[Paper Review] Send Mixed Signals -- Earn More, Work Less

Peter Bro Miltersen, Or Sheffet|arXiv (Cornell University)|Feb 7, 2012

Auction Theory and Applications5 references21 citations

TL;DR

This paper introduces mixed signaling schemes in probabilistic second-price auctions, where the auctioneer uses randomized signals to boost revenue. It proves that optimal mixed signaling can be computed in polynomial time via linear programming and can generate up to twice the revenue of optimal pure signaling schemes, resolving a key computational and revenue gap in prior work.

ABSTRACT

Emek et al. presented a model of probabilistic single-item second price auctions where an auctioneer who is informed about the type of an item for sale, broadcasts a signal about this type to uninformed bidders. They proved that finding the optimal (for the purpose of generating revenue) {\em pure} signaling scheme is strongly NP-hard. In contrast, we prove that finding the optimal {\em mixed} signaling scheme can be done in polynomial time using linear programming. For the proof, we show that the problem is strongly related to a problem of optimally bundling divisible goods for auctioning. We also prove that a mixed signaling scheme can in some cases generate twice as much revenue as the best pure signaling scheme and we prove a generally applicable lower bound on the revenue generated by the best mixed signaling scheme.

Motivation & Objective

• To address the strong NP-hardness of finding optimal pure signaling schemes in second-price auctions with asymmetric information.
• To explore whether mixed signaling—where signals are randomized—can improve revenue and reduce computational complexity.
• To establish a polynomial-time algorithm for computing the optimal mixed signaling scheme.
• To quantify the revenue advantage of mixed over pure signaling, including tight bounds on performance.
• To connect the signaling problem to divisible good bundling and fractional knapsack-like optimization.

Proposed method

• Formulate the optimal mixed signaling problem as a linear program (LP) by modeling signal distributions and expected revenues per type.
• Prove structural properties of the optimal solution, such as the absence of singleton signals in certain optimal configurations, to ensure the LP remains polynomial-sized.
• Use a transformation that maps the signaling problem to a divisible good bundling problem, where goods are types and bidders have linear valuations.
• Apply a procedure to iteratively merge signals and eliminate singleton clusters, proving that revenue does not decrease during this process.
• Leverage duality and benchmark analysis to show that the optimal mixed scheme achieves at least half the revenue of a theoretical benchmark B.
• Draw a connection to the fractional knapsack problem, showing that the problem is solvable in polynomial time due to its convex structure.

Experimental results

Research questions

• RQ1 Can mixed signaling schemes generate significantly higher revenue than pure signaling schemes in second-price auctions?
• RQ2 Is the problem of finding the optimal mixed signaling scheme computationally tractable, unlike the pure signaling case?
• RQ3 What is the worst-case ratio between the revenue of the optimal mixed scheme and the optimal pure scheme?
• RQ4 How does the optimal mixed signaling scheme relate to divisible good bundling and fractional knapsack problems?
• RQ5 Can the LP-based approach scale to settings with exponentially many types, such as those parameterized by d attributes?

Key findings

• The optimal mixed signaling scheme can generate up to twice the revenue of the optimal pure signaling scheme, and this 2x factor is tight in some instances.
• The optimal mixed signaling scheme can be computed in polynomial time using a linear program, unlike the strongly NP-hard pure signaling case.
• The revenue of the optimal mixed signaling scheme is always at least half of the benchmark B, defined as the minimum over bidders of the sum of the second-highest valuations across all types.
• The optimal solution structure avoids singleton signals when possible, and revenue is maximized by bundling types to equalize bidders' expected utilities.
• The problem is structurally equivalent to a fractional knapsack problem, explaining its polynomial-time solvability despite the complexity of the pure signaling variant.
• The mixed signaling model can be extended to Bayesian settings and improves both revenue and computational efficiency compared to pure signaling.