Robust bilateral trade with discrete types

Abstract

Bilateral trade problem is the most common market interaction in which a seller and a buyer bargain over an indivisible object, and the valuation of each agent about the object is private information. We investigate the cases where mechanisms satisfying Dominant Strategy Incentive Compatibility (DIC) and Ex-post Individual Rationality (EIR) properties can exhibit robust performance in the face of imprecision in prior structure. We start with the general mathematical formulation for the bilateral trade problem with DIC, EIR properties. We derive necessary and sufficient conditions for DIC, EIR mechanisms to be Ex-post efficient at the same time. Then, we define a new property—Allocation Maximality—and prove that the Posted Price mechanisms are the only mechanisms that satisfy DIC, EIR and Allocation Maximal properties. We also show that Posted Price mechanism is not the only mechanism that satisfies DIC and EIR properties. The last part of the paper introduces different sets of priors for agents’ types and consequently allows ambiguity in the problem framework. We derive robust counterparts and solve them numerically for the proposed objective function under box and $\phi$-divergence ambiguity specifications. Results suggest that restricting the feasible set to Posted Price mechanisms can decrease the objective value to different extents depending on the uncertainty set.

Appendix

1.1 Proof of Proposition 1

Proof

Assume that there exists a DIC, EIR and Ex-post Efficient mechanism $(p^*,x)$ but convex hull of sets $T_b^*$ and $T_s^*$ have infinite intersection. Then, there exist $b_j \in T_b^*$ and $s_i \in T_s^*$ such that $b_j$ is strictly less than $s_i$. By definition of efficient type sets, there exist types $s_l \in T_s$ and $b_k \in T_b$ satisfying $s_l < b_j$ and $b_k > s_i$. Then, we can write $s_l< b_j< s_i < b_k$ so that $p_{lj} = p_{lk} = p_{ik} =1$ holds. We know from Lemma 1 that $x_{lj} = x_{lk} = x_{ik}$ should also hold in order to satisfy DIC constraints. Given all this information, let us check EIR constraints. We see that $b_j \ge x_{lj} \ge s_l$ and $b_k \ge x_{ik} \ge s_i$ cannot be satisfied together with $x_{lj} = x_{ik}$ since we have $b_j < s_i$. Hence, there is no transfer rule we can use together with $p^*$ to have a DIC, EIR mechanism. This is a contradiction.

Now we start from efficient type sets $T_b^*$ and $T_s^*$ whose convex hulls have finite intersection. If both efficient type sets are empty, we have a trivial case $b_m \le s_1$ where seller always values the good more. Then, any Posted Price mechanism imposes Ex-post Efficiency. In the non-trivial case, both sets are non-empty and minimum type, $\underline{b}$, in $T_b^*$ should be bigger than or equal to maximum type, $\bar{s}$, in $T_s^*$. Here, any Posted Price mechanism with unique price $x \in [\bar{s}, \underline{b}]$ will be Ex-post efficient. Since all Posted Price mechanisms are DIC, EIR, the proof is complete. $\square$