# Is the Constrained Pareto Frontier Linear in Bilateral Trade?

Suppose we have a Myerson-Satterthwaite model of bilateral trade between a buyer and seller. The buyer has private information about their valuation, $$v$$, and the seller has private information about their cost, $$c$$, of providing the good. There is a common prior that $$v\sim F$$ and $$c\sim G$$ for some smooth CDFs $$F,G$$ which are strictly increasing over intervals $$[\underline{v},\bar{v}]$$ and $$[\underline{c},\bar{c}]$$, respectively, and are flat outside of these intervals. Both agents have normalised reservation utilities of $$0$$. We say a mechanism $$\mu$$ is implementable if it is both incentive compatible and individually rational.

Define the most preferred implementable mechanisms of the buyer and seller as $$\mu_B$$ for the former and $$\mu_S$$ for the latter. These will be mechanisms corresponding to take-it-or-leave-it price offers. Define the random dictatorship (RD) mechanism with bargaining parameter $$\beta$$ as the one which selects $$\mu_B$$ with probability $$\beta$$ and $$\mu_S$$ with probability $$1-\beta$$. We write the RD($$\beta$$) mechanism in short as $$\beta \mu_B +(1-\beta) \mu_S$$. This mechanism is implementable.

Fix some $$\beta\in(0,1)$$. I am interested in whether there exists some implementable mechanism $$\mu^*$$ such that the interim utilities of both the buyer and seller are weakly larger under $$\mu^*$$ than the RD($$\beta$$) mechanism and for at least one type, the interim utility is strictly larger.

Effectively, I am asking whether the RD mechanism is incentive-efficient in the terminology of Myerson (1984; “Two-Person Bargaining Problems with Incomplete Information”). If it is not incentive-efficient, what is a good counterexample and what goes wrong with the RD mechanism?

Edit: The uniform case may provide a counterexample. If $$v\sim [0,1]$$ and $$c\sim [0,1]$$, then an example of a Myerson-Satterthwaite (1983) second-best efficient mechanism has allocation rule $$p$$ and transfer (from seller to buyer) function $$x$$ of the form \begin{align*} p(v,c) &= \begin{cases}1 & \text{ if } c\leq v-\frac{1}{4} \\ 0 & \text{o/w} \end{cases} \\ x(v,c) &= \begin{cases} \frac{v+c+0.5}{3}& \text{ if } c\leq v-\frac{1}{4} \\ 0 & \text{o/w} \end{cases} \end{align*} This transfer rule corresponds to the Chatterjee-Samuelson bilateral auction. Clearly, this cannot be an interim efficiency improvement over the RD($$\beta$$) mechanisms since cost types $$c\in (3/4,1]$$ and valuation types $$[0,1/4)$$ never trade so are worse off in this new mechanism. However, I think there must be some way to adjust this transfer rule so as to distribute the efficiency gains in such a way that it is an interim efficiency improvement over the RD($$\beta$$) mechanisms.

• Do you want the mechanism to be budget balanced? Commented Aug 7 at 16:22
• @MichaelGreinecker Yes, I should have said. I think we need to enforce budget balancedness for $\mu_B$ and $\mu_S$ to be well defined. Also, if we don't require $\mu^*$ to be balanced, I think the answer is negative since we could inject money to implement ex-post efficient trade before redistributing the gains to the buyer and seller. Commented Aug 7 at 16:45