# Fair and consistent prize allocation in competitions

Bas J. Dietzenbacher  and  Aleksei Y. Kondratev

###### Abstract.

Given the final ranking of a competition, how should the total prize endowment
be allocated among the competitors? We study consistent prize allocation rules
satisfying elementary solidarity and fairness principles. In particular, we
axiomatically characterize several families of rules satisfying anonymity,
order preservation, and endowment monotonicity, which all fall between the
Equal Division rule and the Winner-Takes-All rule. Specific characterizations
of rules and subfamilies are directly obtained. To our knowledge, this paper
is the first that develops and motivates prize allocation rules for rank-order
competitions directly from axioms.

_Keywords_ : fair allocation, rank-order tournament, prize structure,
tournament design, axiomatic analysis, consistency

<EMAIL_ADDRESS>— Department of Quantitative
Economics, Maastricht University, Maastricht, The Netherlands —
http://orcid.org/0000-0002-8023-0273
<EMAIL_ADDRESS>— corresponding author — HSE University, International
Laboratory of Game Theory and Decision Making, Russia; Institute for Regional
Economic Studies RAS, St. Petersburg, Russia —
http://orcid.org/0000-0002-8424-8198
We are grateful to Alexander Nesterov, Egor Ianovski, Hervé Moulin, and other
colleagues from the International Laboratory of Game Theory and Decision
Making for helpful comments. Special thanks to William Thomson (University of
Rochester) for detailed comments and suggestions. We thank David Connolly (HSE
University) for help with language editing.

##  1\. Introduction

Innovation and crowdsourcing competitions, sales competitions in companies and
sporting events often take the form of rank-order tournaments (Kalra and Shi,,
2001; Szymanski,, 2003; Terwiesch and Xu,, 2008; Archak and Sundararajan,,
2009). Each such tournament is held once, and the participants know the rules
and the prize structure in advance. The absolute result of a participant can
be determined by, for example, the volume of sales in a sales competition, the
number of strokes in golf, or the time of elimination from a poker tournament.
However, a feature of the ranking format is that participants receive prizes
according to their relative results, while tournament organizers are free to
choose the prize structure.

Examples of prize structures are presented in Table 1. The PGA TOUR conducts
many regular golf tournaments with different prize endowments, but the same
rules for distributing prizes. Table 1 shows examples of two such tournaments
with a prize endowment of $9.3 million (The Genesis Invitational) and $6.6
million (Safeway Open). The winner of the tournament receives 18% of the total
prize endowment, the runner-up 10.9%, and so on. WCOOP Poker Tournament has a
comparable prize endowment but follows a different prize-money distribution.
What exactly are the similarities and differences of various prize structures?
What general principles can tournament organizers follow when choosing a prize
structure?

Table 1. Examples of single rank-order tournaments

(a) WCOOP 2019 Main Event poker tournament
---
Position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Top 10 | Total
Prize | 1,666 | 1,188 | 847 | 603 | 430 | 307 | 219 | 156 | 111 | 79 | 5,605 | 11,180
Share | 14.9 | 10.6 | 7.57 | 5.40 | 3.85 | 2.74 | 1.96 | 1.39 | 0.99 | 0.71 | 50.1 | 100

(b) PGA TOUR 2019/20 golf tournaments
---
Position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Top 10 | Total
Genesis | 1,674 | 1,014 | 642 | 456 | 381 | 337 | 314 | 291 | 272 | 253 | 5,633 | 9,300
Safeway | 1,188 | 719 | 455 | 323 | 271 | 239 | 223 | 206 | 193 | 180 | 3,998 | 6,600
Share | 18.0 | 10.9 | 6.9 | 4.9 | 4.1 | 3.63 | 3.38 | 3.13 | 2.93 | 2.73 | 60.6 | 100

_Notes_ : the prize endowment, the total prize for top 10 positions, and the
prize for each position from 1 to 10 are given in thousands of dollars. The
shares are given in percentages of the total prize endowment.

This paper examines prize structures in terms of fairness principles. Since
the basic principles of justice must be universal and clear, they must also be
found in particular examples of tournaments and competitions. Consistent with
the principle of _anonymity_ , is that the participants’ rewards depend only
on their position in the competition. That is, when the organizers award a
prize for a position regardless of which participant takes that position.
According to the principle of _order preservation_ , a higher position does
not correspond to a lower reward. This creates the right incentives for
participants, as the efforts made during preparation for and participation in
the tournament help the competitor to end up higher in the final ranking and
receive a more valuable reward. Thirdly, _endowment monotonicity_ is satisfied
when an increase in the total prize fund does not decrease the reward for any
position. This creates uniform incentives for all participants, as both
leaders and outsiders are interested in increasing the total prize endowment
of the tournament.111E.g., in some poker tournaments, players can re-buy and
add-on chips, which increases the total prize money. Also, during some
e-sports tournaments, viewers and sponsors can add to the total prize money.

To distribute prizes, the organizers need to know three things: the list of
the participants, the results of the competition, and the size of the prize
endowment. For each such triple, the _prize allocation rule_ used should
uniquely distribute the prize endowment among the participants. We call a rule
for the distribution of prizes _‘fair’_ if it satisfies the principles of
anonymity, order preservation, and endowment monotonicity. Anonymity and order
preservation are standard principles for single rank-order tournaments; see
the literature overview in subsection 1.1. Endowment monotonicity is a
standard principle in fair allocation literature; we refer to resource
monotonicity in the book by Moulin, (2003). These three principles of justice
are so undemanding that even together they do not exclude any visible class of
prize structures.

The universal principle of _consistency_ is often applied in fair allocation
problems (Balinski and Young,, 1982; Thomson,, 2012). We apply this principle
to the prize allocation problem as follows. Suppose that the participants have
some ranking in a competition and receive the corresponding prizes. Then some
participants leave with their prizes. The remaining participants decide to
redistribute the remaining prize endowment, taking into account the modified
composition of the participants. The rule is consistent if such a
redistribution does not change the amount the remaining participants receive.

Consistent rules may help to stabilize the set of participants in a
competition. Suppose a coalition of participants are unsatisfied with the
prize money they are receiving. They could agree among themselves to leave the
competition, and organize their own, smaller event, where they will divide the
prize money in a manner they find more beneficial. However, if the organizer
used a consistent rule, the participants would have earned the same amount in
the original competition, and would have no incentive to leave.

Although consistency with respect to single rank-order tournaments is
demanding, it is often observed in reality. For example, the Equal Division
(ED) rule and the Winner-Takes-All (WTA) rule are consistent. As a more
complex example, suppose that in a competition with 100 participants, the
company awards a number of fixed-size cash prizes of $2,000. The limited prize
endowment provides a certain number of equal prizes to the participants with
the highest positions, and any remaining balance goes to the next participant.
So, if the prize fund is $6,000, then the three best participants A, B, and C
win $2,000 each. If the fund grows to $11,000, then the top five participants
A, B, C, D, and E win $2,000 each, and the participant in sixth position, F
wins $1,000.222Similarly, satellites of poker tournaments follow consistent
prize structures. Consistency requires that if the organizer has already
transferred payments to participants B, C, and F, then re-applying the rule to
the competition without participants B, C, and F and with $6,000 prize
endowment will leave everything as before: participants A, D, and E receive
$2,000 each, since they are the best among the remaining participants.

How do _fair and consistent_ prize distribution rules look like? The Equal
Division rule, the Winner-Takes-All rule, and the aforementioned rule
illustrate the main feature of such rules. Our first main result shows that
all fair and consistent rules are some combination of these three rules and we
call them _interval rules_ (Theorem 1). Since this is still a large family, we
can strengthen our requirements for its properties.

In particular, we can strengthen order preservation. We say that the rule
satisfies _strict order preservation_ if prizes for positions from the first
to the last form a strictly decreasing sequence. Unfortunately, there are no
fair and consistent rules for the distribution of prizes that satisfy strict
order preservation. On the other hand, the Winner-Takes-All rule is the only
fair and consistent rule in which the prize for the first position always
exceeds the prize for the last position (Corollary 1).

We can also strengthen endowment monotonicity. To do this, we examine three
more stringent properties. The first strengthening of endowment monotonicity
is _strict endowment monotonicity_. This property requires that with an
increase in the prize endowment, the prize for each position strictly
increases. The Equal Division rule is the only fair and consistent rule that
satisfies strict endowment monotonicity (Corollary 2).

The second strengthening is _winner strict endowment monotonicity_. This
property in addition to endowment monotonicity requires that with an increase
in the prize endowment, the prize for the first position strictly increases.
This leads to the following subfamily of rules. Each such rule sets a maximum
size of an individual prize, which all competitors receive equally regardless
of their position, while all the excess goes to the winner (Corollary 3).
Therefore, we call this a _Winner-Takes-Surplus (WTS) rule_. For example, the
size of a laboratory’s premium fund is often recognized only at the end of the
year. The head of the laboratory with 10 employees can consider a fair premium
of $2,000. Then, with a fund size of $10,000, each employee receives $1,000.
But if the size of the fund is $24,000, then the best worker gets $6,000, and
each of the others receives $2,000.

The third strengthening of endowment monotonicity is also suggested by the
examples from Table 1. The prize structures from golf and poker are _scale
invariant_ , that is, when the prize fund is increased $k$ times, the prize
for each position also increases $k$ times. We show that the only two fair,
consistent, and scale invariant rules for the distribution of prizes are the
Equal Division rule and the Winner-Takes-All rule (Corollary 4). As a result,
we see that the prize structures from poker and golf are not consistent.

Since consistency can be an excessive requirement, we further formulate two
relaxations of consistency.333Our weakening of consistency was inspired by
local stability introduced by Young, (1988). Imagine that a tournament is held
among participants of different skill levels. The organizer could distribute
the prize fund according to the general ranking of all participants; or the
organizer could divide the prize fund into two parts and distribute the prizes
separately among the participants of high and low levels. If both methods lead
to the same distribution of prizes, then the rule satisfies _local
consistency_. If the prize distribution is at least the same for high level
participants, then the rule satisfies _top consistency_. Such rules are
particularly desirable in competitions where losers take their prizes and
leave the competition one by one (as in poker) or in groups (as in golf). For
example, Table 1 shows that the top 10 poker tournament participants receive a
total of $5.605 million of a total prize endowment of $11.18 million. Local
and top consistency require that if you apply the rule separately to the top
10 participants and a prize endowment of $5.605 million, then they will
receive the same prizes.

Our second main result describes the family of _fair and locally consistent_
rules that satisfy winner strict endowment monotonicity. We call them _single-
parametric rules_. Each such rule is generated by some continuous and non-
decreasing function $f$ as follows. The winner of the competition receives
some prize, $x$, the runner-up receives a prize $f(x)$, the third position
receives a prize $f(f(x))$, and so on. The value of $x$ is unambiguously
determined from the condition that the organizer distributes the whole prize
endowment (Theorem 2, Corollary 5). Further, we show that the prize structure
is fair, locally consistent and scale invariant, if and only if the prizes
form a _geometric_ sequence with representation $\lambda$, that is, the prize
for position $r+1$ is the fraction $\lambda$ of the prize for position $r$
(Corollary 6). For example, the prizes in the final stage of the poker
tournament from Table 1 form a geometric sequence with $\lambda=0.713$ and,
therefore, are locally consistent.444The IBU World Cup Biathlon is another
example, where the prizes are almost geometrical (Kondratev et al.,, 2019).
The prizes in golf tournaments, however, do not form a geometric sequence and,
therefore, are not even locally consistent.

Our third main result describes the family of _fair and top consistent_ rules
that satisfy winner strict endowment monotonicity. Each such _parametric rule_
is generated by some sequence of continuous and non-decreasing functions
$f_{2},f_{3},\ldots$, as follows. The winner of the competition receives some
prize, $x$, the runner-up receives a prize $f_{2}(x)$, the third position
receives a prize $f_{3}(x)$, and so on. The value of $x$ is unambiguously
determined from the condition that the organizer distributes the whole prize
endowment (Theorem 3, Corollary 7). Finally, we show that the prize structure
is fair, top consistent and scale invariant, if and only if there exists some
sequence of $\lambda_{1},\lambda_{2},\ldots$, such that the prize for position
$r$ is the fraction $\lambda_{r}/\lambda_{1}$ of the prize for the winner
(Corollary 8). We call them _proportional rules_. For example, the prizes in
the golf tournaments from Table 1 are distributed according to a proportional
rule with $\lambda_{1}=18.0,\lambda_{2}=10.9,\lambda_{3}=6.9$, and so on.

Our results are presented in Tables 2 and 3. Note that no prize allocation
rule satisfies all properties considered. However, the geometric rules with
representation $\lambda$ such that $0<\lambda<1$ satisfy all properties except
for consistency.

Table 2. Properties of consistent rules

|  | Winner- |  | Winner- |
---|---|---|---|---|---
|  | Takes- | Equal | Takes- | WTA
| Interval | All | Division | Surplus | and
| Rules | (WTA) | (ED) | (WTS) | ED
|  Th. 1 |  Cor. 1 |  Cor. 2 |  Cor. 3 |  Cor. 4
| $(a_{k},b_{k}),$ | one | one | $a\in[0,\infty]$ | two
| $k=1,2,\ldots$ | rule | rule |  | rules
Anonymity | + | + | + | + | +
Order Preservation | +* | + | +* | +* | +*
Winner-Loser Strict | only | +* | $-$ | only | only
Order Preservation | WTA |  |  | WTA | WTA
Strict Order Preservation | $-$ | $-$ | $-$ | $-$ | $-$
Endowment Monotonicity | +* | +* | + | + | +
Winner Strict | only | + | + | +* | +
Endowment Monotonicity | WTS |  |  |  |
Strict | only | $-$ | +* | only | only
Endowment Monotonicity | ED |  |  | ED | ED
Scale Invariance | only | + | + | only | +*
(Endowment Additivity) | WTA,ED |  |  | WTA,ED |
Top Consistency | + | + | + | + | +
Local Consistency | + | + | + | + | +
Consistency | +* | +* | +* | +* | +*

_Notes_ :

  * •

\+ indicates that the property is satisfied by each rule in the family, $-$
indicates that the property is not satisfied by each rule in the family, and *
indicates the axiomatic characterizations;

  * •

a WTS rule with representation $a$ is an interval rule with $a_{1}=a$ and
$b_{1}=\infty$;

  * •

the WTA rule is an interval rule with $a_{1}=0$, $b_{1}=\infty$, and a WTS
rule with $a=0$;

  * •

the ED rule is an interval rule with $a_{k}=b_{k}=0$ for all $k$, and a WTS
rule with $a=\infty$.

Table 3. Properties of top and locally consistent rules

| Single- |  |  |
---|---|---|---|---
| Parametric | Geometric | Parametric | Proportional
| Rules | Rules | Rules | Rules
|  Th. 2 |  Cor. 6 |  Th. 3 |  Cor. 8
|  Cor. 5 |  |  Cor. 7 |
|  function $f$: | $\lambda\in[0,1]$ |  functions $f_{k}$: |  $\lambda_{k}\in[0,\infty)$:
| $0\leq f(x)\leq x$ |  | $0\leq f_{k+1}(x)\leq f_{k}(x)\leq x$ | $\lambda_{k+1}\leq\lambda_{k}$
|  |  | $k=1,2,\ldots$ | $k=1,2,\ldots$
Anonymity | +* | +* | +* | +*
Order Preservation | +* | +* | +* | +*
Winner-Loser Strict | $f(x)<x$ | all but | $f_{2}(x)<x$ | $\lambda_{2}<\lambda_{1}$
Order Preservation |  for $x>0$ | ED |  for $x>0$ |
Strict | $0<f(x)<x$ | all but | $f_{k+1}(x)<f_{k}(x)$ | $\lambda_{k+1}<\lambda_{k}$
Order Preservation |  for $x>0$ | WTA,ED |  for $x>0$, for each $k$ |  for each $k$
Endowment Monotonicity | + | + | + | +
Winner Strict | +* | + | +* | +
Endowment Monotonicity |  |  |  |
Strict |  $f$ is | all but |  $f_{k}$ is increasing | $\lambda_{k}>0$
Endowment Monotonicity | increasing | WTA |  for each $k$ |  for each $k$
Scale Invariance | only | +* | only | +*
(Endowment Additivity) | Geometric |  | Proportional |
Top Consistency | + | + | +* | +*
Local Consistency | +* | +* | only | only
|  |  | Single-Parametric | Geometric
Consistency | only | only | only | only
| WTS | WTA,ED | WTS | WTA,ED

_Notes_ :

  * •

\+ indicates that the property is satisfied by each rule in the family, and *
indicates the axiomatic characterizations;

  * •

the Winner-Takes-All rule (WTA) is a single-parametric rule with $f(x)=0$ for
each $x$, a geometric rule with $\lambda=0$, a parametric rule with
$f_{1}(x)=x$ and $f_{k}(x)=0$ for each $k\geq 2$, and a proportional rule with
$\lambda_{1}=1$ and $\lambda_{k}=0$ for each $k\geq 2$;

  * •

the Equal Division rule (ED) is a single-parametric rule with $f(x)=x$ for
each $x$, a geometric rule with $\lambda=1$, a parametric rule with
$f_{k}(x)=x$ for each $k$, and a proportional rule with $\lambda_{k}=1$ for
each $k$;

  * •

a Winner-Takes-Surplus rule (WTS) with representation $a$ is a single-
parametric rule with $f(x)=\min\\{a,x\\}$ for each $x$, and a parametric rule
with $f_{1}(x)=x$ and $f_{k}(x)=\min\\{a,x\\}$ for each $k\geq 2$;

  * •

a geometric rule with representation $\lambda$ is a single-parametric rule
with $f(x)=\lambda x$ for each $x$, a parametric rule with
$f_{k}(x)=\lambda^{k-1}x$ for each $k$, and a proportional rule with
$\lambda_{k}=\lambda^{k-1}$ for each $k$;

  * •

a single-parametric rule with representation $f$ is a parametric rule with
$f_{k}=f^{(k-1)}$ for each $k$;

  * •

a proportional rule with representation $\lambda_{k}$ is a parametric rule
with $f_{k}(x)=\lambda_{k}x/\lambda_{1}$ for each $k$.

###  1.1. Literature overview

In this brief literature overview, we highlight other papers related to our
key topics: fairness, consistency, rank-order tournaments, prize allocation,
and axiomatic characterization.

Francis Galton, (1902) was perhaps the first to write on how the total prize
money should be divided into prizes for each ranking position in a rank-order
tournament. Using a probabilistic approach with order statistics, Galton
concluded that when only two prizes are given, the first prize should
approximately be three times the value of the second. Because of poor economic
motivation, this approach has not received noticeable attention in the
literature.

The seminal paper of Lazear and Rosen, (1981) analyzed rank-order tournaments
from an economic perspective. In their model, a firm assigns a certain prize
to each ranking position. Then each worker chooses a level of effort which
leads to an output. The relative ranking of the outputs determines the prizes
for workers. Lazear and Rosen found the optimal prize structure that maximizes
the worker’s utility in equilibrium. The subsequent literature proposed
similar models and studied optimal prize structures that maximize different
goals, such as the total output (Glazer and Hassin,, 1988), the revenue to the
auctioneer (Barut and Kovenock,, 1998), the total effort of competitors
(Moldovanu and Sela,, 2001), the highest effort among all competitors
(Moldovanu and Sela,, 2006), the weighted total effort of the top $k$
competitors (Archak and Sundararajan,, 2009), the participation of competitors
(Azmat and Möller,, 2009), and the participation of talented competitors
(Azmat and Möller,, 2018).

In the literature on rank-order tournaments, anonymity and order preservation
are standard fairness properties for prize allocation rules. In the model of
Lazear and Rosen, (1981), a firm assigns a certain prize to each position
regardless of the identity of the worker who occupies the position. O’Keeffe
et al., (1984) studied anonymous prize allocation, which they call ‘fair’, and
compared it with non-anonymous prize allocation, which they call ‘unfair’.
Glazer and Hassin, (1988), Barut and Kovenock, (1998), Moldovanu and Sela,
(2001), Moldovanu and Sela, (2006), and Azmat and Möller, (2009) considered
the order preservation of prizes as a model assumption. Archak and
Sundararajan, (2009) call order preserving prizes ‘fair’. Olszewski and
Siegel, (2020) found that total effort maximization, concave prize valuations,
and convex effort costs call for the strict order preservation of prizes.

The Winner-Takes-Surplus (WTS) and geometric prize sequences that we
axiomatically characterize appeared in the literature for other reasons.
Moldovanu et al., (2007) introduced contests for status and found that when
the total prize money is high enough, the WTS prize sequence maximizes the
total effort of competitors. Gershkov et al., (2009) showed that under some
assumptions the WTS prize sequence is an efficient redistribution of output
among partners within teams. Newman and Tafkov, (2014) provided an experiment
and showed that relative performance information has a negative effect on
performance of competitors if the WTS prize sequence is used. Xiao, (2016)
studied geometric prize sequences because under some assumptions it guarantees
the existence and uniqueness of a Nash equilibrium, which is hard to prove for
general prize sequences.

The consistency principle for allocation rules was motivated by Balinski and
Young, (1982) as ‘‘every part of a fair division should be fair’’. Later,
Thomson, (2012) argued that this should be restated as ‘‘every part of every
socially desirable allocation should be socially desirable’’. Consistency has
been applied in several models for allocation problems with infinitely
divisible goods. Moulin, (1985) characterized equal cost allocation rules
using consistency under the name ‘separability’. Inspired by the Talmud,
consistency was applied to bankruptcy problems (Aumann and Maschler,, 1985),
taxation problems (Young,, 1987), and rationing problems (Moulin,, 2000).
Under the name ‘stability’, Lensberg, (1987) studied the implications of
consistency in the context of bargaining problems, in particular for the Nash
solution (Lensberg,, 1988). Thomson, (1988) used consistency to characterize
the Walrasian correspondence for exchange economies. The core and the Shapley
value for cooperative games were characterized in terms of consistency by
Peleg, (1986) and Hart and Mas-Colell, (1989), respectively. Thomson, (1994)
showed that the central uniform rule for allocation problems with single-
peaked preferences admits a characterization based on consistency. Consistent
solutions in atomless economies were analyzed by Thomson and Zhou, (1993).
Recently, Hougaard et al., (2017) studied consistency in the context of
revenue sharing for hierarchical ventures. Osório, (2017) employed consistency
for operational problems (such as river sharing problems, sequential
allocation and rationing problems) and obtained a geometric structure for
resource allocation.

We are aware of two applications of the consistency axiom in the context of
competitions. First, for competitions in which participants put in effort to
increase their probability of winning a single prize, Skaperdas, (1996)
characterized the additive contest success functions, which provide each
player’s probability of winning as a function of all players’ efforts. Though
this model is mathematically equivalent to prize allocation based on cardinal
performance of competitors, we are not aware of real examples of competitions
using such prize allocation schemes. Second, Flores-Szwagrzak and Treibich,
(2020) characterized measures of individual productivity when team membership
and team production are observable but individual contributions to team
production are not. The authors illustrated their approach with the data from
the National Basketball League.

The literature on prize allocation in rank-order competitions has not taken an
axiomatic approach and the fair allocation literature has not focused on
competitions. We connect these two fields by applying the axiomatic approach
to prize allocation in rank-order competitions. Surprisingly, the literature
on this topic is scarce. A rare example is Petróczy and Csató, (2021), who
proposed a parametric rule based on pairwise comparisons and illustrated their
approach with the data from the FIA Formula One racing. However, the authors
did not provide axiomatic characterizations. To our knowledge, our paper is
the first that develops and motivates prize allocation rules for rank-order
competitions directly from axioms.555Recently, Bergantiños and Moreno-Ternero,
2020a ; Bergantiños and Moreno-Ternero, 2020b ; Bergantiños and Moreno-
Ternero, 2021a ; Bergantiños and Moreno-Ternero, 2021b  initiated axiomatic
studies on broadcasting revenue sharing, which is another essential part of
money sharing in competitions (e.g., in soccer leagues).

This paper is structured as follows. Section 2 introduces the model and
characterizes fair and consistent prize allocation rules. Section 3
characterizes fair and locally consistent rules, and fair and top consistent
rules. Section 4 concludes. The proofs are contained in the Appendix.

##  2\. Consistent prize allocation rules

###  2.1. Model

Let $U$ be a countable set of at least three potential _competitors_. On each
given occasion, a finite subset $N\subseteq U$ is in a competition. This
competition results in a _ranking_ , a bijection
$\mathcal{R}:N\to\\{1,\ldots,|N|\\}$ assigning to each competitor a
_position_. Here, competitor $i\in N$ has a higher position in the ranking
than competitor $j\in N$ if $\mathcal{R}(i)<\mathcal{R}(j)$. The _prize
endowment_ $E\in\mathbb{R}_{+}$ is the amount of prize money to be allocated
among the competitors. Thus, a _competition_ is a triple $(N,\mathcal{R},E)$.

Assuming that more money is better for each competitor, the preferences of the
competitors over the feasible prize allocations are in conflict. In order to
select a reasonable compromise for each competition, we study prize allocation
rules $\varphi$ which assign to each competition $(N,\mathcal{R},E)$ an
allocation of the prize endowment among the competitors, i.e.
$\varphi(N,\mathcal{R},E)\in\mathbb{R}_{+}^{N}$ is such that

$\sum_{i\in N}\varphi_{i}(N,\mathcal{R},E)=E.$

Throughout this paper, $\varphi$ denotes a generic prize allocation rule.

Reflecting the opposing principles of egalitarianism and elitism, two extreme
and basic prize allocation rules are the _Equal Division rule_ and the
_Winner-Takes-All rule_. The Equal Division rule divides the prize money
equally among the competitors. The Winner-Takes-All rule allocates all the
prize money to the competitor with the highest position.

Equal Division rule (ED)
The equal division rule ${\rm ED}$ assigns to each competition
$(N,\mathcal{R},E)$ the prize allocation

${\rm ED}_{i}(N,\mathcal{R},E)=\frac{E}{|N|}\quad\text{for each $i\in N$.}$

Winner-Takes-All rule (WTA)
The winner-takes-all rule ${\rm WTA}$ assigns to each competition
$(N,\mathcal{R},E)$ the prize allocation

${\rm WTA}_{i}(N,\mathcal{R},E)=\begin{cases}E&\text{if
$\mathcal{R}(i)=1$;}\\\ 0&\text{otherwise.}\end{cases}$

The ED rule is fair from an egalitarian perspective since it treats each
competitor equally; the WTA rule is fair from an elitist perspective since it
rewards the winner for achieving the highest position in the ranking.
We take an axiomatic approach to study the fundamental differences between
prize allocation rules. This means that we formulate some desirable properties
of rules and analyze their implications. An elementary property imposing a
form of equal treatment of competitors is _anonymity_ , which requires that
the prize allocation does not depend on the identities of the competitors, but
only on the number of competitors, their ranking, and the prize endowment.

Axiom: anonymity
For each pair of competitions $(N,\mathcal{R},E)$ and
$(N^{\prime},\mathcal{R}^{\prime},E)$ with equal numbers of competitors
$|N|=|N^{\prime}|$, and each pair of competitors $i\in N$ and $j\in
N^{\prime}$ with equal positions $\mathcal{R}(i)=\mathcal{R}^{\prime}(j)$, we
have

$\varphi_{i}(N,\mathcal{R},E)=\varphi_{j}(N^{\prime},\mathcal{R}^{\prime},E).$

The fairness property imposing the prize allocation to reflect the ranking is
_order preservation_ , which requires that the prize of a competitor is not
lower than the prize of a competitor with a lower position in the ranking.

Axiom: order preservation
For each competition $(N,\mathcal{R},E)$, if competitor $i\in N$ has a higher
position in the ranking $\mathcal{R}$ than competitor $j\in N$, we have

$\varphi_{i}(N,\mathcal{R},E)\geq\varphi_{j}(N,\mathcal{R},E).$

The solidarity property _endowment monotonicity_ requires that no competitor
is worse off when the prize endowment increases.

Axiom: endowment monotonicity
For each pair of competitions $(N,\mathcal{R},E)$ and
$(N,\mathcal{R},E^{\prime})$ with $E<E^{\prime}$ and each competitor $i\in N$,
we have

$\varphi_{i}(N,\mathcal{R},E)\leq\varphi_{i}(N,\mathcal{R},E^{\prime}).$

Endowment monotonicity is a stronger property than _endowment continuity_ ,
which requires that small changes in the prize endowment have a small impact
on the assigned prize allocation.

Axiom: endowment continuity
For each pair of competitions $(N,\mathcal{R},E)$ and
$(N,\mathcal{R},E^{\prime})$, we have
$\varphi(N,\mathcal{R},E)\rightarrow\varphi(N,\mathcal{R},E^{\prime})$ if
$E\rightarrow E^{\prime}$.

###### Lemma 1

If a prize allocation rule satisfies endowment monotonicity, then it satisfies
endowment continuity.

All proofs are provided in the Appendix.

###  2.2. Joint characterization

Prize allocation may depend on the number of competitors. A criterion for
evaluating whether a prize allocation rule prescribes coherent allocations for
competitions with different numbers of competitors is _consistency_. Consider
an arbitrary competition $(N,\mathcal{R},E)$ and suppose that some competitors
$S\subseteq N$ redistribute their accumulated prizes. This redistribution is
based on their _subranking_ , the bijection
$\mathcal{R}_{S}:S\to\\{1,\ldots,|S|\\}$ such that for each pair of
competitors $i\in S$ and $j\in S$, we have
$\mathcal{R}_{S}(i)<\mathcal{R}_{S}(j)$ if and only if
$\mathcal{R}(i)<\mathcal{R}(j)$. A prize allocation rule is consistent if it
allocates to each competitor $i\in S$ in the corresponding reduced competition
the same prize as in the original competition.

Axiom: consistency
For each competition $(N,\mathcal{R},E)$, each nonempty subset of competitors
$S\subseteq N$, and each competitor $i\in S$, we have666In fact, all results
in this section hold when consistency is weakened to _bilateral consistency_ ,
requiring consistency only for reduced competitions with two competitors.

$\varphi_{i}(N,\mathcal{R},E)=\varphi_{i}\left(S,\mathcal{R}_{S},\sum_{j\in
S}\varphi_{j}(N,\mathcal{R},E)\right).$

The Equal Division rule and the Winner-Takes-All rule both satisfy anonymity,
order preservation, endowment monotonicity, and consistency, but these rules
are not the only ones. The following example provides another rule satisfying
these properties.

###### Example 1

Let a prize allocation rule be defined by allocating the prize endowment in
the following way. Up to the first dollar is allocated to the competitor with
the highest position, the surplus up to the second dollar is allocated to the
competitor with the second highest position, and so on until each competitor
has been allocated one dollar. If there is still money left, the first
additional dollar is allocated to the competitor with the highest position,
the second additional dollar to the competitor with the second highest
position, and so on. This procedure continues until the full prize endowment
is allocated among the competitors. Table 4 illustrates how a prize endowment
from one to six dollars is allocated among two, three, and four competitors.
This rule also satisfies anonymity, order preservation, endowment
monotonicity, and consistency.  $\triangle$

Table 4. Six dollars allocated among two, three, and four competitors in
Example 1

$\varphi_{1}$ | $\varphi_{2}$ | $E$
---|---|---
1 | 0 | 1
1 | 1 | 2
2 | 1 | 3
2 | 2 | 4
3 | 2 | 5
3 | 3 | 6

$\varphi_{1}$ | $\varphi_{2}$ | $\varphi_{3}$ | $E$
---|---|---|---
1 | 0 | 0 | 1
1 | 1 | 0 | 2
1 | 1 | 1 | 3
2 | 1 | 1 | 4
2 | 2 | 1 | 5
2 | 2 | 2 | 6

$\varphi_{1}$ | $\varphi_{2}$ | $\varphi_{3}$ | $\varphi_{4}$ | $E$
---|---|---|---|---
1 | 0 | 0 | 0 | 1
1 | 1 | 0 | 0 | 2
1 | 1 | 1 | 0 | 3
1 | 1 | 1 | 1 | 4
2 | 1 | 1 | 1 | 5
2 | 2 | 1 | 1 | 6

It turns out that anonymity is implied by order preservation, endowment
monotonicity, and consistency. This is captured by the following lemma.

###### Lemma 2

If a prize allocation rule satisfies order preservation, endowment
monotonicity, and consistency, then it satisfies anonymity.

To have a full understanding of the joint implication of order preservation,
endowment monotonicity, and consistency, we only need to focus on the
structure of the corresponding prize allocation rules for competitions with
two competitors. The following lemma shows that each such rule has at most one
consistent extension.777A similar result for consistent extensions in the
context of claims problems was obtained by Aumann and Maschler, (1985).

###### Lemma 3

If two prize allocation rules satisfying endowment monotonicity and
consistency coincide for each competition with two competitors, then the two
rules coincide for each competition with an arbitrary number of competitors.

How do prize allocation rules satisfying order preservation, endowment
monotonicity, and consistency look for competitions with two competitors? To
this end, it helps to graphically illustrate possible _prize allocation paths_
, i.e. draw the prize allocations assigned to competitions with two
competitors when the prize endowment increases from zero.

######  Example 1 (continued)

Let $(N,\mathcal{R},E)$ with $|N|=2$ be a competition with two competitors.
Denote $N=\\{1,2\\}$ such that $\mathcal{R}(1)=1$ and $\mathcal{R}(2)=2$. Let
the prize endowment $E$ gradually increase from zero. Then the prize
allocation paths of the ED rule, the WTA rule, and the prize allocation rule
$\varphi$ described above are illustrated as follows.

$\varphi_{1}$01234$\varphi_{2}$1234${\rm ED}(N,\mathcal{R},E)$${\rm
WTA}(N,\mathcal{R},E)$$\varphi(N,\mathcal{R},E)$

The horizontal axis depicts the prize allocated to competitor one and the
vertical axis depicts the prize allocated to competitor two. The dotted lines
indicate different levels of the prize endowment.  $\triangle$

Each prize allocation rule satisfying order preservation, endowment
monotonicity, and consistency is somehow a combination of the ED rule, the WTA
rule, and the type of rule in Example 1. Each such prize allocation rule can
be described in the following way. Let $N\subseteq U$ be a finite set of
competitors and consider a corresponding ranking and prize endowment. Then
there exist disjoint intervals $(a_{k},b_{k})$ with $a_{k}\in\mathbb{R}_{+}$
and $b_{k}\in\mathbb{R}_{+}\cup\\{+\infty\\}$ for each $k$ such that the
following holds. Each competitor is allocated a prize of $a_{k}$ when the
prize endowment equals $|N|a_{k}$. If the endowment is higher, first the prize
allocated to the competitor with the highest position is increased to $b_{k}$,
then the prize allocated to the competitor with the second highest position is
increased to $b_{k}$, and so on. This means that each competitor is allocated
a prize of $b_{k}$ when the prize endowment equals $|N|b_{k}$. If the average
prize endowment does not belong to one of these intervals, it is equally
divided among the competitors. We call such prize allocation rules _interval
rules_.

Interval rules
There exist disjoint intervals $(a_{1},b_{1}),(a_{2},b_{2}),\ldots$ with
$a_{1},a_{2},\ldots\in\mathbb{R}_{+}$ and
$b_{1},b_{2},\ldots\in\mathbb{R}_{+}\cup\\{+\infty\\}$ such that for each
competition $(N,\mathcal{R},E)$ and each competitor $i\in N$, we have

$\varphi_{i}(N,\mathcal{R},E)=\begin{cases}a_{k}&\text{if $|N|a_{k}\leq
E\leq(|N|-\mathcal{R}(i)+1)a_{k}+(\mathcal{R}(i)-1)b_{k}$};\\\ x&\text{if
$(|N|-\mathcal{R}(i)+1)a_{k}+(\mathcal{R}(i)-1)b_{k}\leq
E\leq(|N|-\mathcal{R}(i))a_{k}+\mathcal{R}(i)b_{k}$};\\\ b_{k}&\text{if
$(|N|-\mathcal{R}(i))a_{k}+\mathcal{R}(i)b_{k}\leq E\leq|N|b_{k}$};\\\
\frac{E}{|N|}&\text{otherwise},\end{cases}$

where $x=E-(|N|-\mathcal{R}(i))a_{k}-(\mathcal{R}(i)-1)b_{k}$.

Note that the ED rule is an interval rule with $a_{k}=b_{k}=0$ for each $k$.
The WTA rule is an interval rule with $a_{1}=0$ and $b_{1}=+\infty$. The prize
allocation rule from Example 1 is an interval rule with $a_{k}=k-1$ and
$b_{k}=k$ for each $k$.

###### Example 2

Let $(N,\mathcal{R},E)$ with $|N|=2$ be a competition with two competitors.
Denote $N=\\{1,2\\}$ such that $\mathcal{R}(1)=1$ and $\mathcal{R}(2)=2$. The
prize allocation path of the interval rule $\varphi$ with $a_{1}=1$,
$b_{1}=a_{2}=2\frac{1}{2}$, $b_{2}=3$, $a_{3}=3\frac{1}{2}$, and
$b_{3}=+\infty$ is illustrated as follows.

$\varphi_{1}$01234$\varphi_{2}$1234$\varphi(N,\mathcal{R},E)$

$\triangle$

###### Theorem 1

A prize allocation rule satisfies order preservation, endowment monotonicity,
and consistency if and only if it is an interval rule.

###  2.3. Strengthening the fairness properties

Theorem 1 shows that many rules satisfy order preservation, endowment
monotonicity, and consistency. This means that there is room for imposing
additional requirements. In particular, it may be possible to strengthen one
of the fairness properties. Unfortunately, order preservation cannot be
strengthened to _strict order preservation_ , which requires that the prize of
a competitor is higher than the prize of a competitor with a lower position.
However, it can be strengthened to the weaker property _winner-loser strict
order preservation_ , which requires in addition to order preservation that
the prize of the competitor with the highest position is higher than the prize
of the competitor with the lowest position. The Winner-Takes-All rule is the
only interval rule satisfying winner-loser strict order preservation.

Axiom: strict order preservation
For each competition $(N,\mathcal{R},E)$ with $E>0$, if competitor $i\in N$
has a higher position in the ranking $\mathcal{R}$ than competitor $j\in N$,
we have

$\varphi_{i}(N,\mathcal{R},E)>\varphi_{j}(N,\mathcal{R},E).$

Axiom: winner-loser strict order preservation
For each competition $(N,\mathcal{R},E)$ with $E>0$, if competitor $i\in N$
has a higher position in the ranking $\mathcal{R}$ than competitor $j\in N$,
we have

$\varphi_{i}(N,\mathcal{R},E)\geq\varphi_{j}(N,\mathcal{R},E)$

and if $\mathcal{R}(i)=1$ and $\mathcal{R}(j)=|N|$, then

$\varphi_{i}(N,\mathcal{R},E)>\varphi_{j}(N,\mathcal{R},E).$

###### Corollary 1

  1. (i)

No prize allocation rule satisfies strict order preservation, endowment
monotonicity, and consistency.

  2. (ii)

The unique prize allocation rule satisfying winner-loser strict order
preservation, endowment monotonicity, and consistency is the Winner-Takes-All
rule.

Instead of strengthening order preservation, it is also possible to strengthen
endowment monotonicity. For instance, _strict endowment monotonicity_ requires
that each competitor is better off when the prize endowment increases, or the
weaker property _winner strict endowment monotonicity_ requires in addition to
endowment monotonicity that the winner is better off when the prize endowment
increases.

Axiom: strict endowment monotonicity
For each pair of competitions $(N,\mathcal{R},E)$ and
$(N,\mathcal{R},E^{\prime})$ with $E<E^{\prime}$ and each competitor $i\in N$,
we have

$\varphi_{i}(N,\mathcal{R},E)<\varphi_{i}(N,\mathcal{R},E^{\prime}).$

Axiom: winner strict endowment monotonicity
For each pair of competitions $(N,\mathcal{R},E)$ and
$(N,\mathcal{R},E^{\prime})$ with $E<E^{\prime}$ and each competitor $i\in N$,
we have

$\varphi_{i}(N,\mathcal{R},E)\leq\varphi_{i}(N,\mathcal{R},E^{\prime})$

and if $\mathcal{R}(i)=1$, then

$\varphi_{i}(N,\mathcal{R},E)<\varphi_{i}(N,\mathcal{R},E^{\prime}).$

The Equal Division rule is the only interval rule which satisfies strict
endowment monotonicity. Moreover, an interval rule satisfies winner strict
endowment monotonicity if and only if it is a _Winner-Takes-Surplus rule_ ,
i.e. it coincides with the ED rule up to some value of the prize endowment,
and it allocates the surplus according to the WTA rule.

###### Corollary 2

The unique prize allocation rule satisfying order preservation, strict
endowment monotonicity, and consistency is the Equal Division rule.

Winner-Takes-Surplus rules (WTS)
There exists $a\in\mathbb{R}_{+}\cup\\{+\infty\\}$ such that for each
competition $(N,\mathcal{R},E)$ and each competitor $i\in N$, we have

$\varphi_{i}(N,\mathcal{R},E)=\begin{cases}E-(|N|-1)a&\text{if $E\geq|N|a$ and
$\mathcal{R}(i)=1$};\\\ a&\text{if $E\geq|N|a$ and $\mathcal{R}(i)\neq 1$};\\\
\frac{E}{|N|}&\text{otherwise}.\end{cases}$

Note that the ED rule is a WTS rule with $a=+\infty$. The WTA rule is a WTS
rule with $a=0$. Table 5 illustrates how a WTS rule $\varphi$ allocates a
prize endowment from $a$ to $6a$ among two, three, and four competitors. The
corresponding prize allocation path is illustrated as follows.

$\varphi_{1}$0$a$$\varphi_{2}$$a$$\varphi(N,\mathcal{R},E)$

Table 5. Winner-Takes-Surplus rule for two, three, and four competitors

$\varphi_{1}$ | $\varphi_{2}$ | $E$
---|---|---
$a/2$ | $a/2$ | $a$
$a$ | $a$ | $2a$
$2a$ | $a$ | $3a$
$3a$ | $a$ | $4a$
$4a$ | $a$ | $5a$
$5a$ | $a$ | $6a$

$\varphi_{1}$ | $\varphi_{2}$ | $\varphi_{3}$ | $E$
---|---|---|---
$a/3$ | $a/3$ | $a/3$ | $a$
$2a/3$ | $2a/3$ | $2a/3$ | $2a$
$a$ | $a$ | $a$ | $3a$
$2a$ | $a$ | $a$ | $4a$
$3a$ | $a$ | $a$ | $5a$
$4a$ | $a$ | $a$ | $6a$

$\varphi_{1}$ | $\varphi_{2}$ | $\varphi_{3}$ | $\varphi_{4}$ | $E$
---|---|---|---|---
$a/4$ | $a/4$ | $a/4$ | $a/4$ | $a$
$a/2$ | $a/2$ | $a/2$ | $a/2$ | $2a$
$3a/4$ | $3a/4$ | $3a/4$ | $3a/4$ | $3a$
$a$ | $a$ | $a$ | $a$ | $4a$
$2a$ | $a$ | $a$ | $a$ | $5a$
$3a$ | $a$ | $a$ | $a$ | $6a$

###### Corollary 3

A prize allocation rule satisfies order preservation, winner strict endowment
monotonicity, and consistency if and only if it is a Winner-Takes-Surplus
rule.

Alternatively, endowment monotonicity can be strengthened to _endowment
additivity_. Suppose that the prize endowment turns out to be larger than
expected. Then there are two ways of proceeding. Either the initial allocation
is cancelled and the rule is applied to the competition with the new prize
endowment, or the rule is applied to the competition with the increment as the
prize endowment and the resulting allocation is added to the initial
allocation. A prize allocation rule satisfies endowment additivity if both
ways of proceeding lead to the same prize allocation.

Axiom: endowment additivity
For each pair of competitions $(N,\mathcal{R},E)$ and
$(N,\mathcal{R},E^{\prime})$ and each competitor $i\in N$, we have

$\varphi_{i}(N,\mathcal{R},E+E^{\prime})=\varphi_{i}(N,\mathcal{R},E)+\varphi_{i}(N,\mathcal{R},E^{\prime}).$

Endowment additivity is equivalent to _scale invariance_ , which requires that
each position in the ranking of a competition is assigned a fixed proportion
of the prize endowment. Even when the prize endowment is not known in advance,
the competitors know which share corresponds to each position. Among all
interval rules, only the ED rule and the WTA rule satisfy endowment additivity
(scale invariance).

Axiom: scale invariance
For each competition $(N,\mathcal{R},E)$ and each competitor $i\in N$, we
have888The scale invariance axiom is also called _homogeneity_.

$\varphi_{i}(N,\mathcal{R},E)=E\varphi_{i}(N,\mathcal{R},1).$

###### Lemma 4

A prize allocation rule satisfies endowment additivity if and only if it
satisfies scale invariance.

###### Corollary 4

The only two prize allocation rules satisfying order preservation, endowment
additivity (scale invariance), and consistency are the Equal Division rule and
the Winner-Takes-All rule.

##  3\. Weakening consistency

###  3.1. Locally consistent prize allocation rules

We know that the Equal Division rule and the Winner-Takes-All rule are two
members of a family of prize allocation rules satisfying anonymity, order
preservation, endowment monotonicity, and consistency. Strengthening order
preservation or endowment monotonicity leads to impossibilities, uniqueness,
or restricted families. However, consistency may be considered a too strong
requirement, since an arbitrary subranking may not reflect the results of the
original competition well. Suppose for instance that two competitors of a
competition with a large number of competitors reevaluate their allocated
prizes on the basis of their subranking. Then the reduced competition tends to
lose significant features of original competition, e.g. it does not take into
account whether the two competitors originally ended up with the highest
position and the second highest position, or with the highest position and the
lowest position.

In fact, it may be desirable to weaken consistency to _local consistency_ ,
which requires only consistent allocations for reduced competitions where for
each two competitors, each other competitor with an intermediate position is
also involved. In other words, local consistency requires invariance under
splitting up the full competition into leagues, where the competitors in the
top segment of the ranking are put in a separate league, the competitors in
the second segment of the ranking are put in a second league, and so on.

Axiom: local consistency
For each competition $(N,\mathcal{R},E)$, each subset of competitors
$S\subseteq N$ with $|\mathcal{R}(i)-\mathcal{R}(j)|\leq|S|-1$ for all $i,j\in
S$, and each competitor $i\in S$, we have999In fact, all results in this
section hold when local consistency is weakened to _bilateral local
consistency_ , requiring local consistency only for reduced competitions with
two competitors.

$\varphi_{i}(N,\mathcal{R},E)=\varphi_{i}\left(S,\mathcal{R}_{S},\sum_{j\in
S}\varphi_{j}(N,\mathcal{R},E)\right).$

In contrast to consistency, the following example shows that a prize
allocation rule satisfying order preservation, endowment monotonicity, and
local consistency does not necessarily satisfy anonymity.

###### Example 3

Let $i\in U$ and $j\in U$ be two potential competitors. Let the prize
allocation rule $\varphi$ be defined in the following way. If $i$ has the
highest position and $j$ has the second highest position in the ranking, then
$\varphi$ divides the prize endowment equally among competitors $i$ and $j$.
Otherwise, $\varphi$ divides the prize endowment according to the WTA rule.
Formally, $\varphi$ assigns to each competition $(N,\mathcal{R},E)$ the prize
allocation such that
if $i,j\in N$, $\mathcal{R}(i)=1$, and $\mathcal{R}(j)=2$, then

$\varphi_{k}(N,\mathcal{R},E)=\begin{cases}\tfrac{1}{2}E&\text{if
$k\in\\{i,j\\}$};\\\ 0&\text{otherwise},\end{cases}$

and otherwise

$\varphi_{k}(N,\mathcal{R},E)=\begin{cases}E&\text{if $\mathcal{R}(k)=1$};\\\
0&\text{otherwise}.\end{cases}$

Then $\varphi$ satisfies order preservation, endowment monotonicity, and local
consistency, but does not satisfy anonymity. By Lemma 2, $\varphi$ does not
satisfy consistency.  $\triangle$

Needless to say, all interval rules satisfy anonymity, order preservation,
endowment monotonicity, and local consistency, but these rules are not the
only ones. The following example provides another rule satisfying these
properties.

###### Example 4

Let a prize allocation rule be defined by allocating the prize endowment in
the following way. Up to the first dollar is allocated to the competitor with
the highest position. The surplus is divided equally among the competitors
with the highest position and the second highest position until they are
allocated two and one dollar, respectively. Then the surplus is divided
equally among the competitors with the three highest positions until they are
allocated three, two, and one dollar, respectively. This procedure continues
until the competitor with the lowest position is allocated one dollar. If
there is still money left, this is equally divided among the competitors.
Table 6 illustrates how a prize endowment from one to eight dollars is
allocated among two, three, and four competitors. This rule also satisfies
anonymity, order preservation, endowment monotonicity, and local consistency.

Let $(N,\mathcal{R},E)$ with $|N|=2$ be a competition with two competitors.
Denote $N=\\{1,2\\}$ such that $\mathcal{R}(1)=1$ and $\mathcal{R}(2)=2$. Then
the prize allocation paths of the ED rule, the WTA rule, and the prize
allocation rule $\varphi$ described above are illustrated as follows.

$\varphi_{1}$01234$\varphi_{2}$1234${\rm ED}(N,\mathcal{R},E)$${\rm
WTA}(N,\mathcal{R},E)$$\varphi(N,\mathcal{R},E)$

$\triangle$

Table 6. Eight dollars allocated among two, three, and four competitors in
Example 4

$\varphi_{1}$ | $\varphi_{2}$ | $E$
---|---|---
1 | 0 | 1
$3/2$ | $1/2$ | 2
2 | 1 | 3
$5/2$ | $3/2$ | 4
3 | 2 | 5
$7/2$ | $5/2$ | 6
4 | 3 | 7
9/2 | 7/2 | 8

$\varphi_{1}$ | $\varphi_{2}$ | $\varphi_{3}$ | $E$
---|---|---|---
1 | 0 | 0 | 1
$3/2$ | $1/2$ | 0 | 2
2 | 1 | 0 | 3
$7/3$ | $4/3$ | $1/3$ | 4
$8/3$ | $5/3$ | $2/3$ | 5
3 | 2 | 1 | 6
10/3 | 7/3 | 4/3 | 7
11/3 | 8/3 | 5/3 | 8

$\varphi_{1}$ | $\varphi_{2}$ | $\varphi_{3}$ | $\varphi_{4}$ | $E$
---|---|---|---|---
1 | 0 | 0 | 0 | 1
$3/2$ | $1/2$ | 0 | 0 | 2
2 | 1 | 0 | 0 | 3
$7/3$ | $4/3$ | $1/3$ | 0 | 4
$8/3$ | $5/3$ | $2/3$ | 0 | 5
3 | 2 | 1 | 0 | 6
13/4 | 9/4 | 5/4 | 1/4 | 7
7/2 | 5/2 | 3/2 | 1/2 | 8

Note that the arithmetic type of rule in Example 4 satisfies winner strict
endowment monotonicity, i.e. the competitor with the highest position is
better off when the prize endowment increases. Each prize allocation rule
satisfying anonymity, order preservation, winner strict endowment
monotonicity, and local consistency admits a compact indirect description. For
each such rule there exists a continuous and non-decreasing function
$f:\mathbb{R}_{+}\to\mathbb{R}_{+}$ with $f(x)\leq x$ for each $x$ such that
the assigned prize allocation for each competition can be described in the
following way. The competitor with the highest position is allocated a prize
of $x\in\mathbb{R}_{+}$. The competitor with the second highest position is
allocated a prize of $f(x)$. The competitor with the third highest position is
allocated a prize of $f(f(x))$, and so on. In total, the full prize endowment
is allocated among the competitors. We call such prize allocation rules
_single-parametric rules_.

Single-Parametric rules
There exists a continuous and non-decreasing function
$f:\mathbb{R}_{+}\to\mathbb{R}_{+}$ with $f(x)\leq x$ for each $x$ such that
for each competition $(N,\mathcal{R},E)$ and each competitor $i\in N$, we have

$\varphi_{i}(N,\mathcal{R},E)=f^{(\mathcal{R}(i)-1)}(x),$

where we denote $f^{(0)}(x)=x$ and $f^{(k)}(x)=f(f^{(k-1)}(x))$ for each $k$,
and $x\in\mathbb{R}_{+}$ is chosen such that $\sum_{k=1}^{|N|}f^{(k-1)}(x)=E$.

Note that the ED rule is a single-parametric rule with $f(x)=x$ for each $x$.
The WTA rule is a single-parametric rule with $f(x)=0$ for each $x$. A Winner-
Takes-Surplus rule (WTS) with representation $a$ is a single-parametric rule
with $f(x)=\min\\{a,x\\}$ for each $x$. The arithmetic type of rule from
Example 4 is a single-parametric rule with $f(x)=\max\\{0,x-1\\}$ for each
$x$.

###### Theorem 2

A prize allocation rule satisfies anonymity, order preservation, winner strict
endowment monotonicity, and local consistency if and only if it is a single-
parametric rule.

Theorem 2 does not describe all prize allocation rules satisfying anonymity,
order preservation, endowment monotonicity, and local consistency. If such
rule does not satisfy winner strict endowment monotonicity, the prize of the
competitor with the second highest position cannot be expressed as a function
of the prize of the competitor with the highest position. Moreover, as the
following example shows, such a rule for competitions with two competitors
does not necessarily have a unique locally consistent extension to
competitions with more competitors.

###### Example 5

Let a prize allocation rule be defined by allocating the prize endowment in
the following way. Up to the first dollar is allocated to the competitor with
the highest position. Subsequently, one dollar is allocated to the competitor
with the second highest position and then another dollar is allocated to the
competitor with the highest position. Thereafter, one dollar is allocated to
the competitor with the third highest position, then another dollar to the
competitor with the second highest position, and then another dollar to the
competitor with the highest position. This procedure continues until the
competitor with the highest position is allocated an amount of dollars equal
to the number of competitors. If there is still money left, first another
dollar is allocated to the competitor with the lowest position, then another
dollar to the competitor with the second lowest position, and so on. This
continues until the full prize endowment is allocated among the competitors.
Table 7 illustrates how a prize endowment from one to eight dollars is
allocated among two, three, and four competitors. This rule satisfies
anonymity, order preservation, endowment monotonicity, and local consistency,
but does not satisfy winner strict endowment monotonicity. Moreover, for
competitions with two competitors, this rule coincides with the prize
allocation rule from Example 1.  $\triangle$

Table 7. Eight dollars allocated among two, three, and four competitors in
Example 5

$\varphi_{1}$ | $\varphi_{2}$ | $E$
---|---|---
1 | 0 | 1
1 | 1 | 2
2 | 1 | 3
2 | 2 | 4
3 | 2 | 5
3 | 3 | 6
4 | 3 | 7
4 | 4 | 8

$\varphi_{1}$ | $\varphi_{2}$ | $\varphi_{3}$ | $E$
---|---|---|---
1 | 0 | 0 | 1
1 | 1 | 0 | 2
2 | 1 | 0 | 3
2 | 1 | 1 | 4
2 | 2 | 1 | 5
3 | 2 | 1 | 6
3 | 2 | 2 | 7
3 | 3 | 2 | 8

$\varphi_{1}$ | $\varphi_{2}$ | $\varphi_{3}$ | $\varphi_{4}$ | $E$
---|---|---|---|---
1 | 0 | 0 | 0 | 1
1 | 1 | 0 | 0 | 2
2 | 1 | 0 | 0 | 3
2 | 1 | 1 | 0 | 4
2 | 2 | 1 | 0 | 5
3 | 2 | 1 | 0 | 6
3 | 2 | 1 | 1 | 7
3 | 2 | 2 | 1 | 8

Subfamilies of single-parametric rules are obtained if order preservation is
strengthened to strict order preservation or winner-loser strict order
preservation, or winner strict endowment monotonicity is strengthened to
strict endowment monotonicity. A single-parametric rule satisfies winner-loser
strict order preservation if and only if it does not coincide with the ED rule
for each positive prize endowment, and satisfies strict order preservation if
and only if it does not coincide with the ED rule nor with the WTA rule for
each positive prize endowment. Moreover, a single-parametric rule satisfies
strict endowment monotonicity if and only if it does not allocate any
additional prize endowment according to the WTA rule.

###### Corollary 5

Let $\varphi$ be a single-parametric rule with representation $f$. Then the
following statements hold.

  1. (i)

$\varphi$ satisfies winner-loser strict order preservation if and only if
$f(x)<x$ for each $x>0$;

  2. (ii)

$\varphi$ satisfies strict order preservation if and only if $0<f(x)<x$ for
each $x>0$;

  3. (iii)

$\varphi$ satisfies strict endowment monotonicity if and only if $f$ is
increasing.

If winner strict endowment monotonicity is strengthened to endowment
additivity (scale invariance), then the interesting family of _geometric
rules_ is obtained. For each geometric rule, there exists $\lambda\in[0,1]$
such that the assigned prize allocation for each competition can be described
in the following way. The competitor with the second highest position is
allocated a prize of $\lambda$ times the prize of the competitor with the
highest position. The competitor with the third highest position is allocated
a prize of $\lambda$ times the prize of the competitor with the second highest
position, i.e. $\lambda^{2}$ times the prize of the competitor with the
highest position. The competitor with the fourth highest position is allocated
a prize of $\lambda^{3}$ times the prize of the competitor with the highest
position, and so on. In total, the full prize endowment is allocated.

Geometric rules
There exists $\lambda\in[0,1]$ such that for each competition
$(N,\mathcal{R},E)$ and each competitor $i\in N$, we have101010Here, we define
$\lambda^{0}=1$ for each $\lambda\in[0,1]$.

$\varphi_{i}(N,\mathcal{R},E)=\frac{\lambda^{\mathcal{R}(i)-1}}{\sum_{k=1}^{|N|}\lambda^{k-1}}E.$

Note that the ED rule is a geometric rule with $\lambda=1$. The WTA rule is a
geometric rule with $\lambda=0$. The prize allocation in the final stage of
the poker tournament presented in Table 1(a) is a geometric rule with
$\lambda=0.713$. The prize allocation paths of the ED rule, the WTA rule, and
the geometric rule $\varphi$ in the poker tournament are illustrated as
follows.

$\varphi_{1}$01234$\varphi_{2}$1234${\rm ED}(N,\mathcal{R},E)$${\rm
WTA}(N,\mathcal{R},E)$$\varphi(N,\mathcal{R},E)$

###### Corollary 6

A prize allocation rule satisfies anonymity, order preservation, endowment
additivity (scale invariance), and local consistency if and only if it is a
geometric rule.

###  3.2. Top consistent prize allocation rules

Theorem 2 shows that many rules satisfy anonymity, order preservation, winner
strict endowment monotonicity, and local consistency. Strengthening order
preservation or winner strict endowment monotonicity leads to direct
restrictions on the representation of the single-parametric rules. Yet, there
are still several rules observed in practice that do not satisfy local
consistency, such as the golf tournament in Table 1. To describe these rules,
it may be desirable to weaken consistency even further to _top consistency_ ,
which requires only consistent allocations for reduced competitions where only
competitors with the highest positions are involved. In other words, top
consistency requires invariance for the competitors in the top segment of the
ranking when they are put in a separate league. In competitions where
competitors are eliminated sequentially, top consistency can be interpreted as
invariance under prize allocation at the moment of elimination.

Axiom: top consistency
For each competition $(N,\mathcal{R},E)$, each subset of competitors
$S\subseteq N$ such that $\mathcal{R}(i)\leq|S|$ for each competitor $i\in S$,
and each competitor $i\in S$, we have

$\varphi_{i}(N,\mathcal{R},E)=\varphi_{i}\left(S,\mathcal{R}_{S},\sum_{j\in
S}\varphi_{j}(N,\mathcal{R},E)\right).$

Needless to say, all single-parametric rules satisfy anonymity, order
preservation, winner strict endowment monotonicity, and top consistency, but
these rules are not the only ones. The following example provides another rule
satisfying these properties.

###### Example 6

Let a prize allocation rule be defined by allocating the prize endowment in
the following way. Up to the first two dollars are allocated to the competitor
with the highest position. The surplus is divided equally among the
competitors with the highest position and the second highest position until
they are allocated three and one dollar, respectively. Then the surplus is
divided equally among the competitors with the three highest positions until
they are allocated four, two, and one dollar, respectively. This procedure
continues until the competitor with the lowest position is allocated one
dollar. If there is still money left, this is equally divided among the
competitors. Table 8 illustrates how a prize endowment from one to eight
dollars is allocated among two, three, and four competitors. This rule also
satisfies anonymity, order preservation, winner strict endowment monotonicity,
and top consistency.  $\triangle$

Table 8. Eight dollars allocated among two, three, and four competitors in
Example 6

$\varphi_{1}$ | $\varphi_{2}$ | $E$
---|---|---
1 | 0 | 1
2 | 0 | 2
5/2 | 1/2 | 3
3 | 1 | 4
7/2 | 3/2 | 5
4 | 2 | 6
9/2 | 5/2 | 7
5 | 3 | 8

$\varphi_{1}$ | $\varphi_{2}$ | $\varphi_{3}$ | $E$
---|---|---|---
1 | 0 | 0 | 1
2 | 0 | 0 | 2
5/2 | 1/2 | 0 | 3
3 | 1 | 0 | 4
10/3 | 4/3 | 1/3 | 5
11/3 | 5/3 | 2/3 | 6
4 | 2 | 1 | 7
13/3 | 7/3 | 4/3 | 8

$\varphi_{1}$ | $\varphi_{2}$ | $\varphi_{3}$ | $\varphi_{4}$ | $E$
---|---|---|---|---
1 | 0 | 0 | 0 | 1
2 | 0 | 0 | 0 | 2
5/2 | 1/2 | 0 | 0 | 3
3 | 1 | 0 | 0 | 4
10/3 | 4/3 | 1/3 | 0 | 5
11/3 | 5/3 | 2/3 | 0 | 6
4 | 2 | 1 | 0 | 7
17/4 | 9/4 | 5/4 | 1/4 | 8

Each prize allocation rule satisfying anonymity, order preservation, winner
strict endowment monotonicity, and top consistency admits a compact indirect
description. For each such rule there exist continuous and non-decreasing
functions $f_{1},f_{2},\ldots:\mathbb{R}_{+}\to\mathbb{R}_{+}$ with
$f_{1}(x)=x$ for each $x$ and $f_{k+1}(x)\leq f_{k}(x)$ for each $k$ such that
the assigned prize allocation for each competition can be described in the
following way. The competitor with the highest position is allocated a prize
of $f_{1}(x)=x$. The competitor with the second highest position is allocated
a prize of $f_{2}(x)$. Generally, the prize for position $k$ is $f_{k}(x)$. In
total, the full prize endowment is allocated among the competitors. We call
such prize allocation rules _parametric rules_.

Parametric rules
There exist continuous and non-decreasing functions
$f_{1},f_{2},\ldots:\mathbb{R}_{+}\to\mathbb{R}_{+}$ with $f_{1}(x)=x$ for
each $x$ and $f_{k+1}(x)\leq f_{k}(x)$ for each $k$ such that for each
competition $(N,\mathcal{R},E)$ and each competitor $i\in N$, we have

$\varphi_{i}(N,\mathcal{R},E)=f_{\mathcal{R}(i)}(x),$

where $x\in\mathbb{R}_{+}$ is such that $\sum_{k=1}^{|N|}f_{k}(x)=E$.

Note that each single-parametric rule with representation $f$ is a parametric
rule with $f_{k}=f^{(k-1)}$ for each $k$. In particular, the ED rule is a
parametric rule with $f_{k}(x)=x$ for each $k$. The WTA rule is a parametric
rule with $f_{1}(x)=x$ and $f_{k}(x)=0$ for each $k\geq 2$. A Winner-Takes-
Surplus rule (WTS) with representation $a$ is a parametric rule with
$f_{1}(x)=x$ and $f_{k}(x)=\min\\{a,x\\}$ for each $k\geq 2$. A geometric rule
with representation $\lambda$ is a parametric rule with
$f_{k}(x)=\lambda^{k-1}x$ for each $k$. The rule from Example 6 is a
parametric rule with $f_{1}(x)=x$ and $f_{k}(x)=\max\\{0,x-k\\}$ for each
$k\geq 2$.

###### Theorem 3

A prize allocation rule satisfies anonymity, order preservation, winner strict
endowment monotonicity, and top consistency if and only if it is a parametric
rule.

Subfamilies of parametric rules are again directly obtained if order
preservation is strengthened to strict order preservation or winner-loser
strict order preservation, or winner strict endowment monotonicity is
strengthened to strict endowment monotonicity.

###### Corollary 7

Let $\varphi$ be a parametric rule with representation $f_{1},f_{2},\ldots$.
Then the following statements hold.

  1. (i)

$\varphi$ satisfies winner-loser strict order preservation if and only if
$f_{2}(x)<x$ for each $x>0$;

  2. (ii)

$\varphi$ satisfies strict order preservation if and only if
$f_{k+1}(x)<f_{k}(x)$ for each $k$ and each $x>0$;

  3. (iii)

$\varphi$ satisfies strict endowment monotonicity if and only if $f_{k}(x)$ is
increasing for each $k$.

If winner strict endowment monotonicity is strengthened to endowment
additivity (scale invariance), then the family of widely used _proportional
rules_ is obtained. For each proportional rule, there exist
$\lambda_{1},\lambda_{2},\ldots\in\mathbb{R}_{+}$ with $\lambda_{1}>0$ and
$\lambda_{k+1}\leq\lambda_{k}$ for each $k$ such that for each competition the
prize for position $k$ is proportional to $\lambda_{k}$. In total, the full
prize endowment is allocated.

Proportional rules
There exist $\lambda_{1},\lambda_{2},\ldots\in\mathbb{R}_{+}$ with
$\lambda_{1}>0$ and $\lambda_{k+1}\leq\lambda_{k}$ for each $k$ such that for
each competition $(N,\mathcal{R},E)$ and each competitor $i\in N$, we have

$\varphi_{i}(N,\mathcal{R},E)=\frac{\lambda_{\mathcal{R}(i)}}{\sum_{k=1}^{|N|}\lambda_{k}}E.$

Note that each geometric rule with representation $\lambda$ is a proportional
rule with $\lambda_{k}=\lambda^{k-1}$ for each $k$. In particular, the ED rule
is a proportional rule with $\lambda_{k}=1$ for each $k$. The WTA rule is a
proportional rule with $\lambda_{1}=1$ and $\lambda_{k}=0$ for each $k\geq 2$.
The prize allocation rule in the golf tournament of Table 1(b) is a
proportional rule with $\lambda_{1}=18.0$, $\lambda_{2}=10.9$,
$\lambda_{3}=6.9$, and so on.

###### Corollary 8

A prize allocation rule satisfies anonymity, order preservation, endowment
additivity (scale invariance), and top consistency if and only if it is a
proportional rule.

##  4\. Concluding remarks

In this paper, we initiate an axiomatic approach to study prize allocation
rules in rank-order competitions. We introduce a model in which the
competitors, their final ranking, and the prize endowment are the primitives
and we axiomatically characterize three families of prize allocation rules:
interval rules (Theorem 1), single-parametric rules (Theorem 2), and
parametric rules (Theorem 3). In the Appendix, we show that the axioms used
are logically independent. Moreover, we have obtained several subfamilies:
Winner-Takes-Surplus rules (Corollary 3), geometric rules (Corollary 6), and
proportional rules (Corollary 8). Each of these families and subfamilies
includes the Equal Division rule and the Winner-Takes-All rule. The relations
of the families are presented in the picture below.

The practical relevance of our paper is that if an organizer wants to choose a
prize allocation rule and accepts some set of axioms, then the problem is
reduced to choosing a single rule from the corresponding family. Since the
families are broad enough, the organizer can choose the rule that maximizes
some desirable goal, such as the total effort of competitors or the
participation of talented competitors. This problem could be studied in future
research using an optimization approach.

Our axiomatic framework can be further developed in many directions. For
single rank-order tournaments, a natural direction is the study of other
desirable axioms. In situations where more data is available, we can take more
details into account for prize allocations. One possibility is to replace the
ordinal ranking in our model by a cardinal ranking, e.g., using finish times,
scores, or the volume of sales. Another possibility is to consider other
competition types, such as knockout tournaments, round-robin tournaments, and
multiple rank-order tournaments.

Multiple rank-order tournaments, consisting of a series of single rank-order
tournaments, are interesting for the following reasons. In many real series, a
competitor gets a prize and a number of points in each single tournament. Then
the total sum of points determines the total ranking and the bonuses for the
entire series. A straightforward question is how to jointly choose a points
system, a prize structure for each single tournament, and a bonus structure
for the entire series. In particular, since Corollary 6 calls for geometric
prize sequences and Kondratev et al., (2019) justify geometric point
sequences, should we choose the same parameter for both geometric sequences?

Another open question is how to apply for competitions the rules developed for
ranking, voting, or budget allocation. For instance, Kreweras, (1965) and
Fishburn, (1984) developed a probabilistic voting rule known as maximal
lotteries. Brandl et al., (2016) noted that ‘the lotteries returned by
probabilistic social choice functions do not necessarily have to be
interpreted as probability distributions. They can, for instance, also be seen
as fractional allocations of divisible objects such as time shares or monetary
budgets.’ Airiau et al., (2019) argued that ‘the maximal lotteries rule, while
attractive according to consistency axioms, spends the entire budget on the
Condorcet winner if it exists. This is often undesirable in a budgeting
context.’ We can conclude from these arguments that any application of well-
known ranking, voting, or allocation rules must be re-motivated and re-
justified in the context of competitions.

## Appendix

###### Lemma 1

If a prize allocation rule satisfies endowment monotonicity, then it satisfies
endowment continuity.

###### Proof.

Let $\varphi$ be a prize allocation rule satisfying endowment monotonicity.
Let $(N,\mathcal{R},E)$ and $(N,\mathcal{R},E^{\prime})$ be two competitions.
Let $i\in N$. By endowment monotonicity,

$\displaystyle|E-E^{\prime}|$ $\displaystyle=|\sum_{j\in
N}\varphi_{j}(N,\mathcal{R},E)-\sum_{j\in
N}\varphi_{j}(N,\mathcal{R},E^{\prime})|$ $\displaystyle=|\sum_{j\in
N}(\varphi_{j}(N,\mathcal{R},E)-\varphi_{j}(N,\mathcal{R},E^{\prime}))|$
$\displaystyle=\sum_{j\in
N}|\varphi_{j}(N,\mathcal{R},E)-\varphi_{j}(N,\mathcal{R},E^{\prime})|$
$\displaystyle\geq|\varphi_{i}(N,\mathcal{R},E)-\varphi_{i}(N,\mathcal{R},E^{\prime})|.$

This means that
$\varphi_{i}(N,\mathcal{R},E)\to\varphi_{i}(N,\mathcal{R},E^{\prime})$ if
$E\to E^{\prime}$. Hence, $\varphi$ satisfies endowment continuity. ∎

###### Lemma 2

If a prize allocation rule satisfies order preservation, endowment
monotonicity, and consistency, then it satisfies anonymity.

###### Proof.

Let $\varphi$ be a prize allocation rule satisfying order preservation,
endowment monotonicity, and consistency. By Lemma 1, $\varphi$ satisfies
endowment continuity.

Suppose for the sake of contradiction that $\varphi$ does not satisfy
anonymity. Then there exist two competitions $(N,\mathcal{R},E)$ and
$(N^{\prime},\mathcal{R}^{\prime},E)$ with equal numbers of competitors
$|N|=|N^{\prime}|$, and two competitors $i_{1}\in N$ and $j_{1}\in N^{\prime}$
with equal positions $\mathcal{R}(i_{1})=\mathcal{R}^{\prime}(j_{1})$ such
that
$\varphi_{i_{1}}(N,\mathcal{R},E)<\varphi_{j_{1}}(N^{\prime},\mathcal{R}^{\prime},E)$.
Since

$\sum_{k\in N}\varphi_{k}(N,\mathcal{R},E)=E=\sum_{k\in
N^{\prime}}\varphi_{k}(N^{\prime},\mathcal{R}^{\prime},E),$

there exist two competitors $i_{2}\in N$ and $j_{2}\in N^{\prime}$ with equal
position $\mathcal{R}(i_{2})=\mathcal{R}^{\prime}(j_{2})$ such that
$\varphi_{i_{2}}(N,\mathcal{R},E)>\varphi_{j_{2}}(N^{\prime},\mathcal{R}^{\prime},E)$.
Suppose without loss of generality that
$\mathcal{R}^{\prime}(j_{1})=\mathcal{R}(i_{1})<\mathcal{R}(i_{2})=\mathcal{R}^{\prime}(j_{2})$.
Denote $x=\varphi(N,\mathcal{R},E)$ and
$y=\varphi(N^{\prime},\mathcal{R}^{\prime},E)$. By consistency,

$(x_{i_{1}},x_{i_{2}})=\varphi(\\{i_{1},i_{2}\\},\mathcal{R}_{\\{i_{1},i_{2}\\}},x_{i_{1}}+x_{i_{2}})$

and

$(y_{j_{1}},y_{j_{2}})=\varphi(\\{j_{1},j_{2}\\},\mathcal{R}_{\\{j_{1},j_{2}\\}}^{\prime},y_{j_{1}}+y_{j_{2}}).$

This is illustrated in the following way.

(1) $N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{1}$ | $1$ | $x_{i_{1}}$
$i_{2}$ | $2$ | $x_{i_{2}}$
$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$j_{1}$ | $1$ | $y_{j_{1}}$
$j_{2}$ | $2$ | $y_{j_{2}}$

By order preservation, $y_{j_{2}}<x_{i_{2}}\leq x_{i_{1}}<y_{j_{1}}$. One of
the following six cases hold.

###### Case 1

$i_{1}=j_{2}$ and $i_{2}=j_{1}$
By (1),

$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{1}=j_{2}$ | $1$ | $x_{i_{1}}$
$i_{2}=j_{1}$ | $2$ | $x_{i_{2}}$
$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$j_{1}$ | $1$ | $y_{j_{1}}$
$j_{2}$ | $2$ | $y_{j_{2}}$

Then there exists another competitor $k\in U\setminus\\{i_{1},i_{2}\\}$. By
order preservation, endowment continuity, and consistency, there exist prize
endowments $E^{\prime}$ and $E^{\prime\prime}$ such that

$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{1}$ | $1$ | $x_{i_{1}}$
$i_{2}$ | $2$ | $x_{i_{2}}$
$k$ | $3$ | $E^{\prime}-x_{i_{1}}-x_{i_{2}}$
$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{1}$ | $1$ | $x_{i_{1}}$
$k$ | $2$ | $E^{\prime\prime}-x_{i_{1}}-x_{i_{2}}$
$i_{2}$ | $3$ | $x_{i_{2}}$

By order preservation, $E^{\prime}-x_{i_{1}}-x_{i_{2}}\leq x_{i_{2}}\leq
E^{\prime\prime}-x_{i_{1}}-x_{i_{2}}$. By consistency,

$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{1}$ | $1$ | $x_{i_{1}}$
$k$ | $2$ | $E^{\prime}-x_{i_{1}}-x_{i_{2}}$
$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{1}$ | $1$ | $x_{i_{1}}$
$k$ | $2$ | $E^{\prime\prime}-x_{i_{1}}-x_{i_{2}}$

By endowment monotonicity,

$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{1}$ | $1$ | $x_{i_{1}}$
$k$ | $2$ | $x_{i_{2}}$

By order preservation, endowment continuity, consistency, and (1), there exist
prize endowments $E^{\prime\prime\prime}$ and $E^{\prime\prime\prime\prime}$
such that

$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{1}$ | $1$ | $x_{i_{1}}$
$i_{2}$ | $2$ | $E^{\prime\prime\prime}-x_{i_{1}}-x_{i_{2}}$
$k$ | $3$ | $x_{i_{2}}$
$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$j_{1}$ | $1$ | $y_{j_{1}}$
$j_{2}$ | $2$ | $y_{j_{2}}$
$k$ | $3$ | $E^{\prime\prime\prime\prime}-y_{j_{1}}-y_{j_{2}}$

By order preservation, $E^{\prime\prime\prime}-x_{i_{1}}-x_{i_{2}}\leq
x_{i_{1}}$ and $E^{\prime\prime\prime\prime}-y_{j_{1}}-y_{j_{2}}\leq
y_{j_{2}}$. By consistency,

$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{2}$ | $1$ | $E^{\prime\prime\prime}-x_{i_{1}}-x_{i_{2}}$
$k$ | $2$ | $x_{i_{2}}$
$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{2}=j_{1}$ | $1$ | $y_{j_{1}}$
$k$ | $2$ | $E^{\prime\prime\prime\prime}-y_{j_{1}}-y_{j_{2}}$

Then $E^{\prime\prime\prime}-x_{i_{1}}-x_{i_{2}}\leq x_{i_{1}}<y_{j_{1}}$ and
$x_{i_{2}}>y_{j_{2}}\geq E^{\prime\prime\prime\prime}-y_{j_{1}}-y_{j_{2}}$.
This contradicts endowment monotonicity.

###### Case 2

$i_{1}=j_{2}$ and $i_{2}\neq j_{1}$
By order preservation, endowment continuity, consistency, and (1), there
exists prize endowment $E^{\prime}$ such that

$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{1}$ | $1$ | $x_{i_{1}}$
$j_{1}$ | $2$ | $E^{\prime}-x_{i_{1}}-x_{i_{2}}$
$i_{2}$ | $3$ | $x_{i_{2}}$

By order preservation, $E^{\prime}-x_{i_{1}}-x_{i_{2}}\geq x_{i_{2}}$. By
consistency,

$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{1}=j_{2}$ | $1$ | $x_{i_{1}}$
$j_{1}$ | $2$ | $E^{\prime}-x_{i_{1}}-x_{i_{2}}$

Then $x_{i_{1}}<y_{j_{1}}$ and $E^{\prime}-x_{i_{1}}-x_{i_{2}}\geq
x_{i_{2}}>y_{j_{2}}$. By (1), a contradiction follows from Case 1.

###### Case 3

$i_{1}\neq j_{2}$ and $i_{2}=j_{1}$
By order preservation, endowment continuity, consistency, and (1), there
exists a prize endowment $E^{\prime}$ such that

$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{1}$ | $1$ | $x_{i_{1}}$
$j_{2}$ | $2$ | $E^{\prime}-x_{i_{1}}-x_{i_{2}}$
$i_{2}$ | $3$ | $x_{i_{2}}$

By order preservation, $E^{\prime}-x_{i_{1}}-x_{i_{2}}\leq x_{i_{1}}$. By
consistency,

$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$j_{2}$ | $1$ | $E^{\prime}-x_{i_{1}}-x_{i_{2}}$
$i_{2}=j_{1}$ | $2$ | $x_{i_{2}}$

Then $E^{\prime}-x_{i_{1}}-x_{i_{2}}\leq x_{i_{1}}<y_{j_{1}}$ and
$x_{i_{2}}>y_{j_{2}}$. By (1), a contradiction follows from Case 1.

###### Case 4

$i_{1}=j_{1}$ and $i_{2}\neq j_{2}$
By order preservation, endowment continuity, consistency, and (1), there
exists a prize endowment $E^{\prime}$ such that

$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{1}$ | $1$ | $x_{i_{1}}$
$j_{2}$ | $2$ | $E^{\prime}-x_{i_{1}}-x_{i_{2}}$
$i_{2}$ | $3$ | $x_{i_{2}}$

By order preservation, $E^{\prime}-x_{i_{1}}-x_{i_{2}}\geq x_{i_{2}}$. By
consistency,

$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{1}=j_{1}$ | $1$ | $x_{i_{1}}$
$j_{2}$ | $2$ | $E^{\prime}-x_{i_{1}}-x_{i_{2}}$

Then $x_{i_{1}}<y_{j_{1}}$ and $E^{\prime}-x_{i_{1}}-x_{i_{2}}\geq
x_{i_{2}}>y_{j_{2}}$. By (1), this contradicts endowment monotonicity.

###### Case 5

$i_{1}\neq j_{1}$ and $i_{2}=j_{2}$
By order preservation, endowment continuity, consistency, and (1), there
exists a prize endowment $E^{\prime}$ such that

$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{1}$ | $1$ | $x_{i_{1}}$
$j_{1}$ | $2$ | $E^{\prime}-x_{i_{1}}-x_{i_{2}}$
$i_{2}$ | $3$ | $x_{i_{2}}$

By order preservation, $E^{\prime}-x_{i_{1}}-x_{i_{2}}\leq x_{i_{1}}$. By
consistency,

$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$j_{1}$ | $1$ | $E^{\prime}-x_{i_{1}}-x_{i_{2}}$
$i_{2}=j_{2}$ | $2$ | $x_{i_{2}}$

Then $E^{\prime}-x_{i_{1}}-x_{i_{2}}\leq x_{i_{1}}<y_{j_{1}}$ and
$x_{i_{2}}>y_{j_{2}}$. By (1), this contradicts endowment monotonicity.

###### Case 6

$i_{1}\neq j_{1}\neq i_{2}$ and $i_{1}\neq j_{2}\neq i_{2}$
By order preservation, endowment continuity, consistency, and (1), there exist
$z_{j_{1}}$ and $z_{j_{2}}$ such that

$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$i_{1}$ | $1$ | $x_{i_{1}}$
$j_{1}$ | $2$ | $z_{j_{1}}$
$j_{2}$ | $3$ | $z_{j_{2}}$
$i_{2}$ | $4$ | $x_{i_{2}}$

By order preservation, $z_{j_{1}}\leq x_{i_{1}}$ and $z_{j_{2}}\geq
x_{i_{2}}$. By consistency,

$N$ | $\mathcal{R}$ | $\varphi$
---|---|---
$j_{1}$ | $1$ | $z_{j_{1}}$
$j_{2}$ | $2$ | $z_{j_{2}}$

Then $z_{j_{1}}\leq x_{i_{1}}<y_{j_{1}}$ and $z_{j_{2}}\geq
x_{i_{2}}>y_{j_{2}}$. By (1), this contradicts endowment monotonicity.

∎

###### Lemma 3

If two prize allocation rules satisfying endowment monotonicity and
consistency coincide for each competition with two competitors, then the two
rules coincide for each competition with an arbitrary number of competitors.

###### Proof.

Let $\varphi$ and $\varphi^{\prime}$ be two prize allocation rules satisfying
endowment monotonicity and consistency such that
$\varphi(N,\mathcal{R},E)=\varphi^{\prime}(N,\mathcal{R},E)$ for each
competition with two competitors. Suppose for the sake of contradiction that
there exists a competition $(N,\mathcal{R},E)$ such that
$\varphi(N,\mathcal{R},E)\neq\varphi^{\prime}(N,\mathcal{R},E)$. Since

$\sum_{i\in N}\varphi_{i}(N,\mathcal{R},E)=E=\sum_{i\in
N}\varphi_{i}^{\prime}(N,\mathcal{R},E),$

there exist two competitors $i\in N$ and $j\in N$ such that

(2) $\varphi_{i}(N,\mathcal{R},E)<\varphi_{i}^{\prime}(N,\mathcal{R},E)\mbox{
and }\varphi_{j}(N,\mathcal{R},E)>\varphi_{j}^{\prime}(N,\mathcal{R},E).$

By consistency,

$(\varphi_{i}(N,\mathcal{R},E),\varphi_{j}(N,\mathcal{R},E))=\varphi(\\{i,j\\},\mathcal{R}_{\\{i,j\\}},\varphi_{i}(N,\mathcal{R},E)+\varphi_{j}(N,\mathcal{R},E))$

and

$\displaystyle(\varphi_{i}^{\prime}(N,\mathcal{R},E),\varphi_{j}^{\prime}(N,\mathcal{R},E))$
$\displaystyle=\varphi^{\prime}(\\{i,j\\},\mathcal{R}_{\\{i,j\\}},\varphi_{i}^{\prime}(N,\mathcal{R},E)+\varphi_{j}^{\prime}(N,\mathcal{R},E))$
$\displaystyle=\varphi(\\{i,j\\},\mathcal{R}_{\\{i,j\\}},\varphi_{i}^{\prime}(N,\mathcal{R},E)+\varphi_{j}^{\prime}(N,\mathcal{R},E)).$

By endowment monotonicity,

$\varphi_{i}(N,\mathcal{R},E)\leq\varphi_{i}^{\prime}(N,\mathcal{R},E)\mbox{
and }\varphi_{j}(N,\mathcal{R},E)\leq\varphi_{j}^{\prime}(N,\mathcal{R},E),$

or

$\varphi_{i}(N,\mathcal{R},E)\geq\varphi_{i}^{\prime}(N,\mathcal{R},E)\mbox{
and }\varphi_{j}(N,\mathcal{R},E)\geq\varphi_{j}^{\prime}(N,\mathcal{R},E).$

This contradicts (2). Hence,
$\varphi(N,\mathcal{R},E)=\varphi^{\prime}(N,\mathcal{R},E)$ for each
competition $(N,\mathcal{R},E)$. ∎

###### Theorem 1

A prize allocation rule satisfies order preservation, endowment monotonicity,
and consistency if and only if it is an interval rule.

###### Proof.

It is readily checked that each interval rule satisfies order preservation,
endowment monotonicity, and consistency.

Let $\varphi$ be a prize allocation rule satisfying order preservation,
endowment monotonicity, and consistency. By Lemma 1, $\varphi$ satisfies
endowment continuity. By Lemma 2, $\varphi$ satisfies anonymity. By Lemma 3,
we only need to show that $\varphi$ is an interval rule for each competition
with two competitors, since each such rule has a unique consistent extension
to competitions with more competitors. Let $N\subseteq U$ with $|N|=2$ be two
competitors and let $\mathcal{R}$ be a ranking. Denote $N=\\{1,2\\}$ such that
$\mathcal{R}(1)=1$ and $\mathcal{R}(2)=2$. The proof consists of two steps.

Step 1. For each prize endowment $E$, if $E=x_{1}+x_{2}$ such that

(3) $\varphi(N,\mathcal{R},x_{1}+x_{2})=(x_{1},x_{2}),$

then

(4) $\varphi(N,\mathcal{R},x_{1}+x_{1})=(x_{1},x_{1})\ \text{or}\
\varphi(N,\mathcal{R},x_{2}+x_{2})=(x_{2},x_{2}).$

_Proof of Step 1._ Let $E$ be a prize endowment and denote
$\varphi(N,\mathcal{R},E)=(x_{1},x_{2})$. Then $E=x_{1}+x_{2}$ and (3) holds.
By order preservation, $x_{1}\geq x_{2}$. If $x_{1}=x_{2}$, then (4) follows
immediately from (3).

Suppose that $x_{1}>x_{2}$. Let $N^{\prime}=\\{1,2,3\\}$ and let
$\mathcal{R}^{\prime}$ be a ranking of $N^{\prime}$ such that
$\mathcal{R}^{\prime}(1)=1$, $\mathcal{R}^{\prime}(2)=2$, and
$\mathcal{R}^{\prime}(3)=3$. By order preservation and endowment continuity,
there exists a prize endowment $E^{\prime}$ such that
$\varphi_{1}(N^{\prime},\mathcal{R}^{\prime},E^{\prime})+\varphi_{3}(N^{\prime},\mathcal{R}^{\prime},E^{\prime})=x_{1}+x_{2}$.
By anonymity, consistency, and (3),

(5)
$\varphi(N^{\prime},\mathcal{R}^{\prime},E^{\prime})=(x_{1},E^{\prime}-x_{1}-x_{2},x_{2}).$

By order preservation, $x_{1}+x_{2}+x_{2}\leq E^{\prime}\leq
x_{1}+x_{1}+x_{2}$. By anonymity and consistency,

(6) $\displaystyle\varphi(N,\mathcal{R},E^{\prime}-x_{1})$
$\displaystyle=(E^{\prime}-x_{1}-x_{2},x_{2})$ (7) and
$\displaystyle\varphi(N,\mathcal{R},E^{\prime}-x_{2})$
$\displaystyle=(x_{1},E^{\prime}-x_{1}-x_{2}).$

If $E^{\prime}=x_{1}+x_{2}+x_{2}$, then (4) follows immediately from (6). If
$E^{\prime}=x_{1}+x_{1}+x_{2}$, then (4) follows immediately from (7).

Suppose that $x_{1}+x_{2}+x_{2}<E^{\prime}<x_{1}+x_{1}+x_{2}$. Then
$E^{\prime}-x_{1}<x_{1}+x_{2}<E^{\prime}-x_{2}$. By endowment monotonicity,
(3), (6), and (7),

(8)
$\varphi(N,\mathcal{R},E^{\prime\prime})=\begin{cases}(E^{\prime\prime}-x_{2},x_{2})&\text{if
$E^{\prime}-x_{1}\leq E^{\prime\prime}\leq x_{1}+x_{2}$;}\\\
(x_{1},E^{\prime\prime}-x_{1})&\text{if $x_{1}+x_{2}\leq E^{\prime\prime}\leq
E^{\prime}-x_{2}$.}\end{cases}$

Denote
$(y_{1},y_{2},y_{3})=\varphi(N^{\prime},\mathcal{R}^{\prime},x_{1}+x_{2}+x_{2})$
and
$(z_{1},z_{2},z_{3})=\varphi(N^{\prime},\mathcal{R}^{\prime},x_{1}+x_{1}+x_{2})$.
By endowment monotonicity and (5),

$\displaystyle y_{1}$ $\displaystyle\leq$ $\displaystyle x_{1}$
$\displaystyle\leq z_{1},$ $\displaystyle y_{2}$ $\displaystyle\leq
E^{\prime}-$ $\displaystyle x_{1}-x_{2}$ $\displaystyle\leq z_{2},$
$\displaystyle y_{3}$ $\displaystyle\leq$ $\displaystyle x_{2}$
$\displaystyle\leq z_{3}.$

Then $y_{1}+y_{2}\leq E^{\prime}-x_{2}$ and $E^{\prime}-x_{1}\leq
z_{2}+z_{3}$. Since $y_{3}\leq x_{2}$ and
$y_{1}+y_{2}+y_{3}=x_{1}+x_{2}+x_{2}$, we have $x_{1}+x_{2}\leq
x_{1}+x_{2}+x_{2}-y_{3}=y_{1}+y_{2}$. Since $x_{1}\leq z_{1}$ and
$z_{1}+z_{2}+z_{3}=x_{1}+x_{1}+x_{2}$, we have
$z_{2}+z_{3}=x_{1}+x_{1}+x_{2}-z_{1}\leq x_{1}+x_{2}$. This means that

$\displaystyle x_{1}+x_{2}\leq y_{1}+y_{2}\leq E^{\prime}-x_{2},$ and
$\displaystyle E^{\prime}-x_{1}\leq z_{2}+z_{3}\leq x_{1}+x_{2}.$

By anonymity, consistency, and (8),

$\displaystyle\varphi_{1}(N,\mathcal{R},y_{1}+y_{3})=y_{1}=\varphi_{1}(N,\mathcal{R},y_{1}+y_{2})=x_{1},$
and
$\displaystyle\varphi_{2}(N,\mathcal{R},z_{1}+z_{3})=z_{3}=\varphi_{2}(N,\mathcal{R},z_{2}+z_{3})=x_{2}.$

Since $y_{1}\leq z_{1}$ and $y_{3}\leq z_{3}$, we have $y_{1}+y_{3}\leq
z_{1}+z_{3}$. By endowment monotonicity and (8), for each $E^{\prime\prime}$
such that $\varphi_{1}(N,\mathcal{R},E^{\prime\prime})\geq x_{1}$, we have
$E^{\prime\prime}\geq x_{1}+x_{2}$. Since
$\varphi_{1}(N,\mathcal{R},y_{1}+y_{3})=x_{1}$, we have $y_{1}+y_{3}\geq
x_{1}+x_{2}$. By endowment monotonicity and (8), for each $E^{\prime\prime}$
such that $\varphi_{2}(N,\mathcal{R},E^{\prime\prime})\leq x_{2}$, we have
$E^{\prime\prime}\leq x_{1}+x_{2}$. Since
$\varphi_{2}(N,\mathcal{R},z_{1}+z_{3})=x_{2}$, we have $z_{1}+z_{3}\leq
x_{1}+x_{2}$. Hence,

$x_{1}+x_{2}\leq y_{1}+y_{3}\leq z_{1}+z_{3}\leq x_{1}+x_{2}.$

Then $y_{1}+y_{3}=x_{1}+x_{2}$ and $z_{1}+z_{3}=x_{1}+x_{2}$. Since
$y_{1}=x_{1}$ and $z_{3}=x_{2}$, we have $y_{3}=x_{2}$ and $z_{1}=x_{1}$. This
means that $y=(x_{1},x_{2},x_{2})$ and $z=(x_{1},x_{1},x_{2})$. By anonymity
and consistency, $\varphi(N,\mathcal{R},x_{1}+x_{1})=(x_{1},x_{1})$ and
$\varphi(N,\mathcal{R},x_{2}+x_{2})=(x_{2},x_{2})$. Hence, (4) holds.

Step 2. There exist disjoint intervals $(a_{1},b_{1}),(a_{2},b_{2}),\ldots$
with $a_{1},a_{2},\ldots\in\mathbb{R}_{+}$ and
$b_{1},b_{2},\ldots\in\mathbb{R}_{+}\cup\\{+\infty\\}$ such that

(9) $\varphi(N,\mathcal{R},E)=\begin{cases}(E-a_{k},a_{k})&\text{if
$a_{k}+a_{k}\leq E\leq b_{k}+a_{k}$};\\\ (b_{k},E-b_{k})&\text{if
$b_{k}+a_{k}\leq E\leq b_{k}+b_{k}$};\\\
\frac{E}{2}&\text{otherwise}.\end{cases}$

_Proof of Step 2._ If
$\varphi_{1}(N,\mathcal{R},E)=\varphi_{2}(N,\mathcal{R},E)$ for each prize
endowment $E$, then (9) follows immediately by defining $a_{k}=b_{k}=0$ for
each $k$.

Let $E$ be a prize endowment with
$\varphi_{1}(N,\mathcal{R},E)>\varphi_{2}(N,\mathcal{R},E)$. Denote
$\varphi(N,\mathcal{R},E)=(x_{1},x_{2})$. Then $E=x_{1}+x_{2}$, $x_{1}>x_{2}$,
and

$\varphi(N,\mathcal{R},x_{1}+x_{2})=(x_{1},x_{2}).$

By Step 1, $\varphi(N,\mathcal{R},x_{1}+x_{1})=(x_{1},x_{1})$ or
$\varphi(N,\mathcal{R},x_{2}+x_{2})=(x_{2},x_{2})$.

Suppose that $\varphi(N,\mathcal{R},x_{1}+x_{1})=(x_{1},x_{1})$. Define

(10) $\displaystyle b_{E}$ $\displaystyle=x_{1}$ and $\displaystyle a_{E}$
$\displaystyle=\min_{E^{\prime}\in\mathbb{R}_{+}}\\{\varphi_{2}(N,\mathcal{R},E^{\prime})\mid\varphi_{1}(N,\mathcal{R},E^{\prime})=b_{E}\\}.$

Then $a_{E}<b_{E}$ since $a_{E}\leq x_{2}<x_{1}=b_{E}$. Moreover,
$\varphi(N,\mathcal{R},b_{E}+a_{E})=(b_{E},a_{E})$ and
$\varphi(N,\mathcal{R},b_{E}+b_{E})=(b_{E},b_{E})$. By endowment monotonicity,

(11) $\varphi(N,\mathcal{R},E^{\prime})=(b_{E},E^{\prime}-b_{E})\quad\text{if
$b_{E}+a_{E}\leq E^{\prime}\leq b_{E}+b_{E}$.}$

This also means that

(12) $b_{E}+a_{E}<E^{\prime}<b_{E}+b_{E}\quad\text{if
$a_{E}<\varphi_{2}(N,\mathcal{R},E^{\prime})<b_{E}$}.$

Let $E^{\prime}$ be a prize endowment with
$a_{E}+a_{E}<E^{\prime}<b_{E}+a_{E}$. Denote
$(y_{1},y_{2})=\varphi(N,\mathcal{R},E^{\prime})$. By endowment monotonicity
and (10), $y_{1}<b_{E}$ and $y_{2}\leq a_{E}$. Then $a_{E}<y_{1}$ since
$a_{E}\leq a_{E}+a_{E}-y_{2}<E^{\prime}-y_{2}=y_{1}$. By Step 1,
$\varphi(N,\mathcal{R},y_{1}+y_{1})=(y_{1},y_{1})$ or
$\varphi(N,\mathcal{R},y_{2}+y_{2})=(y_{2},y_{2})$. If
$\varphi(N,\mathcal{R},y_{1}+y_{1})=(y_{1},y_{1})$, then
$a_{E}<y_{1}=\varphi_{2}(N,\mathcal{R},y_{1}+y_{1})<b_{E}$, (12) implies
$b_{E}+a_{E}<y_{1}+y_{1}<b_{E}+b_{E}$ and (11) implies
$y_{1}=\varphi_{1}(N,\mathcal{R},y_{1}+y_{1})=b_{E}$, which is a
contradiction. This means that
$\varphi(N,\mathcal{R},\varphi_{2}(N,\mathcal{R},E^{\prime})+\varphi_{2}(N,\mathcal{R},E^{\prime}))=(\varphi_{2}(N,\mathcal{R},E^{\prime}),\varphi_{2}(N,\mathcal{R},E^{\prime}))$
for each prize endowment $E^{\prime}$ with
$a_{E}+a_{E}<E^{\prime}<b_{E}+a_{E}$. Since
$\varphi_{2}(N,\mathcal{R},b_{E}+a_{E})=a_{E}$, we have
$\varphi(N,\mathcal{R},a_{E}+a_{E})=(a_{E},a_{E})$ by endowment continuity. By
endowment monotonicity,

$\varphi(N,\mathcal{R},E^{\prime})=(E^{\prime}-a_{E},a_{E})\quad\text{if
$a_{E}+a_{E}\leq E^{\prime}\leq b_{E}+a_{E}$.}$

Suppose that $\varphi(N,\mathcal{R},x_{2}+x_{2})=(x_{2},x_{2})$. Define

(13) $\displaystyle a_{E}$ $\displaystyle=x_{2}$ and $\displaystyle b_{E}$
$\displaystyle=\sup_{E^{\prime}\in\mathbb{R}_{+}}\\{\varphi_{1}(N,\mathcal{R},E^{\prime})\mid\varphi_{2}(N,\mathcal{R},E^{\prime})=a_{E}\\}.$

Then $a_{E}<b_{E}$ since $a_{E}=x_{2}<x_{1}\leq b_{E}$. Moreover,
$\varphi(N,\mathcal{R},a_{E}+a_{E})=(a_{E},a_{E})$ and
$\varphi(N,\mathcal{R},b_{E}+a_{E})=(b_{E},a_{E})$. By endowment monotonicity,

(14) $\varphi(N,\mathcal{R},E^{\prime})=(E^{\prime}-a_{E},a_{E})\ \text{if
$a_{E}+a_{E}\leq E^{\prime}\leq b_{E}+a_{E}$.}$

This also means that

(15) $a_{E}+a_{E}<E^{\prime}<b_{E}+a_{E}\ \text{if
$a_{E}<\varphi_{1}(N,\mathcal{R},E^{\prime})<b_{E}$}.$

Let $E^{\prime}$ be a prize endowment with
$b_{E}+a_{E}<E^{\prime}<b_{E}+b_{E}$. Denote
$(y_{1},y_{2})=\varphi(N,\mathcal{R},E^{\prime})$. By endowment monotonicity
and (13), $b_{E}\leq y_{1}$ and $a_{E}<y_{2}$. Then $y_{2}<b_{E}$ since
$y_{2}=E^{\prime}-y_{1}<b_{E}+b_{E}-y_{1}\leq b_{E}$. By Step 1,
$\varphi(N,\mathcal{R},y_{1}+y_{1})=(y_{1},y_{1})$ or
$\varphi(N,\mathcal{R},y_{2}+y_{2})=(y_{2},y_{2})$. If
$\varphi(N,\mathcal{R},y_{2}+y_{2})=(y_{2},y_{2})$, then
$a_{E}<y_{2}=\varphi_{1}(N,\mathcal{R},y_{2}+y_{2})<b_{E}$, (15) implies
$a_{E}+a_{E}<y_{2}+y_{2}<b_{E}+a_{E}$ and (14) implies
$y_{2}=\varphi_{2}(N,\mathcal{R},y_{2}+y_{2})=a_{E}$, which is a
contradiction. This means that
$\varphi(N,\mathcal{R},\varphi_{1}(N,\mathcal{R},E^{\prime})+\varphi_{1}(N,\mathcal{R},E^{\prime}))=(\varphi_{1}(N,\mathcal{R},E^{\prime}),\varphi_{1}(N,\mathcal{R},E^{\prime}))$
for each prize endowment $E^{\prime}$ with
$b_{E}+a_{E}<E^{\prime}<b_{E}+b_{E}$. Since
$\varphi_{1}(N,\mathcal{R},b_{E}+a_{E})=b_{E}$, we have
$\varphi(N,\mathcal{R},b_{E}+b_{E})=(b_{E},b_{E})$ by endowment continuity. By
endowment monotonicity,

$\varphi(N,\mathcal{R},E^{\prime})=(b_{E},E^{\prime}-a_{E})\ \text{if
$b_{E}+a_{E}\leq E^{\prime}\leq b_{E}+b_{E}$.}$

The set of intervals
$\\{(a_{E},b_{E}):\varphi_{1}(N,\mathcal{R},E)>\varphi_{2}(N,\mathcal{R},E)\\}$
is countable because the intervals are disjoint and we can construct a one-to-
one correspondence between each interval and a rational number from the
interval. Hence, (9) holds.

By anonymity, (9) holds for each competition with two competitors. By Lemma 3,
the description of the corresponding interval rule is the unique consistent
extension of (9) to competitions with more competitors. ∎

###### Remark

The axioms order preservation, endowment monotonicity, and consistency (used
in Theorem 1) are logically independent.

###### Proof.

The rule which allocates all the prize money to the competitor with the lowest
position satisfies endowment monotonicity and consistency, but does not
satisfy order preservation. The rule which coincides with the Equal Division
rule for competitions with a prize endowment of at most one dollar, and
coincides with the Winner-Takes-All rule for competitions with a higher prize
endowment, satisfies order preservation and consistency, but does not satisfy
endowment monotonicity. The rule from Example 3 satisfies order preservation
and endowment monotonicity, but does not satisfy consistency. ∎

###### Lemma 4

A prize allocation rule satisfies endowment additivity if and only if it
satisfies scale invariance.

###### Proof.

Let $\varphi$ be a prize allocation rule satisfying scale invariance. Let
$(N,\mathcal{R},E)$ and $(N,\mathcal{R},E^{\prime})$ be two competitions and
let $i\in N$ be a competitor. Then

$\displaystyle\varphi_{i}(N,\mathcal{R},E+E^{\prime})$
$\displaystyle=(E+E^{\prime})\varphi_{i}(N,\mathcal{R},1)$
$\displaystyle=E\varphi_{i}(N,\mathcal{R},1)+E^{\prime}\varphi_{i}(N,\mathcal{R},1)$
$\displaystyle=\varphi_{i}(N,\mathcal{R},E)+\varphi_{i}(N,\mathcal{R},E^{\prime}).$

Hence, $\varphi$ satisfies endowment additivity.

Now, let $\varphi$ be a prize allocation rule satisfying endowment additivity.
Then $\varphi$ satisfies endowment monotonicity. By Lemma 1, $\varphi$
satisfies endowment continuity. Let $(N,\mathcal{R},E)$ be a competition and
let $i\in N$ be a competitor. If $E$ is a rational number, then there exist
two natural numbers $p\in\mathbb{N}$ and $q\in\mathbb{N}$ such that
$E=\frac{p}{q}$. By endowment additivity,

$\displaystyle\varphi_{i}(N,\mathcal{R},E)$
$\displaystyle=\varphi_{i}(N,\mathcal{R},\tfrac{p}{q})$
$\displaystyle=p\varphi_{i}(N,\mathcal{R},\tfrac{1}{q})$
$\displaystyle=\tfrac{p}{q}q\varphi_{i}(N,\mathcal{R},\tfrac{1}{q})$
$\displaystyle=\tfrac{p}{q}\varphi_{i}(N,\mathcal{R},1)$
$\displaystyle=E\varphi_{i}(N,\mathcal{R},1).$

By endowment continuity,
$\varphi_{i}(N,\mathcal{R},E)=E\varphi_{i}(N,\mathcal{R},1)$ for each real
number $E$. Hence, $\varphi$ satisfies scale invariance. ∎

###### Theorem 2

A prize allocation rule satisfies anonymity, order preservation, winner strict
endowment monotonicity, and local consistency if and only if it is a single-
parametric rule.

###### Proof.

It is readily checked that each single-parametric rule satisfies anonymity,
order preservation, winner strict endowment monotonicity, and local
consistency.

Let $\varphi$ be a prize allocation rule satisfying anonymity, order
preservation, winner strict endowment monotonicity, and local consistency.
Then $\varphi$ satisfies endowment monotonicity. By Lemma 1, $\varphi$
satisfies endowment continuity. Let $N\subseteq U$ with $|N|=2$ be a set of
two competitors and let $\mathcal{R}$ be the corresponding ranking. Denote
$N=\\{1,2\\}$ such that $\mathcal{R}(1)=1$ and $\mathcal{R}(2)=2$. For each
$x\in\mathbb{R}_{+}$, define $f(x)=y$ if and only if
$\varphi_{1}(N,\mathcal{R},x+y)=x$ and $\varphi_{2}(N,\mathcal{R},x+y)=y$. By
winner strict endowment monotonicity, endowment continuity, and order
preservation, $f:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a well-defined,
continuous and non-decreasing function with $f(x)\leq x$ for each $x$. By
anonymity, $\varphi$ is a single-parametric rule with representation $f$ for
each competition with two competitors. Let $\varphi^{\prime}$ be a single-
parametric rule with representation $f$ for each competition with arbitrary
number of competitors.

Suppose for the sake of contradiction that there exists a competition
$(N,\mathcal{R},E)$ such that
$\varphi^{\prime}(N,\mathcal{R},E)\neq\varphi(N,\mathcal{R},E)$. Denote
$N=\\{1,\ldots,|N|\\}$ such that $\mathcal{R}(k)=k$ for each $k\in N$. Let
$i\in N$ be such that
$\varphi_{i}^{\prime}(N,\mathcal{R},E)\neq\varphi_{i}(N,\mathcal{R},E)$ and
$\varphi_{j}^{\prime}(N,\mathcal{R},E)=\varphi_{j}(N,\mathcal{R},E)$ for each
$j\in N$ with $j<i$. Suppose without loss of generality that
$\varphi_{i}^{\prime}(N,\mathcal{R},E)>\varphi_{i}(N,\mathcal{R},E)$. By local
consistency and anonymity,

$\displaystyle\varphi_{i+1}^{\prime}(N,\mathcal{R},E)$
$\displaystyle=\varphi_{i+1}^{\prime}(\\{i,i+1\\},\mathcal{R}_{\\{i,i+1\\}},\varphi_{i}^{\prime}(N,\mathcal{R},E)+\varphi_{i+1}^{\prime}(N,\mathcal{R},E))$
$\displaystyle=f(\varphi_{i}^{\prime}(N,\mathcal{R},E))$ $\displaystyle\geq
f(\varphi_{i}(N,\mathcal{R},E))$
$\displaystyle=\varphi_{i+1}(\\{i,i+1\\},\mathcal{R}_{\\{i,i+1\\}},\varphi_{i}(N,\mathcal{R},E)+\varphi_{i+1}(N,\mathcal{R},E))$
$\displaystyle=\varphi_{i+1}(N,\mathcal{R},E).$

In a similar way, this implies that
$\varphi_{i+2}^{\prime}(N,\mathcal{R},E)\geq\varphi_{i+2}(N,\mathcal{R},E)$.
Continuing this reasoning,
$\varphi_{j}^{\prime}(N,\mathcal{R},E)\geq\varphi_{j}(N,\mathcal{R},E)$ for
each $j\in N$ with $j>i$. This means that

$\sum_{j\in N}\varphi_{j}^{\prime}(N,\mathcal{R},E)>\sum_{j\in
N}\varphi_{j}(N,\mathcal{R},E)=E.$

This is a contradiction. Hence,
$\varphi^{\prime}(N,\mathcal{R},E)=\varphi(N,\mathcal{R},E)$ for each
competition $(N,\mathcal{R},E)$. ∎

###### Remark

The axioms anonymity, order preservation, winner strict endowment
monotonicity, and local consistency (used in Theorem 2) are logically
independent.

###### Proof.

The rule from Example 3 satisfies order preservation, winner strict endowment
monotonicity, and local consistency, but does not satisfy anonymity. The
geometric rule with representation $\lambda=2$ satisfies anonymity, winner
strict endowment monotonicity, and local consistency, but does not satisfy
order preservation. The rule from Example 1 satisfies anonymity, order
preservation, and local consistency, but does not satisfy winner strict
endowment monotonicity. The rule which coincides with the ED rule for
competitions with two competitors and coincides with the WTA rule for
competitions with more than two competitors satisfies anonymity, order
preservation, and winner strict endowment monotonicity, but does not satisfy
local consistency. ∎

###### Theorem 3

A prize allocation rule satisfies anonymity, order preservation, winner strict
endowment monotonicity, and top consistency if and only if it is a parametric
rule.

###### Proof.

It is readily checked that each parametric rule satisfies anonymity, order
preservation, winner strict endowment monotonicity, and top consistency.

Let $\varphi$ be a prize allocation rule satisfying anonymity, order
preservation, winner strict endowment monotonicity, and top consistency. Then
$\varphi$ satisfies endowment monotonicity. By Lemma 1, $\varphi$ satisfies
endowment continuity. We show that $\varphi$ is a parametric rule by induction
on the number of competitors. Clearly,
$\varphi_{i}(N,\mathcal{R},E)=E=f_{1}(x)$ holds for each competition
$(N,\mathcal{R},E)$ with $|N|=1$ and each competitor $i\in N$. Let
$n\in\mathbb{N}$ and assume that there exist continuous and non-decreasing
functions $f_{1},f_{2},\ldots,f_{n}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ with
$f_{1}(x)=x$ for each $x$ and $f_{k+1}(x)\leq f_{k}(x)$ for each $k$ such that
for each competition $(N,\mathcal{R},E)$ with $|N|\leq n$ and each competitor
$i\in N$, we have $\varphi_{i}(N,\mathcal{R},E)=f_{\mathcal{R}(i)}(x)$, where
$x\in\mathbb{R}_{+}$ is such that $\sum_{k=1}^{|N|}f_{k}(x)=E$.

Let $N=\\{1,\ldots,n+1\\}$ be a set of $n+1$ competitors and let $\mathcal{R}$
be the corresponding ranking such that $\mathcal{R}(i)=i$ for all $i\in N$. By
order preservation, endowment continuity, and winner strict endowment
monotonicity, $\varphi_{1}(N,\mathcal{R},E)$ is unbounded, continuous and
increasing function of $E$. Hence, for each $x\in\mathbb{R}_{+}$, there is a
unique $E$ such that $\varphi_{1}(N,\mathcal{R},E)=x$, we can define
$f_{n+1}(x)=\varphi_{n+1}(N,\mathcal{R},E)$ and, by top consistency and
inductive hypotheses,
$\varphi_{2}(N,\mathcal{R},E)=f_{2}(x),\ldots,\varphi_{n}(N,\mathcal{R},E)=f_{n}(x)$.
By endowment continuity, winner strict endowment monotonicity, and order
preservation, $f_{n+1}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a continuous and
non-decreasing function with $f_{n+1}(x)\leq f_{n}(x)$ for each $x$. By
anonymity, $\varphi$ is a parametric rule with representation
$f_{1},f_{2},\ldots,f_{n+1}$ for each competition with $n+1$ competitors. ∎

###### Remark

The axioms anonymity, order preservation, winner strict endowment
monotonicity, and top consistency (used in Theorem 3) are logically
independent.

###### Proof.

The rule from Example 3 satisfies order preservation, winner strict endowment
monotonicity, and top consistency, but does not satisfy anonymity. The
geometric rule with representation $\lambda=2$ satisfies anonymity, winner
strict endowment monotonicity, and top consistency, but does not satisfy order
preservation. The rule from Example 1 satisfies anonymity, order preservation,
and top consistency, but does not satisfy winner strict endowment
monotonicity. The rule which coincides with the ED rule for competitions with
two competitors and coincides with the WTA rule for competitions with more
than two competitors satisfies anonymity, order preservation, and winner
strict endowment monotonicity, but does not satisfy top consistency. ∎

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