On the equivalence of Bayesian and dominant strategy implementation for environments with nonlinear utilities

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 convex-valued assumption. The new equivalence result produces novel implications to the literature on the principal-agent problem with allocative externalities, environmental mechanism design, and public good provision.

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 truth-telling Bayes–Nash equilibrium if and only if

$$\begin{aligned} U_i(x_i)&\ge E_{\mathbf {x}_{-i}}\left( v_i(q(x_i', \mathbf {x}_{-i}), x_i)+t_i(x_i', \mathbf {x}_{-i})\right) \nonumber \\&=U_i(x_i')+E_{\mathbf {x}_{-i}}\left( v_{i}(q(x_i', \mathbf {x}_{-i}), x_i)-v_{i}(q(x_i', \mathbf {x}_{-i}), x_i')\right) \end{aligned}$$

(A.1)

for all $x_i, x_i'\in X_i$. Using (2), this is equivalent to require that for all $x_i, x_i'\in X_i$,

$$\begin{aligned}&\int _{x_i'}^{x_i}E_{\mathbf {x}_{-i}}\left( v_{ix}(q(s, \mathbf {x}_{-i}), s)\right) \mathrm{d}s\\&\quad \ge \,E_{\mathbf {x}_{-i}}\left( v_{i}(q(x_i', \mathbf {x}_{-i}), x_i)\right) -E_{\mathbf {x}_{-i}}\left( v_{i}(q(x_i', \mathbf {x}_{-i}), x_i')\right) , \end{aligned}$$

which is true under the condition that $E_{\mathbf {x}_{-i}}(v_{ix}(q(s, \mathbf {x}_{-i}), x_i))$ is non-decreasing in s for all $x_i\in X_i$.

For the only if statement, suppose mechanism (q, t) is BIC. We then have

$$\begin{aligned} U_i(x_i) = \max _{x_i'\in X_i} \bigl (E_{\mathbf {x}_{-i}}\left( v_i(q(x_i', \mathbf {x}_{-i}), x_i)+t_i(x_i', \mathbf {x}_{-i})\right) ). \end{aligned}$$

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 monotone-expected-marginal 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

$$\begin{aligned}&E_{\mathbf {x}_{-i}}\left( v_{i}(q(y, \mathbf {x}_{-i}), x)-v_{i}(q(y, \mathbf {x}_{-i}), y)\right) \\&\quad >E_{\mathbf {x}_{-i}}\left( v_{i}(q(x, \mathbf {x}_{-i}), x)-v_{i}(q(x, \mathbf {x}_{-i}), y)\right) . \end{aligned}$$

At the same time, the incentive compatibility implies

$$\begin{aligned} E_{\mathbf {x}_{-i}}\left( v_{i}(q(y, \mathbf {x}_{-i}), x)-v_{i}(q(y, \mathbf {x}_{-i}), y)\right) \le U_i(x)-U_i(y) \end{aligned}$$

and

$$\begin{aligned} E_{\mathbf {x}_{-i}}\left( v_{i}(q(x, \mathbf {x}_{-i}), x)-v_{i}(q(x, \mathbf {x}_{-i}), y)\right) \ge U_i(x)-U_i(y). \end{aligned}$$

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

$$\begin{aligned} v_{i}(a,x')-v_i(a,y')&=\alpha (a,x',y') \bigl (v_{i}(\overline{a},x')-v_i(\overline{a},y')\bigr )\\&\quad +(1-\alpha (a,x',y'))\bigl (v_{i}(\underline{a},x')-v_i(\underline{a},y')\bigr ). \end{aligned}$$

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

$$\begin{aligned} \int v_{i}(a,x')dG-\int v_{i}(a,x')dF=\int v_{i}(a,y')dG-\int v_{i}(a,y')\mathrm{d}F, \end{aligned}$$

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,

$$\begin{aligned} v_i(a,x)&=\alpha (a,x',y') v_{i}(\overline{a},x)+(1-\alpha (a,x',y'))v_{i}(\underline{a},x)+g_i(a)\\&=f_i(a) M_i(x)+m_i(x)+g_i(a) \end{aligned}$$

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 non-decreasing 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

$$\begin{aligned} {\mathbf f}(\tilde{q}(S)) =2^{nI}\int _{S}{\mathbf f}(\tilde{q}(\mathbf {x}))\mathrm{d}\mathbf {x}. \end{aligned}$$

Note allocation $\tilde{q}(S)\in A$ is well defined, because mapping ${\mathbf f}$ is convex-valued. In addition, discrete allocation $\tilde{q}(S)$ inherits non-decreasing expected marginals from $\tilde{q}$. Lemma 1 then ensures that there exists an allocation q(S) with non-decreasing 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 non-decreasing 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 non-decreasing expected marginals. For allocation $\tilde{q}'$, we then apply the results of Lemmas 1 and A1 to obtain an allocation $q'$ with non-decreasing marginals defined on uniformly distributed types. We then use transformation $\lambda _i$ to recover an allocation q with non-decreasing 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 convex-valued is generally indispensable for the equivalence result of Theorem 1.

Consider a two-agent 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 convex-valued, where $f_1(q)=q$ and $f_2(q)=q^2$. Let us consider the following allocation rule:

$$\begin{aligned} q(x_1, x_2)={\left\{ \begin{array}{ll}1 &{}\text {if}~\max \{x_1, x_2\}\le \frac{1}{2}~\text {or}~\min \{x_1, x_2\}>1/2,\\ 0 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

This allocation rule is Bayesian implementable because its expected marginals $\int _{0}^1f_i(x_i, x_j)dx_j$ are non-decreasing 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 $\forall x_i, x_i'\in X_i$,

$$\begin{aligned} U_i(x_i)&=U_i(x_i')+\int _{x_i'}^{x_i}\int _{0}^1 f_i(q(s, x_j))\mathrm{d}x_j\mathrm{d}s\\&=\hat{U}_i(x_i')+\int _{x_i'}^{x_i}\int _{0}^1 f_i(\hat{q}(s, x_j))\mathrm{d}x_j\mathrm{d}s=\hat{U}_i(x_i). \end{aligned}$$

Therefore, we have for almost all $x_i\in [0, 1]$, and for all $i, j\in \{1,2\}$, $i\ne j$,

$$\begin{aligned} \int _{0}^1f_i(\hat{q}(x_i, x_j))\mathrm{d}x_j=\int _{0}^1f_i(q(x_i, x_j))\mathrm{d}x_j=\frac{1}{2}. \end{aligned}$$

(A.2)

Integrating (A.2) over $x_j$, we have for all $i\in \{1, 2\}$,

$$\begin{aligned} \int _0^1\int _0^1f_i(\hat{q}(x_1, x_2))dx_1\mathrm{d}x_2=\frac{1}{2}, \end{aligned}$$

(A.3)

which further implies that

$$\begin{aligned} 0&=\int _0^1\int _0^1[f_1(\hat{q}(x_1, x_2))-f_2(\hat{q}(x_1, x_2))]\mathrm{d}x_1\mathrm{d}x_2\nonumber \\&=\int _0^1\int _0^1\left[ \hat{q}(x_1, x_2)-(\hat{q}(x_1, x_2))^2\right] \mathrm{d}x_1\mathrm{d}x_2. \end{aligned}$$

(A.4)

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 non-decreasing in $x_2$. Equal-expected-marginal 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 $x_2\in (1/2, 1]$.²⁰

This allocation rule, however, does not satisfy equal-expected-marginal 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

$$\begin{aligned} E_{\mathbf {x}}\left( \sum \limits _i g_i(q(\mathbf {x}))\right) \ge E_{\mathbf {x}}\left( \sum \nolimits _i g_i(\tilde{q}(\mathbf {x}))\right) . \end{aligned}$$

(A.5)

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 non-decreasing, 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 equal-marginal conditions in (4) and delivering strictly smaller value to objective $E_\mathbf{x}||\mathbf{f}(\cdot )||^2$. Since function $\check{f}_j$ is non-decreasing and concave (or non-increasing and convex), we also have

$$\begin{aligned} \hat{q}_j(x_j, \mathbf {x}_{-j})=\hat{q}_j(x'_j, \mathbf {x}_{-j})&\le \frac{1}{2}\tilde{q}_j(x_j,\mathbf {x}_{-j})+\frac{1}{2}\tilde{q}_j(x_j',\mathbf {x}_{-j}),\\ \hat{q}_j(x_j, \mathbf {x}'_{-j})&\le (1-\delta )\tilde{q}_j(x_j, \mathbf {x}'_{-j})+\delta \tilde{q}_j(x_j', \mathbf {x}_{-j}'),\\ \hat{q}_j(x_j', \hat{\mathbf {x}}_{-j})&\le (1-\delta )\tilde{q}_j(x_j', \mathbf {x}'_{-j})+\delta \tilde{q}_j(x_j, \mathbf {x}'_{-j}). \end{aligned}$$

Since $g_i$ is non-increasing and concave in each component, this further implies

$$\begin{aligned} g_i(\hat{q}(x_j, \mathbf {x}_{-j}))+g_i(\hat{q}(x_j', \mathbf {x}_{-j}))&\ge g_i(\tilde{q}(x_j, \mathbf {x}_{-j}))+g_i(\tilde{q}(x_j', \mathbf {x}_{-j})),\\ g_i(\hat{q}(x_j, \mathbf {x}_{-j}'))+g_i(\hat{q}(x_j', \mathbf {x}_{-j}'))&\ge g_i(\tilde{q}(x_j, \mathbf {x}_{-j}'))+g_i(\tilde{q}(x_j', \mathbf {x}_{-j}')), \end{aligned}$$

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 non-decreasing 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 non-decreasing 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

$$\begin{aligned} E_{\mathbf x}\left( \sum \limits _iv_{i}(q({\mathbf x}), x_i)\right)&=E_{\mathbf x}\left( \sum \limits _i f_i(q({\mathbf x}))M_i(x_i)+m_i(x_i)+g_i(q({\mathbf x}))\right) \\&\ge E_{\mathbf x} \left( \sum \limits _i f_i(\tilde{q}({\mathbf x}))M_i(x_i)+m_i(x_i)+g_i(\tilde{q}({\mathbf x}))\right) \\&= E_{\mathbf x}\left( \sum \limits _iv_{i}(\tilde{q}({\mathbf x}), x_i)\right) , \end{aligned}$$

where the inequality follows from the equal-marginal 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 non-decreasing and convex, the expected cost of allocations $q^n$ is non-increasing 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 non-decreasing 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

$$\begin{aligned} \left\{ \begin{array}{l} v_i(a, x)-v_i(a', x)\le v_i(a, y)-v_i(a', y)\\ v_i(a, y)-v_i(a', y)\ge v_i(a, z)-v_i(a', z) \end{array}\right. , \end{aligned}$$

(A.6)

with at least one inequality being strict, or

$$\begin{aligned} \left\{ \begin{array}{l} v_i(a, x)-v_i(a', x)\ge v_i(a, y)-v_i(a', y)\\ v_i(a, y)-v_i(a', y)\le v_i(a, z)-v_i(a', z) \end{array}\right. , \end{aligned}$$

(A.7)

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 $\forall s\ne x, y, z$,

$$\begin{aligned} q(s)=\left\{ \begin{array}{ll} a~&{}\text {if}~v_i(a, s)-v_i(a', s)\ge \bar{t}_i\\ a'~&{}\text {otherwise} \end{array}\right. , \end{aligned}$$

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 non-decreasing. Suppose, in contradiction, that $v_{ix}(q(\cdot ), s)$ is non-decreasing for all $s\in X_i$. Then, we have

$$\begin{aligned} v_{ix}(q(x), s)\le v_{ix}(q(y), s)\le v_{ix}(q(z), s),~\forall s\in X_i \end{aligned}$$

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

$$\begin{aligned} v_i(a', y)-v_i(a', x)=v_i(a, y)-v_i(a, x)~\text {and}~ v_i(a', z)-v_i(a', y)=v_i(a, z)-v_i(a, y), \end{aligned}$$

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 non-decreasing 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

$$\begin{aligned} \varDelta (G, F, s)=\int v_i(a, s)\mathrm{d}G-\int v_i(a, s)\mathrm{d}F. \end{aligned}$$

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

$$\begin{aligned} \varDelta (G, F, x)\le \varDelta (G, F, y)~\text {and}~\varDelta (G, F, y)\ge \varDelta (G, F, z) \end{aligned}$$

(A.8)

with at least one inequality being strict, or

$$\begin{aligned} \varDelta (G, F, x)\ge \varDelta (G, F, y)~\text {and}~\varDelta (G, F, y)\le \varDelta (G, F, z) \end{aligned}$$

(A.9)

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

$$\begin{aligned} \varDelta (G_{\alpha }, F_{\beta }, x)\le \varDelta (G_{\alpha }, F_{\beta }, y)~\text {and}~\varDelta (G_{\alpha }, F_{\beta }, y)\ge \varDelta (G_{\alpha }, F_{\beta }, z) \end{aligned}$$

(A.10)

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) $v_i(a, x)-v_i(a'', x)\ne v_i(a, y)-v_i(a'', y)\ne v_i(a, z)-v_i(a'', z)$.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

$$\begin{aligned} v_i(a, x)-v_i(a'', x)= v_i(a, y)-v_i(a'', y)=v_i(a, z)-v_i(a'', z), \end{aligned}$$

or there exists $\lambda _{aa'a''}\in \mathbb {R}$ such that

$$\begin{aligned}&(v_i(a, y)-v_i(a', y))-(v_i(a, x)-v_i(a', x)\\&\quad =\lambda _{aa'a''} [(v_i(a, y)-v_i(a'', y))-(v_i(a, x)-v_i(a'', x))] \end{aligned}$$

and

$$\begin{aligned}&(v_i(a, z)-v_i(a', z))-(v_i(a, y)-v_i(a', y))\\&\quad =\lambda _{aa'a''} [(v_i(a, z)-v_i(a'', z))-(v_i(a, y)-v_i(a'', y))]. \end{aligned}$$

Fix any $a', a''\in A$ such that $v_i(a', x)-v_i(a'', x)\ne v_i(a', y)-v_i(a'', y)\ne v_i(a', z)-v_i(a'', z).$²² Let us consider a set

$$\begin{aligned} A_{a''}=\{a\in A: v_i(a, x)-v_i(a'', x)=v_i(a, y)-v_i(a'', y)=v_i(a, z)-v_i(a'', z)\}, \end{aligned}$$

and $\bar{A}_{a''}=A\setminus A_{a''}$. Note that $\forall s, s'\in X_i$, we have

$$\begin{aligned}&\varDelta (G, F, s)-\varDelta (G, F, s')\\&\quad =\int _{A_{a''}\cup \bar{A}_{a''}}\int [(v_i(a, s)-v_i(\tilde{a}, s))-(v_i(a, s')-v_i(\tilde{a}, s'))]\mathrm{d}F_{\tilde{a}}\mathrm{d}G_{a}. \end{aligned}$$

Hence,

$$\begin{aligned}&\varDelta (G, F, y)-\varDelta (G, F, x)\\&\quad =\int _{\bar{A}_{a''}}\int \lambda _{a\tilde{a}a''}[(v_i(a, y)-v_i(a'', y))-(v_i(a, x)-v_i(a'', x))]\mathrm{d}F_{\tilde{a}}dG_{a}\\&\qquad +\int _{A_{a''}}\int [(v_i(a'', y)-v_i(\tilde{a}, y))-(v_i(a'', x)-v_i(\tilde{a}, x))]\mathrm{d}F_{\tilde{a}}\mathrm{d}G_{a}\\&\quad =\int _{\bar{A}_{a''}}\int -\lambda _{a\tilde{a}a''}\lambda _{a''aa'}[(v_i(a'', y)-v_i(a', y))-(v_i(a'', x)-v_i(a', x))]\mathrm{d}F_{\tilde{a}}\mathrm{d}G_a\\&\qquad +\int _{A_{a''}}\int \lambda _{a''\tilde{a}a'}[(v_i(a'', y)-v_i(a', y))-(v_i(a'', x)-v_i(a', x))]\mathrm{d}F_{\tilde{a}}\mathrm{d}G_{a}\\&\quad =[(v_i(a'', x)-v_i(a', x))-(v_i(a'', y)-v_i(a', y))]K, \end{aligned}$$

where

$$\begin{aligned} K=\int _{\bar{A}_{a''}}\int \lambda _{a\tilde{a}a''}\lambda _{a''a a'}\mathrm{d}F_{\tilde{a}}\mathrm{d}G_a-\int _{A_{a''}}\int \lambda _{a'' \tilde{a} a'}\mathrm{d}F_{\tilde{a}}\mathrm{d}G_a, \end{aligned}$$

and, similarly,

$$\begin{aligned}&\varDelta (G, F, z)-\varDelta (G, F, y)\\&\quad =\int _{\bar{A}_{a''}}\int \lambda _{a\tilde{a}a''}[(v_i(a, z)-v_i(a'', z))-(v_i(a, y)-v_i(a'', y))]\mathrm{d}F_{\tilde{a}}\mathrm{d}G_a\\&\qquad +\int _{A_{a''}}\int [(v_i(a'', z)-v_i(\tilde{a}, z))-(v_i(a'', y)-v_i(\tilde{a}, y))]\mathrm{d}F_{\tilde{a}}\mathrm{d}G_a\\&\quad =\int _{\bar{A}_{a''}}\int -\lambda _{a\tilde{a}a''}\lambda _{a''a a'}[(v_i(a'', z)-v_i(a', z))-(v_i(a'', y)-v_i(a', y))]\mathrm{d}F_{\tilde{a}}\mathrm{d}G_a\\&\qquad +\int _{A_{a''}}\int \lambda _{a''\tilde{a}a'}[(v_i(a'', z)-v_i(a', z))-(v_i(a'', y)-v_i(a', y))]\mathrm{d}F_{\tilde{a}}\mathrm{d}G_a\\&\quad =\left[ (v_i(a'', y)-v_i(a', y))-(v_i(a'', z)-v_i(a', z))\right] K. \end{aligned}$$

Since $v_i(a'', s)-v_i(a', s)$ is monotone in s $\forall s\in X_i$, we have

$$\begin{aligned} sign\left[ \varDelta (G, F, y)-\varDelta (G, F, x)\right] =sign\left[ \varDelta (G, F, z)-\varDelta (G, F, y)\right] , \end{aligned}$$

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

$$\begin{aligned}&v_i(a, x)-v_i(a', x)\ne v_i(a, y)-v_i(a', y)\ne v_i(a, z)-v_i(a', z) ~\text {and}~\\&v_i(a'', x)-v_i(a', x)\ne v_i(a'', y)-v_i(a', y)\ne v_i(a'', z)-v_i(a', z), \end{aligned}$$

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

$$\begin{aligned} v_i(a, y)-v_i(a, x)<\min \left\{ v_i(a', y)-v_i(a', x), v_i(a'', y)-v_i(a'', x)\right\} . \end{aligned}$$

(A.11)

Next, for all $s\in X_i$, let us denote

$$\begin{aligned} \varDelta (\alpha , \beta , s)&=[\alpha v_i(a'', s)+(1-\alpha )v_i(a, s)]-[\beta v_i(a', s)+(1-\beta )v_i(a, s)]\\&=\alpha [v_i(a'', s)-v_i(a, s)]+\beta [v_i(a, s)-v_i(a', s)] \end{aligned}$$

and

$$\begin{aligned} \hat{\varDelta } (\alpha , \beta , s)&=[\alpha v_i(a', s)+(1-\alpha )v_i(a, s)]-[\beta v_i(a'', s)+(1-\beta )v_i(a, s)]\\&=\alpha [v_i(a', s)-v_i(a, s)]+\beta [v_i(a, s)-v_i(a'', s)]. \end{aligned}$$

Given (A.11) and the increasing differences property, $\varDelta (\alpha , \beta , y)-\varDelta (\alpha , \beta , x)\ge 0$ if and only if

$$\begin{aligned} \alpha \ge \beta \left[ \frac{(v_i(a', y)-v_i(a, y))-(v_i(a', x)-v_i(a, x))}{(v_i(a'', y)-v_i(a, y))-(v_i(a'', x)-v_i(a, x))}\right] , \end{aligned}$$

(A.12)

while $\varDelta (\alpha , \beta , y)-\varDelta (\alpha , \beta , z)\ge 0$ if and only if

$$\begin{aligned} \alpha \le \beta \left[ \frac{(v_i(a', y)-v_i(a, y))-(v_i(a', z)-v_i(a, z))}{(v_i(a'', y)-v_i(a, y))-(v_i(a'', z)-v_i(a, z))}\right] . \end{aligned}$$

(A.13)

Similarly, we have $\hat{\varDelta } (\alpha , \beta , y)-\hat{\varDelta } (\alpha , \beta , x)\ge 0$ if and only if

$$\begin{aligned} \alpha \ge \beta \left[ \frac{(v_i(a'', y)-v_i(a, y))-(v_i(a'', x)-v_i(a, x))}{(v_i(a', y)-v_i(a, y))-(v_i(a', x)-v_i(a, x))}\right] , \end{aligned}$$

(A.14)

while $\hat{\varDelta } (\alpha , \beta , y)-\varDelta (\alpha , \beta , z)\ge 0$ if and only if

$$\begin{aligned} \alpha \le \beta \left[ \frac{(v_i(a'', y)-v_i(a, y))-(v_i(a'', z)-v_i(a, z))}{(v_i(a', y)-v_i(a, y))-(v_i(a', z)-v_i(a, z))}\right] . \end{aligned}$$

(A.15)

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

$$\begin{aligned}&\frac{(v_i(a', y)-v_i(a, y))-(v_i(a', x)-v_i(a, x))}{(v_i(a'', y)-v_i(a, y))-(v_i(a'', x)-v_i(a, x))}\\&\quad <\frac{(v_i(a', y)-v_i(a, y))-(v_i(a', z)-v_i(a, z))}{(v_i(a'', y)-v_i(a, y))-(v_i(a'', z)-v_i(a, z))}, \end{aligned}$$

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 monotone-expected-marginal 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

$$\begin{aligned} q(x_i, \mathbf {x}_{-i})={\left\{ \begin{array}{ll}a_G~~~~~&{}\text {if}~x_i=y, ~\text {and}~x_j\in X_j^{G},\\ a_G'~~~~~&{}\text {if}~x_i=y, ~\text {and}~x_j\in X_j^{G'},\\ a_F~~~~&{}\text {if}~x_i\in \{x,z\}, ~\text {and}~x_{j}\in X_{j}^{F},\\ a_F'~~~~&{}\text {if}~x_i\in \{x,z\},~\text {and}~x_{j}\in X_{j}^{F'}, \end{array}\right. } \end{aligned}$$

and $\forall s\ne x, y, z$,

$$\begin{aligned} q(s, \mathbf {x}_{-i})={\left\{ \begin{array}{ll}a_G~~~~~&{}\text {if}~ \varDelta (G_{\alpha }, F_{\beta }, s) \ge \bar{t}_i~\text {and}~x_j\in X_j^{G},\\ a_G'~~~~~&{}\text {if}~ \varDelta (G_{\alpha }, F_{\beta }, s)\ge \bar{t}_i~\text {and}~x_j\in X_j^{G'},\\ a_F~~~~&{}\text {if}~\varDelta (G_{\alpha }, F_{\beta }, s)<\bar{t}_i~\text {and}~x_{j}\in X_{j}^{F},\\ a_F'~~~~&{}\text {if}~\varDelta (G_{\alpha }, F_{\beta }, s)<\bar{t}_i~\text {and}~x_{j}\in X_{j}^{F'}, \end{array}\right. } \end{aligned}$$

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 non-decreasing. In contradiction, suppose $E_{\mathbf {x}_{-i}}v_{ix}(q(\cdot , \mathbf {x}_{-i}), s)$ is non-decreasing for all $s\in X_i$. Then, for any $s\in X_i$ we have

$$\begin{aligned} E_{\mathbf {x}_{-i}}[v_{ix}(q(x, \mathbf {x}_{-i}), s)]\le E_{\mathbf {x}_{-i}}[v_{ix}(q(y, \mathbf {x}_{-i}), s)]\le E_{\mathbf {x}_{-i}}[v_{ix}(q(z, \mathbf {x}_{-i}), s)], \end{aligned}$$

or, equivalently,

$$\begin{aligned} \int v_{ix}(a, s)\mathrm{d}F_{\beta }\le \int v_{ix}(a, s)\mathrm{d}G_{\alpha }\le \int v_{ix}(a, s)\mathrm{d}F_{\beta }, \end{aligned}$$

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

$$\begin{aligned} \varDelta (G_{\alpha }, F_{\beta }, x)= \varDelta (G_{\alpha }, F_{\beta }, y)~\text {and}~\varDelta (G_{\alpha }, F_{\beta }, y)= \varDelta (G_{\alpha }, F_{\beta }, z), \end{aligned}$$

which contradicts to (A.10). $\square$