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In an incomplete information setting, we analyze the sealed bid auction proposed by Knaster (cf. Steinhaus (1948)). This procedure was designed to efficiently and fairly allocate multiple indivisible items when participants report their valuations truthfully. In equilibrium, players do not follow truthful bidding strategies. We find that, ex-post, the equilibrium allocation is still efficient but may not be fair. However, on average, participants receive the same outcome they would have received if everyone had reported truthfully—

This paper conducts an equilibrium analysis of a sealed-bid auction proposed by famed mathematician Bronislaw Knaster. This auction was designed to

Suppose Ann, Bob, and Carol are heirs to an estate containing four indivisible objects

Knaster’s procedure allocates each item to the highest bidder and uses these bid vectors to determine side payments for each player. In particular, side payments are constructed so that each heir receives an equal “surplus” over their initial fair share.

Since items go to the high bidder, heir

In summary, Knaster’s procedure awards each of the items to the high bidder and the bidders “ pay” their final excess valuation (players with negative excess valuation receive a payment). Heirs leave the auction with value equal to their adjusted fair share. In the example,

Knaster’s procedure generates efficient proportional outcomes when the heirs report truthfully.

“ The numbers in this example have been chosen only to exhibit the advantages that can accrue to a player who falsely portrays his own valuations with a knowledge of the other player’s true valuations. It points up a clear need for an analysis of the strategic opportunities of this situation.”

It is unclear, however, if such manipulation could or would take place when bidder information is incomplete. We therefore seek to answer Kuhn’s call for strategic analysis of Knaster’s auction, but provide the analysis in an incomplete information setting. In

In this section, we formalize Knaster’s procedure as an auction.

Knaster’s procedure solicits bids from each player and uses this information to make an allocation decision. Specifically, each player submits a bid vector

First, each player

Second,

Third, a player

Last, player

It is easy to verify that

This concludes the description of the mechanism. Note that our definition of initial fair share and surplus is for each item. If we sum up a player’s initial fair share (surplus) for each item, we arrive at that player’s total initial fair share (total surplus) as in the example found in the introduction. Player

Knaster’s procedure is a mechanism that induces a Bayesian game between

We begin with a heuristic derivation of a Bayes–Nash equilibrium when only one object is being auctioned.

Suppose players

Equilibrium Bidding Strategy for u(0,1) and

In general, the equilibrium bidding strategy prescribed in (3) recommends players shade their bids when their type is higher than the threshold type and to pad their bid when their type is lower than the threshold. This is intuitive. When a player is not likely to win the auction, he can gain compensation by increasing his bid. Similarly, a high type player who is more likely to win the item can gain by lowering his bid to reduce the compensation he must pay others. As in other auctions with shading/padding, there is a marginal benefit/cost to such actions—

Since the items are unrelated, we can treat each item independently when searching for the optimal bid. The next theorem follows immediately.

Knaster’s auction was designed to achieve an efficient and proportional outcome when all players report their true valuations—

We are interested in the impact that strategic behavior has on the

Let

Knaster’s auction, when players follow truth telling strategies, is an allocation rule that satisfies all three of the above properties. However, we are interested in whether Knaster’s auction, when players follow equilibrium strategies.

Several welfare results are immediate. First, the ex-post assignment of the items in equilibrium is the same as when players report truthfully—

Our next result concerns ex-ante fairness—

While this result is nice, Knaster was interested in ex-post fairness—

Proposition 4 is discouraging, but expected given the form of the equilibrium bid function.

Now, we show that the bid function does not converge to the 45

Bidding Strategies for u(0,1)

The results in this paper contribute to several literatures: auctions, dissolving a partnership, bargaining, and fair division. In particular, we have used techniques frequently used in the auctions literature to analyze a well-known fair division procedure. Specifically, we have modeled Knaster’s fair division procedure as a sealed bid auction, computed the symmetric Bayes–Nash equilibrium in increasing bidding strategies, analyzed the welfare consequences of strategic behavior, and then performed some simple comparative statics of the equilibrium bidding functions.

Knaster’s auction remains efficient at the Bayes–Nash equilibrium outcome. However, the expected side payments made by bidders are typically different than under truth telling. As a consequence, the auction is no longer ex-post proportional.

Fair division mechanisms, such as Knaster’s auction, are appealing when all individuals involved have a claim to an object or objects. Divorce, inheritance, and dissolving a partnership are natural contexts to apply such mechanisms. The later topic has been well studied in economics under the guise of efficiency when agents are strategic. Crampton, Gibbons, and Klemperer (1987), McAfee (1992), Morgan (2004), Moldovanu (2002) all study mechanisms for dissolving a partnership in an incomplete information environment.

Brams and Taylor (1999) analyze the fairness properties of several simple fair division mechanisms and discuss how these procedures could be applied to bargaining scenarios.

In this appendix, we provide the details for solving the differential equation (2). First, putting (2) in the standard form we have:

First, suppose

This auction first appeared in Steinhaus’s now classic 1948 article on fair division. Steinhaus credits the auction to Knaster. Subsequently, descriptions of the procedure have appeared in Luce and Raiffa (1957), Raiffa (1982), Young (1994), and Brams and Taylor (1996) among others. Kuhn (1967) demoststrates how Knanster’s procedure could be “ discovered” using linear programming.

Several such auctions are studied in Morgan (2004). In this paper, Morgan analyzes auctions that could be used to “ fairly” dissolve a partnership. He does not consider Knaster’s auction.

An outcome is proportional if each of the

The following is adopted from Luce and Raiffa (1957). The numbers have been adapted to ease some of the calculations.

In addition, when

See Krishna (2010) for an introduction to auction theory.

Ties are broken via random assignment.

Written out, the side payment rule is:

The steps used to solve equation (2) are provided in the Appendix.

This is easy to check using equation (2).

For notational simplicity, our results will be for the one item case. The generalization is straightforward and left to the reader.

There is also a case where players evaluate the outcome when they know their type, but not the types of the other players (

It straightforward to verify that, when

For

Proportional allocations satisfy a basic notion of fairness, but stronger concepts have been developed since Steinhaus’s paper. Concepts such as envy-freeness, egalitarian, consistency, population monotonicity, and transparent inequity have all been studied in the fair division literature. See, for instance, Varian (1974), Crawford (1977), Crawford and Heller (1979), Crawford (1980), Demange (1984), Takenuma and Thomson (1993), Moulin (1990b), and Alkan, Demange, and Gale (1991). Young (1994) and Moulin (1988, 1990a, and 2003) survey this large literature.

Also related is Segal and Whinston (2011) who provide general conditions under which efficient bargaining is possible.

See Krishna, Chapter 5, for a streamlined discussion of this result.

There is a large body of work on fair division mechanisms presented throughout the mathematics, economics, and political science literature. For instance, the problem of how to fairly divide a cake has generated a significant body of interest and can be applied to both divisible good and indivisible good allocation problems. Introductions to this cake cutting literature can be found in Brams and Taylor (1996), Robertson and Webb (1998), or Su (1999).

In particular, our integrating factor