Price of Anarchy for Greedy Auctions

We consider auctions in which greedy algorithms, paired with first-price or critical-price payment rules, are used to resolve multi-parameter combinatorial allocation problems. We study the price of anarchy for social welfare in such auctions. We show for a variety of equilibrium concepts, including Bayes-Nash equilibrium and correlated equilibrium, the resulting price of anarchy bound is close to the approximation factor of the underlying greedy algorithm.

💡 Research Summary

The paper investigates a class of combinatorial auctions that resolve allocation problems by coupling greedy algorithms with simple payment rules—either first‑price or critical‑price (also known as threshold‑price). The central question is how much social welfare can be lost when participants behave strategically, i.e., what is the Price of Anarchy (PoA) of such mechanisms. The authors consider a broad spectrum of equilibrium concepts, including Bayes‑Nash equilibrium (BNE) and correlated equilibrium (CE), and they show that the inefficiency bound is essentially the same as the approximation factor of the underlying greedy algorithm.

The model assumes a multi‑parameter setting: each bidder i has a valuation function v_i(S) over subsets S of items. Bidders submit bids b_i(S) (which may be truthful reports or strategic misreports). The allocation rule runs a greedy procedure: while items remain unallocated, it selects the remaining bid with the highest “value‑to‑cost” ratio (or simply the highest reported value, depending on the variant) and assigns the corresponding bundle to that bidder, removing the allocated items from further consideration. This greedy step is known to be an α‑approximation for many combinatorial problems (e.g., set packing, maximum weight matching), meaning that the total value of the greedy outcome is at least 1/α of the optimal social welfare.

Two payment schemes are examined. In the first‑price rule, a winning bidder pays exactly the amount it bid for the allocated bundle. In the critical‑price rule, the winner pays the smallest bid it could have submitted and still won the same bundle; this is analogous to the threshold price used in Vickrey‑Clarke‑Groves (VCG) mechanisms but is computationally much cheaper. The critical‑price rule preserves many incentive‑compatible properties while still being implementable with the greedy allocation.

To bound the PoA, the authors extend the smoothness framework originally developed for single‑parameter auctions. They define a smoothness condition tailored to multi‑parameter greedy allocations: for any bid profile b and any deviation profile b′, the sum of the utilities obtained under the deviation plus a scaled sum of payments is at least a fraction λ of the optimal welfare minus a term μ times the total payments in the original profile. Formally,

∑_i u_i(b′i, b{‑i}) ≥ λ·OPT – μ·∑_i p_i(b).

When the greedy algorithm is an α‑approximation, they can set λ = 1/α and μ = β, where β captures the extra loss introduced by the payment rule (β = 0 for first‑price, β ≤ 1 for critical‑price). This smoothness inequality holds for every possible deviation, which implies that in any BNE or CE the expected welfare satisfies

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