An Uncertain Prognosis for Medicare Auctions: Part I

May 18, 2012

Erica Klarreich

In 1997, in an attempt to rein in the spiraling costs of health care, the U.S. Congress instructed Medicare to implement a competitive bidding process for suppliers of durable medical equipment---things like hospital beds, walkers, and oxygen tanks. Decades of research and practical demonstrations had suggested that the government could get a lot more bang for its buck by replacing its longstanding fee schedules with competitive auctions.

Accordingly, in 1999 the Centers for Medicare and Medicaid Services began small-scale tests of auction processes for setting reimbursement prices and identifying suppliers. In November 2009, CMS unrolled the selected auction in nine major metropolitan areas; it plans to expand the auction to 91 more cities in 2012.

There's just one problem: The design CMS chose for its auctions is "a train wreck waiting to happen," according to Peter Cramton, an economist at the University of Maryland.

Based on a mathematical analysis of the auction design that he conducted with mathematician Sean Ellermeyer and economist Brett Katzman of Kennesaw State University in Atlanta, Georgia, Cramton concludes that the auctions are likely to result in shortages, corner-cutting on quality and service, increased fraud, and ultimately, increased costs in the form of hospitalizations that could have been prevented with the right equipment.

In a draft letter to the chair of the House Subcommittee on Health, Cramton outlined the auction's flaws. Circulated to auction theory experts in September 2010, the letter attracted 167 signatures in less than two days.

The auction design chosen by CMS collects bids from all would-be suppliers; it chooses the winners by working its way up the bids, starting with the lowest, until it has enough suppliers to meet the projected demand for the item. But these auction winners do not get the price they bid: Instead, CMS sets a single reimbursement price, the median of the winning bids, regarding that price as the consensus among the winning bidders.

One obvious objection to a median-price auction is that some of the winners will be offered a price lower than their bids, and possibly lower than their actual costs. To deal with this issue, CMS made the bids non-binding: A winner who is unhappy with the price is not obligated to sign a contract.

To the uninitiated, this design might seem fairly reasonable. However, Cramton, Ellermeyer, and Katzman write in an August 10, 2011, preprint, "the evidence is overwhelming that a fundamental change in auction procedure is necessary to avoid catastrophic failure of the Medicare auctions."

A Simple Strategy
A rich mathematical theory of auctions has developed over the last half-century, starting with a seminal 1961 paper by William Vickrey of Columbia University. Vickrey used game theory to analyze bidding strategies in what economists call "private-value" auctions, in which each bidder's value for the item being sold is independent of the other bidders' values. Vickrey compared three common auction types (English, Dutch, and first-price), together with a fourth type of his own design, now known as the "Vickrey auction."

An English auction is the familiar "going, going, gone!" auction, in which the auctioneer keeps raising the price until all bidders but one drop out. In a Dutch auction, the auctioneer starts at a high price and lowers it until someone is willing to pay. In a first-price auction, bidders submit sealed bids; the highest bidder wins and pays the amount bid. In a Vickrey auction, sometimes called a "second-price" auction, bidders submit sealed bids; again, the highest bidder wins, but now pays only the amount of the second-highest bid.

At first glance, the Vickrey auction design might seem bizarre. Yet it offers an exceptionally simple optimal bidding strategy: Simply bid the value of the item to you.

Suppose, for example, that you are willing to pay any amount below $1000 for a painting being sold in a Vickrey auction. If you try to economize and bid only $900, there are three possible outcomes. If the highest rival bid is lower than $900---say, $800---you will win and pay $800, but the same thing would have happened if you had bid $1000. If the highest rival bid is above $1000, you will lose the auction, again the same thing that would have happened if you had bid $1000. But if the highest rival bid is between $900 and $1000---say, $950---you will lose the auction, whereas if you had bid your true value, $1000, you would have won and happily paid $950 for the painting. By bidding $900 instead of $1000 you never improve your lot, and sometimes lose an auction you would have liked to win. By similar logic, by bidding more than $1000, you never improve your lot, and sometimes win an auction that, because the price is too high, you would have liked to lose.

Despite the elegantly simple optimal strategy, it might seem unlikely that a seller would agree to use a Vickrey auction. After all, if the seller were to opt for a first-price auction instead, the winner would have to pay his own bid, not the second-highest bid---resulting in more money for the seller, right?

Astonishingly, Vickrey proved that this is not the case: If all bidders follow their optimal strategies, the seller can expect the same amount of money no matter which of the four auction formats is used. (In 1981, game theorist Roger Myerson of the University of Chicago extended this result by showing that all auction formats that award the item to the bidder who values it most will bring in the same expected revenue, with the exception of auctions that involve some kind of fee or reward for taking part.)

It's not hard to see that a Vickrey auction will bring in essentially the same amount as an English auction. In an English auction, the bidding ends when the second-highest bidder drops out, and the winning bidder pays the second-highest bidder's price (perhaps with a small increment, depending on the auction rules). And a Dutch auction, in which the winning bidder pays the highest price he is willing to bid, is equivalent to a first-price auction.

But why does a first-price auction generate the same expected price as a Vickrey auction? The answer lies in the optimal strategy for a bidder in a first-price auction: to bid not the true value of the item to him, but something lower---what economists call "shading" a bid. For example, in the painting scenario, if the highest rival bid is $900 and you bid your true value, $1000, you will win, but you could have won and paid much less by bidding $901.

In a Vickrey auction, bidding your true value is the "dominant" strategy---it's the best strategy no matter what the other bidders do. In a first-price auction, there is no dominant strategy: Your best move will depend very much on what everyone else is doing.

To analyze bidding behavior in a first-price auction, Vickrey computed the players' Bayesian Nash equilibrium strategies. In game theory, a Nash equilibrium is a collection of strategies, one for each player, such that no player has anything to gain from unilaterally switching to a different strategy. In situations in which the players have incomplete information about each others' payoffs (the case in most auctions, in which bidders don't know each others' values for the item), the corresponding concept is the Bayesian Nash equilibrium: a collection of strategies for the players such that any player who unilaterally switches to a different strategy will lower his expected payoff, given what he believes about the other players' hidden information.

In a first-price auction, these equilibrium strategies will depend on the number of participating bidders; the larger the number of bidders, the less each one should shade his bid, because the gap between the highest and the second-highest bidders' values is likely to be smaller. But in every case, Vickrey showed that if the players follow their equilibrium strategies, the seller's expected payoff will be the same as for a Vickrey auction.

Common Values
Vickrey's and Myerson's work offered a clear picture of the "private-value" setting, in which each bidder's value for the item is independent of all other bidders' values. But the Medicare auctions don't quite fall into that category. It's true that a bidder's value for a Medicare contract is simply equal to his cost of providing the item; in theory, then, knowing the other bidders' values should not affect the price the bidder is willing to accept. But a Medicare bidder may not be certain that he has correctly estimated his cost, and knowing the other bidders' costs could give him valuable insight into whether his estimate is in the ballpark. The Medicare auctions fall somewhere between the private-value setting and what is known as the "common-value" setting, in which the item has some intrinsic value that each bidder has tried to estimate.

Bidding strategies are more complex in the common-value setting than in the private-value setting. For participants in a Vickrey auction, the optimal strategy is no longer to bid their true values. In fact, if all bid their true values, the result would be what's called the "winner's curse": The winner is the bidder who most grossly overestimated the value of the item, and he gets stuck buying the item for the second-highest bidder's value, which is probably also an overestimate. To guard against the winner's curse, bidders should shade their bids.

In 1982, auction theorists Paul Milgrom of Stanford University and Robert Weber of Northwestern University showed that in the common-value setting, an English auction usually raises the most revenue. Because the bidders can see each others' bids, they develop something of a consensus on the value of the item and don't have to shade their bids as cautiously as when submitting one-shot sealed bids.

A Spectrum of Bids
Until about 20 years ago, the use of game theory in the design of auctions was predominantly an academic pursuit. That changed in the early 1990s, when the Federal Communications Commission harnessed auction theory for a series of wildly successful auctions of electromagnetic spectrum licenses for wireless communication services.

In consultation with auction theory experts, the FCC designed a variant of the open English auction, conducted in rounds. In each round, bidders could place sealed bids on a variety of licenses. The highest price for each license was then announced, and another round of bidding would commence, with the price for each license rising until only one bidder for that license remained.

This design reflected not only Milgrom and Weber's result about common-value auctions, but also research on "complementarities"---in which a bidder's value for one item might depend on whether he can also acquire another item (a bidder in the spectrum auctions, for example, might want the northern California license only if he could also get the southern California license). By auctioning off all the licenses together, the design gave the participants the most information possible about their options, allowing more aggressive bidding than if the licenses had been auctioned off in sequence. (This consideration also applies in the Medicare auctions---a contract to supply oxygen, say, might be valuable to a supplier only if accompanied by a contract to supply breathing equipment.)

The spectrum auctions surpassed all expectations, generating $42 billion by early 2001---more than four times the FCC's estimate of the worth of the entire spectrum. The New York Times dubbed one of the earliest spectrum auctions, which garnered more than $7 billion, "the greatest auction ever."

An auction design along the lines of the spectrum auctions would probably work well for the Medicare auctions, Cramton says. But as discussed in Part II of this article, CMS, unlike the FCC, does not appear to have availed itself of the rich body of auction theory research. The deeply flawed auction design it chose, Cramton, Ellermeyer, and Katzman argue, offers bidders no way to come up with a sensible strategy. If CMS continues to expand its auctions, Cramton says, "the efficient providers are going to be put in a terrible position, whipsawed around by the craziness."

Erica Klarreich writes from Berkeley, California.

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