Complexity of Auctions with Interdependence

Abstract

We study auction design in the celebrated interdependence model introduced by Milgrom and Weber [1982], where a mechanism designer allocates a good, maximizing the value of the agent who receives it, while inducing truthfulness using payments. In the lesser-studied procurement auctions, one allocates a chore, minimizing the cost incurred by the agent selected to perform it. Most of the past literature in theoretical computer science considers designing truthful mechanisms with constant approximation for the value setting, with restricted domains and monotone valuation functions. In this work, we study the general computational problems of optimizing the approximation ratio of truthful mechanism, for both value and cost, in the deterministic and randomized settings. Unlike most previous works, we remove the domain restriction and the monotonicity assumption imposed on value functions. We provide theoretical explanations for why some previously considered special cases are tractable, reducing them to classical combinatorial problems, and providing efficient algorithms and characterizations. We complement our positive results with hardness results for the general case, providing query complexity lower bounds, and proving the NP-Hardness of the general case.

Figures (7)

• Figure 1: Bipartite graph $G_\gamma$ with $\gamma = 2$. First, at each vertex ${{\mathbf{s}}}$ we strike-out non acceptable agents $i\notin A_\gamma({{\mathbf{s}}})$, for which $\rho_i({{\mathbf{s}}}) < 1/\gamma$. Second, we draw the edges of the cube, oriented using $\boldsymbol\sigma$ (in the special case of increasing value functions), dotted if the corresponding agent is not acceptable at one of the two endpoints. Third, we color gray the must-match vertices ${{\mathbf{s}}}$ with $A_\gamma({{\mathbf{s}}})\subseteq C({{\mathbf{s}}})$, for which all acceptable agents correspond to outgoing edges. Finally we dash edges which are not incident to a must-match vertex. The final graph is the set of plain edges.
• Figure 2: Example of graph $G(\boldsymbol\sigma)$, obtained from the boolean formula with $n=2$ agents. For each $i\in[n]$ and ${{\mathbf{s}_{-i}}}\in[k]$ the monotonicity constraints are given by the (partial) strict order $\sigma_i({{\mathbf{s}_{-i}}})$. There exists a blocking chain of implications, represented in red. To compute efficiently the existence of a blocking chain, one could remove redundant edges that can be deduced using transitivity, and compute the strongly connected components, represented in gray.
• Figure 3: Gadget for a variable $a$ located at $(s_2,s_3) = (1,1)$. We have $n=4$ agents, but we do not represent the fourth dimension (we set $\boldsymbol\rho$ to be constant over $s_4$). At three vertices, we set some performance ratio at $\varepsilon$ to prevent the items being allocated to, and we leave the other ratios at $1$.
• Figure 4: Gadget for a clause $\ell_1\vee\ell_2\vee\ell_3$. We create one cycle per literal, which all intersect in one signal profile ${{\mathbf{s}}}$. We set $\boldsymbol\rho({{\mathbf{s}}}) = (1,1,1,\varepsilon)$, and the choice of winner at ${{\mathbf{s}}}$ decides which of the three literal is true.
• Figure 5: Gadget for a XOR-connector at height $h$. It connects two horizontal lines $(s_2, s_3)$ and $(s_2', s_3')$ such that $s_2 < s_2'$ and $s_3 > s_3'$, and such that no other horizontal line shares a coordinate.

Theorems & Definitions (67)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary
• Theorem 5
• Lemma 2.0
• proof
• Lemma 2.0: adapted from RoughgardenT16
• proof