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Games 2013, 4(1), 21-37;

An Equilibrium Analysis of Knaster’s Fair Division Procedure
Department of Economics, Finance, and Legal Studies, University of Alabama, Tuscaloosa, AL 35487, USA
Received: 19 October 2012; in revised form: 8 January 2013 / Accepted: 8 January 2013 / Published: 18 January 2013


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—i.e., the mechanism is ex-ante fair.
fair division; auction
JEL Codes:
C72; C78; C82

1. Introduction

This paper conducts an equilibrium analysis of a sealed-bid auction proposed by famed mathematician Bronislaw Knaster. This auction was designed to efficiently and fairly allocate multiple indivisible items and has played an important role in the early development of the fair division literature.1 As in standard auctions, players in Knaster’s auction simultaneously submit a bid for each of the items, where each item is allocated to the bidder who submitted the highest bid for that item. Unlike standard auctions, however, the winning bidder of each item compensates the losing bidders in the form of side payments.2 When players are truthful, Knaster’s process generates an efficient assignment of the indivisible items and side payments so that each person receives, in their view, a proportional outcome.3 This is a basic notion of fairness. The workings of auction are best introduced via a simple “inheritance” example.4
Suppose Ann, Bob, and Carol are heirs to an estate containing four indivisible objects Games 04 00021 i001, Games 04 00021 i002, Games 04 00021 i003, and Games 04 00021 i004. The heirs have an equal clam to the objects and are looking to “ fairly” divide these items using Knaster’s procedure. In particular, each heir submits a bid for each item. In the table below, we display bid vectors for the three heirs, the sum of these bids (or total reported value), and each bidder’s initial fair sharei.e., Games 04 00021 i005 of their total reported value.
Games 04 00021 i006
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 Games 04 00021 i007 receives Games 04 00021 i001, heir Games 04 00021 i008 receives Games 04 00021 i004, and heir Games 04 00021 i009 receives items Games 04 00021 i002 and Games 04 00021 i003. These items received create value for the recipient. The amount of value created above the initial fair share for the recipient is the bidder’s initial excess valuation. If we add up the initial excess values for the three heirs we get surplus. In this example, the surplus is 6000.
Games 04 00021 i010
Next, we divide the surplus equally among the participants and add this amount to each participant’s initial fair share to compute an adjusted fair share. An individual’s side payment is their value received minus their total adjusted fair share.
Games 04 00021 i011
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, Games 04 00021 i007 ends up with item 1 and pays $3700 to compensate the other two heirs, Games 04 00021 i008 gets item 4 and receives $2700, Games 04 00021 i009 receives items Games 04 00021 i012 and Games 04 00021 i013 and receives $1000. By construction, the side payment made by Games 04 00021 i007 balances with the amounts paid to Games 04 00021 i008 and Games 04 00021 i009.
Knaster’s procedure generates efficient proportional outcomes when the heirs report truthfully.5 Despite this nice property, Knaster’s auction is vulnerable to manipulation if the players have knowledge of one another’s preferences. Suppose, for instance, Heir B increases his bid for item 1 from $4000 to $9400. Each bidder still wins the same items, however Games 04 00021 i008’s side payment (compensation) increases from to $2700 to $3900, which is a clear improvement for Games 04 00021 i008. Kuhn (1967) provides a similar example in his analysis of Knaster’s procedure and concludes by saying,
“ 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 Section 2, we model Knaster’s procedure as a sealed bid auction. In Section 3, we find equilibrium bidding strategies for the Bayesian game induced by Knaster’s auction. Welfare consequences of strategic behavior are explored in Section 4.

2. Knaster’s Fair Division Procedure

In this section, we formalize Knaster’s procedure as an auction.6 Let Games 04 00021 i014 be a set of Games 04 00021 i015 unrelated items to be allocated among Games 04 00021 i016 heirs, whom we shall henceforth refer to as players. Each player Games 04 00021 i017 assigns the value Games 04 00021 i018 to item Games 04 00021 i019, for Games 04 00021 i020. The Games 04 00021 i021 private values for each item Games 04 00021 i019 are independently distributed according to cumulative distribution function Games 04 00021 i022 with support Games 04 00021 i023, where the probability density function of Games 04 00021 i022 by Games 04 00021 i024 Individual values are private, but the distribution function for each item is known to all players. Last, since objects are unrelated, the value to a player of receiving multiple items is simply the sum of each item’s value.
Knaster’s procedure solicits bids from each player and uses this information to make an allocation decision. Specifically, each player submits a bid vector Games 04 00021 i025 to a mediator who awards each item to the high bidder. Thus, player Games 04 00021 i017 is awarded item Games 04 00021 i026 if Games 04 00021 i027.7 Next, side payments are computed for each item Games 04 00021 i026 as follows:
  • First, each player Games 04 00021 i017’s initial fair share of item Games 04 00021 i019 (defined as Games 04 00021 i028-th of their reported value) is computed—i.e.,
    Games 04 00021 i029
  • Second, surplus value for unit Games 04 00021 i019 (defined as the difference between the high bid and the average bid) is computed—i.e.,
    Games 04 00021 i030
  • Third, a player Games 04 00021 i017’s adjusted fair share is computed from the bids. This is the player’s initial fair share of item Games 04 00021 i026 plus an even share of the surplus—i.e.,
    Games 04 00021 i031
    Games 04 00021 i032
  • Last, player Games 04 00021 i017’s side payment for item Games 04 00021 i019 is their reported value received (if they win the item) minus their adjusted fair share—i.e.,
    Games 04 00021 i033
    It is easy to verify that Games 04 00021 i034.8
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 Games 04 00021 i017’s payoff for the Games 04 00021 i019-th item as a function of submitted bids Games 04 00021 i035 is therefore
Games 04 00021 i036
Since Knaster’s auction can be analyzed item by item, a player’s total payoff is just the sum of the player’s payoffs from each individual item—i.e., Games 04 00021 i037.

3. Equilibrium

Knaster’s procedure is a mechanism that induces a Bayesian game between Games 04 00021 i021 players. In this section, we find the Bayes–Nash equilibrium for this induced game in symmetric and increasing bidding strategies.

3.1. Knaster’s Procedure with a Single Object

We begin with a heuristic derivation of a Bayes–Nash equilibrium when only one object is being auctioned.
Suppose players Games 04 00021 i038 follow the symmetric and increasing strategy Games 04 00021 i039. We consider player 1’s best response problem. Player 1 wins the object if he has the high bid—i.e., Games 04 00021 i040 Games 04 00021 i041. He receives compensation if one of the other players is the high bidder. For instance, Player 2 wins if Games 04 00021 i042, Games 04 00021 i043, Games 04 00021 i044, ...., and Games 04 00021 i045. It is convenient to define the following functions:
Games 04 00021 i046
Games 04 00021 i047
The best response problem for player 1 is to choose a bid Games 04 00021 i048 to maximize his expected payoff—i.e., to solve:
Games 04 00021 i049
The first order condition for the problem is found using Leibniz’s Rule:
Games 04 00021 i050
Games 04 00021 i051
Games 04 00021 i052
Games 04 00021 i053
Since the function Games 04 00021 i054 is symmetric in its last Games 04 00021 i055 arguments and the partial derivative of Games 04 00021 i056 is
Games 04 00021 i057
the first order condition simplifies to:
Games 04 00021 i058
Games 04 00021 i059
Games 04 00021 i060
Games 04 00021 i061
In a symmetric Bayes–Nash equilibrium Games 04 00021 i062, which, after inputting into (1), combining terms and simplifying, yields:
Games 04 00021 i063
Games 04 00021 i064
Games 04 00021 i065
Games 04 00021 i066
Thus, we are left with the differential equation
Games 04 00021 i067
In standard auctions, the boundary condition Games 04 00021 i068 is used to solve the differential equation. However, in Knaster’s auction, this condition is not optimal. Fortunately, there is a unique value Games 04 00021 i069 such that Games 04 00021 i070. At this value, the differential equation reduces to the expression Games 04 00021 i071. This is our boundary condition. The solution to Equation (2) is found to be
Games 04 00021 i072
where Games 04 00021 i073 is defined such that Games 04 00021 i074.9 It remains to show that strategy profile where every player follows (3) forms a Bayes–Nash equilibrium.
Proposition 1 Symmetric equilibrium strategies in the Bayesian game induced Knaster’s Fair Division Procedure with Games 04 00021 i021 players and one object are given by (3).
Proof. We need to check that following Games 04 00021 i039 is an equilibrium. First, it is easily checked that the bidding strategy Games 04 00021 i039 is increasing and continuous. Second, a bidder will never want to bid above Games 04 00021 i075 or below Games 04 00021 i076. Bidder 1, for instance, should not submit a bid Games 04 00021 i077 Games 04 00021 i075. A bid equal to Games 04 00021 i075 wins the item with probability one, thus increasing one’s bid only decreases bidder 1’s expected payoff. Similarly, Bidder 1 should not submit a bid Games 04 00021 i078 Games 04 00021 i076. A bid equal to Games 04 00021 i076 is guaranteed to lose the item with probability one (but win the compensation), so decreasing one’s bid only decreases 1’s expected payoff since it lowers the compensation. Finally, the expected payoff of bidder 1 whose type is Games 04 00021 i079 but bids as if his type were Games 04 00021 i080 is
Games 04 00021 i081
Games 04 00021 i082
Games 04 00021 i083
Differentiating with respect to Games 04 00021 i080 and simplifying the resulting expression yields:
Games 04 00021 i084
Now setting Games 04 00021 i085 we have
Games 04 00021 i086
Games 04 00021 i087
If Games 04 00021 i088, then Games 04 00021 i089. If Games 04 00021 i090, then Games 04 00021 i091. Hence, Games 04 00021 i092 is maximized at Games 04 00021 i093. Therefore bidding truthfully according to Games 04 00021 i039 is a best response.
Example 1 Suppose Games 04 00021 i094 and each player Games 04 00021 i017’s private value is distributed according to the uniform distribution—i.e., Games 04 00021 i095 for Games 04 00021 i096 and Games 04 00021 i097, then the equilibrium bidding strategy for each player is given by Games 04 00021 i098.10
The equilibrium bidding strategy prescribes over reporting when the player is less likely to win the item and more likely to earn compensation (i.e., when Games 04 00021 i099), truth telling at Games 04 00021 i100, and for a player to shade his bid when more likely to win the item and less likely to earn compensation (i.e., Games 04 00021 i101).
Figure 1. Equilibrium Bidding Strategy for u(0,1) and N = 2.
Figure 1. Equilibrium Bidding Strategy for u(0,1) and N = 2.
Games 04 00021 g001
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—i.e., increasing one’s bid increases the probability a player will win the auction and decreasing one’s bid increases the probability a player will lose the auction. In the optimal bid, a player continues shading/padding until the marginal benefit falls short of the marginal cost.

3.2. Multiple Objects

Since the items are unrelated, we can treat each item independently when searching for the optimal bid. The next theorem follows immediately.
Proposition 2 Symmetric equilibrium strategies in the Bayesian game induced Knaster’s Fair Division Procedure with Games 04 00021 i016 players and multiple objects are given by the vector valued bid function Games 04 00021 i104, where for Games 04 00021 i105, Games 04 00021 i106 is defined as
Games 04 00021 i107
whereeach Games 04 00021 i108 is defined as the Games 04 00021 i109 such that Games 04 00021 i110.

4. Welfare and Comparative Statics

Knaster’s auction was designed to achieve an efficient and proportional outcome when all players report their true valuations—i.e., allocations where the items ended up with the people who valued them the most and each player receives, in their estimation, at least Games 04 00021 i028-th of the item’s value. However, in equilibrium, bidders typically do not report truthfully. We now check to see if this behavior has welfare consequences.11
We are interested in the impact that strategic behavior has on the fairness and the efficiency of the equilibrium allocation. However, in an incomplete information environment, the notion of fairness is slightly ambiguous. Specifically, we need to clarify what information players have at the time they are evaluating the outcome or expected outcome. In particular, we need to know if the players are evaluating the outcome before they know their type (i.e., ex-ante) or evaluating the outcome after the auction is done (i.e., ex-post).12 While Knaster was clearly interested in the ex-post case, we mention some results from the ex-ante case, which is of interest. It is useful to define these notions in terms of general allocation rules.
Let Games 04 00021 i111 be an allocation rule—i.e., a function that assigns to each realization of types a specific allocation, then the following properties are of interest:
Definition 1 The item assignment of an allocation rule Games 04 00021 i111 is ex-post efficient if, for each realization of types, the object in the allocation prescribed is assigned to the player with the highest realized type for that object.
Definition 2 An allocation rule Games 04 00021 i111 is ex-post proportional if, for each realization of types, after the allocation rule has been applied, each player Games 04 00021 i017 with realized type Games 04 00021 i112 gets a utility of at least Games 04 00021 i113.
Definition 3 An allocation rule Games 04 00021 i111 is ex-ante proportional if, prior to observing types, each player Games 04 00021 i017’s expected utility from his part of the allocation rule is greater than Games 04 00021 i114.
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—i.e., the item assignment is ex-post efficient. This follows since equilibrium bidding strategies are increasing. Second, aggregate welfare is the same in equilibrium as it is under truth telling. This follows from the fact that the ex-post assignment of items is the same and that side payments always sum to zero. Third, although aggregate welfare is the same, there are bidders whose expected utility in equilibrium is smaller than their expected utility when everyone tells the truth. This is easily demonstrated when Games 04 00021 i094, where the Games 04 00021 i115 type always prefers the truth telling outcome. The reason is intuitive. This player’s bid in equilibrium is his true value—i.e., Games 04 00021 i116. At the Bayes–Nash equilibrium, relative to the truth telling outcome, the Games 04 00021 i115 player wins the object with the same probability, has to pay more in compensation if he wins (since types lower than Games 04 00021 i115 bid above their value), and receives less in compensation if he loses (since types above Games 04 00021 i115 bid below their value).13 In contrast, low and high types, relative to the truth telling outcome, are better off at the equilibrium outcome.
Our next result concerns ex-ante fairness— i.e., a player’s belief about the outcome he will receive in the auction before he knows his value. Denote the expected utility of a player with type Games 04 00021 i112 in the truth telling and equilibrium outcomes by Games 04 00021 i117 and Games 04 00021 i118 respectively. Specifically, we show that, on average, there is no difference in the truth telling outcome and the equilibrium outcome. Since the truth telling outcome is known to be proportional, the expectation is that the equilibrium allocation must also be proportional.
Proposition 3 The equilibrium outcome of Knaster’s auction is ex-ante proportional. In particular, the expected difference in the truth telling outcome and the equilibrium outcome is zero—i.e.,
Games 04 00021 i119
Proof. We illustrate the proof for Games 04 00021 i094, the general case is similar and is left to the reader. Since the probability of winning the item is the same in equilibrium as in truth telling, the expected difference in Games 04 00021 i120 is just the expected difference in the side payments. By design, the transfers always sum to zero regardless of whether we are at the Bayes–Nash equilibrium or the truth telling outcome—i.e., Games 04 00021 i121 and Games 04 00021 i122. Thus,
Games 04 00021 i123
The expected difference is
Games 04 00021 i124
Games 04 00021 i125
The transfer function Games 04 00021 i126 is symmetric—i.e., Games 04 00021 i127. So, the above equality can be re-written as:
Games 04 00021 i128
This implies
Games 04 00021 i215
Since we know the truth telling outcome is proportional, the result follows.
Example 2 Consider Games 04 00021 i120 when types are uniformly distributed over the interval Games 04 00021 i129 and there are only two bidders. Using the bidding strategies computed in Example 1, this difference simplifies to Games 04 00021 i130. From this expression we can see that high types and low types both prefer the outcome under strategic behavior whereas middle types prefer the outcome under truth telling. The expression is maximized at Games 04 00021 i131. In addition, the expected difference is
Games 04 00021 i132
While this result is nice, Knaster was interested in ex-post fairness—i.e., the values people had after the auction was finished. Unfortunately, Knaster’s auction does not yield an ex-post proportional outcome in equilibrium as our next result demonstrates.
Proposition 4 The equilibrium outcome of Knaster’s auction is not ex-post proportional.
Proof. Suppose types are uniformly distributed over the interval Games 04 00021 i129 and there are only two bidders. Specifically, let Player 1 have the type Games 04 00021 i133 and Player 2 have the type Games 04 00021 i134. The symmetric equilibrium bid function, as given in Example 1, for each player Games 04 00021 i017 is Games 04 00021 i098. So, Player 1’s bid is Games 04 00021 i135 and Player 2’s bid is Games 04 00021 i136. Therefore, Player 1 wins the object and pays Player 2 a compensation of Games 04 00021 i137. The outcome results in a profit of Games 04 00021 i138, which is worse than the ex-post proportional outcome for Player 1 of Games 04 00021 i139
Proposition 4 is discouraging, but expected given the form of the equilibrium bid function.14 Our last result explores if competition might eliminate this negative feature of Knaster’s auction. In particular, we want to know whether the equilibrium bid functions converge to truth telling as the number of players increases. Why? If this were true, as it is in the first price sealed bid auction, we would know that in the limit Knaster’s auction is ex-post proportional. Alas, this is not the case. The bid function in Knaster’s auction diverges from the 45 Games 04 00021 i140 line as Games 04 00021 i021 increases. To establish this claim, we demonstrate that the bid of the lowest type player is diverging from the truth as the number of players increases. First, however, we need the following lemma concerning the threshold type.
Lemma 1 Threshold type Games 04 00021 i073 is strictly increasing in Games 04 00021 i021 for Games 04 00021 i141.
Proof. Games 04 00021 i142 if and only if Games 04 00021 i143. At Games 04 00021 i144, Games 04 00021 i145. Taking the derivative of the left hand side and right hand side yields Games 04 00021 i146 and Games 04 00021 i147 respectively. Now Games 04 00021 i148 for Games 04 00021 i141, which implies Games 04 00021 i149. Thus, the right hand side is decreasing at a slower rate than the left hand side for all Games 04 00021 i141. So, Games 04 00021 i143 for Games 04 00021 i141. Hence, Games 04 00021 i150 is increasing in Games 04 00021 i021. Now Games 04 00021 i151 is a cdf that , by assumption, is differentiable and strictly increasing. Thus, Games 04 00021 i151 has an inverse Games 04 00021 i152 that is strictly increasing. Thus, Games 04 00021 i153 is also increasing in Games 04 00021 i021.
Now, we show that the bid function does not converge to the 45 Games 04 00021 i140 line by demonstrating that the bid of the lowest type bidder is moving in the wrong direction.
Proposition 5 Equilibrium bid functions do not converge to the truth telling function as the number of bidders increases. In particular, the bid of the player with a type of zero, Games 04 00021 i154, is strictly increasing in Games 04 00021 i021 for Games 04 00021 i016.
Proof. Games 04 00021 i155, where Games 04 00021 i156, Games 04 00021 i157, and Games 04 00021 i158.
Games 04 00021 i159
Games 04 00021 i160
The last equality follows from Leibniz’s rule and the fact that Games 04 00021 i161 by definition of Games 04 00021 i073. Now from the chain rule:
Games 04 00021 i162
Games 04 00021 i163
Since Games 04 00021 i164 we have Games 04 00021 i165. Thus, a sufficient condition for Games 04 00021 i166 to be positive is for
Games 04 00021 i167
Games 04 00021 i168
Games 04 00021 i169
Games 04 00021 i170
Now since Games 04 00021 i080 is less than or equal to Games 04 00021 i073 we can form a bound on Games 04 00021 i171. In particular, the largest Games 04 00021 i171 can get is
Games 04 00021 i172
Games 04 00021 i173
We now show Games 04 00021 i174.
Games 04 00021 i175
Games 04 00021 i176
Games 04 00021 i177
The last inequality was established in Lemma 1. Therefore Games 04 00021 i178 for all Games 04 00021 i080. It follows
Games 04 00021 i179
Example 3 The following diagram plots the graph of the bid function Games 04 00021 i180 for the uniform distribution case when Games 04 00021 i094, Games 04 00021 i013, Games 04 00021 i181, Games 04 00021 i182, and Games 04 00021 i183. The bid functions displayed are each increasing in Games 04 00021 i021.
Figure 2. Bidding Strategies for u(0,1) N = 2,...,6.
Figure 2. Bidding Strategies for u(0,1) N = 2,...,6.
Games 04 00021 g002

5. Discussion

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.15 Additionally, since bidding strategies do not approach truth telling with competition, the distortions from truth telling created by strategic behavior do diminish with the number of players. However, despite these distortions, the auction does maintain some semblance of fairness. Specifically, the expected difference between the truth telling outcome and the Bayes–Nash equilibrium outcome is zero.
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.16 Specifically, Crampton, Gibbons, and Klemperer, working an independent private values framework, find a simple and efficient way to dissolve a partnership that is interim individually rational. This is in contrast to the well known impossibility theorem of Myerson and Satterthwaite.17 McAfee and Morgan’s papers are of interest to us because both papers consider fair division mechanisms. In particular, McAfee looks at an independent private values model and examines several simple auction mechanisms (including a simple cake cutting algorithm). In contrast, Morgan looks at several simple mechanisms in a two player common values setting and compares the outcomes of these mechanisms based on a fairness criterion. Clearly, Knaster’s auction could also be applied to any of these applied settings.
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.18 One of these mechanisms, the Adjusted Winner Procedure, for two players generates allocations that have several nice properties. Brams and Taylor (1996) compare Adjusted Winner with Knaster’s auction in several examples. Finally, we note that it is still an open question whether any of these mechanisms (or to what extent) are effective resource allocation mechanisms in practice. This question is well posed for future research.

6. Appendix

In this appendix, we provide the details for solving the differential equation (2). First, putting (2) in the standard form we have:
Games 04 00021 i185
This can be solved using the integrated factor Games 04 00021 i186.19 Multiplying both sides of equation (4) by the integrating factor, we have:
Games 04 00021 i187
Therefore, from the Fundamental Theorem of Calculus,
Games 04 00021 i188
To dispense with the absolute value signs, we look at the function in two cases: Games 04 00021 i189 and Games 04 00021 i190.
First, suppose Games 04 00021 i189, then by definition of absolute value Games 04 00021 i191. So, if Games 04 00021 i192 , then
Games 04 00021 i193
Games 04 00021 i194
Now solving for the bid function yields:
Games 04 00021 i195
Applying integration be parts, the last equation can be alternatively stated as
Games 04 00021 i196
Second, suppose Games 04 00021 i197 , then
Games 04 00021 i198
Games 04 00021 i199
Next, applying integration by parts on the two integrals on the right hand side of the equation gives us:
Games 04 00021 i200
Games 04 00021 i201
Solving for the bid function yields:
Games 04 00021 i202
Finally, the two cases give us bidding functions (5) and (6). Applying the “initial” condition Games 04 00021 i071 defines the constant of integration to be Games 04 00021 i203 for both functions. The equilibrium bid function (3) is found by combining (5) and (9) and inserting the value of Games 04 00021 i009.
  • 1 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.
  • 2 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.
  • 3 An outcome is proportional if each of the Games 04 00021 i204 participating players receive at least Games 04 00021 i205-th of their value for the whole collection of items.
  • 4 The following is adopted from Luce and Raiffa (1957). The numbers have been adapted to ease some of the calculations.
  • 5 In addition, when Games 04 00021 i206 the procedure generates an envy-free allocation.
  • 6 See Krishna (2010) for an introduction to auction theory.
  • 7 Ties are broken via random assignment.
  • 8 Written out, the side payment rule is:
    Games 04 00021 i207
  • 9 The steps used to solve equation (2) are provided in the Appendix.
  • 10 This is easy to check using equation (2).
  • 11 For notational simplicity, our results will be for the one item case. The generalization is straightforward and left to the reader.
  • 12 There is also a case where players evaluate the outcome when they know their type, but not the types of the other players (i.e., interim).
  • 13 It straightforward to verify that, when Games 04 00021 i206, the difference Games 04 00021 i208 is maximized at Games 04 00021 i209.
  • 14 For Games 04 00021 i206, when values are uniformly distributed over the interval Games 04 00021 i210, it is straightforward to verify that Knaster’s procedure is interim proportional. However, it is unknown whether this is true in general.
  • 15 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.
  • 16 Also related is Segal and Whinston (2011) who provide general conditions under which efficient bargaining is possible.
  • 17 See Krishna, Chapter 5, for a streamlined discussion of this result.
  • 18 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).
  • 19 In particular, our integrating factor Games 04 00021 i211 is found by solving the differential equation
    Games 04 00021 i212
    Games 04 00021 i213
    Games 04 00021 i214
    We do not need the most general solution to this differential equation. Hence, we dispense with the arbitrary constant.


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