<p>Suppose we have a Myerson-Satterthwaite model of bilateral trade between a buyer and seller. The buyer has private information about their valuation, <span class="math-container">$v$</span>, and the seller has private information about their cost, <span class="math-container">$c$</span>, of providing the good. There is a common prior that <span class="math-container">$v\sim F$</span> and <span class="math-container">$c\sim G$</span> for some smooth CDFs <span class="math-container">$F,G$</span> which are strictly increasing over intervals <span class="math-container">$[\underline{v},\bar{v}]$</span> and <span class="math-container">$[\underline{c},\bar{c}]$</span>, respectively, and are flat outside of these intervals. Both agents have normalised reservation utilities of <span class="math-container">$0$</span>. We say a mechanism <span class="math-container">$\mu$</span> is implementable if it is both incentive compatible and individually rational.</p>
<p>Define the most preferred implementable mechanisms of the buyer and seller as <span class="math-container">$\mu_B$</span> for the former and <span class="math-container">$\mu_S$</span> 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 <span class="math-container">$\beta$</span> as the one which selects <span class="math-container">$\mu_B$</span> with probability <span class="math-container">$\beta$</span> and <span class="math-container">$\mu_S$</span> with probability <span class="math-container">$1-\beta$</span>. We write the RD(<span class="math-container">$\beta$</span>) mechanism in short as <span class="math-container">$\beta \mu_B +(1-\beta) \mu_S$</span>. This mechanism is implementable.</p>
<p>Fix some <span class="math-container">$\beta\in(0,1)$</span>. I am interested in whether there exists some implementable mechanism <span class="math-container">$\mu^*$</span> such that the interim utilities of both the buyer and seller are weakly larger under <span class="math-container">$\mu^*$</span> than the RD(<span class="math-container">$\beta$</span>) mechanism and for at least one type, the interim utility is strictly larger.</p>
<p>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?</p>
<hr />
<p><strong>Edit:</strong> The uniform case may provide a counterexample. If <span class="math-container">$v\sim [0,1]$</span> and <span class="math-container">$c\sim [0,1]$</span>, then an example of a Myerson-Satterthwaite (1983) second-best efficient mechanism has allocation rule <span class="math-container">$p$</span> and transfer (from seller to buyer) function <span class="math-container">$x$</span> of the form <span class="math-container">$$\begin{align*} p(v,c) &amp;= \begin{cases}1 &amp; \text{ if } c\leq v-\frac{1}{4} \\
0 &amp; \text{o/w}
 \end{cases} \\
x(v,c) &amp;= \begin{cases} \frac{v+c+0.5}{3}&amp; \text{ if } c\leq v-\frac{1}{4} \\
0 &amp; \text{o/w}
 \end{cases}  \end{align*}$$</span> This transfer rule corresponds to the Chatterjee-Samuelson bilateral auction. Clearly, this cannot be an interim efficiency improvement over the RD(<span class="math-container">$\beta$</span>) mechanisms since cost types <span class="math-container">$c\in (3/4,1]$</span> and valuation types <span class="math-container">$[0,1/4)$</span> 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(<span class="math-container">$\beta$</span>) mechanisms.</p>

CC BY-SA 4.0
2024-08-07T16:22:53.997Z
Do you want the mechanism to be budget balanced?
Michael Greinecker
CC BY-SA 4.0
2024-08-07T16:45:02.507Z
@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.
Joseph Basford
<microeconomics><mathematical-economics><pareto-efficiency><mechanism-design>
Is the Constrained Pareto Frontier Linear in Bilateral Trade?
Joseph Basford
# Is the Constrained Pareto Frontier Linear in Bilateral Trade?

(Asked by: username_0 on 2024-08-07)

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?

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**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.

**Tags:** microeconomics, mathematical-economics, pareto-efficiency, mechanism-design

> **Comments:**
> **username_1**: Do you want the mechanism to be budget balanced?  
> **username_0**: @username_1 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.  

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## 0 answers

