# An Equilibrium Analysis of Knaster’s Fair Division Procedure

## Abstract

**:**

## 1. Introduction

“ 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.”

## 2. Knaster’s Fair Division Procedure

- First, each player ’s initial fair share of item (defined as -th of their reported value) is computed—i.e.,
- Second, surplus value for unit (defined as the difference between the high bid and the average bid) is computed—i.e.,
- Third, a player ’s adjusted fair share is computed from the bids. This is the player’s initial fair share of item plus an even share of the surplus—i.e.,
- Last, player ’s side payment for item is their reported value received (if they win the item) minus their adjusted fair share—i.e.,It is easy to verify that .8

## 3. Equilibrium

#### 3.1. Knaster’s Procedure with a Single Object

**Proposition 1**Symmetric equilibrium strategies in the Bayesian game induced Knaster’s Fair Division Procedure with players and one object are given by (3).

**Proof.**We need to check that following is an equilibrium. First, it is easily checked that the bidding strategy is increasing and continuous. Second, a bidder will never want to bid above or below . Bidder 1, for instance, should not submit a bid . A bid equal to 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 . A bid equal to 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 but bids as if his type were is

**Example 1**Suppose and each player ’s private value is distributed according to the uniform distribution—i.e., for and , then the equilibrium bidding strategy for each player is given by .10

#### 3.2. Multiple Objects

**Proposition 2**Symmetric equilibrium strategies in the Bayesian game induced Knaster’s Fair Division Procedure with players and multiple objects are given by the vector valued bid function , where for , is defined as

## 4. Welfare and Comparative Statics

**Definition 1**The item assignment of an allocation rule 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 is ex-post proportional if, for each realization of types, after the allocation rule has been applied, each player with realized type gets a utility of at least .

**Definition 3**An allocation rule is ex-ante proportional if, prior to observing types, each player ’s expected utility from his part of the allocation rule is greater than .

**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.,

**Proof.**We illustrate the proof for , 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 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., and . Thus,

**Example 2**Consider when types are uniformly distributed over the interval and there are only two bidders. Using the bidding strategies computed in Example 1, this difference simplifies to . 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 . In addition, the expected difference is

**Proposition 4**The equilibrium outcome of Knaster’s auction is not ex-post proportional.

**Proof.**Suppose types are uniformly distributed over the interval and there are only two bidders. Specifically, let Player 1 have the type and Player 2 have the type . The symmetric equilibrium bid function, as given in Example 1, for each player is . So, Player 1’s bid is and Player 2’s bid is . Therefore, Player 1 wins the object and pays Player 2 a compensation of . The outcome results in a profit of , which is worse than the ex-post proportional outcome for Player 1 of

**Lemma 1**Threshold type is strictly increasing in for .

**Proof.**if and only if . At , . Taking the derivative of the left hand side and right hand side yields and respectively. Now for , which implies . Thus, the right hand side is decreasing at a slower rate than the left hand side for all . So, for . Hence, is increasing in . Now is a cdf that , by assumption, is differentiable and strictly increasing. Thus, has an inverse that is strictly increasing. Thus, is also increasing in .

**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, , is strictly increasing in for .

**Proof.**, where , , and .

**Example 3**The following diagram plots the graph of the bid function for the uniform distribution case when , , , , and . The bid functions displayed are each increasing in .

## 5. Discussion

## 6. Appendix

^{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 participating players receive at least -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 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:^{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 , the difference is maximized at .^{14 }For , when values are uniformly distributed over the interval , 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 is found by solving the differential equation

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Van Essen, M.
An Equilibrium Analysis of Knaster’s Fair Division Procedure. *Games* **2013**, *4*, 21-37.
https://doi.org/10.3390/g4010021

**AMA Style**

Van Essen M.
An Equilibrium Analysis of Knaster’s Fair Division Procedure. *Games*. 2013; 4(1):21-37.
https://doi.org/10.3390/g4010021

**Chicago/Turabian Style**

Van Essen, Matt.
2013. "An Equilibrium Analysis of Knaster’s Fair Division Procedure" *Games* 4, no. 1: 21-37.
https://doi.org/10.3390/g4010021