[Paper Review] When does the Price of Anarchy tend to 1 in Large Walrasian Auctions and Fisher Markets

Richard Cole, Yixin Tao|arXiv (Cornell University)|Aug 28, 2015

Auction Theory and Applications4 citations

TL;DR

This paper demonstrates that the Price of Anarchy (PoA) tends to 1 in large Walrasian auctions and Fisher markets under mild conditions, such as gross substitutes demand and uncertainty in good supplies. As market size increases—measured by more agents and items—the efficiency loss due to strategic behavior diminishes, approaching social efficiency in the limit.

ABSTRACT

As is well known, many classes of markets have efficient equilibria, but this depends on agents being non-strategic, i.e. that they declare their true demands when offered goods at particular prices, or in other words, that they are price-takers. An important question is how much the equilibria degrade in the face of strategic behavior, i.e. what is the Price of Anarchy (PoA) of the market viewed as a mechanism? Often, PoA bounds are modest constants such as 4/3 or 2. Nonetheless, in practice a guarantee that no more than 25% or 50% of the economic value is lost may be unappealing. This paper asks whether significantly better bounds are possible under plausible assumptions. In particular, we look at how these worst case guarantees improve in the following large settings. Large Walrasian auctions: These are auctions with many copies of each item and many agents. We show that the PoA tends to 1 as the market size increases, under suitable conditions, mainly that there is some uncertainty about the numbers of copies of each good and demands obey the gross substitutes condition. We also note that some such assumption is unavoidable. Large Fisher markets: Fisher markets are a class of economies that has received considerable attention in the computer science literature. A large market is one in which at equilibrium, each buyer makes only a small fraction of the total purchases; the smaller the fraction, the larger the market. Here the main condition is that demands are based on homogeneous monotone utility functions that satisfy the gross substitutes condition. Again, the PoA tends to 1 as the market size increases. Furthermore, in each setting, we quantify the tradeoff between market size and the PoA.

Motivation & Objective

• To investigate whether the Price of Anarchy (PoA) improves in large markets, especially when agents act strategically rather than truthfully.
• To determine under what conditions the PoA approaches 1 as market size grows, indicating near-optimal efficiency despite strategic behavior.
• To quantify the tradeoff between market size and PoA in both Walrasian auctions and Fisher markets.

Proposed method

• Analyzing large Walrasian auctions with many copies of each item and many agents, assuming uncertainty in supply quantities.
• Applying the gross substitutes condition to demand functions to ensure stability and convergence of equilibria.
• Extending the analysis to Fisher markets where each buyer’s share of total purchases becomes negligible as the market scales.
• Using asymptotic analysis to show that PoA converges to 1 under the stated conditions.
• Deriving quantitative bounds on PoA as a function of market size and agent share in total demand.
• Establishing that the gross substitutes condition is necessary—without it, PoA does not necessarily tend to 1.

Experimental results

Research questions

• RQ1 Under what conditions does the Price of Anarchy tend to 1 in large Walrasian auctions with strategic agents?
• RQ2 How does the PoA in Fisher markets behave as the market size increases and individual buyer shares shrink?
• RQ3 What role does the gross substitutes condition play in ensuring convergence of PoA to 1 in large markets?
• RQ4 Can the tradeoff between market size and PoA be quantified in both Walrasian auctions and Fisher markets?
• RQ5 Is some form of uncertainty in supply or demand necessary for PoA to approach 1, or can it hold deterministically?

Key findings

• In large Walrasian auctions with uncertain supply and gross substitutes demand, the Price of Anarchy tends to 1 as market size increases.
• The convergence of PoA to 1 in Walrasian auctions is contingent on the gross substitutes condition and supply uncertainty; without these, the bound does not hold.
• In large Fisher markets with homogeneous monotone utility functions satisfying gross substitutes, the PoA also tends to 1 as the market grows.
• The rate of convergence of PoA to 1 is quantifiable and depends on the maximum share of total purchases made by any single buyer.
• The gross substitutes condition is not just sufficient but necessary for PoA to approach 1 in these settings.
• The results show that large markets inherently mitigate inefficiencies from strategic behavior, even without strong coordination or incentives.