Abstract
We extend the equivalence between Bayesian and dominant strategy implementation (Manelli and Vincent in Econometrica 78:1905–1938, 2010; Gershkov et al. in Econometrica 81: 197–220, 2013) to environments with nonlinear utilities satisfying a property of increasing differences over distributions and a convexvalued assumption. The new equivalence result produces novel implications to the literature on the principalagent problem with allocative externalities, environmental mechanism design, and public good provision.
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Notes
 1.
Bergemann and Morris (2005)’s results apply only to quasilinear environments with unrestricted transfers and when the mechanism designer seeks to implement a singlevalued social choice function that only depends on agents’ payoff types. We are thankful to a referee for bringing to our attention a more precise statement.
 2.
Goeree and Kushnir (2017) provide an alternative proof of this equivalence result using a novel geometric approach to mechanism design. Kushnir (2015) extends the result to environments with correlated types. Kushnir and Liu (2017) explain how the BIC–DIC equivalence problem reduces to a purely mathematical question of when a linear transformation of intersection of two closed convex sets coincides with the intersection of their images.
 3.
 4.
 5.
Throughout the paper, “increasing” (“decreasing”) refers to a strictly increasing (decreasing).
 6.
The result below follows from Propositions 1, 2, and 3 of Mookherjee and Reichelstein (1992), because when \(v_i\) is differentiable with respect to its second argument, the increasing differences property is equivalent to the weak singlecrossing property used in their paper.
 7.
See also Milgrom and Segal (2002). The sufficiency part holds even without imposing the increasing differences.
 8.
As in Proposition 1, the sufficiency part holds even without imposing increasing differences over distributions.
 9.
 10.
We thank Navin Kartik, SangMok Lee, and Daniel Rappoport for pointing out this connection to us.
 11.
 12.
The necessity part actually holds only under an additional mild condition. If we denote the matrix transforming \(\mathbf{f}\) to \(\mathbf{g}\) as A and the diagonal matrix with elements \(M_i(x_i)\) as \(M(\mathbf{x})\) with \(\mathbf{x}=(x_1,\ldots ,x_I)\), the additional condition states that the sum of matrices \(M(\mathbf{x})+A\) has a full rank.
 13.
Gershkov et al. (2013) use a minimization problem to find a feasible allocation that satisfies (4) and that has nondecreasing marginals. Their approach could also be adapted to our settings. We use an algorithmic proof because of its convenience in the proofs of our Theorem 2 and the applications presented in Sect. 5.
 14.
For general \(\mathbf {g}\), the constructed DIC mechanism still delivers the same interim expected utilities.
 15.
Similar to condition (ii) in Theorem 2, the result also extends to settings when \(\check{f}_i\) is nonincreasing and concave (or nondecreasing and convex) and G is continuous, nondecreasing, and concave in each component.
 16.
The constructed DIC mechanism satisfies even a stronger notion of ex post individual rationality.
 17.
The result of Corollary 6 extends without any change to nonsymmetric settings with mapping \(\mathbf{f}=(f_1(\cdot ),\ldots ,f_I(\cdot ))\) being convexvalued and functions \(f_i\), \(i\in \mathcal{I}\), being continuous, nondecreasing, and concave.
 18.
For some applications, it is natural to require mechanisms to be ex post budget balanced, i.e., inequality (8) holds for each \(\mathbf {x}\in \mathbf {X}\). Börgers and Norman (2009) show that for every ex ante budget balanced DIC mechanism (q, t) there exist transfers \(t'\) such that \((q, t')\) is (i) BIC for all agents and DIC for all but one agent and (ii) ex post budget balanced. Agents also have the same interim expected payments in both mechanisms (see also Börgers 2015).
 19.
We also establish that the standard increasing differences property is necessary for dominant strategy incentive compatibility to be conveniently characterized in terms of a monotonemarginal condition and an envelope formula.
 20.
Because of the monotonicity of the allocation rule and \(\hat{q}\in \{0,1\}\), the only indeterminacy in \(\hat{q}(x_1,x_2)\) could happen at \(x_2=1/2\).
 21.
Note that because of the increasing differences property, \(\forall a, a''\in A\) we can only have either \(v_i(a, x)v_i(a'', x)\ne v_i(a, y)v_i(a'', y)\ne v_i(a, z)v_i(a'', z)\) or \(v_i(a, x)v_i(a'', x)= v_i(a, y)v_i(a'', y)=v_i(a, z)v_i(a'', z)\).
 22.
If such allocations do not exist, we will have \(\varDelta (G, F, x)=\varDelta (G, F, y)=\varDelta (G, F, z)\), which violates (A.8).
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Acknowledgements
We are very grateful to JeanMichel Benkert, Tilman Börgers, Satoshi Fukuda, Navin Kartik, SangMok Lee, Alejandro Manelli, Laurent Mathevet, Nick Netzer, Harry Di Pei, Daniel Rappoport, Philipp Strack, Mehmet Bumin Yenmez, seminar participants at Brown University, Johns Hopkins University, New York University, New Economic School, University of California, Berkeley, University of Pittsburgh, University of Zurich, as well as participants at various conferences and workshops for discussions and useful suggestions. We are also very thankful to the precious help of the editor Nicholas Yannelis and two anonymous referees. Due to their efforts the paper has significantly improved.
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The author Shuo Liu would like to acknowledge the hospitality of Columbia University, where some of this work was carried out, and the financial support by the Swiss National Science Foundation (Doc. Mobility grant P1ZHP1_168260) and the Forschungskredit of the University of Zurich (Grant No. FK17018).
Appendix
Appendix
Proof of Proposition 2
Our proof essentially extends the proof of Propositions 1, 2, and 3 in Mookherjee and Reichelstein (1992) to Bayesian settings. For the if statement, note that agent i does not deviate from the truthtelling Bayes–Nash equilibrium if and only if
for all \(x_i, x_i'\in X_i\). Using (2), this is equivalent to require that for all \(x_i, x_i'\in X_i\),
which is true under the condition that \(E_{\mathbf {x}_{i}}(v_{ix}(q(s, \mathbf {x}_{i}), x_i))\) is nondecreasing in s for all \(x_i\in X_i\).
For the only if statement, suppose mechanism (q, t) is BIC. We then have
Since \(v_i\) is absolutely continuous in \(x_i\) and has a bounded derivative with respect to type \(x_i\), equation (2) follows from the envelope theorem (Milgrom and Segal 2002). It remains to show that BIC also implies the monotoneexpectedmarginal condition. Suppose, in contradiction, we have \(E_{\mathbf {x}_{i}}v_{ix}(q(y, \mathbf {x}_{i}), z)>E_{\mathbf {x}_{i}}v_{ix}(q(x, \mathbf {x}_{i}), z)\) for some agent i and \(x, y, z\in X_i\), with \(y<x\). Since \(v_i\) satisfies the increasing differences over distribution property, this implies that the difference \(E_{\mathbf {x}_{i}}v_{ix}(q(y, \mathbf {x}_{i}), z')E_{\mathbf {x}_{i}}v_{ix}(q(x, \mathbf {x}_{i}), z')\) is strictly positive for all \(z'\in X_i\). It then follows that \(E_{\mathbf {x}_{i}}v_{i}(q(y, \mathbf {x}_{i}), z')  E_{\mathbf {x}_{i}}v_{i}(q(x, \mathbf {x}_{i}), z')\) is increasing in \(z'\) for all \(z'\in X_i\). Therefore, we have
At the same time, the incentive compatibility implies
and
We thus reach a contradiction. \(\square \)
Proof of Proposition 3
The sufficiency part is straightforward. Let us prove the necessity part. Consider some \(x',y' \in X_i\) such that \(x'>y'\) and let \(\underline{a}\in \arg \min _{a\in A}\, (v_{i}(a,x')v_i(a,y'))\) and \(\overline{a}\in \arg \max _{a\in A}\, (v_{i}(a,x')v_i(a,y'))\). Given our assumption that \(v_i(a,x_i)\) is continuous in a, such \(\underline{a}\) and \(\overline{a}\) are guaranteed to exist. For each \(a\in A\), we can then always find \(\alpha (a,x',y')\in [0,1]\) such that
Let us consider distribution G that puts the unit mass on allocation a and distribution F that puts probability \(\alpha (a,x',y')\) on \(\overline{a}\) and probability \(1\alpha (a,x',y')\) on \(\underline{a}\). By construction, we have
and the increasing differences over distributions implies that the difference \(\int v_{i}(a,x)\mathrm{d}G\int v_{i}(a,x)\mathrm{d}F\) is a constant function in x, which we denote as \(g_i(a)\). Hence,
where \(f_i(a)=\alpha (a,x',y')\), \(M_i(x)=v_{i}(\overline{a},x)v_{i}(\underline{a},x)\), and \(m_i(x)=v_i(\underline{a},x)\). The increasing differences over distributions and \(v_{i}(\overline{a},x')v_{i}(\underline{a},x')\ge v_{i}(\overline{a},y')v_{i}(\underline{a},y')\) then implies that \(M_i(x)\) is either an increasing or constant function. For the latter case, we redefine \(\tilde{f}_i(a)=0\), \(\tilde{M}_i(x)\) to be any increasing function, and \(\tilde{g}_i(a)=g_i(a)+f_i(a)M_{i}(x')\) to obtain expression (3). \(\square \)
Proof of Theorem 1. Below, we will prove two lemmas, which completes the proof of Theorem 1 as explained in the main text.
Lemma A1
Suppose, for all \(i\in \mathcal {I}\), \(X_{i}=[0,1]\) and \(\lambda _{i}\) is the uniform distribution on \(X_{i}\). Then, for any BIC allocation \(\tilde{q}\) there exists a feasible allocation q satisfying (4) with \(f_{i}(q(\cdot ,\mathbf{x}_{i}))\) being nondecreasing for all \(i\in \mathcal {I}\) and \(\mathbf {x}_{i}\in \mathbf{X}_{i}\).
Proof
The proof essentially repeats the proof of Lemma 2 in Gershkov et al. (2013), and we only sketch it here. We consider a partition \([0,1]^{I}\) to \(2^{nI}\) cubes of equal size. For each cube S in this partition, we approximate \(\mathbf{f}(\tilde{q}(\mathbf{x}))\), \(\mathbf{x}\in S\), by its average defined by
Note allocation \(\tilde{q}(S)\in A\) is well defined, because mapping \({\mathbf f}\) is convexvalued. In addition, discrete allocation \(\tilde{q}(S)\) inherits nondecreasing expected marginals from \(\tilde{q}\). Lemma 1 then ensures that there exists an allocation q(S) with nondecreasing marginals that can also be extended to piecewise constant functions over \([0,1]^{I}\). Taking the limit with respect to the size of partition, we obtain the result of the lemma. For the details of the construction, we refer to Gershkov et al. (2013). \(\square \)
Lemma A2
Suppose, for all \(i\in \mathcal {I}\), \(X_{i}\subseteq \,\mathbb {R}\,\) and \(\lambda _{i}\) is some distribution on \(X_{i}\). Then, for any BIC allocation \(\tilde{q}\) there exists a feasible allocation q satisfying (4) with \(f_{i}(q(\cdot ,\mathbf{x}_{i}))\) being nondecreasing for all \(i\in \mathcal {I}\) and \(\mathbf {x}_{i}\in \mathbf{X}_{i}\).
Proof
The proof repeats the proof of Lemma 3 in Gershkov et al. (2013). Its main idea is to relate the uniform distribution covered by Lemma A1 to the case of a general distribution. In particular, if random variable \(Z_{i}\) is uniformly distributed, then \(\lambda _{i}^{1}(Z_{i})\) is distributed according to \(\lambda _{i}\), where \(\lambda _{i}^{1}(z_{i})=\inf \{x_{i}\in X_{i}\lambda _{i}(x_{i})\ge z_{i}\}\). Hence, for a given BIC allocation \(\tilde{q}\) we use transformation \(\lambda _{i}^{1}\) to construct an allocation \(\tilde{q}'\) defined on uniformly distributed types that also has a nondecreasing expected marginals. For allocation \(\tilde{q}'\), we then apply the results of Lemmas 1 and A1 to obtain an allocation \(q'\) with nondecreasing marginals defined on uniformly distributed types. We then use transformation \(\lambda _i\) to recover an allocation q with nondecreasing marginals defined on types distributed according to \(\lambda _{i}\). For the details of the construction, we refer to Gershkov et al. (2013). \(\square \)
Example A1
. We now show that the assumption that mapping \(\mathbf{f}\) being convexvalued is generally indispensable for the equivalence result of Theorem 1.
Consider a twoagent example with the set of possible allocations \(A=[0, 1]\). Each agent i’s type \(x_i\) is drawn independently from the uniform distribution over [0, 1]. For an allocation \(q\in A\) and transfers \(t_1, t_2\in \mathbb {R}\), agent 1’s utility equals to \(qx_1+t_1\), and agent 2’s utility is \(q^2x_2+t_2\). This environment satisfies all conditions of Theorem 1 except for the assumption that mapping \((f_1, f_2)\) is convexvalued, where \(f_1(q)=q\) and \(f_2(q)=q^2\). Let us consider the following allocation rule:
This allocation rule is Bayesian implementable because its expected marginals \(\int _{0}^1f_i(x_i, x_j)dx_j\) are nondecreasing everywhere. It is, however, not dominant strategy implementable because marginals \(f_i(x_1, x_2)\) are strictly decreasing for some \((x_1, x_2)\in \mathbf{X}\). We now show that there does not exist an equivalent DIC mechanism for any BIC mechanism with allocation rule q.
Suppose, in contradiction, that for some BIC mechanism (q, t) there exists an equivalent DIC mechanism \((\hat{q}, \hat{t})\). Let \(U_i(x_i)\) and \(\hat{U}_i(x_i)\) be agent i’s interim expected utilities in mechanisms (q, t) and \((\hat{q}, \hat{t})\), respectively. Since the two mechanisms are equivalent, we have \(U_i(x_i)=\hat{U}_i(x_i)\) for all \(x_i\in X_i\) and \(i=1, 2\). The envelope formula then implies that \(\forall x_i, x_i'\in X_i\),
Therefore, we have for almost all \(x_i\in [0, 1]\), and for all \(i, j\in \{1,2\}\), \(i\ne j\),
Integrating (A.2) over \(x_j\), we have for all \(i\in \{1, 2\}\),
which further implies that
Since \(q(x_1,x_2)\in A = [0, 1]\), Eq. (A.4) implies that \(\hat{q}(x_1, x_2)\in \{0, 1\}\) for almost every type profile \((x_1,x_2)\in \mathbf{X}\). In addition, allocation \(\hat{q}\) being dominant strategy implementable implies that \(f_2(\hat{q}(x_1,x_2))=(\hat{q}(x_1,x_2))^2\) must be nondecreasing in \(x_2\). Equalexpectedmarginal condition (A.2) for agent 1 then implies that for almost all \(x_1\in [0, 1]\), \(\hat{q}(x_1, x_2)=0\) for \(x_2\in [0, 1/2]\) and \(\hat{q}(x_1, x_2)=1\) for \(x_2\in (1/2, 1]\).^{Footnote 20}
This allocation rule, however, does not satisfy equalexpectedmarginal condition (A.2) for agent 2. In particular, \(\int _{0}^1(\hat{q}(x_1, x_2))^2\mathrm{d}x_1=0\) for all \(x_2\in [0, 1/2]\), and \(\int _{0}^1(\hat{q}(x_1, x_2))^2\mathrm{d}x_2=1\) for all \(x_2\in (1/2, 1]\). We thus reach a contradiction. \(\square \)
Proof of Theorem 2
Consider an arbitrary BIC mechanism \((\tilde{q}, \tilde{t})\) and the corresponding DIC mechanism (q, t) constructed in Theorem 1. Since equation (7) holds for any \(\mathbf {g}\), the first part of Theorem 2 immediately follows. The idea behind the proof of the second part of the theorem is to show that if functions \(\check{f}_i\) and \(g_i\) satisfy conditions (i) or (ii), the DIC mechanism constructed in Theorem 1 also satisfies
Suppose condition (i) is satisfied. Let us first consider the case where types are discrete and uniformly distributed (as in Lemma 1). If the marginals of allocation \(\tilde{q}\) are not nondecreasing, then \(\check{f}_j(\tilde{q}_j(x_j', \mathbf {x}_j))<\check{f}_j(\tilde{q}_j(x_j, \mathbf {x}_{j}))\) for some j, \(x_j'>x_j\), and \(\mathbf {x}_{j}\). Using the construction of the algorithm in Lemma 1, we then obtain an allocation \(\hat{q}\in A\) satisfying the equalmarginal conditions in (4) and delivering strictly smaller value to objective \(E_\mathbf{x}\mathbf{f}(\cdot )^2\). Since function \(\check{f}_j\) is nondecreasing and concave (or nonincreasing and convex), we also have
Since \(g_i\) is nonincreasing and concave in each component, this further implies
for each \(i \in \mathcal{I}\) and, hence, \(E_{\mathbf {x}}(\sum _i g_i(\hat{q}(\mathbf {x})))\ge E_{\mathbf {x}}(\sum _ig_i(\tilde{q}(\mathbf {x})))\). We iterate this procedure to obtain a sequence of allocations \(q^n\in A\) and a decreasing numerical sequence \(s^n=E_\mathbf{x}\mathbf{f}(q^n(\mathbf{x}))^2\), \(n=1, 2,\ldots \). If we find that \(\check{f}_j(q_j^n(\cdot , \mathbf {x}_{j}))\) is nondecreasing for all j and \(\mathbf {x}_{j}\), we set \(q^{n+1}\equiv q^n\) and \(s^{n+1}\equiv s^n\). Since \(s^n\) is a weakly decreasing sequence bounded below by 0, it has a limit, which we denote as s. Since set A is compact, there also exists a convergent subsequence \(q^n\) with a limit q such that \(q(\mathbf{x})\in A\) for all \(\mathbf{x}\in X\). Clearly, \(s=E_{\mathbf {x}}(\mathbf{f}(q(\mathbf {x})^2)\) and \(\check{f}_j(q_j(\cdot , \mathbf {x}_{j}))\) is nondecreasing for each j and \(\mathbf {x}_{j}\). Since functions \(g_i\) are continuous, we also have \(E_{\mathbf {x}}\bigl (\sum _ig_i(q(\mathbf {x}))\bigr )\ge E_{\mathbf {x}}\bigl (\sum _i g_i(\tilde{q}(\mathbf {x}))\bigr )\).
The result can then be further extended to continuous space with an arbitrary distribution similar to Lemmas A1 and A2. We then use equation (6) to define payment rule t delivering the same interim expected utilities. Finally, we derive that the social surplus in the constructed allocation
where the inequality follows from the equalmarginal conditions in (4) and inequality (A.5). This establishes the claim of the theorem. The proof is analogous when condition (ii) is satisfied. \(\square \)
Proof of Corollaries 1 and 5
The statements follow from Theorem 1. \(\square \)
Proof of Corollaries 2, 3, and 4
The statements follow from Theorem 2. \(\square \)
Proof of Corollary 6
Consider any BIC mechanism \((\tilde{q}, \tilde{t})\) and the equivalent DIC mechanism (q, t), constructed in Theorem 1. Since we have \(g_i(q)=0\) for each \(i \in \mathcal{I}\) in the public good provision setting, the same ex ante expected utilities in both mechanisms implies that both mechanism yield the same expected transfers, i.e., \(E_{\mathbf {x}}\left( \sum _{i\in \mathcal {I}}t_i(\mathbf {x})\right) =E_{\mathbf {x}}\left( \sum _{i\in \mathcal {I}}\tilde{t}_i(\mathbf {x})\right) \).
To prove the claim of the corollary, we need to show that the expected costs for the DIC mechanism is lower than the expected costs for the BIC mechanism, i.e., \(E_{\mathbf {x}}(K(q(\mathbf {x})))\le E_{\mathbf {x}}(K(\tilde{q}(\mathbf {x})))\). This statement follows from applying the argument of the proof of Theorem 2 to function \(K\) instead of functions \(g_i\), \(i\in \mathcal{I}\). In particular, consider the sequence of allocation \(q^n\) constructed in the algorithm of Theorem 1. Since function K is nondecreasing and convex, the expected cost of allocations \(q^n\) is nonincreasing in n, i.e., \(E_{\mathbf {x}}(K(q^{n+1}(\mathbf {x})))\le E_{\mathbf {x}}(K(q^n(\mathbf {x})))\le E_{\mathbf {x}}(K(\tilde{q}(\mathbf {x})))\). The continuity of function K then implies that the inequality holds in the limit. Finally, the result further extends to continuous type space with an arbitrary distribution similar to Lemmas A1 and A2. \(\square \)
Proposition A1
If function \(v_i\) violates the increasing differences property for some agent \(i\in \mathcal {I}\), then there exists a dominant strategy incentive compatible mechanism (q, t) that does not have nondecreasing marginals \(v_{ix}(q(\cdot ,\mathbf {x}_{i}), x_i)\) for all \(\mathbf{x}_{i} \in \mathbf {X}_{i}\) and \(x_i \in X_i\).
Proof
Suppose \(v_i(a, x)\) does not satisfy the increasing differences property. There must exist \(a, a'\in A\), and \(x, y, z\in X\) with \(x<y<z\) such that either
with at least one inequality being strict, or
with at least one strict inequality. We consider only case (A.6). Case (A.7) can be treated similarly.
Let us assume that the utility of agent i satisfies (A.6). We consider a mechanism with an allocation rule q and a payment rule t that are functions of agent i’s reports only, i.e., \(q: X_i\rightarrow A\) and \(t: X_i\rightarrow \mathbb {R}^I\). In particular, we assign \(q(x)=q(z)=a', q(y)=a\), and \(\forall s\ne x, y, z\),
where \(\bar{t}_i=v_i(a, y)v_i(a', y)\). Agent i receives no transfers if allocation a is chosen and \(\bar{t}_i\) otherwise, i.e., \(t_i(s)=0\) if \(q(s)=a\) and \(t_i(s)=\bar{t}_i\) if \(q(s)=a'\). All other agents receive no transfers, i.e., \(t_j(s)\equiv 0\) for all \(j\ne i\) and \(s\in X_i\). It is straightforward to check that (q, t) is dominant strategy incentive compatible.
We now show that agent i’s marginals induced by allocation rule q cannot be all nondecreasing. Suppose, in contradiction, that \(v_{ix}(q(\cdot ), s)\) is nondecreasing for all \(s\in X_i\). Then, we have
or, equivalently, \(v_{ix}(a', s)\le v_{ix}(a, s)\le v_{ix}(a', s),~\forall s\in X_i\). But then \(v_{ix}(a', s)=v_{ix}(a, s), \forall s\in X_i\), and by integration over s we have
which contradicts (A.6). \(\square \)
Proposition A2
Suppose that there exist two agents whose type distributions are absolutely continuous. If function \(v_i\) violates the increasing differences over distributions property for some agent \(i\in \mathcal {I}\), then there exists a Bayesian incentive compatible mechanism (q, t) that does not have nondecreasing expected marginals \(E_{\mathbf {x}_{i}}[v_{ix}(q(\cdot ,\mathbf {x}_{i}), x_i)]\) for all \(x_i \in X_i\).
Proof. For any \(G, F\in \varDelta (A)\) and any \(s\in X_i\), let
Suppose \(v_i(a, x)\) does not satisfy the increasing differences over distributions property. Then, there must exist \(G, F\in \varDelta (A\)), and \(x, y, z\in X\) with \(x<y<z\) such that either
with at least one inequality being strict, or
with at least one strict inequality. We consider only case (A.8). Case (A.9) can be treated similarly.
Assume that the utility of agent i satisfies (A.8). Let \(a_G, a_G', a_F, a_F'\in A\), and \(G_{\alpha }\) (\(F_{\beta }\)) be the binary probability distribution that puts a weight \(\alpha \) (\(\beta \)) on the allocation \(a_G\) (\(a_F\)) and the remaining weight \(1\alpha \) (\(1\beta \)) on the allocation \(a_G'\) (\(a_F'\)), where \(\alpha , \beta \in [0, 1]\). We establish the following lemma.
Lemma A3
There exists a pair of binary distributions \(G_{\alpha }, F_{\beta }\) such that
with at least one inequality being strict.
Proof
Since both \(G_{\alpha }\) and \(F_{\beta }\) can be deterministic, the claim of the lemma is clearly true if the increasing differences property is violated. Thus, it is without loss to assume that this property is satisfied by \(v_i\). We want to first show that \(\exists a, a', a''\in A\) that satisfy the two following conditions simultaneously:

(i)
\(v_i(a, x)v_i(a'', x)\ne v_i(a, y)v_i(a'', y)\ne v_i(a, z)v_i(a'', z)\).^{Footnote 21}

(ii)
\(\not \exists \lambda \in \mathbb {R}\) such that
$$\begin{aligned}&(v_i(a, y)v_i(a', y))(v_i(a, x)v_i(a', x))\\&\quad =\lambda [(v_i(a, y)v_i(a'', y))(v_i(a, x)v_i(a'', x))], \end{aligned}$$and
$$\begin{aligned}&(v_i(a, z)v_i(a', z))(v_i(a, y)v_i(a', y))\\&\quad =\lambda [(v_i(a, z)v_i(a'', z))(v_i(a, y)v_i(a'', y))]. \end{aligned}$$
From the contrary, suppose that such a triple of allocations does not exist. Then, \(\forall a, a', a''\in A\), either
or there exists \(\lambda _{aa'a''}\in \mathbb {R}\) such that
and
Fix any \(a', a''\in A\) such that \( v_i(a', x)v_i(a'', x)\ne v_i(a', y)v_i(a'', y)\ne v_i(a', z)v_i(a'', z). \)^{Footnote 22} Let us consider a set
and \(\bar{A}_{a''}=A\setminus A_{a''}\). Note that \(\forall s, s'\in X_i\), we have
Hence,
where
and, similarly,
Since \(v_i(a'', s)v_i(a', s)\) is monotone in s \(\forall s\in X_i\), we have
which violates (A.8). Hence, there must exist \(a, a', a''\in A\) that satisfy both (i) and (ii). Note that for any such a triple of allocations \((a, a', a'')\), we must also have
since otherwise the two equations of (ii) will hold for either \(\lambda =0\) or \(\lambda =1\). Consequently, any triple of allocations that is a permutation of \((a, a', a'')\) will also satisfy conditions (i) and (ii), which suggests that the order of selecting a, \(a'\) and \(a''\) does not matter. Hence, without loss of generality, we can assume further that
Next, for all \(s\in X_i\), let us denote
and
Given (A.11) and the increasing differences property, \(\varDelta (\alpha , \beta , y)\varDelta (\alpha , \beta , x)\ge 0\) if and only if
while \(\varDelta (\alpha , \beta , y)\varDelta (\alpha , \beta , z)\ge 0\) if and only if
Similarly, we have \(\hat{\varDelta } (\alpha , \beta , y)\hat{\varDelta } (\alpha , \beta , x)\ge 0\) if and only if
while \(\hat{\varDelta } (\alpha , \beta , y)\varDelta (\alpha , \beta , z)\ge 0\) if and only if
Note that again because of the increasing differences property, the R.H.S. of inequalities (A.12), (A.13), (A.14) and (A.15) are all positive. Hence, if
one can always find \(\alpha , \beta \in [0, 1]\) such that both (A.12) and (A.13) are satisfied, and with at least one of them holds strictly. Otherwise, if the above strict inequality holds the other way round, then one can always find \(\alpha , \beta \in [0, 1]\) such that both (A.14) and (A.15) are satisfied, and with at least one of them holds strictly. In conclusion, we can always construct a pair of binary probability distributions \(G_{\alpha }, F_{\beta }\) that satisfy \( \varDelta (G_{\alpha }, F_{\beta }, x)\le \varDelta (G_{\alpha }, F_{\beta }, y)~\text {and}~\varDelta (G_{\alpha }, F_{\beta }, y)\ge \varDelta (G_{\alpha }, F_{\beta }, z) \), with at least one inequality being strict. \(\square \)
Lemma A3 shows that if \(v_i\) violates the property of increasing differences over distributions for some probability distributions (G, F), it must also violate this property for some binary probability distributions \((G_{\alpha }, F_{\beta })\). Given this important observation, we now construct a Bayesian incentive compatible mechanism that violates the monotoneexpectedmarginal condition.
Let \((G_{\alpha }, F_{\beta })\) be a pair of binary distributions that satisfies (A.10). By assumption, there must exist an agent \(j\ne i\) whose type distribution is absolutely continuous (and hence atomless). By continuity, we can always find transfers \(t_j^G, t_j^F\in \mathbb {R}\), and partitions \(X_j^{G}\cup X_j^{G'}=X_j\) and \(X_{j}^{F}\cup X_{j}^{F'}=X_{j}\) such that

(i)
\(\Pr \left( x_j\in X_j^{G}\right) =1\Pr \left( x_j\in X_j^{G'}\right) =\alpha \), \(\Pr \left( x_j\in X_j^{F}\right) =1\Pr \left( x_j\in X_j^{F'}\right) =\beta \);

(ii)
\(v_j(a_G, x_j)\ge v_j(a_G', x_j)+t_j^G ~\forall x_j\in X_j^{G},\)\(v_j(a_G, x_j)\le v_j(a_G', x_j)+t_j^{G} ~\forall x_j\in X_j^{G'}\);

(iii)
\(v_j(a_F, x_j)\ge v_j(a_F', x_j)+t_j^F ~\forall x_j\in X_j^{F},\) \(v_j(a_F, x_j)\le v_j(a_F', x_j)+t_j^{F} ~\forall x_j\in X_j^{F'}.\)
Consider a mechanism with an allocation rule q and a payment rule t that are functions of the reports of agents i and j. In particular, we let
and \(\forall s\ne x, y, z\),
where \( \bar{t}_i=\varDelta (G_{\alpha }, F_{\beta }, y). \) Agent i receives \(\bar{t}_i\) if either allocation \(a_F\) or \(a_F'\) is chosen, and \(t_i=0\) otherwise. Agent j receives \(t_j^G\) (\(t_j^F\)) if allocation \(a_G'\) (\(a_F'\)) is chosen, and \(t_j=0\) otherwise. For all agents \(k\ne i, j,\)\(t_k(\mathbf {x})=0\)\(\forall \mathbf {x}\in \mathbf {X}\). It is straightforward to check that (q, t) is a Bayesian incentive compatible mechanism.
We now show that agent i’s expected marginals induced by allocation rule q cannot be always nondecreasing. In contradiction, suppose \(E_{\mathbf {x}_{i}}v_{ix}(q(\cdot , \mathbf {x}_{i}), s)\) is nondecreasing for all \(s\in X_i\). Then, for any \(s\in X_i\) we have
or, equivalently,
which implies \(\int v_{ix}(a, s)dG_{\alpha }=\int v_{ix}(a, s)\mathrm{d}F_{\beta }\) for all \(s\in X_i.\) Then, by the integration over s we have
which contradicts to (A.10). \(\square \)
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Kushnir, A., Liu, S. On the equivalence of Bayesian and dominant strategy implementation for environments with nonlinear utilities. Econ Theory 67, 617–644 (2019). https://doi.org/10.1007/s001990181124x
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Keywords
 Bayesian implementation
 Dominant strategy implementation
 Mechanism design
 Nonlinear utilities
 Increasing differences over distributions
JEL Classification
 D82