In this post I describe an alternative proof of a nifty result that appears in a forthcoming paper by Goeree, Kushnir, Moldovanu and Xi. They show (under private values) given any Bayesian incentive compatible mechanism, M, there is a dominant strategy mechanism that gives each agent the same expected surplus as M provides.

For economy of exposition only, suppose 2 agents and a finite set of possible outcomes, ${A}$. Suppose, also the same type space, ${\{1, \ldots, m\}}$ for both. Let ${f(\cdot)}$ be the density function over types. To avoid clutter, assume the uniform distribution, i.e., ${f(\cdot) = \frac{1}{m}}$. Nothing in the subsequent analysis relies on this.

When agent 1 reports type ${s}$ and agent 2 reports type ${t}$, denote by ${z_r(s,t) }$ the probability that outcome ${r \in A}$ is selected. The ${z}$‘s must be non-negative and satisfy ${\sum_{r \in A}z_r(s,t) = 1.}$

Associated with each agent ${i}$ is a vector ${\{a_{ir}\}_{r \in A}}$ that determines, along with her type, the utility she enjoys from a given allocation. In particular, given the allocation rule ${z}$, the utility that agent ${1}$ of type ${t}$ enjoys when the other agent reports type ${s}$ is ${\sum_{r \in A}ta_{1r}z_r(t,s).}$ A similar expression holds agent 2.

Let ${q_i(s,t) = \sum_{r \in A}a_{ir}z_r(s,t).}$ Interpret the ${q}$‘s as the quantity’ of goods that each agent receives. Dominant strategy implies that ${q_1(s,t)}$ should be monotone increasing in ${s}$ for each fixed ${t}$ and ${q_2(s,t)}$ should be monotone increasing in ${t}$ for each fixed ${s}$. The interim quantities will be: ${mQ_i(s) = \sum_t\sum_{r \in A}a_{ir}z_r(s,t).}$ Bayesian incentive compatibility (BIC) implies that the ${Q_i}$‘s should be monotone. To determine if given BIC interim `quantities’ ${Q_i}$‘s can be implemented via dominant strategies, we need to know if the following system is feasible

$\displaystyle \sum_{r \in A}a_{1r}[z_r(i,j) - z_r(i-1,j)] \geq 0\,\, \forall \,\, i = 2, \ldots, m \ \ \ \ \ (1)$

$\displaystyle \sum_{r \in A}a_{2r}[z_r(i,j) - z_r(i,j-1)] \geq 0\,\, \forall \,\, j = 2, \ldots, m \ \ \ \ \ (2)$

$\displaystyle \sum_t\sum_{r \in A}a_{1r}z_r(s,t) = mQ_1(s) \ \ \ \ \ (3)$

$\displaystyle \sum_s\sum_{r \in A}a_{2r}z_r(s,t) = mQ_2(t) \ \ \ \ \ (4)$

$\displaystyle \sum_{r \in A}z_r(s,t) = 1\,\, \forall s,t \ \ \ \ \ (5)$

$\displaystyle z_r(i,j) \geq 0\,\, \forall i,j,r \ \ \ \ \ (6)$

System (1-6) is feasible iff the optimal objective function value of the following program is zero:

$\displaystyle \min \sum_{i,j}w_{ij} + \sum_{i,j}h_{ij}$

subject to
$\displaystyle \sum_{r \in A}a_{1r}[z_r(i,j) - z_r(i-1,j)] + w_{ij} - u_{ij} = 0\,\, \forall \,\, i = 2, \ldots, m\,\, (\mu_{ij}) \ \ \ \ \ (7)$

$\displaystyle \sum_{r \in A}a_{2r}[z_r(i,j) - z_r(i,j-1)] + h_{ij} - v_{ij} = 0\,\, \forall \,\, j = 2, \ldots, m\,\, (\nu_{ij}) \ \ \ \ \ (8)$

$\displaystyle \sum_j\sum_{r \in A}a_{1r}z_r(i,j) = mQ_1(i)\,\, (\alpha_i) \ \ \ \ \ (9)$

$\displaystyle \sum_i\sum_{r \in A}a_{2r}z_r(i,j) = mQ_2(j)\,\, (\beta_j) \ \ \ \ \ (10)$

$\displaystyle \sum_{r \in A}z_r(i,j) = 1\,\, \forall i,j\,\, (\lambda(i,j)) \ \ \ \ \ (11)$

$\displaystyle w_{ij},\,\, h_{ij}\,\,z_r(i,j) \geq 0\,\, \forall i,j,r \ \ \ \ \ (12)$

Let ${(z^*, w^*, h^*, u^*, v^*)}$ be an optimal solution to this program.
Suppose, for a contradiction there is a pair ${(p,q)}$ such that ${w^*_{pq} > 0}$. I will argue that there must exist an ${s}$ such that ${u^*_{ps} > 0}$. Suppose not, then for each ${j \neq q}$, either ${w^*_{pj} > 0}$ or ${w^*_{pj} = 0}$ and ${u^*_{pj} = 0}$ (at optimality, it cannot be that ${w^*_{pq}}$ and ${u^*_{pq}}$ are both non-zero). In this case

$\displaystyle m[Q_1(p)-Q_1(p-1)] = \sum_j\sum_{r \in A}a_{1r}z^*_r(p,j) - \sum_j\sum_{r \in A}a_{1r}z^*_r(p-1,j)$

$= \sum_{j: w^*_{pj} > 0}a_{1r}[z^*_r(p,j) - z^*_r(p-1,j)]$ $= -\sum_j w^*_{pj}$. This last term is negative, a contradiction. Therefore, there is a $s$ such that $w^*_{ps} = 0$ but $u^*_{ps} > 0$.

Let ${Z_1(i,j) = \sum_{r \in A}a_{1r}z^*_r(i,j)}$, ${Z_2(i,j) = \sum_{r \in A}a_{2r}z^*_r(i,j)}$ and denote by ${Z(i,j)}$ the point ${(Z_1(i,j), Z_2(i,j))}$. Observe that ${Z(i,j)}$ is in the convex hull, ${K}$ of ${\{(a_{1r}, a_{2r}\}_{r \in A}}$ for all ${i,j}$. Thus choosing ${\{z_r(i,j)\}_{r \in A}}$‘s amounts to choosing a point ${Z(i,j) \in K}$. Equivalently, choosing a point ${Z(i,j) \in K}$ gives rise to a set of ${\{z_r(i,j)\}_{r \in A}}$‘s. For convenience assume that ${Z(i,j)}$ is in the strict interior of ${K}$ for all ${(i,j)}$ and that $K$ is full dimensional. This avoids having to deal with secondary arguments that obscure the main idea.

Recall, ${w^*_{pq} > 0}$ implies ${Z_1(p,q) 0}$ implies that ${Z_1(p,s) > Z_1(p,s)}$. Take all points ${\{Z(i,q)\}_{i \geq p}}$ and shift them horizontally to the right by ${\delta}$. Call these new points ${\{Z'(i,q)\}_{i \geq p}}$. Observe that ${Z'(i,q) \in K}$ for all ${i \geq p}$. Next, take all points ${\{Z(i,s)\}_{i \geq p}}$ and shift them horizontally to the left by ${\delta}$ to form new points ${\{Z'(i,s)\}_{i \geq p}}$. These points are also in ${K}$. Leave all other points ${\{Z(p,j)\}_{j \neq, q,s}}$ unchanged.

Because the vertical coordinates of all points were left unchanged, (8) and (10) are satisfied by this choice of points. Because ${\{Z(i,q)\}_{i \geq p}}$ and ${\{Z(i,s)\}_{i \geq p}}$ were shifted in opposite directions along the horizontal, (9) is still true. Finally, because all points in ${\{Z(i,q)\}_{i \geq p}}$ and ${\{Z(i,s)\}_{i \geq p}}$ were shifted by the same amount, (7) continues to hold.

The shift leftwards of ${Z(p,s)}$ reduces ${u_{ps}}$ while the rightward shift of ${Z(p,q)}$ reduces ${w_{pq}}$. Thus, we get a new solution with higher objective function value, a contradiction.

If ${Z(p,q)}$ and ${Z(p,s)}$ are not the interior of ${K}$ but on the boundary, then horizontal shifts alone may place them outside of ${K}$. In the case of ${Z(p,q)}$ this can only happen if ${Z_2(p,q) > Z_2(p-1,q)}$. In this case, shift ${Z(p,q)}$ across and to the right by ${\delta}$ as well and then downwards by the same amount. This would have to be matched by a corresponding upward shift by some point ${Z_2(h,q)}$. Similarly with ${Z(p,s)}$.