1. Carnegie Mellon University
School of Computer Science
Senior Thesis
Staying Fair: Axiomatic Fairness for Recurrent Allocation
Author:
Elias Szabo-Wexler
School of Computer Science
szabowexler@cmu.edu
Adviser:
Dr. Ariel Procaccia
Assistant Professor
School of Computer Science
arielpro@cs.cmu.edu
April 28, 2015
2. Wir m¨ussen wissen. Wir werden wissen.
We must know. We will know.
David Hilbert
1
3. Abstract
Fairness in the context of the allocation of goods is a universal construct whose violation elicits
extremely strong reactions. It has nonetheless historically been mathematically ill-defined. In
recent years, the situation has improved as economists, mathematicians, and computer scientists
have tackled the issue of an axiomatic treatment of fairness. These axioms enable researchers to
qualify algorithms for fair allocation and to meaningfully compare different mechanisms. None
have yet extended this axiomatic approach to fairness over time: there is no axiomatic treatment
of situations with recurrent allocation events whose outcomes are linked beyond the naive iterated
application of existing (ill-suited) axioms. The current core axioms for time invariant fair allocation
are extended herein to account for recurrency. In particular, the axioms of efficiency, envyfreeness,
and strategyproofness are extended to be historically aware. Using the extended axioms, the seminal
probabilistic serial allocation mechanism of Bogomolnaia and Moulin is generalized to attain a
strictly superior allocation mechanism in the canonical and iterated fair allocation settings.
2
4. Acknowledgments
As is usually the case, I am here because I worked tremendously hard to get here, and I had a
fantastic group of people supporting me. In particular, I would like to thank my adviser Dr. Ariel
Procaccia for his patience with me and for not giving up on me, even when the project seemed
doomed. Without him, I would not understand what it means to be a researcher. I would also like
to thank my academic advisers, Drs. Tom Cortina and David Eckhardt, for offering an abundance
of support and helpful feedback throughout the entire process.
I would also like to thank my family and friends for holding me up through the bleak uncertainty
of a first foray into research. In particular, my father, Bernard Szabo, and mother, Tamar Wexler,
for their endless patience in listening to me go on (at length) about obscure technical details, day
after day, and my friend Grant Della Silva for conversations about and technical contributions to
my definition of historical strategyproofness. And finally, my girlfriend Ariana Weinstock for her
endless patience, and seeming imperturbability in the face of the obsessed.
Without you all, I would most certainly not be where I am today. Thank you.
3
7. List of Figures
1 Comparing allocations of players 1 and 2 for weak envyfreeness. . . . . . . . . . . . . 15
2 Comparing allocations of players 1 and 2 for weak strategyproofness. . . . . . . . . . 17
3 General mechanism framework for current mechanisms. . . . . . . . . . . . . . . . . 26
4 General mechanism framework with feedback between rounds. . . . . . . . . . . . . . 26
5 IHPS and SHPS in the context of our general framework. . . . . . . . . . . . . . . . 28
6 Envy graph (based on welfare). Players in the same level (having the same welfare)
are ensured to be envyfree with respect to each other, and advantaged relative to
every player “beneath” them. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
List of Tables
1 Summary of PS and RSD properties adopted from [11]. . . . . . . . . . . . . . . . . 19
2 Summary of impossibility results, adopted from [15]. . . . . . . . . . . . . . . . . . . 21
3 Relaxed strategyproofness mechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Modifying symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6
8. 1 Introduction
As long as there are at least two people and something they both want on this Earth, there will be
arguments of fairness. It is almost inevitable that groups of people will want similar things, and
when that happens, deciding who gets what is a delicate process. Though we would na¨ıvely hope
that people could decide amicably among themselves, in practice this doesn’t always pan out. And
after some consideration, it isn’t surprising that this is the case. While there are situations where
it is easy to decide, as when Alice and Bob divide an apple and an orange when Alice wants the
apple, and Bob the orange, there are also situations where it is actually impossible for everyone to
be satisfied, as when Alice and Bob divide a single guitar.
In such settings, there isn’t really a clean way for the involved parties to decide for themselves
how to resolve the situation. In fact, the example above actually illustrates that even armed a
priori with some agreed upon deterministic algorithm to figure out who should get what good,
Alice and Bob would still be at an impasse with the guitar, because one of them would (rather
justifiably) complain that the game was rigged, since they walk away with nothing and never had
any chance of getting anything.
The underlying problem is that any deterministic procedure we can give will admit scenarios
where some parties are discriminated against. Most people, when considering a division problem,
will either identify with the “lucky” parties, or the “unlucky” ones. Those in the former camp
would be perfectly happy with an “unfair” procedure, but those in the latter camp who identify
with the disadvantaged underdogs would be justifiably quite upset.1
And so we come to our main dilemma! We need a way to resolve such situations, so that people
are minimally upset. In other words, we need to remove the quotations from our categorization of
a process as “fair,” and then we need options for truly fair mechanisms for dividing goods. It is
the pursuit and categorization of such mechanisms that motivate this thesis.
1.1 What’s Wrong Now?
The most compelling argument for the need for this work comes from observing an example.
Consider a situation in which three people need to divide three fruits among themselves. Say Alice,
Bob, and Charlie need to decide who gets which of an apple, an orange, and a banana every morning
for breakfast. Let’s assume that all three of them strongly prefer bananas to apples, and apples to
oranges. Using the current state of the art mechanism (see Section 2.6.1 for the technical details),
each person has an equal shot at each fruit on each day. But what happens over multiple days?
Consider Alice’s experience over the course of two breakfasts. If we just apply the same mechanism
on both days, there is a 1/9 chance that Alice will receive oranges both days, while Bob receives
bananas - this outcome is equiprobable with every other possible assignment of goods over both
breakfasts. However, there is a significant unfairness here: it is unfair when a single person gets the
short end of the stick both days. While it’s reasonable that this might have a chance of happening,
it shouldn’t be as probable as other, more fair arrangements. Fortunately, we propose a mechanism
which addresses this issue (see Chapter 4 for the technical details). Using our new mechanism,
there is no chance that Alice (or, indeed, any of the players) could receive oranges both days. I
1
This mirrors the general approach to taxation in the United States. Interestingly, the wealthy tend to pay
disproportionately low tax rates for their wealth. One can conjecture that the reason is that everyone hopes to
someday become rich, and so is reluctant to enact harsher tax rates, in case those rates should one day apply to
them. It is an optimistic outlook, assuming that one will always be the advantaged party, and so seeking to maximize
that advantage. The irony is that in reality, only a small minority actually benefit from the optimism of the masses.
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9. will reiterate this point for emphasis: in this example, our mechanism prevents the occurrence of
an unfair situation which is not even understood to be unfair by any existing mechanism.
1.2 Canonical Fair Allocation
This particular type of problem of assigning goods to players is called fair allocation. In the
canonical problem, there are exactly as many goods as there are players, and each player is to
receive exactly one good [4]. From there, slight modifications spawn a host of related, but slightly
different problems. Of particular interest are cases where there are more goods than players [4];
where there are duplicates of some goods; where players require more than one good [17]; where
players may receive no goods at all [6]; or where there are constraints on acceptable allocations [6].
1.3 Recurrent Motivation
The canonical problem and all of its related variants share a major feature: they are concerned
with a situation where goods are to be allocated once. For many settings this is appropriate, e.g.
when allocating one-time jobs to servers or an estate to heirs. However, there are a host of real
world instances where some set of players are allocated some set of indivisible goods on a recurring
basis (e.g. postings to medical students2, or shares of a contested resource among long running
processes). The example in Section 1.1 demonstrates that even current mechanisms with highly
attractive fairness properties may be significantly unfair in settings with recurring allocations. This
thesis is the first work (to the author’s knowledge) which incorporates recurrent allocations into
fairness properties, updating them for the new domain and modifying a current mechanism to
achieve stronger fairness properties over time.
1.4 Contribution
The main contributions of this thesis are threefold:
1. We update existing (sequential) allocation fairness axioms to be exact in a historical sense;
2. We propose a generic framework for building conformant allocation mechanisms;
3. We use the framework to propose two such distinct allocation mechanisms.
The reader will excuse a brief detour explaining the significance of the term axiomatic, which
will play a major role in our exposition. Fair allocation is now defined by a set of axioms, or prop-
erties, which are (intended to be) expressive enough to completely encapsulate the mathematically
informal notion of fairness. These axioms let us, as researchers, completely characterize allocation
mechanisms, and compare them directly with contenders. They are the building blocks from which
all current solutions are built, and as such, it is no trivial thing to modify them. It is generally not
done unless in the case of some serious deficiency, as I identify in this thesis. Because we follow
this field’s insistence that mechanisms conform to axioms, the effect of tightening the axioms is
fundamental: what used to pass for a good mechanism prior to focusing on historical fairness no
longer does; the bar is raised. Moreover, the increased strength in the axioms derives from use of
stronger abstractions, and these stronger abstractions buttress the foundations for future work.
2
In Israel, doctors-to-be are required to serve year-long residencies, and may rank available residencies, adding some
constraints. They serve multiple residencies, and the openings are divided among their class each year. Allocating
residencies to doctors fairly sparked the current research.
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10. 1.5 Organization
The thesis is laid out as follows. In Chapter 2, we present an overview of the current state of the art.
We introduce the fundamental axioms of fairness for efficiency, envyfreeness, and strategyproofness.
We discuss the trade-offs inherent between deterministic and randomized mechanisms, as well as
different methods for representing player preferences, and the best-in-class mechanisms for each
common approach. Finally, we offer a discussion of the relative merits of the axioms, and some
thoughts on choosing intelligently in mechanism design. In Chapter 3, we extend the axioms to
incorporate recurrent allocation. We then use these axioms to categorize the updated mechanisms
proposed in Chapter 4. We identify future areas for research in Chapter 5, and provide our proofs
in Chapter 7. We also include the pseudocode necessary to implement our algorithm in Chapter 8.
As a reference, we collect all of our notation in Chapter 9, for convenience.
9
11. 2 Background and Related Work
Consider some people who must divide some goods among themselves. Assume there are as many
goods as players, and each player is to receive exactly one good. How should we allocate the
goods, so the result is “fair”? What does it mean for an allocation to be “fair?” We will begin
with a discussion of fairness, and representation of preferences, before moving into a more technical
discussion of axioms and trade-offs in mechanism design. The mathematical notation used is
collected in Chapter 9, and we do not explicitly introduce it here, except as absolutely necessary.
The more technical reader is advised to begin by reading that chapter, or referring to it as needed.
2.1 Player Preference Types
To begin a discussion of fairness, we need some metric for evaluating outcomes on a per-player
basis. This topic has received significant attention, since the seminal work of von Neumann and
Morgenstern, [16], published in 19443. What is truly remarkable about [16] is that it presents a clear
argument for assuming that people have numerically comparable “valuations” for goods. These
can be compared, and may induce orderings on arbitrary sets (and so, saying that Alice preferred
bananas to apples to oranges in the example from Section 1.1 was a reasonable statement). These
valuations are called von Neumann Morgenstern utilities, or just VNM utilities.
Definition 1. Any function U : G → R is a valid VNM utility function, if:
1. U defines a complete ordering on G
2. U satisfies ordering properties (loosely speaking, transitivity is obeyed)
3. U satisfies some combining properties (loosely speaking, utilities can be combined in a reason-
able fashion, though not, generally, additively)
For the full and rigorous mathematical definition, see [16, p. 26].
Three types of preference profile are prevalent in the literature layered on top of these utilities.
The least expressive is the dichotomous preference profile [3].
Definition 2. A dichotomous preference profile is a binary profile distinguishing between acceptable
and unacceptable goods.
The simplest way to distinguish between outcomes (for a player) is to label some outcomes
successful, and others not. Using dichotomous preference profiles, an outcome is said to be successful
if a player receives an acceptable good, and not otherwise. We use the example from Section 1.1
as a running demonstration of the different preference profiles.
Example 1. Dichotomous preference profiles. is acceptable, is not.
Banana Apple Orange
Alice
Bob
Charlie
3
Titled The Theory of Games and Economic Behavior, it is widely accepted to be the foundational work of the
entire field of game theory. Brilliantly formulated (though dense), it is a fascinating read, even seven decades later.
10
12. More information is conveyed using ordinal (or individual) preference profiles [4].
Definition 3. An ordinal preference profile is induced by the underlying VNM utility functions of
the players. The profile is merely the ordering imposed by the utility functions.
An ordinal preference profile reflects an implicit inequality in goods from the perspective of an
individual player hidden by a dichotomous profile, but hides the extent to which a particular good
may be preferred (or despised).
Example 2. Ordinal preference profiles (lower is better).
Banana Apple Orange
Alice 1 2 3
Bob 1 2 3
Charlie 1 2 3
This information is available in the most expressive profile scheme: the cardinal preference
profile [6].
Definition 4. A cardinal preference profile is simply the VNM utility of each good, for each player.
A cardinal preference profile places a numeric value on each good - this allows strong statements
comparing how much a good is worth, in terms of other goods.
Example 3. Cardinal preference profiles (higher is better).
Banana Apple Orange
Alice 10.0 2.0 0.01
Bob 5.0 1.0 0.5
Charlie 3.0 2.0 1.0
Each of these three systems for representing preferences offers trade-offs. Dichotomous prefer-
ence profiles are extremely simple, and people do not find it difficult to provide a yes/no decision.
Unfortunately, they hide a great deal of information, since yes is extremely inexpressive. Ordinal
preference profiles offer a good trade-off between complexity to elicit (people can generally rank
a list of goods by preference) and expressiveness (rankings generally give more power than binary
reports). Cardinal preference profiles give a complete description of the relation between players
and goods, but are very difficult to obtain. People do not usually assign numeric values to their
feelings, and find it very hard to evaluate how much more they might prefer a given good to another.
Whichever is used, settling on a scheme for representing preferences permits an easy comparison
between allocations. From a player’s perspective, the best allocation is the one which gives them
the optimal result (or just an acceptable one, for dichotomous preferences), under their preference
profile.
2.2 Deterministic vs. Random Mechanisms
This lays out the task for the mechanism designer: given a collective preference profile for several
players and a set of goods, design a mechanism which assigns goods to players.
The first decision faced by the designer is whether to create a deterministic or random mech-
anism. Unfortunately, when dealing with indivisible goods, deterministic algorithms are often
11
13. perceived as asymmetric or unfair [6]. The easiest answer is to then favor random mechanisms,
with favorable properties. But what does it mean for a mechanism to be “random?”
In this context, it means that the mechanism generates some lottery (either via a deterministic
or a nondeterministic procedure). The lottery defines a distribution over players and goods - loosely
speaking, it defines the probabilities that each player will get each good. A sample lottery is given
in Example 4, below.
Example 4.
Banana Apple Orange
Alice 1/3 1/3 1/3
Bob 1/3 1/3 1/3
Charlie 1/3 1/3 1/3
Notice, however, that the lottery does not actually provide a description of how to select which
goods go to which player. Classically, we rely on the famous Birkhoff von Neumann theorem
(provided in Algorithm 2), which holds that any bistochastic n × n matrix (i.e. any lottery) can
be expressed as a convex combination over (at most (n − 1)2 + 1) permutation matrices.
Definition 5. A convex combination C of the set of matrices X is a linear combination where all
coefficients are nonnegative and sum to 1. That is:
C =
x∈X
αxx (1)
αx ≥ 0, ∀x ∈ X (2)
x∈X
αx = 1 (3)
Translated, this means that any lottery can be expressed as a probability distribution over at
most (n−1)2+1 deterministic assignments, which explicitly assign a particular player to a particular
good. We call the process of converting a lottery to an actual distribution over deterministic
assignments instantiating that lottery. The algorithm for the procedure is given in Algorithm 2.
Applying it to this setting, we might instantiate the lottery from Example 4 as in Example 5, below.
Example 5.
0.3333
1 0 0
0 1 0
0 0 1
+ 0.3333
0 0 1
1 0 0
0 1 0
+ 0.3333
0 1 0
0 0 1
1 0 0
The interpretation of this lottery is that with probability 1/3, we assign Alice the banana, Bob
the apple, and Charlie the orange, OR, with probability 1/3 we assign Alice the orange, Bob the
banana, and Charlie the apple, OR with probability 1/3 we assign Alice the apple, Bob the orange,
and Charlie the banana. As promised, each player is equiprobable to get each good.
When evaluating these lotteries, we try to identify properties which are intrinsic to the lotteries,
which we can use to compare them. If we can identify properties which all lotteries produced by a
particular mechanism will share, we assign that property to the mechanism itself.
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14. 2.3 Defining Fairness
Indeed, the properties guaranteed by a mechanism provide a convenient way to compare mecha-
nisms, and also provide a basis for claiming fairness. Modern fair allocation literature has largely
settled into an axiomatic approach. Axiomatic characterizations of mechanisms as in [9] are increas-
ingly common, and extremely useful for comparing different mechanisms. These characterizations
frequently characterize mechanisms along three axes: efficiency, envyfreeness, and strategyproof-
ness.
Before defining them, we briefly explain what these axioms convey. Efficiency and envyfreeness
axioms capture properties of lotteries themselves (as will be explained later), while strategyproof-
ness axioms capture properties of mechanisms. Mechanisms are said to attain either efficiency
or envyfreeness axioms when they are guaranteed to produce lotteries which are themselves either
efficient or envyfree. The axioms are contracts for mechanisms: an efficient mechanism produces ef-
ficient lotteries, an envyfree mechanism produces envyfree lotteries, and a strategyproof mechanism
operates in a fashion which is strategyproof.
2.3.1 Efficiency
Axioms related to efficiency attempt to capture the large scale behavior of a particular alloca-
tion. As such, most axioms are built around the concept of Pareto efficiency, which captures a
fundamental non-wasteful property of an allocation.
Definition 6. A deterministic allocation is Pareto efficient if no player’s outcome may be improved
without harming another player’s outcome.
Phrased differently, an allocation is Pareto efficient if nothing has been left on the table that
didn’t need to be, and no set of swaps can be made that do not harm some player. The same
concept is extended to the randomized context (where allocations are probability distributions,
instead of actual assignments).
Ex post efficiency ensures the deterministic allocation resulting from a lottery is itself Pareto
efficient, while ordinal efficiency guarantees that certain types of “inferior” random allocations
cannot occur. Finally, ex ante efficiency ensures that an allocation is societally optimal for a fixed
set of utility functions, i.e. that the expected utility over all players is maximized. We keep the
definitions of efficiency, envyfreeness, and strategyproofness as in [4].
Definition 7. A random allocation is ex post efficient if it may be expressed as a convex combination
over Pareto efficient deterministic allocations.
In order to define a stronger notion of efficiency, we must define a new mathematical construct.
Specifically, the notion of stochastic dominance, mathematically defined in Definition 24. Loosely
speaking, stochastic dominance captures the notion of universal preference - if some allocation
stochastically dominates another, then every player (at least weakly) prefers the former over the
latter.
Definition 8. A random allocation is ordinally efficient if it is not stochastically dominated by any
other random allocation.
This distinction is subtle, but important. An ex post efficient allocation is, in some sense,
“efficient” in that it cannot be directly improved. But there may be other lotteries over equally
13
15. efficient allocations which are preferred by every player. The ordinal efficiency axiom captures
this notion: a lottery is ordinally efficient if there are no other lotteries that are (at least weakly)
preferred by every player. We might say that an ordinally efficient random allocation is, in some
sense, an optimum in the set of efficient lotteries.
Definition 9. A random allocation is ex ante efficient if the expected utility of the allocation for
the set of VNM utility functions the players are endowed with is at least as great as that of any
other random allocation for the same utility functions.
2.3.2 Envyfreeness
On the other hand, envyfreeness axioms deal with jealousy. In particular, with whether (and how
much) players wish they had received other players’ allocations, or lottery tickets. The deterministic
envyfreeness guarantee is rarely used, but stipulates that every player prefers their own good to the
good received by every other player. Unfortunately, any deterministic mechanism admits situations
where true envyfreeness is impossible to attain (see the introduction example about two players
and a single guitar, for proof).
To cope with this, we define randomized envyfreeness, in terms of the lottery from the perspective
of a given player. There are two conventional forms of envyfreeness in the literature: weak, and
strong, as per [4]. Both are defined in terms of the stochastic dominance relation. We write
Pi sd( i) Pj to indicate that player i’s allocation stochastically dominates player j’s allocation,
under lottery P with (ordinal) preferences . We interpret this to mean that player i (at least
weakly) prefers their allocation to that of player j.
Definition 10. An allocation P is weakly envyfree if for all players i, j ∈ P
Pj sd( i) Pi ⇒ Pi = Pj (4)
Definition 11. An allocation P is envyfree if, for all players i, j ∈ P
Pi sd( i) Pj (5)
These axioms are of markedly different strengths. Strong envyfreeness effectively ensures that
each player is most satisfied with their lottery, and no other player has a better lottery. However,
weak envyfreeness merely ensures that each player’s lottery is not stochastically dominated by any
other’s. This admits a fair bit of envy, prohibiting only certain extreme types (see Example 6).
Example 6. Below, we present the set of players and goods on the left, and their preferences on
the right. Then, we present a viable lottery.
14
16. P = {1, 2, 3, 4}
1 A B C D
2 A B C D
G = {A, B, C, D}
3 B C D A
4 D C B A
A B C D
1 0.75 0.20 0.02 0.03
2 0.01 0.01 0.96 0.02
3 0.04 0.75 0.01 0.20
4 0.20 0.04 0.01 0.75
Note that this random allocation is weakly envyfree, since no player’s allocation stochastically
dominates any other’s. However, consider the tension between players 1 and 2. Both have the same
preference profiles - both would prefer the same goods. However, player 1 seems to be clearly advan-
taged. We can see this in Figure 1, below. Player 1 has a right-skewed distribution of probability
(which is good, according to their preference profile, since most of their distribution is to the left,
or on their favored goods), while player 2 has a left-skewed distribution of probability (which is bad,
according to their preference profile, since most of their distribution is to the right, or on their less
preferred goods). The casual observer can easily deduce that, given a choice between players 1 and
2, they want player 1. And yet, this allocation is still weakly envyfree.
Figure 1: Comparing allocations of players 1 and 2 for weak envyfreeness.
2.3.3 Strategyproofness
Finally, strategyproofness axioms deal with whether it is possible for a player to gain by misre-
porting their preferences, and so are necessarily axioms which apply to mechanisms (rather than
individual allocations). As with envyfreeness, there exist a common weak and strong form of strat-
egyproofness [4]. Interpreting P( ) to be the random allocation provided by mechanism P when
given preference profile , we write P( |i ∗
i ) to indicate the random allocation returned by
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17. mechanism P when given preference profile , with player i’s true preferences replaced by some
inaccurate preferences, ∗
i .
Definition 12. A mechanism P is weakly strategyproof if:
Pi( |i ∗
i ) sd( i) Pi( ) ⇒ Pi( |i ∗
i ) = Pi( ) (6)
Definition 13. A mechanism P is strategyproof if, for all possible ∗
i :
Pi ( ) sd( i) Pi( |i ∗
i ) (7)
Notice that the definitions for strategyproofness closely mirror those of envyfreeness. This does
mean that weak stategyproofness is also an extremely permissive property. To demonstrate this,
we modify Example 6.
Example 7. Below, we present the set of players and goods on the left, and their preferences on
the right. Then, we present a viable lottery. All are the same as in Example 6.
P = {1, 2, 3, 4}
1 A B C D
2 A B C D
G = {A, B, C, D}
3 B C D A
4 D C B A
A B C D
1 0.75 0.20 0.02 0.03
2 0.01 0.01 0.96 0.02
3 0.04 0.75 0.01 0.20
4 0.20 0.04 0.01 0.75
However, consider the case where player 2 knows that this outcome will occur, and so lies. They
misreport their preferences, as:
2∗
| B A C D (8)
Under these new preferences, the resulting lottery is as follows.
A B C D
1 0.75 0.02 0.20 0.03
2 0.01 0.96 0.01 0.02
3 0.04 0.01 0.75 0.20
4 0.20 0.01 0.04 0.75
The resulting allocation remains weakly envyfree, but player 2 has clearly managed to benefit
significantly by lying. Consider the graph of their allocations below, in Figure 2.
16
18. Figure 2: Comparing allocations of players 1 and 2 for weak strategyproofness.
The choice between player 1 and player 2 is no longer quite so obvious. This suggests that weak
strategyproofness doesn’t really protect against manipulation, since it is quite possible for a player
to gain significantly by misreporting, without violating weak strategyproofness.
2.4 Random Serial Dictatorship
The random serial dictatorship (RSD) is the golden standard against which a new mechanism is
first compared (and is applicable under any preference reporting scheme). A formalization of the
biblical concept of allocation by random lottery, RSD is simply explained. An ordering of the
players is chosen uniformly at random, and players choose goods in order, until there are either no
more goods, or every player has chosen.
RSD is a fairly compelling mechanism because it is easily explained, easily implemented, and
not altogether unfair. Indeed, in [4] it is proven that RSD is ex post efficient, weakly envyfree, and
strategyproof. Though not excessively strong, ex post efficiency is a reasonable efficiency guarantee,
and strategyproofness seems compelling, unless we accept that strategyproofness is the lesser of
the three axes.
2.5 Dichotomous Mechanisms
Over the past decade, all three preference profile schemes have received attention (as evidenced
by the incredible body of literature that has been created in response to e.g. [13, 18, 4, 3]).
The common vision is that a mechanism designer chooses the reporting preference profile and the
(realizable) combination of axioms they need, and then picks among existing mechanisms which
are well suited for that setting.
In this regard, the dichotomous preference profile has the decided advantage of admitting a
single, elegant solution attaining extremely strong efficiency guarantees, strategyproofness, and
envyfreeness. The leximin (egalitarian) mechanism of [3] is Lorenz dominant4, strategyproof (even
to coalitions of colluding players) and envyfree.
4
An extremely strong property mentioned in [3] - in every other setting it is so strong as to be considered
unattainable
17
19. 2.5.1 Leximin
The egalitarian solution proposed in [3] categorizes the set of players into three sets: overdemanded,
perfect, and disposable players. The matching itself is achieved via a recursive procedure to max-
imize the leximin ordering. In particular, the lowest utility is first maximized. Then, subject to
this fixed value, the second lowest utility is maximized, and so on.
While [3] originally formulated the egalitarian solution with a recursive definition, [14] proposes
an elegant linear programming solution, and applies it to an allocation problem faced by a Cali-
fornia school district. In terms of axiomatic completeness, the leximin mechanism for the case of
dichotomous preferences is the most satisfying - it can believably be said to be the fairest known
algorithm for allocation.
2.6 Ordinal Mechanisms
Unfortunately, there is no such satisfying all-encompassing solution once we move out of the domain
of dichotomous preferences, nor can there be such. We have a strong impossibility result; as shown
in [4], there exists no mechanism in the ordinal preference domain which is ordinally efficient,
strategyproof, and satisfies equal treatment of equals5.
While disappointing, this impossibility result mirrors the general shape of fair allocation. At
every level of the design process, trade offs are made between the expressiveness and power of the
mechanism in a particular way against the quality and quantity of the fairness guarantees it can
ensure. This tension is most pronounced when selecting preference profiles, or when selecting which
of efficiency, envyfreeness, and strategyproofness to make strongest, and how strong to make them.
2.6.1 Probabilistic Serial
The probabilistic serial mechanism (PS) proposed by Bogomolnaia and Moulin in [4] is now the
golden standard against which a new ordinal mechanism is compared. It is built on a beautifully
simple and elegant model for the act of allocation. Specifically, each good is viewed as an infinitely
divisible unit-good. Players are endowed with an “eating function,” and proceed to eat goods (as
available and in order of preference) in parallel until all goods are gone. The resulting shares of
goods eaten are treated as probabilities which define the lottery according to which the deterministic
allocation is drawn. It is shown to be ordinally efficient (and is the first mechanism to be shown
to be ordinally efficient), envyfree, and weakly strategyproof.
This represents a significant trade-off between axioms. PS (respectively RSD) is ordinally ef-
ficient (resp ex post efficient), envyfree (resp weakly envyfree), and weakly strategyproof (resp
strategyproof). Efficiency and envyfreeness both see significant upgrades, but strategyproofness
takes a hit. The upgrades are both very large, and make the PS mechanism very exciting indeed.
However, they come at a cost - PS is as much weaker than RSD in terms of strategyproofness as
it is stronger than RSD in efficiency and envyfreeness. This does make it a trade-off with regards
to strategyproofness, rather than a strict improvement over all axioms, but it is very significant,
as it presents one of the largest improvements over RSD in millenia in terms of efficiency and
envyfreeness (see [4]).
However, the PS mechanism is quite flexible, and extremely expressive (as we will explore in
depth, later). Budish, Che, Jojima, and Milgrom generalize the PS mechanism in [6], allowing for
5
A weaker form of envyfreeness, stipulating that players with identical preference profiles must be treated identi-
cally.
18
20. Table 1: Summary of PS and RSD properties adopted from [11].
n ≥ m n m
PS RSD PS RSD
strategyproofness weak
ordinal efficiency
envyfreeness weak weak
lower and upper piecewise quotas. They characterize the conditions for universal implementability,
and design a powerful mechanism for fair allocation under ordinal preferences. Alio˘gullari, Barlo
and Tuncay take the mechanism in an altogether different direction in [1]. They incorporate a
fairness clause of the Turkish government concerning the random assignment of new doctors into
the mechanism, to produce the R1 (reservation-1) mechanism, which improves performance with
regards to a new aggregate efficiency property designed to reflect the aggregate welfare. Rather
than permitting players to eat strictly in order of preferences, they restrict players from eating goods
which are available, and still the first choice of any other players. The new mechanism remains
ordinally efficient, weakly strategyproof, and (weakly) envyfree, but is an improvement over PS on
the grounds of aggregate efficiency. This presents a trade-off: by accepting a weaker envyfreeness
property, the efficiency of the mechanism may be improved. Aziz extends the mechanism in yet a
third way in [2], to allow for multi-unit demands. The resulting mechanism is still envyfree and
weakly strategyproof, but it is not even ex post efficient (instead, it attains a much weaker property
called unanimity).
2.7 Cardinal Mechanisms
However, neither dichotomous nor ordinal preference profiles convey a complete picture of a partic-
ular player’s true preferences. Cardinal preferences are superior in this regard, in that they convey
the full relation over items, for each player. While this is an improvement for the designer, it is
quite difficult for most people to achieve.6 However, if we accept that cardinal preferences can
actually be collected, there is a compelling mechanism which attains very significant efficiency and
envyfreeness results in this space, presented in Section 2.7.1, below.
2.7.1 Market Clearing
If the probabilistic serial mechanism is the golden standard of ordinal mechanisms, then the implicit
market clearing mechanism of Hylland and Zeckhauser (HZ) [13] is the golden standard of cardinal
ones. Achieving strong efficiency and envyfreeness properties7, the mechanism is quite compelling
in settings where the set of players is markedly larger than the set of goods, and was also extended
to form a more general mechanism in [6].
An adaptation of the competitive equilibrium with equal incomes solution for the fair division
of unproduced commodities (CEEI) to the random assignment model ([4]), it takes a somewhat
different approach from PS. Instead of eating probability, players are endowed with some amount
of virtual money and an initial allocation. They are then permitted to purchase and sell goods
6
Consider trying to assign numeric values to each of a banana, an apple, and an orange. While it’s probably easy
to rank order them, assigning values is much harder.
7
Ex ante efficient (resp envyfree)
19
21. (which are actually probability shares), until a stable equilibrium can be reached. The result defines
the probability vectors of each player, which in turn define the lottery instantiated to allocate the
goods, and [13] shows that there always exists a stable equilibrium (i.e. that the mechanism always
produces a lottery with the desired properties).
2.8 Axiomatic Trade-offs
So far we have discussed mechanisms in terms of determinism and player preference profiles, merely
reporting the fairness properties achieved without offering any judgment on the relative importance
of each property. When surveying the literature this is appropriate, as it gives us a starting point
and an appreciation for what is possible. To fill out the picture and guide our design, we offer a
brief discussion of emerging trends in mechanisms, and lay out some impossibility results.
The foremost (and most crushing) impossibility result dates back to 1990, when Zhou showed
in [18] the incompatibility of equal treatment of equals, strategyproofness, and ex ante efficiency
for one sided matching problems. Extending his work, Nesterov shows in [15] the mutual in-
compatibility of envyfreeness, strategyproofness, and ex post efficiency, and of weak envyfreeness,
strategyproofness, and ordinal efficiency. This shapes our mechanism design process strongly: in
general, we must choose two of the three axioms to focus on, as one will be necessarily quite weak.
Theorem 1. (Zhou) There exists no mechanism for n 3 which is ex ante efficient, satisfies equal
treatment of equals, and is strategyproof.
Theorem 2. (Nesterov) There exists no mechanism for n 3 which is ordinally efficient, weakly
envyfree, and strategyproof.
Theorem 3. (Nesterov) There exists no mechanism for n 3 which is ex post efficient, envyfree,
and strategyproof.
But which axiom is least important? We do not have a compelling (theoretic) answer to the
question. However, evidence begins to mount suggesting that strategyproofness is the lesser of the
three. In particular, Budish’s survey of the Harvard Business School course allocation mechanism
(HBS) in [5] presents an interesting case study of the real world performance of a mechanism
which is not strategyproof. The mechanism analyzed is a modified draft form of the random serial
dictatorship. And, indeed, the students at Harvard have figured out how to game the system,
and do so. However, Budish uses a series of surveys to reveal true preferences, and simulates
a strategyproof mechanism. Surprisingly, the resulting mechanism is less efficient (having lower
societal utility over allocations) than the non-strategyproof draft!
In the survey, Budish makes the implicit assumption that players will tell the truth when dealing
with a strategyproof mechanism8, and shows that the strategyproof mechanism under performs
from an efficiency standpoint. Unfortunately, this assumption is not well founded. Hugh-Jones
et. al. perform a laboratory experiment investigating incentive (strategyproof) properties of PS in
[12]. They find that in situations where a player ought to lie, they rarely will, whereas in situations
where a player ought not to lie, they frequently will. This is extremely distressing: it suggests
that average people who might make use of fair allocation mechanisms will not necessarily act even
in their own best interests (whether for reasons of confusion, or because of deep seated societal
prejudices against reporting what they really want). A flip side of this is that mechanisms which
8
After all, what rational individual will lie, when that lie can only harm them? Unfortunately, we cannot help
but read the preceding sentence as sarcastic, precisely because humans are generally not entirely rational, and the
concept of a purely rational individual evokes Spock.
20
22. depend on accurate reporting for optimal results (i.e. efficiency or envyfreeness depend on truthful
reporting) are likely to perform poorly in practice, simply because they will rarely (if ever) be given
fully truthful profiles.
Reinforcing this, Guillen and Hakimov summarize current work in measuring truthfulness in [8],
concluding that:
...it seems clear that the majority of participants in laboratory experiments don’t un-
derstand strategyproofness, but instead respond to changes in the environment and are
somehow guided by their own risk attitudes.
They show the presence of an effect they dub “Monkey See, Monkey Do,” revealing that infor-
mation about the preferences revealed by other players has an effect on truth-telling rates. They
vary the information available to players from full (i.e. players know what other players report) to
none whatsoever, and find that in settings with less than complete information, truthful preference
revelation is greatly decreased. Simply put: when people don’t know what everyone else is saying,
they lie to compensate.
Even when full information is available, advice also seems to motivate lying. In [7], Guillen finds
that any advice whatsoever, whether correct, incorrect, or a confusing mixture, causes people to lie
dramatically more. The conclusion drawn is that telling the truth is sometimes a default (in the
absence of a better strategy), but in the presence of advice which seems to indicate the optimal lies
to tell, players will throw honesty out the window in the hopes of getting a slightly better outcome.
While this all seems like a pretty depressing characterization of human nature, the only real
takeaway is that strategyproofness is likely the least important of the three axioms. Indeed, if
we relax our requirements on that front, the picture looks a fair bit rosier than portrayed by the
collection of impossibility results in Table 2, below. Indeed, our results in Table 3 are cause for some
jubilee. Of particular note, by relaxing the requirement that mechanisms be strictly strategyproof,
we are able to consider substantially more powerful efficiency and envyfreeness axioms, which is no
small thing.
Table 2: Summary of impossibility results, adopted from [15].
Strategyproof Mechanisms
Envyfree Weakly envyfree Equal division lower
bound
Equal
treatment
of equals
Ex post efficient
N = 3 ∅ RSD! RSD RSD!
N 3 ∅ RSD RSD RSD
Ordinally efficient N 3 ∅ ∅ ∅ ∅
Exclamation mark denotes uniqueness.
21
23. Table 3: Relaxed strategyproofness mechanisms.
Weakly strategyproof Not strategyproof
Historic envyfree Envyfree Historic envyfree Envyfree
Ordinally efficient HPS PS
Ex ante efficient ∅∗ ∅ ! HZ
Historically efficient ∅∗ ! ∅∗
Exclamation mark denotes conjectured existence, empty set with asterisk denotes conjectured impossibility
result. HPS is the historic probabilistic serial mechanism proposed herein. Blank cells indicate superceded
set of axioms.
22
24. 3 Extending the Axioms
We have discussed the existing axioms and mechanisms at length, and are now prepared to address
the issue at hand. In particular, the current formulations of efficiency, envyfreeness, and strat-
egyproofness do not address recurrent allocation. We will remedy this by extending each axiom
naturally to the domain of recurring allocation.
3.1 Historic Efficiency
We begin by defining a rank mapping function π : D → [n]m. Given some particular deterministic
allocation d ∈ D, πi(d) is the rank of the good player i receives under d. So if i receives their most
preferred good, πi(d) = |n|, and if they receive their second most preferred good, πi(d) = |n| − 1,
and so on. Using π, we define the windowed rank mapping function Π : Dk → [n]m×k. Consider
some sequence of allocations, H ∈ H. Then Πi(H) is the lexicographically sorted outcome vector
for player i. We use this to define historic efficiency. In particular, let F, G ∈ Hk be two sequences
of k allocations. Then define historic lexicographic dominance ( H) as:
F H G ⇔ Π(F) Π(G) (9)
Definition 14. A sequence of k allocations H ∈ Hk is weakly historically efficient if it is not
historic lexicographically dominated by any other sequence of k allocations.
Definition 15. A sequence of k allocations H ∈ Hk is historically efficient if it historic lexico-
graphically dominates every other sequence of k allocations.
Theorem 4. Historical efficiency implies weakly historical efficiency, but the converse does not
hold. Weakly historical efficiency is a necessary condition for historical efficiency, but not a suffi-
cient one. We omit the easy proofs.
3.2 Historic Envyfreeness
In general, we would like players who are doing poorly to be favored, relative to players who are
doing better than them. Informally, we want the following conditions to hold: (1) if two players
have the same welfare (under a given definition of welfare), then conventional envyfreeness ought to
hold (i.e. no player should prefer another’s allocation), and (2) if two players do not have the same
welfare, then the disadvantaged player should have some sort of edge over the advantaged one. We
define Φk to be the welfare of all players over k rounds, with Φk
i the welfare of player i over the
last k rounds. There are multiple meaningful forms of welfare - we consider average rank of good
received, median rank of good received, and minimum rank of good received to all be justifiable
functions for welfare. We will use mean rank of good received unless otherwise specified.
Intuitively, the weak form of historic envyfreeness is simply that the disadvantaged player is
strictly advantaged (without any regard to quantifiers). The strong form provides a lower bound to
the advantage, in terms of the ratio of welfare. Intuitively, this ensures that the more disadvantaged
a player is, the more they will be advantaged in a given round. To assist in clean definitions, we
define Ei = {k | ˆpik 0} as the set of goods in allocation P for which player i has a nonzero
probability share, with m0 the maximal value in Ei, or the least preferred good for which i has
a probability share. We let γk
i = {j | Φk
j Φk
i } to be the set of players who have outperformed
player i, and βk
i = {j | Φk
j = Φk
i } to be the set of players who have performed equivalently to player
i (including i). The remaining players are denoted by δk
i = {j | j ∈ βk
i ∧ j ∈ γk
i }. Using the strict
stochastic dominance relation defined in Equation (37), we write the axioms as follows.
23
25. Definition 16. We say an allocation is weakly k-historically envyfree if on round r for r ≥ k, for
every player i,
Pi sd( i) Pj, ∀j ∈ βk
i (10)
Pi ssd( i) Pj ∀j ∈ γk
i (11)
Definition 17. We say an allocation is k-historically α-envyfree for k 0 and α ≥ 1 if on round
r ≥ k, for every player i,
Pi sd( i) Pj, ∀j ∈ βk
i (12)
t
k=1 ˆpi
ik
t
k=1 ˆpi
jk
≥ α
Φ(Hj)
Φ(Hi)
t ∈ Ei {m0}, ∀j ∈ γk
i (13)
Notice that the strong form is parameterized in (α, k). The former reflects how much handicap a
player attains as they are more disadvantaged, while the latter reflects how much of history effects
a particular round. If a mechanism is ∞-historically α-envyfree, then we simply call it historically
α-envyfree.
Theorem 5. k-historical α-envyfreeness implies k-historical weak envyfreeness, but the converse
does not hold. Indeed, k-historical weak envyfreeness is a necessary condition for k-historical α-
envyfreeness, but it is not a sufficient one. We omit the easy proofs.
3.3 Historic Strategyproofness
There are two ways we can go about extending strategyproofness to account for multiple rounds.
We can either generalize the concept of expected utility to multiple rounds, and then use the
same stochastic dominance guarantees, or we may strengthen the requirements on sets of rounds,
culminating in a very strong guarantee over arbitrary windows. We do both, calling the former
aggregate strategyproofness, and the latter historic strategyproofness.
Aggregate strategyproofness will ensure that the outcome over some number of rounds cannot
be improved by lying, but may permit individual round tampering. We must first update our
notation. Denote by ˜ the set of preference profiles reported over a set of rounds. Then ˜r
is the
complete preference profile reported on round r, and ˜r
i is the preference profile for player i on
round r. We denote the preference profile over a set of rounds with a single misreport by player i
by ˜ |i ˜∗
i . That is, ˜ with player i’s preferences replaced by ˜∗
i .
Definition 18. Let R ∈ Rk
i be the random assignments for player i over k rounds. Then define
the multi-round utility matrix V k, by:
V k
ix =
k
r=1
rrx, ∀x ∈ G (14)
Informally, V k
ix is the expected utility player i will derive from good x, over k rounds, and V k
i is
the expected utility vector for player i over k rounds.
We are now prepared to define our extensions on regular strategyproofness. In this context, let
Pk
i (˜) = V k
i .
24
26. Definition 19. A mechanism is aggregate strategyproof if, for all preference profiles ˜:
Pk
i (˜) sd( i) Pk
i (˜ |i ˜∗
i ) (15)
Informally, this captures the idea that over a window of k rounds, false reporting on any set
of rounds will not improve the outcome in the end. It is in some sense quite permissive, as on
any particular round, false reporting may improve the outcome. If we are concerned with this
permissiveness, we can strengthen the axiom, as follows:
Definition 20. A mechanism is historically strategyproof if each round is individually strategyproof,
and for all preference profiles ˜:
Pn
i (˜) sd( i) Pn
i (˜ |i ˜∗
i ), ∀n = 1, . . . , k (16)
The first condition forces each round to be individually strategyproof: that means that no lie on
any single round can improve the outcome on that round for a given player. The second condition
ensures that no lie can improve a player’s outcome over the first n rounds, for all n ≤ k. Succinctly
put: no lie can improve a player’s outcome on a given round, or in aggregate up to and including
that round.
Theorem 6. Any historically strategyproof mechanism is also aggregate strategyproof, but the con-
verse does not necessarily hold. That is, aggregate strategyproofness is a necessary condition for
historical strategyproofness, but is not a sufficient one. We omit the easy proofs.
25
27. 4 Historical Probabilistic Serial
We have now updated the axioms in Chapter 3, and recognize that existing mechanisms do not
achieve these recurrent fairness properties. We diagram the process current mechanisms follow, and
propose an updated process for building historically fair mechanisms. Having identified the problem
and proposed the framework, we use it to create two mechanisms attaining historic envyfreeness, per
round ordinal efficiency, and per round weak strategyproofness. The fundamental idea underpinning
the modifications is that a mechanism ought to incorporate some form of feedback between rounds.
At present, mechanisms have a very regular, and linear process, diagrammed in Figure 3, below.
Figure 3: General mechanism framework for current mechanisms.
Until now, all approaches have begun by defining the mechanism (A) used to produce the
lottery (1), have then invoked the Birkhoff von Neumann theorem (B) to instantiate the lottery
(2), and have used some randomization (C) to arrive at a final allocation (3). If multiple rounds are
necessary, the process can be repeated as many times as necessary, but no feedback is present - in
particular, there is no means at present for incorporating the outcome represented by a particular
allocation from one round into the lottery for the next. We propose introducing the notion of
welfare over time to bridge the gap. Using welfare, the control flow (with feedback) would instead
be a cycle, as diagrammed in Figure 4, below.
Figure 4: General mechanism framework with feedback between rounds.
26
28. The fundamental change is that allocation is no longer a stateless process. Welfare, now (1),
is a tangible metric for a player’s state at a given point in time (i.e. Φk), which is utilized by
the mechanism and may influence the resulting lottery. Any mechanism defined in terms of this
framework has the possibility for feedback, and so has a hope at attaining historical notions of
efficiency, envyfreeness, and strategyproofness.
The framework we have defined ties together the steps that mechanism designers have historically
handled themselves. In particular, a mechanism designer using our framework need only provide a
definition for Φk to define (D), and their mechanism (accepting as input P, G, , and Φk) to define
(A). The framework implements all other steps. The framework decomposes the emitted lottery
via Algorithm 2, and converts it to a deterministic allocation via sampling of a random variable.
The framework then converts the deterministic allocation to welfare for feedback purposes via Φk,
and fed back to the designer’s mechanism, as necessary for recurring allocation. To illustrate the
process, we propose two such mechanisms, and identify each of the components in terms of our
framework. Both are built on the probabilistic serial mechanism of [4], mentioned in Section 2.6.1
and outlined in Algorithm 1. We explain how probabilistic serial may be updated to be historically
aware in Section 4.1, below.
4.1 Probabilistic Serial, Expanded
Recall the probabilistic serial mechanism proposed by Bogomolnaia and Moulin in [4]. Each good
is viewed as an infinitely divisible set of probability shares of size 1, and each player is endowed
with some eating speed. Players consume shares of their most preferred available good according to
their eating speed, until all shares of all goods are completely consumed. The resulting distribution
of probability is used to instantiate a lottery, and produce a deterministic assignment. In [4], the
eating speed of each player is taken to be 1. We provide the complete algorithm (with arbitrary
eating speed) in Chapter 8. We propose two parameterized extensions to the probabilistic serial
mechanism via the introduction of historic welfare functions, which we use to provide a feedback
mechanism for per-round adjustment of eating speeds.
Consider the set of possible outcomes for player i over k rounds, Hk
i . Then let Φk
i : Hk
i → R
be the welfare function of player i, and let Φk = Φk
1, Φk
2, . . . , Φk
m be the set of historic welfares
over all players for the preceding k rounds. Denote by ei(t) the amount eaten by player i at time
t. Finally, let ωi : R+ → R+ be the eating speed function of each player, given by:
ωi(t) =
g(Φk), ei(t) 1
0, else
(17)
Let g be a strictly decreasing function9, and let the minimal eating speed be given by
ωm(t) = min
i∈P
ωi(t). (18)
The round will continue until all players have eaten probability exactly one. The total time
elapsed will thus be that time t , such that:
t
0
ωm(x)dx = 1 (19)
9
The requirement that g be strictly decreasing captures the intuitive notion that handicaps (though not necessarily
linear) work in the expected fashion: worse performance in previous rounds yields better performance in future rounds.
27
29. If the round runs until time t , then all players will eat probability exactly one. That is, as
defined,
t
0
ωi(x)dx = 1 (20)
Once the eating phase has ended, the resulting lottery is instantiated, yielding our final deter-
ministic allocation. Notice that if we set g(x) = 1 (thereby ignoring the requirement that g be
strictly decreasing), then the conventional PS obtains.
We now turn to defining g(Φk), as this is the linchpin for the entire mechanism. We have a
choice: we may either focus on a definition based on an individual’s welfare, e.g. gi(Φk) = h(Φk
i ),
for an appropriately chosen h, or we may focus on a definition based on societal welfare, e.g.
gi(Φk) = hi(Φk), for an appropriately chosen h. The fundamental difference between the two is in
the amount of information we choose to expose when determining eating speed - we may either look
at a player independently of others, with the intent to ensure that poorly performing players are
advantaged in future rounds, or we may look at a player in the context of others, with the intent
to ensure that poorly performing players relative to others should be advantaged in future rounds.
We do both, calling the former definition the individually historical probabilistic serial mechanism,
and the latter the societally historical probabilistic serial mechanism. Notice that once we define
g, each mechanism (filling role (A) in Figure 4, above) is a function mapping (P, G, , Φk) to a
lottery in A. We have already explicitly defined Φk as the average rank of good received over the
preceding k rounds, thus filling in (E) in that same diagram.
Both follow the same fundamental process, which we are now prepared to diagram. In particular,
in Figure 5, we show how the welfare function Φk (E) is mapped to eating speeds via g (A), which are
then used to run the probabilistic serial algorithm described above. In this figure, the mechanism
defined as arc (A) in Figure 4 is expanded into the welfare mapping function (A), producing the
eating speed (2) which is fed into the probabilistic serial mechanism (B). These steps together
comprise the “mechanism” required for our framework.
Figure 5: IHPS and SHPS in the context of our general framework.
28
30. 4.2 Individually Historical Probabilistic Serial
Definition 21. The individually historical probabilistic serial (IHPS) assignment attains when for
player i:
g(Φk
) =
1
Φk
i
. (21)
This captures the fundamental principle that poor performance over time should increase a
player’s luck, and excellent performance over time should decrease a player’s luck.
Example 8. We provide an example of running the individually historical probabilistic serial mech-
anism. Consider the following scenario, letting k = 1. For the first round, we take every player to
have previously attained their least favorite good (and thus have minimal welfare).
P = {1, 2, 3, 4}
1 A B C D
2 A B C D
G = {A, B, C, D}
3 B C D A
4 D C B A
On the first round, r = 1, all players have the same welfare, so they have the same eating speed,
namely 1. The resulting lottery is thus:
A B C D
1 1/2 1/6 1/3 0
2 1/2 1/6 1/3 0
3 0 2/3 1/3 0
4 0 0 0 1
The lottery is instantiated, via the following distribution:
0.1667
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
+ 0.1667
0 1 0 0
1 0 0 0
0 0 1 0
0 0 0 1
+ 0.3333
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
+
0.3333
0 0 1 0
1 0 0 0
0 1 0 0
0 0 0 1
The first allocation is chosen, namely:
Player Good Welfare
1 A 4
2 B 3
3 C 3
4 D 4
The resulting values for g are thus:
29
31. Player gi(Φk
i )
1 1/4
2 1/3
3 1/3
4 1/4
Running the mechanism for round 2, the lottery is then:
A B C D
1 3/7 9/77 5/11 0
2 4/7 12/77 3/11 0
3 0 8/11 3/11 0
4 0 0 0 1
This lottery is instantiated, yielding the following distribution over deterministic assignments:
0.1558
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
+ 0.1169
0 1 0 0
1 0 0 0
0 0 1 0
0 0 0 1
+ 0.2727
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
+
0.4545
0 0 1 0
1 0 0 0
0 1 0 0
0 0 0 1
Notice that the single most probable outcome (occurring with 45% chance) is now:
Player Good Welfare
1 C 2
2 A 4
3 B 4
4 D 4
Of particular note, those players who did not achieve their favorite good are more likely to get
a better good on this round.
Theorem 7. The individually historical probabilistic serial mechanism is per-round ordinally ef-
ficient, per-round weakly strategyproof, and k-historically α-envyfree for any choice of k and for
α = 1.
Theorem 8. The individually historical probabilistic serial mechanism is not k-historically α-
envyfree for α = 1.
4.3 Societally Historical Probabilistic Serial
Definition 22. The societally historical probabilistic serial (SHPS) attains when for player i:
gi(Φk
) =
α
Φk
i
max
j∈γk
i
gj(Φk
)Φk
j , |γi| 0
1, else
(22)
30
32. The recursive definition suggests the relationship between players, where players are at the
same recursive depth if they have the same welfare, above all players who have performed better
than them (historically), and below all players who have performed worse than them (historically).
Eating speeds decrease with increasing recursive depth, by a ratio fixed by α.
Example 9. We provide an example of running the societally historical probabilistic serial mech-
anism. Consider the following scenario, letting k = 1, and α = 3.
P = {1, 2, 3, 4}
1 A B C D
2 A B C D
G = {A, B, C, D}
3 B C D A
4 D C B A
On the first round, r = 1, all players have the same welfare, so they have the same eating speed,
namely 1. The resulting allocation is thus:
A B C D
1 1/2 1/6 1/3 0
2 1/2 1/6 1/3 0
3 0 2/3 1/3 0
4 0 0 0 1
The lottery is instantiated as in Example 8, and the same assignments are chosen.
Player Good Welfare
1 A 4
2 B 3
3 C 3
4 D 4
The resulting welfare graph (implicit to the definition of g) is presented in Figure 6, below.
Figure 6: Envy graph (based on welfare). Players in the same level (having the same welfare)
are ensured to be envyfree with respect to each other, and advantaged relative to every player
“beneath” them.
31
33. Intuitively, players B and C would be envious of players A and D in this situation if the lottery
were the same as in the preceding round, because A and D outperformed B and C last time. The
possible envy is captured via our definition of the sets βi and γi, presented below.
Player βk
i γk
i δk
i
1 4 2,3
2 3 1,4
3 2 1,4
4 1 2,3
The resulting values for g are thus:
Player gi(Φk)
1 1
2 4
3 4
4 1
Running the mechanism for round 2, the lottery is then:
A B C D
1 1/5 1/45 7/9 0
2 4/5 4/45 1/9 0
3 0 8/9 1/9 0
4 0 0 0 1
This lottery is instantiated, and the distribution samples from is:
0.0889
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
+ 0.0222
0 1 0 0
1 0 0 0
0 0 1 0
0 0 0 1
+ 0.1111
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
+
0.7778
0 0 1 0
1 0 0 0
0 1 0 0
0 0 0 1
Note that the most probable assignment is the same as in Example 8, but that it is even more
likely, occurring with probability 78%:
Player Good Welfare
1 C 2
2 A 4
3 B 4
4 D 4
32
34. Of particular note, both of the disadvantaged players are now quite likely to receive goods they
prefer, relative to the players who outperformed them (as was our goal). As the welfares fluctuate,
varying eating speeds automatically adjust handicaps to try and be as fair as possible across rounds.
Theorem 9. The societally historical probabilistic serial mechanism is per-round ordinally efficient,
per-round weakly strategyproof, and k-historically α-envyfree for any choice of (k, α).
Unfortunately, neither of these two modified mechanisms is historically efficient.
Theorem 10. Both individually historical probabilistic serial and societally historical probabilistic
serial may produce allocations which are not even weakly historically efficient.
Corollary 1. Per-round ex post efficiency is a necessary, but not a sufficient condition for weak
historic efficiency.
33
35. 5 Future Work
This work serves as a first foray into the space of recurrent fair allocation. We have selected the
simplest possible model (where m = n, and the sets P and G are fixed over time), and even so our
analysis raises questions. We have shown a mechanism which is ∞-historically α-envyfree, for any
choice of α. However, it remains open whether there are any mechanisms which are historically
efficient, or either historically strategyproof or aggregate strategyproof. The following conjectures
are extremely interesting areas for future work (and are statements reflecting Table 3).
Conjecture 1. There exists no mechanism which is per round ex ante efficient, k-historically
α-envyfree, and per round weakly strategyproof.
Conjecture 2. There exists no mechanism which is historically efficient, k-historically α-envyfree,
and per round weakly strategyproof.
Conjecture 3. There exists no mechanism which is even just historically efficient and k-historically
α-envyfree.
Conjecture 4. There exists some mechanism which is historically efficient, per round envyfree,
and per round weakly strategyproof.
Conjecture 5. There exists some mechanism which is per round ex ante efficient and k-historically
α-envyfree.
Beyond this model, we think extending the available mechanisms in this space to be an extremely
interesting challenge. In particular, we would like to realize mechanisms for the attainable recurrent
axioms which do not require that m = n, and indeed for which P and G may change each round.
Adding per-round minimum and maximum quotas, and aggregate minimum and maximum quotas
is also an interesting area for research.
Finally, we believe there is further work to be done in the analysis of the performance of recurrent
allocation mechanisms. No one has yet examined the expected utility over multiple rounds of
popular mechanisms, nor their variances.
Conjecture 6. The historical mechanisms proposed (IHPS and SHPS) effectively reduce the vari-
ance of the utility in each round, relative to that of the naive probabilistic serial.
34
36. 6 Conclusion
In this thesis, we offered a broad treatment of the state of fair allocation. We discussed options for
representing preferences and the available mechanisms for each of the three accepted choices. In
particular, we mentioned the leximin mechanism for the dichotomous preference profile setting, the
probabilistic serial mechanism for the ordinal preference profile setting, and the market clearing
mechanism for the cardinal preference profile setting. We examined the set of axioms currently used
to classify allocations and mechanisms, and identified a deficiency with regards to recurrent allo-
cation. We repaired the axioms to account for recurrent allocation, proposed a general framework
for creating historically fair mechanisms, and proposed a new mechanism built on that framework
which is strictly superior to the probabilistic serial mechanism. We conjecture the existence of two
new mechanisms, and the impossibility of any mechanisms attaining particular combinations of
axioms.
35
37. 7 Appendix: Proofs
Theorem 7. The individually historical probabilistic serial mechanism is per-round ordinally ef-
ficient, per-round weakly strategyproof, and k-historically α-envyfree for any choice of k and for
α = 1.
Proof.
Ordinally efficient:
It is proven in [4] that PS is ordinally efficient for any profile of eating speeds. Since we do
not change the mechanism, but instead adjust the eating speeds, each round is individually
ordinally efficient.
Weakly strategyproof:
It is proven in [4] that PS is weakly strategyproof, with the assumption that a round lasts for
time 1. The proof is easily updated to account for variable time, and applies in our setting.
k-historically 1-envyfree:
By construction, for each player i, all players in βk
i will have the same eating speed. Among
these players, IHPS is simply the conventional PS mechanism, and so the resulting allocation
is envyfree, among βk
i . This means that for all players j ∈ βk
i , in the resulting allocation P,
Pi sd( i) Pj, and so (12) holds.
We then turn to showing that (13) holds. Consider some player j ∈ γk
i , and let x = Φk(Hk
i )
and y = Φk(Hk
j ). Then by construction, ωi = g(Φk(Hk
i )) = 1
x and ωj = g(Φk(Hk
j )) = 1
y .
Substituting into (13), we have that:
t
k=1 ˆpi
ik
t
k=1 ˆpi
jk
≥
y
x
t ∈ Ei {m0}, ∀j ∈ γk
i (23)
Consider some k ∈ Ei {m0}. Then since good k is less preferred than all goods n, for n k,
i will eat k only after all preceding k − 1 goods are exhausted. Moreover, i will continue to
eat k until either i has eaten 1 probability total, or k is exhausted. Since k = m, the former
cannot occur.
So, k is exhausted at time t∗. In that case, player i has eaten at rate 1
x for total time t∗,
accumulating t∗
x total probability shares. Similarly, player j has eaten at rate 1
y , accumulating
t∗
y total probability shares. Thus, we have that
t
k=1 ˆpi
ik
t
k=1 ˆpi
jk
=
t∗
x
t∗
y
=
y
x
≥ 1 ·
y
x
It follows that for all k ∈ Ei {m0}, we have that (13) is upheld.
Thus, IHPS is per-round ordinally efficient, per-round weakly strategyproof, and k-historically
1-envyfree for any choice of k.
36
38. Theorem 8. The individually historical probabilistic serial mechanism is not k-historically α-
envyfree for α = 1.
Proof. As argued above, for some k ∈ Ei {m0}, good k is exhausted at time t∗. At this time, the
ratio of probability shares consumed is given by:
t
k=1 ˆpi
ik
t
k=1 ˆpi
jk
=
t∗g(x)
t∗g(y)
=
g(x)
g(y)
For this to be historically α-envyfree, we need
≥ α
y
x
Rearranging, we need that, for all ∆ 0,
g(x) ≥ α
g(x + ∆)(x + ∆)
x
Let ∆ be arbitrarily chosen to be 1. Then
g(x) ≥ α
g(x + 1)(x + 1)
x
Since this condition holds for all x, y, we can expand the recurrence, by
≥ α
αg(x+2)(x+2)
x+1 (x + 1)
x
≥ α2 g(x + 2)(x + 2)
x
Inductively, the scalar grows to the power ∆, as we continue to expand. Since we require the
condition to hold for any two x, y with x = y, we have that α = α2 = . . . = α∆. The only number
satisfying this property is 1, and so there is no α = 1, such that IHPS is historically α-envyfree,
for any choice of g.
Theorem 9. The societally historical probabilistic serial mechanism is per-round ordinally efficient,
per-round weakly strategyproof, and k-historically α-envyfree for any choice of (k, α).
Proof. Let k ≥ 1 and α ≥ 1 be fixed.
Consider two players, i, j ∈ P. One of the following three conditions holds. Either Φk
i Φk
j ,
Φk
i = Φk
j , or Φk
i Φk
j . No restrictions govern the former case, and so we ignore it.
In the second case, j ∈ βk
i , and so we must show that Pi sd( i) Pj. Observe that i and j have
the same eating speed by construction, since γk
i = γk
j . It follows that SHPS is simply the regular
PS mechanism, with respect to these two players, and therefore envyfree, so (12) holds.
37
39. In the third case, j ∈ γk
i , and so we must show that (13) holds. Since j ∈ γk
i , we have by
construction that:
gi(Φk
) ≥
α
Φk
i
gj(Φk
)Φk
j . (24)
Rearranging (24), we get:
gi(Φk)
gj(Φk)
≥ α
Φk
j
Φk
i
(25)
We can then calculate the probability share eaten by each player at a given time. In particular,
if we take the time that player i finished eating all goods in Ei {m0} to be t∗, then we have that
for all times t ≤ t∗,
t
0 ωi(x)dx
t
0 ωj(x)dx
≥ α
Φk
j
Φk
i
(26)
This follows from our guarantee that i eats faster than j, and that neither finishes (i.e. reaches
probability 1 consumed) during this interval. We may thus write this in terms of probabilities
eaten, as:
t
k=1 ˆpi
ik
t
k=1 ˆpi
jk
≥ α
Φk
j
Φk
i
∀t ∈ Ei {m0} (27)
But this is exactly condition (13). As to efficiency and strategyproofness, the exact same
arguments from Theorem 7 apply to this context as well. Therefore, SHPS is k-historically α-
envyfree for any round count k ≥ 1, and α ≥ 1, as well as per-round ordinally efficient and weakly
strategyproof.
Theorem 10. Neither individually historical probabilistic serial nor societally historical probabilis-
tic serial is weakly historically efficient.
Proof. We offer a counterexample: a case for each mechanism where the allocations over two rounds
are weakly historically inefficient. We take k = 1, α = 1, and we give the setting below.
P = {1, 2, 3, 4, 5}
1 A B C D E
2 A B C D E
G = {A, B, C, D, E}
3 A B C D E
4 D A C E B
5 A D C B E
The resulting lottery is:
A B C D E
1 1/4 1/5 1/3 1/60 1/5
2 1/4 1/5 1/3 1/60 1/5
3 1/4 1/5 1/3 1/60 1/5
4 0 1/5 0 3/5 1/5
5 1/4 1/5 0 7/20 1/5
38
42. Player Good Round 1 Good Round 2 Lexicographically Sorted Welfare
1 E A 15
2 C B 34
3 A C 35
4 D E 25
5 B D 24
Critically, observe that this alternative allocation lexicographically dominates the one which
occurred, and is indeed viable. Since both IHPS and SHPS may produce allocations which are
historically dominated, neither is even weakly historically efficient.
Corollary 1. Per-round ex post efficiency is a necessary, but not a sufficient condition for weak
historic efficiency.
Proof. We have shown with Theorem 10 that ex post efficiency is not a sufficient condition for
weak historic efficiency. In particular, since both the proposed mechanisms are per-round ex post
efficient, and not weakly historically efficient, the property is itself insufficient. We now show that
while not sufficient, per round ex post efficient is a necessary condition for any weakly historically
efficient mechanism. In particular, observe that if a mechanism is not per-round ex post efficient,
it may produce an allocation on a round which is not Pareto efficient. In that case, some set of
players may trade allocations, and all benefit. By definition, the resulting allocation after their
trades will lexicographically dominate the original allocation, since all players will be at least as
well off on all rounds, and some players will be strictly better off on the inefficient round.
41
43. 8 Appendix: Algorithms
Algorithm 1 The seminal probabilistic serial algorithm proposed in [4]. As input, we take the set
of players, P, the set of goods, G, the preference profiles of all players over all goods, , and the
eating speed functions, ω (assuming constant eating speeds). The result is a probability distribution
over deterministic allocations, calculated in O(n3√
n).
1: function PS(P, G, , ω)
2: E ← P E := the set of eating players
3: A ← G A := the set of unexhausted goods
4: t ← 0
5: while E = ∅ ∧ A = ∅ do Continue as long as a player can still eat
6: tg ← min
g∈A
E(g) E(g) := time to exhaust g assuming that is next event
7: tp ← min
p∈E
D(p) D(p) := time for p to eat 1, assuming that is next event
8: tn ← min(tg, tp) tn := the time of the next event
9: δ ← tn − t δ := size of next time step
10: step(E , ω, δ)
11: updateGoods(A )
12: updatePlayers(E )
13: t ← t + δ Step forward in time
14: end while
15: R ← allocation(P, G) rij = i.probEaten(j), i ∈ P, j ∈ G
16: return bvnDecomposition(R)
17: end function
1: function step(E , ω, δ)
2: for p ∈ E do
3: p.eat(ωp · δ)
4: end for
5: end function
1: function updateGoods(A )
2: for g ∈ A do
3: if g.consumed() = 1 then Check if g is exhausted
4: A ← A {g}
5: end if
6: end for
7: end function
42
44. 1: function updatePlayers(E )
2: for p ∈ E do
3: if p.consumed() = 1 then Check if p is done eating
4: E ← E {p}
5: else
6: p.updateEating() Set p to eat best available good, or remove from E if none left
7: end if
8: end for
9: end function
Algorithm 2 Algorithm for decomposing an arbitrary n × n bistochastic matrix as a convex
combination of at most (n−1)2 +1 permutation matrices. The loop body is executed O(n2) times,
and the perfect matching can be computed in O(n
√
n) via the Hopcroft-Karp perfect maximum
matching algorithm of [10]. Overall runtime is O(n3√
n).
1: function bvnDecomposition(R) R := n × n bistochastic matrix
2: C ← ∅ C := convex combination over permutation matrices
3: while R = 0 do
4: V ← [2n] Vertices are rows and columns
5: E ← {eij | ri(j−n) 0} Edges from row to column for nonzero entries
6: G ← (E, V ) G is a bipartite graph
7: M ← matching(G) M := perfect matching on G (via e.g. Hopcroft-Karp algorithm)
8: P ←fromMatching(M) Permutation matrix, pij =
1 ei(j+n) ∈ M
0 else
9: λ ← min
eij∈M
Ri(j−n)
10: C ← C ∪ {{λ, P}}
11: R ← R − λP
12: end while
13: return C
14: end function
43
45. 9 Appendix: Notation
We begin by presenting our notation, which we will make use of consistently throughout. We denote
the set of players P, and denote the number of players |P| = m. We denote the set of goods G, and
denote the number of goods |G| = n. We denote an ordinal preference profile for all players P over
the goods G by the symbol . We denote the preference profile of a particular player p ∈ P over
the goods G by the symbol p.
A deterministic assignment of goods to players is denoted by a matrix of dimension m × n.
Letting d represent a deterministic assignment, a particular element dij is 1 if and only if player i
receives good j, and 0 if and only if player i does not receive good j. We denote by D the set of
deterministic assignment matrices. In the case where m = n, as is the case in our setting, D is the
set of all binary bistochastic m × m matrices. That is, for any deterministic assignment d, we have
the following:
j
dij = 1, ∀i ∈ P (28)
i
dij = 1, ∀j ∈ G (29)
dij ∈ {0, 1} ∀i ∈ P, ∀j ∈ G (30)
These have straightforward interpretations. That is, (28) stipulates that each player is allocated
exactly one good, and (29) stipulates that each good is allocated exactly once.
A random assignment of goods to players is also denoted by a matrix of dimension m × n.
Letting r represent a random assignment, a particular element rij represents the probability that
player i receives good j. We denote by A the set of random assignments matrices. In our setting, A
is the set of all bistochastic m×n matrices r, such that r may be written as the convex combination
of deterministic allocations. That is, for any random allocation r, we have the following:
j
rij = 1, ∀i ∈ P (31)
i
rij = 1, ∀j ∈ G (32)
r =
k
αkdk,
k
αk = 1, dk ∈ D, αk 0 (33)
These also have easy translations. That is, (31) stipulates that each player’s chances of being
assigned a particular good define a probability distribution, ensuring that each player is allocated
some good. Similarly, (32) stipulates that each good’s chances of being assigned a particular player
define a probability distribution, ensuring that each good is allocated some player. These two
conditions ensure that every player should be allocated some good. That the resulting allocation be
feasible is captured by (33), which requires that the random allocation be a probability distribution
over a set of (feasible) deterministic allocations.
We will frequently be concerned with permuted row and column vectors of assignment matrices.
In particular, given some matrix M, we will denote by mi the ith row-vector of that matrix. We will
use mi to denote the ascending row-sorted ith vector of matrix M, mij to denote the jth element
44
46. of the ascending row-sorted ith vector of matrix M, and M to denote the ascending row-sorted
matrix M. We will use ˆmi to denote the row-sorted ith vector of matrix M, sorted according to a
particular i, and ˆM to denote the row-sorted matrix M, sorted according to a particular . We
will also use ˆmij to denote the jth element of the ith row-vector of ˆM. This notation is collected
in Table 4, below.
Table 4: Modifying symbols.
Symbol Interpretation
Preference Sorted ˆx Sort by preference profile (ascending).
ˆM Matrix row-sorted by profile.
ˆmi Player i’s vector, sorted by i’s preferences.
ˆmi
j Player j’s vector, sorted by i’s preferences.
ˆmij Player i’s allocation for their jth favorite good.
ˆmk
ij Player i’s allocation for player k’s jth favorite good.
Lexicographically Sorted x Sort numerically (ascending).
M M, row-sorted
mi ith sorted row of M.
mij jth element of ith sorted row of M.
When comparing random allocations, we will make heavy use of the stochastic dominance rela-
tion defined in [4].
Definition 23. The stochastic dominance of random allocation Q by random allocation P for
player i is defined as:
Pi sd( i) Qi ⇔
t
k=1
ˆpi
ik ≥
t
k=1
ˆqi
ik, ∀t = 1 . . . |ˆpi| (34)
An allocation P stochastically dominates another allocation Q from the perspective of player
i if i is at least as likely to receive their favorite k goods under P as under Q, for all values of k.
This can be loosely understood to mean that Q is no better than P for player i. This suggests the
full stochastic dominance relation:
Definition 24. The stochastic dominance of random allocation Q by random allocation P over
some set of players X is defined as:
P sd( , X) Q ⇔ Pi sd( i) Qi, ∀i ∈ X (35)
When X = P, we will simply write P sd( ) Q for brevity.
If every player is no better off under Q than under P, then P stochastically dominates Q. We
introduce a stronger notion, called strict stochastic dominance. In particular,
Definition 25. The strict stochastic dominance of random allocation Q by random allocation P
for player i is defined as:
Pi ssd( i) Qi ⇔
t
k=1
ˆpi
ik
t
k=1
ˆqi
ik, ∀t = 1 . . . |ˆpi| − 1 (36)
45
47. Definition 26. The strict stochastic dominance of random allocation Q by random allocation P
for some set of players X is defined as:
P ssd( , X) Q ⇔ Pi ssd( i) Qi, ∀i ∈ X (37)
Again, when X = P we will write P ssd( ) Q for brevity.
If we instead sort numerically (as opposed to by preference) and take regular stochastic domi-
nance as the model, we get the Lorenz dominance relation. In particular, we define the lexicographic
dominance (or Lorenz dominance) relation for a given player i as:
Definition 27. The lexicographic dominance of random allocation Q by random allocation P for
player i is defined as:
P i Q ⇔
t
k=1
pik ≥
t
k=1
qik, ∀t = 1 . . . |pi| (38)
Row i of a matrix P lexicographically dominates row i of another matrix Q if the partial sum of
the first k smallest elements of P is at least that of the partial sum of the first k smallest elements
of Q, for all k. We may generalize this exactly as for stochastic dominance, to define the full
dominance relation.
Definition 28. The lexicographic dominance of random allocation Q by random allocation P is
defined as:
P Q ⇔ P i Q, ∀i ∈ P (39)
We denote a sequence of k deterministic assignments by a set of k m×n matrices. We denote by
Hk the set of all such sequences. That is, Hk = Dk. Given some sequence of assignments H ∈ Hk,
a particular player’s assignment history is denoted by Hi, and is intuitively the set of rows from the
individual assignments which concern that player. Correspondingly, Hi is an k × n matrix, with
the element at row r and column c 1 if and only if player i received good c on round r, and 0 else.
The set of all possible His is denoted Hk
i .
The random assignment parallel is a sequence of k random assignments, also represented by a
set of k m × n matrices. We denote by Rk the set of all such sequences. That is, Rk = Ak. Given
some sequence of assignments R ∈ Rk, a particular player’s random allocation history is denoted
by Ri, and is defined as in the deterministic case. Correspondingly, Ri is a k × n matrix, with the
element at row r and column c the probability that player i receives good c on round r. The set of
all possible Ris is denoted Rk
i .
46
48. 10 Glossary
aggregate strategyproof A player cannot do better over the course of all considered rounds by
lying in any subset of the rounds. 25
Birkhoff von Neumann theorem Any n×n bistochastic matrix is expressible as a convex com-
bination of permutation matrices. 12, 26
cardinal preference profile VNM utilities of goods. 11, 35
convex combination Linear combination of terms. 12, 13, 43, 44
dichotomous preference profile Binary preference profile. 10, 17, 35
envyfree Each player’s lottery ticket is the best, from their perspective. 14, 17–20, 34
equal treatment of equals A weaker form of envyfreeness, stipulating that players with identical
preference profiles must be treated identically. 18, 20
ex ante efficient Maximizing expected societal welfare. 14, 19, 20, 34
ex post efficient Random allocation with efficient support. 13, 14, 17, 19, 20, 41
framework Algorithmic scaffolding for complete process of assigning goods to players. 6, 8, 26–28
historic lexicographic dominance Sorted outcomes over window of rounds at least weakly pre-
ferred by every player. 23
historically efficient This allocation is historically optimal (as good as any other). 23, 34
historically strategyproof A player cannot lie on any round (or any subset of rounds) to improve
the outcome on that round. 25
individually historical probabilistic serial PS, modified by a player’s individual welfare over
preceding rounds. 28–30, 36–38
instantiating Converting random lottery to weighted choice of deterministic assignments. 12
k-historically envyfree Based on preceding k rounds, envyfree among players who did equiva-
lently well, and players do better than those who outperformed them. 24
k-historically α-envyfree Based on preceding k rounds, envyfree among players who did equiva-
lently well, and players do better than those who outperformed them by a function determined
by α. 24, 30, 33, 34, 36, 37
lottery Probability each player has of getting each good on a particular round. 12
ordinal preference profile Ranked preference profile. 11, 35, 44
ordinally efficient Random allocation with very efficient support. 13, 20, 30, 33, 36, 37
Pareto efficient No waste. 13
47
49. probabilistic serial Players eat probability at certain speeds according to preferences to define
the lottery. 18, 19, 27, 28, 34, 35, 42
random serial dictatorship Players choose in randomized order. 17
societally historical probabilistic serial PS, modified by a player’s individual welfare relative
to societal welfare over preceding rounds. 28, 30, 37, 38
stochastic dominance Partial sum dominance. 13, 14, 23, 24, 45, 46
strategyproof A player’s optimal strategy is to tell the truth. 16, 17, 20, 25
VNM utilities von Neumann Morgenstern utilities. 10
weakly envyfree No other player’s lottery ticket is better. 14, 17, 20
weakly historically efficient No alternative sequence of allocations historically lexicographically
dominates this one. 23, 33, 38
weakly strategyproof A player cannot do strictly better by lying. 16, 18, 30, 33, 34, 36, 37
48
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