price for their good. The buyer accepts or rejects the first offer before the second seller proposes her price. There are no satiation effects and the buyer might buy zero, one, or two goods. The important feature of the model is that the subsequent seller might additionally receive information about the course of the preceding negotia1In numerous other cases, severe fines were imposed on firms accused of having exchanged commercially sensitive information. See European Commission (2020), where ethylene purchasers were fined 260 million euros, and European Commission (2021), where investment banks were fined 371 million euros. 2See, for example, fines imposed on the Association of the German Confectionery Industry for encouraging an anti-competitive exchange of information (German Federal Cartel Office, 2019). 3See IRS (2024), the IRS refers to professional and trade associations as business leagues. 2 tion. Two scenarios are compared: a benchmark scenario in which the second seller does not receive any additional information and an information exchange scenario in which the first seller reveals to the second seller both her offered price and whether or not it was accepted. In both scenarios, the buyer is unaware of the possibility of an information exchange between the two sellers. The welfare effects caused by this information exchange are evaluated under general conditions. Most notably, it is shown that the buyer benefits if a rejection is revealed but might be harmed if an acceptance is revealed. From an ex-ante point of view the information exchange can be positive or negative for the consumer. Societal welfare is always increased by the information exchange as it reduces inefficiencies and promotes additional trade. The remainder of this paper is structured as follows. In Section 2some related literature is presented. Section 3introduces the bilateral trading stage game before Section 4 sets up the sequential trading model and analyses the equilibria following an acceptance or rejection. In Section 5the welfare effects caused by the information exchange are analysed from different timing perspectives. Section 6concludes. All proofs are relegated to the appendix. 2 Related Literature While competition law in the United States and the European Union clearly prohibits firms from colluding on prices, there is no unified position on the sole exchange of prices among firms. Harrington (2022) criticizes this lack of a common treatment and attributes it, at least in part, to the “absence of a well-established theory of harm”. He establishes a theoretical foundation that shows how a private exchange of prices between competitors results in higher consumer prices in a standard duopoly setting. Our paper differs as it assesses the economic effects of information exchanges between firms, who act as information providers and receivers.4Albeit from a different perspective, our paper hence contributes to the establishment of a solid theory of harm of information exchanges by distinguishing general conditions under which third parties are harmed by or benefit from a private exchange of sensitive information. 4See Kirby (1988) for a discussion of the incentives of oligopolistic firms to use trade associations as an information exchange mechanism. 3 Information exchanges and their impact on competition have been thoroughly examined within oligopolistic models in which firms ex-ante voluntarily disclose private information. K¨uhn and Vives (1995) survey this strand of literature. Our paper differs from this perspective as the analyzed information exchange can not alter the welfare of the information provider. This also distinguishes the current paper from the information design literature as surveyed by Bergemann and Morris (2019). The bilateral trading stage game in this paper resembles that of Myerson and Satterthwaite (1983) with independently distributed valuations and costs and asymmetric information. They show that no incentive-compatible and individually rational trading mechanism can be ex-post efficient. The minimal efficiency loss in this setting when jointly using a mechanism and the optimal information structure is quantified by Schottm¨uller (2023). In our paper, however, the trading mechanism is fixed, and we compare the welfare effects that result from the information exchange. Bergemann, Brooks, and Morris (2015) evaluate the welfare consequences of additional information that is provided to a monopolist. The monopolist uses this information about the valuation of consumers and engages in third degree price discrimination. They characterize the possible consumer and producer surplus that can be implemented in this setting. The limits of these characterizations are also the limits for the welfare of the buyer and the second seller in this model. Ali, Kleiner, and K. Zhang (2024) provide an equivalence result between the achievable payoff profiles given by these limits and the payoffs supported by an equilibrium in a game of voluntary disclosure. Glode and Opp (2016) is the closest paper in terms of the model setting. They also analyze sequential bilateral trading games with sellers making take-it-or-leave-it offers but focus on the effect of intermediation chains between a seller and a buyer. They show that the addition of moderately informed intermediaries improves trade efficiency by reducing the incentive for the seller to inefficiently screen the buyer for his valuation. This counterincentive to screen for the buyer’s valuation is also at the core of the welfare-enhancing effects of the information exchange in our paper. However, their analysis focuses on implementing first-best trade and assumes that trade is always mutually beneficial. In a very similar setting, Glode, Opp, and X. Zhang (2018) analyze the incentives for a buyer to voluntarily disclose private information both in an ex-ante and in an interim stage of the game. Yet again, their focus is on the implementation of first-best trade. Our paper hence differs by scrutinizing offers that might be rejected with positive probability. 4 Within this paper, players consider information about their negotiations by default as private to avoid distorting the analysis by effects that may arise from the anticipation of disclosed information. On the one hand side, this allows us to achieve an unbiased assessment of the buyer’s worst-case welfare effects. On the other hand side, the buyer’s unawareness ensures that we do not need to be concerned about reputation building as in Milgrom and Roberts (1982) or Kreps and Wilson (1982). 3 The Bilateral Trading Game To analyze the welfare effects of an information exchange in a sequential bargaining context, it is necessary to first consider the individual stage game, which is played repeatedly. This section defines this stage game and analyses its equilibrium strategies and payoffs. 3.1 Notation and Preliminaries In a bilateral trading game with incomplete information, a risk-neutral seller (she) offers a single indivisible good at a price p≥0 to a risk-neutral buyer (he) who either accepts or rejects that offer. The buyer’s private valuation for the good is denoted by v≥0, and the seller’s (opportunity) cost for producing it is given by c≥0. If the buyer accepts, the good is exchanged, and payoffs are v−pfor the buyer and p−cfor the seller. In case of a rejection, players obtain zero payoffs. Both vand care independently distributed according to commonly known probability distributions Fand Gwith densities fand g. Thus, any bilateral trading game is uniquely determined by the pair (f, g), and, with slight abuse of notation, we identify the game by this pair. We denote the (set-theoretic) supports of fand gby V:= {v∈R≥0:f(v)>0}and C:= {c∈R≥0:g(c)>0}, respectively. To ease notation, let v:= inf(V), v:= sup(V), c:= inf(C), and c:= sup(C). Throughout this paper, all densities are assumed to be continuous on their supports.5 A pure strategy for the seller is a map σ:C → Rthat maps any realized cost cto a take-it-or-leave-it price p. A pure strategy for the buyer assigns to any pair (v, p) of a valuation and an offered price a rejection or acceptance. 5For technical reasons, fis w.l.o.g. assumed to be right-continuous at v, i.e., f(v) = limv&vf(v). 5 The following standard assumption is imposed on the buyer’s distribution.6 Assumption 1. The hazard rate h:V \ {v} → R≥0, defined by h(v) := f(v) 1−F(v), is non-decreasing for all v∈ V \ {v}. Note that Assumption 1ensures the convexity of Vand, therefore, guarantees that F(v) is strictly increasing for all v∈[v,v). 3.2 Equilibrium Strategies In a perfect Bayesian equilibrium (henceforth, just equilibrium), the buyer will accept any offer p < v and reject any offer p > v.7Given this strategy, the seller’s expected payoff from offering p, given her own cost c, is Π(p;c) = (1 −F(p)) (p−c).(1) Hence, if c > v, there is no scope for trade in equilibrium; and if cis sufficiently low such that it is not worthwhile for the seller to risk a rejection, she will simply offer vand make a profit of v−cwith probability 1. In the middle ground, the seller faces the tradeoff that higher prices are rejected with higher probability. One can easily check that an interior point pthat maximizes (1) must satisfy the first order condition h(p)(p−c) = 1.(2) We denote the solution of (2) by ˆp(c). Then, there exists a unique cwith p= ˆp(c) for all p∈(v,v) by Assumption 1. Indeed, solving (2) for cquickly gives ˆp−1(v) = v−1 h(v).8 6The assumption is, for instance, satisfied by a log-concave density (Bagnoli and Bergstrom, 2005, Corollary 2) and therefore by a wide range of probability distributions, e.g., the uniform distribution on any convex set, the exponential distribution, and the gamma distribution for shape parameters greater or equal to one. It is also satisfied by the Kumaraswamy distribution K(a, b) with parameters a≥1 and b > 0 that allows for a wide range of densities and, in contrast to the related Beta-distribution, can be expressed in closed form (Kumaraswamy, 1980). Assumption 1is also tightly connected to the omnipresent regularity assumption in auction theory, which ensures strictly increasing virtual valuations (Myerson, 1981) as well as to Assumption 1 in Glode and Opp (2016) and equivalent to the second hazard rate condition in McAfee (2002). 7In order to avoid uninteresting case distinctions, we assume that for p=vthe offer will be accepted. 8It might be worth mentioning here that the inverse ˆp−1takes the form of a virtual valuation in 6 Thus, define τ=τ(f)≡lim v&vˆp−1(v) =    v−1 h(v),if f(v)>0 −∞,if f(v)=0. (3) This provides us with a lower bound of the domain of ˆp. The following Lemma summarizes all properties of ˆpand is used for later reference. Lemma 3.1. The map ˆp: (τ(f), v)→(v, v)is a continuous and strictly increasing bijection with inverse ˆp−1(v) = v−1 h(v)for all v∈(v, v). In particular, ˆp(c)> c for all c∈(τ(f), v). With slight abuse of notation, we shall write ˆp−1(v) = τ(f) and ˆp−1(v) = v, keeping in mind that τ(f) and vmight be −∞ and +∞, respectively. After having defined ˆp, we can now describe the seller’s equilibrium strategies in the area where they are unique. Lemma 3.2. Every bilateral trading game (f, g)has an equilibrium. In any equilibrium, the seller’s strategy ˆσsatisfies for all c∈ C ˆσ(c) = ˆσ(c;f, g) =    v, if c≤τ(f) ˆp(c),if τ(f)< c < v. Moreover, ˆσis continuous in cfor all c∈ C with c < v. The monotonicity of ˆpimplies, unsurprisingly, that higher cost lead to higher proposed prices. So far, no assumptions about the relation of fand gwere made. The following remark captures the trivial cases. Remark 3.3. In a bilateral trading game (f, g) the following hold: i) If c < τ(f), then ˆσ≡vand trade happens with certainty at price v. ii) If c > v, then mutually beneficial trade is ex-ante impossible. In equilibrium, any price p≥ccan be chosen, and any offer will be rejected.  classical auction theory (see Myerson, 1981). As in this strand of literature, the strict monotonicity of ˆp and ˆp−1is ensured by Assumption 1. 7 The results of Proposition 4.6 and Corollary 4.7 are not that surprising: if the second seller has observed a rejection, she will offer a lower price; and the lower the rejected price she observed, the lower her price will be. This has two effects: First, there is a higher chance that her offer will be accepted. Second, in case her offer is accepted, the buyer will pay a lower price than in the benchmark. As the first effect only appears for buyer valuations that are sufficiently small, the buyer’s gain depends on v. 5 Welfare Effects We shall now investigate the gains and losses of the buyer and the second seller from the unilateral information disclosure by the first seller. We consider four stages: ex post, i.e., after all parameters have been revealed; interim, i.e., after all players know their own parameters, both before and after the information of the first seller has been disclosed; and ex ante, i.e., before players even know their own parameter. As there is no effect on the first seller, we only investigate the effects on the buyer’s and the second seller’s payoff. 5.1 Ex-post Welfare Effects In this subsection, we investigate welfare effects after the buyer’s valuation and the sellers’ costs have realized. Proposition 4.5 and Proposition 4.6 show that the buyer can be harmed by the information disclosure only in case of an acceptance, and that he can profit from the information disclosure only in case of a rejection in the first game. The following proposition formalizes these insights. Proposition 5.1. Let (f, g1, g2)be a sequential trading game, and let v∈ V,c1∈ C1, and c2∈ C2. 1. If v≥ˆp(c1), then ˆuv, c2;fA c1, g2≤ˆu(v, c2;f, g2). The inequality is strict if and only if c2< c1. Moreover, ˆuv, c2;fA c1, g2is decreasing in c1and strictly decreasing in c1if c2< c1. 2. If v < ˆp(c1), then ˆuv, c2;fR c1, g2≥ˆu(v, c2;f, g2). The inequality is strict if and only if τ(f)< c2<ˆpR c1−1(v). Moreover, ˆuv, c2;fR c1, g2is decreasing in c1and strictly decreasing in c1if τfR c1< c2<ˆp(c1). 14 The overall welfare that is generated by trade is v−c2and independent of the information disclosure. Thus, the information exchange only affects overall welfare if it changes whether or not a trade occurs. By Proposition 4.5, an accepted offer in the first round does not influence whether or not the buyer and the seller in the second round trade. By Proposition 4.6, a rejected offer in the first round either has no effect or increases the amount of trade. Thus, we obtain the following results and omit the proofs. Theorem 5.2. Let (f, g1, g2)be a sequential trading game, and let v∈ V,c1∈ C1, and c2∈ C2. 1. If v≥ˆp(c1), then ˆuv, c2;fA c1, g2+ ˆπv, c2;fA c1, g2= ˆu(v, c2;f, g2) + ˆπ(v, c2;f, g2). 2. If v < ˆp(c1), then ˆuv, c2;fR c1, g2+ ˆπv, c2;fR c1, g2≥ˆu(v, c2;f, g2) + ˆπ(v, c2;f, g2). This inequality is strict if and only if τ(f)< c2<ˆpR c1−1(v). Interestingly, the information disclosure by the first seller might also harm the second seller from an ex-post perspective. This happens if she adapts and lowers her price after observing a rejection, although her benchmark offer would have been accepted. Corollary 5.3. Let (f, g1, g2)be a sequential trading game, and let v∈ V,c1∈ C1, and c2∈ C2. If τ(f)< c2<ˆp−1(v)< c1, then ˆπv, c2;fR c1, g2<ˆπ(v, c2;f, g2). 5.2 Interim Welfare Effects After Information Disclosure We next consider interim effects after the information disclosure, i.e., effects after both the buyer and the second seller know their own type and the seller has updated her belief about the buyer’s type. As the outcome of the first game is not affected by the disclosure of information, the effect on the buyer’s payoff after an accepted or rejected offer are ∆ˆ UA(v;c1)≡   ˆ Uv;fA c1, g2−ˆ U(v;f, g2),if v≥ˆp(c1) 0,if v < ˆp(c1), ∆ˆ UR(v;c1)≡   0,if v≥ˆp(c1) ˆ Uv;fR c1, g2−ˆ U(v;f, g2),if v < ˆp(c1), respectively. (Recall that ˆ U(v;f, g) denotes the buyer’s interim expected payoff in the bilateral trading game (f, g), defined in Equation (6).) 15 Proposition 5.4. Let (f, g1, g2)be a sequential trading game, let v∈ V and c1∈ C1. 1. Let v≥ˆp(c1). Then ∆ˆ UA(v;c1)≤0. The inequality is strict if and only if c1> c2. Moreover, ∆ˆ UA(v;c1)does not depend on v. 2. Let v < ˆp(c1). Then ∆ˆ UR(v;c1)≥0. The inequality is strict if and only if c2> τ(f) and G2ˆpR c1−1(v)> G2(τ(f)). Moreover, ∆ˆ UR(v;c1)is weakly increasing in v. At the interim stage after the information disclosure revealed an acceptance in the first game, the first part of Proposition 5.4 shows that the buyer can only suffer from the information disclosure. However, maybe surprisingly, the size of the damage is independent of his own valuation v. The reasoning is that an acceptance of the first offer has no effect on the likelihood that the second offer will be accepted as well. Indeed: (i) If c2≤τ(f), then in the benchmark the seller would have offered vand this would have been accepted with certainty. With the additional information she offers ˆp(c1), which is accepted with certainty as well; (ii) If τ(f)< c2< c1, then in the benchmark she would have offered ˆp(c2)<ˆp(c1) and the offer would have been accepted. After obtaining the additional information she offers ˆp(c1), which is still accepted with certainty. (iii) If c2≥c1, the behavior of the second seller is not affected at all. In contrast, the realized cost of the first seller c1has, conditional on an acceptance in the first round, two effects: First, if c1increases, the probability that c1exceeds c2increases as well, which is exactly the probability that the second seller increases her price due to the obtained information. Second, it increases the (conditional) expected difference ˆp(c1)−ˆp(c2), which, if the second offer is accepted as well, increases the buyer’s loss due to the information disclosure. At the interim stage after the information disclosure revealed a rejection, the second part of Proposition 5.4 shows that the buyer can only profit: (i) If c2≤τ(f), the second seller will offer v, either way, so the information disclosure makes no difference. (ii) If τ(f)< c2≤τ(fR c1), the seller would have offered ˆp(c2) in the benchmark compared to the lower and certainly accepted vafter observing a rejection. Since ˆp(c2) is 16 rejected with strictly positive probability, the buyer profits in two ways: a higher probability of trade and lower prices given trade. (iii) If τ(fR c1)< c2≤ˆp−1(v), the seller offers ˆpR c1(c2)<ˆp(c2) after observing a rejection. As v≥ˆp(c2), no additional trade (compared to the benchmark) emerges, nevertheless the buyer profits from the lower prices. (iv) If ˆp−1(v)< c2≤(ˆpR c1)−1(v) the seller again offers ˆpR c1(c2)<ˆp(c2), but now inducing additional trade, as the benchmark offer would have been rejected while the actual offer is not. This additionally generated trade is the reason that the interim expected profit ∆ ˆ UR(v;c1) after observing a rejection, is increasing in v. (v) If c2>(ˆpR c1)−1(v), then the second seller’s offer will be rejected with and without information disclosure. In Proposition 5.4 no assumptions on the second seller’s distribution G2were made. If G2is strictly increasing, i.e., if the support of g2is convex, then the condition in part 2 simplifies significantly. Corollary 5.5. Let (f, g1, g2)be a sequential trading game, let v∈ V,c1∈ C1with ˆp(c1)> v, and let C2be convex. Then ∆ˆ UR(v;c1)>0if and only if c2> τ(f)and c2<ˆpR c1−1(v). We close this subsection with the following example, which illustrates the findings in Proposition 5.4. Example 5.6. Consider the sequential trading game (f, g1 1, g2), where f=g1 1=g2are the uniform distribution on [0,1]. In this case we have ˆp(c1) = 1+c1 2and ∆ˆ UA(v;c1) =    −c2 1 4if v≥1+c1 2 0 if v < 1+c1 2, ∆ˆ UR(v;c1) =          0 if v≥1+c1 2 (4v−c1−1)2 16 if 1+c1 4< v < 1 2 (1−c1)(8v−c1−3) 16 ,if 1 2≤v < 1+c1 2. 17 -0.10 -0.05 0.00 0.05 0.10 v ∆ˆ UA(v;c1)+∆ˆ UR(v;c1) 0 0.25 0.5 0.75 1 c1= 0 c1=1 4 c1=1 2 Figure 2: Interim effect for the buyer after result of first trading game has been disclosed. The overall effect of the information disclosure on the buyer’s welfare is given by ∆ ˆ UA+ ∆ˆ URand depicted in Figure 2for several values of c1. As shown in Proposition 5.4, the effect is positive for v < ˆp(c1), where a discontinuity occurs. For v≥ˆp(c1), the effect is negative (strictly if c1>0) and independent of v. Moreover, a buyer with fixed valuation vwho is harmed by the information exchange will be harmed more severely for larger c1. Similarly, a buyer (with fixed v) who benefits from the information exchange will profit less for larger c1. 5.3 Interim Welfare Effects Before Information Disclosure We next consider the expected welfare effect on the buyer before the cost of the first seller has been revealed, which is given by ∆ˆ U(v)≡EG1h∆ˆ UA(v;·)+∆ˆ UR(v;·)i=Z∞ −∞ g1(c1)∆ˆ UA(v;c1)+∆ˆ UR(v;c1)dc1. In terms of Figure 2, this function is a weighted average of the functions depicted there. The following example shows this function for different distributions G1. 18 Example 5.7. Recall the sequential trading game (f, g1 1, g2) from Example 5.6, where all distributions are uniform on [0,1]. Then, ∆ˆ U1(v)≡EG1 1h∆ˆ UA(v;·)+∆ˆ UR(v;·)i=         0 if v < 1 4 (4v−1)3 48 if v < 1 4≤v < 1 2 v3 6−v2+v−1 4if v≥1 2 , which is depicted in Figure 3b. Figure 3a depicts the corresponding density g1 1(= g2=f). Observe that ∆ ˆ U1(v)≡0 for sufficiently small v. More specifically, if v < ˆpR c1c2=1 4, there is no trade at all and the interim welfare in the benchmark and the information exchange scenario are both zero. On the other hand, ∆ ˆ U1(v)>0 for intermediary values of vand ∆ ˆ U1(v)<0 for sufficiently high valuations. This follows as for all vsuch that ˆpR c1c2< v < ˆpc1=1 2, the benchmark offer will be rejected with certainty and the buyer expects a strictly positive profit. For vslightly above ˆpc1, the expected gain following a rejection still outweighs the expected loss following an acceptance, in particular as the probability of an acceptance is low. However, even though, by Proposition 5.4, the expected gain after a rejection increases in v, for large vthis effect is outweighed by the increasing probability that the first offer is accepted, which would result in a loss. Figure 3c depicts the density function g2 1with g2 1(c1)=4−8c1for all c1∈0,1 2and 0 otherwise. The corresponding function ∆ ˆ U2is shown in Figure 3d. For v < ˆpc1=1 2, the interim expected gains are ordered: smaller c1, which are more likely given g2 1than g1 1, imply lower prices by the second seller after a rejection (Corollary 4.7) and, hence, higher gains (Figure 2). Thus, ∆ ˆ U2(v) increases faster than ∆ ˆ U1(v) for v < 1 2. A third, maybe a bit pathological, density is given by g3 1(c1) = 1 + sin 6πc1+π 12for all c1∈0,1 2and 0 otherwise, which is depicted in Figure 3e. The corresponding function ∆ˆ U3in Figure 3f illustrates two points: First, given ∆ ˆ U3<0, it is not monotonic. That is, unlike Figures 3b and 3d suggest, buyers with larger vdo not automatically suffer more. Furthermore, ∆ ˆ U3has several roots. Hence, even if some buyer with valuation v suffers at the interim level, there might be another buyer with a larger valuation v0who gains from the information disclosure. All functions ∆ ˆ Uj(v) for j∈ {1,2,3}are not differentiable for v= ˆp(c1), indeed, differentiability at that point would require that g1c1= 0. Both ∆ ˆ U2and ∆ ˆ U3can be derived analytically. The respective expressions can be found in Appendix B. 19 n n 0 1 0 0.5 1 (a) PDF g1 1as in Example 5.7 n n 0 0.25 0.5 0.75 1 -0.05 0 0.05 (b) ∆ˆ U1(v) given PDF g1 1as in Example 5.7. n n 0 1 0 2 4 (c) PDF g2 1as in Example 5.7 n n 0 0.25 0.5 0.75 1 -0.05 0 0.05 (d) ∆ˆ U2(v) given PDF g2 1as in Example 5.7. v n 0 1 0 1 2 (e) PDF g3 1as in Example 5.7 v ∆U(v) 0 0.25 0.5 0.75 1 -0.05 0 0.05 0 (f) ∆ˆ U3(v) given PDF g3 1as in Example 5.7. Figure 3: Interim buyer welfare effects before information disclosure for different distributions. 20 As the previous example shows, there is little that can be said about ∆ ˆ Uin terms of comparative statics. There are certain areas though, where crisp results can be obtained. Proposition 5.8. Let (f, g1 1, g2)and (f, g2 1, g2)be two sequential trading games such that C1 1=C2 1and g1 1first-order stochastically dominates g2 1. 1. For all v < ˆp(c1 1)it holds that 0≤∆ˆ U1(v)≤∆ˆ U2(v). 2. For all v > ˆp(c1 1)it holds that ∆ˆ U1(v)≤∆ˆ U2(v)≤0. 5.4 Ex-Ante Welfare Effects From an overall welfare perspective, we have seen in Theorem 5.2 that the disclosure of information by the first seller is beneficial, even at the ex-post level. From the buyer’s perspective this is not the case: ex post, the effect can be both positive or negative, and the same is true at the interim stage. The buyer’s ex-ante effect is given by EFh∆ˆ U(·)i=Z∞ −∞ f(v)∆ ˆ U(v)dv, which again can be positive or negative as the following example illustrates. Example 5.9. Recall the sequential trading games (f, g1 1, g2) and (f, g2 1, g2) from Example 5.7. We find EFh∆ˆ U1(·)i=−1 768 <0 and EFh∆ˆ U2(·)i=67 15360 >0 From an ex-ante perspective, there is also not much that can be said with respect to changes in the distributions. Figure 2from Example 5.6 illustrates that ∆ ˆ UA+ ∆ ˆ UR only depends on the values for c1but not their likelihood. It can also be seen that if g0 1first order stochastically dominates g1the effect on ∆ ˆ U(v) can be positive or negative depending on v. Hence, we can construct probability distributions Fsuch that a first order stochastic shift from g1to g0 1can have a positive or a negative effect on EFh∆ˆ U(v)i. 5.5 When Costs are not Independent Throughout this paper, we have assumed that the costs of the two sellers are independent. As they produce a homogeneous good, this might not be the case. Let ˜g(c1, c2) be a joint probability density function with marginal densities ˜g1(c1) and ˜g2(c2), and allow for the possibility that ˜g(c1, c2)6= ˜g1(c1) ˜g2(c2). Consider now the sequential trading game 21 (f, ˜g1,˜g2) as before. Clearly, the equilibrium strategies do not change compared to the previous analysis, as they do not depend on any dependency structure between the two costs. The following corollary follows directly from Proposition 5.1 and Theorem 5.2. Corollary 5.10. Let (f, ˜g1,˜g2)be a sequential trading game with ˜g(c1, c2)=0for all c2< c1. Then for all v∈ V,c1∈ C1, and c2∈ C2it holds that 1. ˆuv, c2;fA c1,˜g2= ˆu(v, c2;f, ˜g2)and ˆπv, c2;fA c1,˜g2= ˆπ(v, c2;f, ˜g2). 2. ˆuv, c2;fR c1,˜g2≥ˆu(v, c2;f, ˜g2)with strict inequality if and only if both v < ˆp(c1) and τ(f)< c2<ˆpR c1−1(v)hold. Corollary 5.10 shows that it is not possible for the buyer to get harmed by the information exchange, if the cost of the second seller is (almost) always higher than the cost of the first seller. Since, it is guaranteed that the buyer can not be harmed by the information exchange from this ex-post perspective, there is no possibility of damage for the buyer from all other timing perspectives as well. Note that, analogous to Corollary 5.3, it is still possible that the second seller is harmed by the information exchange from an ex-post perspective. 6 Discussion and Conclusion The objective of this paper was to analyze the welfare effects of private and unilateral disclosure of sensitive information in a sequential bargaining context. We develop a sequential trading model in which one buyer bargains with two different sellers. Our results show that the buyer profits from the private and unilateral disclosure among the sellers if it reveals a rejection but might be harmed by a revealed acceptance. It is shown that the buyer cannot be harmed but might profit from the information exchange, if the cost of the seller that receives the message is higher than the cost of the information providing seller. This result holds from an ex-post, interim, and ex-ante perspective. It is also shown that the exchange of information improves the societal welfare by reducing inefficiencies and potentially generating additional trade. Additionally, the buyer can only be harmed by the information exchange if the seller in the second game holds the entire bargaining power as assumed in the paper. To see 22 this, consider a reversion of the bargaining power such that the buyer proposes a take-itor-leave-it price in the second game. The second seller accepts every cost-covering offer irrespective of any information exchange between the two sellers. Thus, the unilateral disclosure of information has no effect on any players’ welfare. Overall, this paper strengthens the theoretical framework for assessing the welfare effects of information exchanges by offering new insights and providing tools to assess causality for alleged damages. This does not only question existing economic principles as described above, but also contributes to questions that are at the core of follow-on damages claims, in which consumers must demonstrate a causal link between alleged anticompetitive practices and their claimed damages. References Ali, S.Nageeb, Kleiner, Andreas, and Zhang, Kun (2024): “From Design to Disclosure”. In: arXiv 2411.03608. Bagnoli, Mark and Bergstrom, Ted (2005): “Log-Concave Probability and Its Applications”. In: Economic Theory 26(2), pp. 445–469. Bergemann, Dirk, Brooks, Benjamin, and Morris, Stephen (2015): “The Limits of Price Discrimination”. In: American Economic Review 105(3), pp. 921–957. Bergemann, Dirk and Morris, Stephen (2019): “Information Design: A Unified Perspective”. In: Journal of Economic Literature 57(1), pp. 44–95. Blackwell, David (1953): “Equivalent comparisons of experiments”. In: The annals of mathematical statistics 24(2), pp. 265–272. de Oliveira, Henrique (2018): “Blackwell’s Informativeness Theorem using Diagrams”. In: Games and Economic Behavior 109, pp. 126–131. European Commission (2020): Antitrust: Commission fines ethylene purchasers €260 million in cartel settlement.url:https://ec.europa.eu/commission/presscorner/detail/ en/ip_20_1348 (visited on 03/10/2025). 23 +Zˆp−1(v) τ(fR c1) g2(c2)ˆp(c2)−ˆpR c1(c2)dc2 +Z(ˆpR c1)−1(v) ˆp−1(v) g2(c2)v−ˆpR c1(c2)dc2 ≥0, where the non-negativity of the three integrands follows from ˆp(c2)≥vfor all c2≥τ(f), Proposition 4.6, and ˆpR c1(c2)≤vfor all c2≤ˆpR c1−1(v), respectively. Since all integrands are strictly positive in the interior of the respective intervals, ∆ ˆ UR(v;c1)>0 if and only if G2ˆpR c1−1(v)> G2(τ(f)). Finally, the first integral does not depend on v, and the sum of the last two integrals is weakly increasing by the same reasoning as before.  Proof of Proposition 5.8.1. Let v < ˆp(c1 1) = ˆp(c2 1). Then ∆ ˆ UA(v;c1) = 0 for all c1∈ C1 1=C2 1. Therefore, ∆ˆ Uj(v) = Z∞ −∞ gj 1(c1)∆ ˆ UR(v;c1)dc1≥0 for j∈ {1,2}. Consider now c1, c0 1∈ C1 1=C2 1with c1< c0 1and note that by Proposition 5.1 ˆu(v, c2;fR c1, g2)≥ˆu(v, c2;fR c0 1, g2) for all c2∈ C2and, hence, ∆ ˆ UR(v;c1)≥∆ˆ UR(v;c0 1). Since g1 1first-order stochastically dominates g2 1, we thus have ∆ˆ U1(v) = Z∞ −∞ g1 1(c1)∆ ˆ UR(v;c1)dc1≤Z∞ −∞ g2 1(c1)∆ ˆ UR(v;c1)dc1= ∆ ˆ U2(v). 2. Let v > ˆp(c1 1). Then ∆ ˆ UR(v;c1) = 0 for all c1∈ C1 1and, thus, ∆ˆ Uj(v) = Z∞ −∞ gj 1(c1)∆ ˆ UA(v;c1)dc1≤0 for j∈ {1,2}. Again, Proposition 5.1 implies that ∆ ˆ UA(v;c1)≥∆ˆ UA(v;c0 1) for all c1, c0 1∈ C1 1with c1< c0 1and, hence, ∆ ˆ U1(v)≤∆ˆ U2(v) by first-order stochastic dominance.  30 B Complementary Functions Example 5.7 ∆ˆ U2(v;f, g2 1) =                      0,if v∈0,1 4 (3−4v)(4v−1)3 24 ,if v∈1 4,3 8 v2−7 12v+11 128,if v∈3 8,1 2 −2 3v4+22 3v3−12v2+27 4v−463 384,if v∈1 2,3 4 −1 96,if v∈3 4,1 . ∆ˆ U3(v;f, g3 1) =                  0,if v∈0,1 4 cosπ(288v−71) 12 +6π(4v−1)(3π(4v−1)(2π(4v−1)+cos(π 12 ))+sin(π 12 ))−cos(π 12 ) 1728π3,if v∈1 4,1 2 (72π2v2−5)cosπ(144v−71) 12 +288π3v3−1728π3v2−12π(3v−2) sinπ(144v−71) 12  1728π3 +12πsin(73π 12 )−24π(sin(73π 12 )−72π2)v+cos(73π 12 )+4 cos(π 12 )−432π3 1728π3,if v∈1 2,1 31