Fast Algorithms for Computing Interim Allocations in Single-Parameter Environments

Abstract

Myerson’s seminal work characterized optimal auctions; applied naively, however, his approach yields exponential-time algorithms. Using Border’s theorem, in contrast, one can solve mechanism design problems in polynomial time. This latter approach relies on linear programming machinery, the mechanics of which are significantly more complicated than Myerson’s. Motivated by the simplicity and transparency of Myerson’s analysis, we present fast algorithms for computing interim allocations in simple auction settings. These methods apply to both surplus and revenue maximization, and yield ex-ante symmetric solutions.

Appendices

Appendix

A Proofs

1.1 A.1 Proof of Lemma 1

Proof

Consider any set of virtual values $\psi _{1}> \psi _{2} > \psi _{3}$. For , let for $c < d$. Let $r_c$ be any number in [0, 1]. The difference between any $r_c$ and $r_d$ cannot exceed 1, so . Let . Then we have , so $\psi _{c} + r_c \epsilon _A > \psi _{d} + r_d \epsilon _A$. See Fig. 1.

Fig. 1.

Graphical depiction of the proof of Lemma 1. Any change of the virtual values by $\epsilon _A$ preserves the ordering of virtual values.

1.2 A.2 Proof of Lemma 2

Proof

Let $\epsilon _B^{U}$ be the minimum of the absolute value of the set of all non-zero virtual values: . Let $0< \epsilon _B < \epsilon _B^{U}$. Any virtual value $\psi _{i} (z_{i,k}) \ge 0$ can have $\epsilon _B$ added to it and remain non-negative. Similarly, any virtual value $\psi _{i} (z_{i,k}) < 0$ can have $\epsilon _B$ added to it and remain negative, since . See Fig. 2.

Fig. 2.

Graphical depiction of the proof of Lemma 2. Any change of the virtual values by $\epsilon _B$ does not change whether virtual values are negative or not.

1.3 A.3 Proof of Theorem 3

Proof

Lemma 1 shows the existence of an $\epsilon _A$ which satisfies the first property, and Lemma 2 shows the existence of an $\epsilon _B$ which satisfies the latter properties. These values are not unique: for any $\epsilon _A, \epsilon _B$, we can construct an $\epsilon _{A}' < \epsilon _{A}$ and $\epsilon _{B}' < \epsilon _{B}$. Thus, the minimum of $\epsilon _A$ and $\epsilon _B$ satisfies all three properties of the theorem.

1.4 A.4 Proof of Theorem 4

Proof

For any value vector $\mathbf {v_{}}_{}\in T_{}$, Algorithm 1 will allocate only to bidders with the highest non-negative virtual value. As defined earlier, let $w (\mathbf {v_{}}_{})$ be the set of bidders with the highest virtual value. Suppose instead of virtual values, we used perturbed virtual values. Let $\tilde{w} (\mathbf {v_{}}_{})$ be the set of bidders with the highest perturbed virtual values that have met their reserve price:

Using a valid $\epsilon$ guarantees that the intersection of $w (\mathbf {v_{}}_{})$ and $\tilde{w} (\mathbf {v_{}}_{})$ is nonempty. If there are no ties, then $w (\mathbf {v_{}}_{}) = \tilde{w} (\mathbf {v_{}}_{})$.

The interesting case is when there are ties. Since all perturbed virtual values are unique, , and the unique bidder $i^* \in w (\mathbf {v_{}}_{}) \cap \tilde{w} (\mathbf {v_{}}_{})$ contributes $\psi _{i^*} (v_{i^*})$ to the total expected virtual surplus. The probability that $i \in \tilde{w} (\mathbf {v_{}}_{})$ is allocated depends on the perturbations. Since perturbations are drawn independently and uniformly at random, the $r_{i,k}$ values act as tie-breaking rules, where the probability that any $j \in w (\mathbf {v_{}}_{})$ wins is uniform over the cardinality of $w (\mathbf {v_{}}_{})$, just as in Algorithm 2. The maximum virtual surplus attained from any convex combination of winners in $w (\mathbf {v_{}}_{})$ where is , which is the outcome of Algorithm 2. In Algorithm 1, the virtual surplus given by $\mathbf {v_{}}_{}$ is $\psi _{i^*} (v_{i^*})$. Since $\max _{j \in w(\mathbf {v_{}}_{})} \psi _{j}(v_{j}) = \psi _{i^*} (v_{i^*})$, the contribution any $\mathbf {v_{}}_{}\in T_{}$ has on total expected revenue is equivalent in both algorithms.

B Algorithms

1.1 B.1 Pointwise Maximization

Algorithm 2 describes the pointwise approach in detail for the one-good setting, where, when maximizing total expected surplus is the objective, each $L_i$ is the identity function. When maximizing total expected revenue, each $L_i$ is the virtual value function. Further notice that Algorithm 2 preserves tie-breaking probabilities in allocation terms, thus preserving symmetry in the final outcome.

Algorithm 2 is exponential in runtime. This is because determining the interim allocation takes exponential time. Indeed, if we are given interim allocations, we can solve for interim payments, total expected surplus, and total expected revenue in polynomial time. Thus, we see that the bottleneck is computing interim allocations, and is the main subject of this paper. We first analyze the one-good setting in detail, and then analyze other single-parameter settings. There are value vectors in $T_{}$. For each value vector, determining allocations is , so determining all allocations is . Each is computed in , so determining all interim allocations takes . Each is computed in , so determining all interim payments takes . Computing S is done in . Computing R is done in . Therefore, the complexity of Algorithm 2 is .