Fair and consistent prize allocation in competitions
aa r X i v : . [ c s . G T ] A ug 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 endowmentbe allocated among the competitors? We study consistent prize allocation rules satisfyingelementary solidarity and fairness principles. In particular, we axiomatically characterizetwo families of rules satisfying anonymity, order preservation, and endowment monotonicity,which all fall between the Equal Division rule and the Winner-Takes-All rule. Specificcharacterizations of rules and subfamilies are directly obtained.
Keywords : fair allocation, rank-order tournament, prize structure, tournament design, ax-iomatic analysis, [email protected] — National Research University Higher School of Economics, 16, Soyuza Pechatnikovst., St. Petersburg, 190121, Russia — http://orcid.org/[email protected] — corresponding author — National Research University Higher School of Economics,16, Soyuza Pechatnikov st., St. Petersburg, 190121, Russia; Institute for Regional Economic Studies RAS,38, Serpuhovskaya st., St. Petersburg, 190013, Russia — http://orcid.org/0000-0002-8424-8198We are grateful to Victor Mironov (Karelian Research Center RAS, Petrozavodsk), Alexander Nesterov,Egor Ianovski, Herv´e Moulin, and other colleagues from the International Laboratory of Game Theory andDecision Making for helpful comments. Special thanks to William Thomson (University of Rochester) fordetailed comments and suggestions. We thank David Connolly (Higher School of Economics, Moscow) forhelp with language editing. . Introduction
Innovation and crowdsourcing competitions, sales competitions in companies and sport-ing 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 isheld once, and the participants know the rules and the prize structure in advance. Theabsolute result of a participant can be determined by, for example, the volume of sales ina sales competition, the number of strokes in golf, or the time of elimination from a pokertournament. However, a feature of the ranking format is that participants receive prizesaccording to their relative results, while tournament organizers are free to choose the prizestructure.Examples of prize structures are presented in Table 1. The PGA TOUR conducts manyregular golf tournaments with different prize endowments, but the same rules for distribut-ing prizes. Table 1 shows examples of two such tournaments with a prize endowment of$9.3 million and $6.6 million. The winner of the tournament receives 18% of the total prizeendowment, the runner-up 10.9%, and so on. WCOOP Poker Tournament has a comparableprize endowment but follows a different prize-money distribution. What exactly are the sim-ilarities and differences of various prize structures? What general principles can tournamentorganizers follow when choosing a prize structure?
Table 1.
Examples of single rank-order tournaments
WCOOP 2019 Main Event Poker TournamentPosition 1 2 3 4 5 6 7 8 9 10 Top 10 Endow.Main 1,666 1,188 847 603 430 307 219 156 111 79 5,605 11,180Share 14.9 10.6 7.57 5.40 3.85 2.74 1.96 1.39 0.99 0.71 50.1 100PGA TOUR 2019/20 Golf TournamentsPosition 1 2 3 4 5 6 7 8 9 10 Top 10 Endow.Genesis 1,674 1,014 642 456 381 337 314 291 272 253 5,633 9,300Safeway 1,188 719 455 323 271 239 223 206 193 180 3,998 6,600Share 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 to10 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 prin-ciples of justice must be universal and clear, they must also be found in particular examplesof tournaments and competitions. Consistent with the principle of anonymity , is that theparticipants’ rewards depend only on their position in the competition. That is, when theorganizers 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 alower reward. This creates the right incentives for participants, as the efforts made duringpreparation 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 eachposition. This creates uniform incentives for all participants, as both leaders and outsidersare interested in increasing the total prize endowment of the tournament.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 theparticipants. We call a rule for the distribution of prizes ‘fair’ if it satisfies the principles ofanonymity, order preservation, and endowment monotonicity. Anonymity and order preser-vation are standard principles for single rank-order tournaments; see the literature overviewin subsection 1.1. Endowment monotonicity is a standard principle in fair allocation liter-ature; see Moulin (2003). These three principles of justice are so undemanding that eventogether they do not exclude any visible class of prize structures. Therefore, we are free toadd another principle.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 corre-sponding prizes. Then some participants leave with their prizes. The remaining participantsdecide to redistribute the remaining prize endowment, taking into account the modified com-position of the participants. The rule is consistent if such a redistribution does not changethe amount the remaining participants receive.Although consistency with respect to single rank-order tournaments looks demanding, it isoften 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 competitionwith 100 participants, the company awards a number of fixed-size cash prizes of $2,000. Thelimited prize endowment provides a certain number of equal prizes to the participants withthe highest positions, and any remaining balance goes to the next participant. So, if theprize fund is $6,000, then the three best participants A, B, and C win $2,000 each. If thefund grows to $11,000, then the top five participants A, B, C, D, and E win $2,000 each, andthe participant in sixth position, F wins $1,000. Consistency requires that if the organizerhas already transferred payments to participants B, C, and F, then re-applying the rule tothe competition without participants B, C, and F and with $6,000 prize endowment willleave everything as before: participants A, D, and E receive $2,000 each, since they are thebest among the remaining participants.What do the fair and consistent prize distribution rules look like? The Equal Divisionrule, the Winner-Takes-All rule, and the aforementioned rule illustrate the main feature ofsuch rules. Our first main result shows that all fair and consistent rules are some combination of these three rules (Theorem 1). Since this is still a large family, we can strengthen ourrequirements for its properties.In particular, we can strengthen order preservation. We say that the rule satisfies strictorder preservation if prizes for positions from the first to the last form a strictly decreasingsequence. There are no fair and consistent rules for the distribution of prizes that satisfystrict order preservation. On the other hand, the Winner-Takes-All rule is the only fair andconsistent rule in which the prize for the first position always exceeds the prize for the lastposition (Corollary 1).We can also strengthen endowment monotonicity. To do this, we examined three morestringent properties. The first strengthening of endowment monotonicity is winner strictendowment monotonicity . This property additionally requires that with an increase in theprize endowment, the prize for the first position strictly increases. This leads to the followingone-parameter family. Each rule from this family sets a maximum size of an individual prize,which all competitors receive equally regardless of their position, while all the excess goesto the winner (Corollary 2(i)). 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 endof 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 ofthe fund is $24,000, then the best worker gets $6,000, and each of the others receives $2,000.The second strengthening of endowment monotonicity is strict endowment monotonicity .This property requires that with an increase in the prize endowment, the prize for eachposition strictly increases. The Equal Division rule is the only fair and consistent rule thatsatisfies strict endowment monotonicity (Corollary 2(ii)).The third strengthening of endowment monotonicity is also suggested by the examplesfrom Table 1. The prize structures from golf and poker are proportional , that is, when theprize fund is increased k times, the prize for each position also increases k times. We showthat the only two fair, consistent, and proportional rules for the distribution of prizes arethe Equal Division rule and the Winner-Takes-All rule (Corollary 3). As a result, we seethat the prize structures from poker and golf are not consistent.Since consistency can be an excessive requirement, we further formulate a weaker principleof local consistency . Imagine that a tournament is held among participants of different skilllevels. The organizer could distribute the prize fund according to the general ranking of allparticipants; or the organizer could divide the prize fund into two parts and distribute theprizes separately among the participants of high and low levels. If both methods lead to thesame distribution of prizes, then the rule is locally consistent. For example, Table 1 showsthe top 10 poker tournament participants receive a total of $5.605 million of a total prize We were inspired by local stability introduced by Young (1988). endowment of $11.18 million. Local consistency requires that if you apply the rule separatelyto the top 10 participants and a prize endowment of $5.605 million, then they will receivethe same prizes.Our second main result describes the class of fair and locally consistent rules that satisfywinner strict endowment monotonicity. Each such rule is generated by some continuous andnon-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 soon. The value of x is unambiguously determined from the condition that the organizerdistributes the whole prize endowment (Theorem 2, Corollary 4). This class contains, forinstance, arithmetic type of prize allocation rules (Example 6 in Section 3). Further, weshow that the prize structure is fair, locally consistent and proportional, iff the prizes forma geometric sequence with parameter λ , that is, the prize for position r + 1 is the fraction λ of the prize for position r (Corollary 5). For example, the prizes in the final stage of thepoker tournament from Table 1 form a geometric sequence with λ = 0 . and, therefore,are locally consistent. The prizes in golf tournaments, however, do not form a geometricsequence and, therefore, are not locally consistent.Our results are presented in Table 2. Note that no prize allocation rule satisfies all 10properties considered. However, the geometric rules with a parameter λ such that < λ < satisfy all properties except for consistency.The paper is structured as follows. The next subsection reviews the literature. Section 2introduces the model and characterizes fair and consistent prize allocation rules. Section 3characterizes fair and locally consistent prize allocation rules. Section 4 concludes. Theproofs are contained in the Appendix.1.1. Literature overview.
In this brief literature overview, we highlight other papers re-lated to our key topics: fairness, consistency, rank-order tournaments, prize allocation, andaxiomatic characterization.Francis Galton (1902) was the first to ask how the total prize money should be divided intoprizes for each ranking position in a rank-order tournament. Using a probabilistic approachwith order statistics, Galton concluded that when only two prizes are given, the first prizeshould approximately be three times the value of the second.The seminal paper of Lazear and Rosen (1981) analyzed rank-order tournaments froman economic perspective. In their model, a firm assigns a certain prize to each rankingposition. Then each worker chooses a level of effort which leads to an output. The rela-tive ranking of the outputs determines the prizes for workers. Lazear and Rosen found theoptimal prize structure that maximizes the worker’s utility in equilibrium. The subsequentliterature proposed similar models and studied optimal prize structures that maximize differ-ent goals, such as the total output (Glazer and Hassin, 1988), the revenue to the auctioneer
Table 2.
Properties of prize allocation rules
Winner- Winner- Fair andFair and Takes- Takes- Equal WTA LocallyConsistent All Surplus Division and Consistent GeometricRules (WTA) (WTS) (ED) ED Rules RulesTh. 1 Cor. 1 Cor. 2(i) Cor. 2(ii) Cor. 3 Th. 2 Cor. 5Cor. 4 a k , b k , one a ∈ [0 , ∞ ] one two function f : λ ∈ [0 , k =1 , , . . . rule rule rules ≤ f ( x ) ≤ x Anonymity + + + + + +* +*Order Pres. +* + +* +* +* +* +*Win.-Los. St. only +* only − only f ( x ) < x all butOrd. Pres. WTA WTA WTA for x > EDStrict − − − − −
Model.
Let U be a countable set of at least three potential competitors . On each givenoccasion, a finite subset N ⊆ U is in a competition. This competition results in a ranking ,a bijection R : N → { , . . . , | N |} assigning to each competitor a position . Here, competitor i ∈ N has a higher position in the ranking than competitor j ∈ N if R ( i ) < R ( j ) . The prizeendowment E ∈ R + is the amount of prize money to be allocated among the competitors.Thus, a competition is a triple ( N, R , E ) .Assuming that more money is better for each competitor, the preferences of the com-petitors over the feasible prize allocations are in conflict. In order to select a reasonablecompromise for each competition, we study prize allocation rules ϕ which assign to eachcompetition ( N, R , E ) an allocation of the prize endowment among the competitors, i.e. ϕ ( N, R , E ) ∈ R N + is such that X i ∈ N ϕ i ( N, R , E ) = E. Throughout this paper, ϕ denotes a generic prize allocation rule.Reflecting the opposing principles of egalitarianism and elitism, two extreme and basicprize allocation rules are the Equal Division rule and the
Winner-Takes-All rule . The EDrule divides the prize money equally among the competitors, i.e. it assigns to each competition ( N, R , E ) the prize allocation ED i ( N, R , E ) = E | N | for each i ∈ N .The WTA rule allocates all the prize money to the competitor with the highest position, i.e.it assigns to each competition ( N, R , E ) the prize allocation WTA i ( N, R , E ) = E if R ( i ) = 1 ; otherwise.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 thehighest position in the ranking. We take an axiomatic approach to study the fundamental differences between prize allo-cation rules. This means that we formulate some desirable properties of rules and analyzetheir implications. An elementary property imposing a form of equal treatment of competi-tors is anonymity , which requires that the prize allocation does not depend on the identitiesof the competitors, but only on the number of competitors, their ranking, and the prizeendowment.
Anonymity
For each pair of competitions ( N, R , E ) and ( N ′ , R ′ , E ) with equal numbers of competitors | N | = | N ′ | , and each pair of competitors i ∈ N and j ∈ N ′ with equal positions R ( i ) = R ′ ( j ) ,we have ϕ i ( N, R , E ) = ϕ j ( N ′ , R ′ , E ) . The fairness property imposing the prize allocation to reflect the ranking is order preserva-tion , which requires that the prize of a competitor is not lower than the prize of a competitorwith a lower position in the ranking.
Order preservation
For each competition ( N, R , E ) , if competitor i ∈ N has a higher position in the ranking R than competitor j ∈ N , we have ϕ i ( N, R , E ) ≥ ϕ j ( N, R , E ) . The solidarity property endowment monotonicity requires that no competitor is worse offwhen the prize endowment increases.
Endowment monotonicity
For each pair of competitions ( N, R , E ) and ( N, R , E ′ ) with E < E ′ and each competitor i ∈ N , we have ϕ i ( N, R , E ) ≤ ϕ i ( N, R , E ′ ) . Endowment monotonicity is a stronger property than endowment continuity , which requiresthat small changes in the prize endowment have a small impact on the assigned prize allo-cation.
Endowment continuity
For each pair of competitions ( N, R , E ) and ( N, R , E ′ ) , we have ϕ ( N, R , E ) → ϕ ( N, R , E ′ ) if E → E ′ . Lemma 1
If a prize allocation rule satisfies endowment monotonicity, then the rule satisfies endowmentcontinuity.
All proofs are provided in the Appendix. Joint characterization.
Prize allocation may depend on the number of competitors.A criterion for evaluating whether a prize allocation rule prescribes coherent allocationsfor competitions with different numbers of competitors is consistency . Consider an ar-bitrary competition ( N, R , E ) and suppose that some competitors S ⊆ N redistributetheir accumulated prizes. This redistribution is based on their subranking , the bijection R S : S → { , . . . , | S |} such that for each pair of competitors i ∈ S and j ∈ S , we have R S ( i ) < R S ( j ) iff R ( i ) < R ( j ) . A prize allocation rule is consistent if it allocates to eachcompetitor i ∈ S in the corresponding reduced competition the same prize as in the originalcompetition. Consistency
For each competition ( N, R , E ) , each nonempty subset of competitors S ⊆ N , and eachcompetitor i ∈ S , we have ϕ i ( N, R , E ) = ϕ i S, R S , X j ∈ S ϕ j ( N, R , E ) ! . The Equal Division rule and the Winner-Takes-All rule both satisfy anonymity, orderpreservation, endowment monotonicity, and consistency, but these rules are not the onlyones. 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 upto the second dollar is allocated to the competitor with the second highest position, and soon until each competitor has been allocated one dollar. If there is still money left, the firstadditional dollar is allocated to the competitor with the highest position, the second addi-tional dollar to the competitor with the second highest position, and so on. This procedurecontinues until the full prize endowment is allocated among the competitors. This rule alsosatisfies anonymity, order preservation, endowment monotonicity, and consistency. △ 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 consis-tency, then the rule satisfies anonymity.
To have a full understanding of the joint implication of order preservation, endowmentmonotonicity, and consistency, we only need to focus on the structure of the corresponding In fact, all results in this section hold when consistency is weakened to bilateral consistency , requiringconsistency only for reduced competitions with two competitors. prize allocation rules for competitions with two competitors. The following lemma showsthat each such rule has at most one consistent extension. Lemma 3
If two prize allocation rules satisfying endowment monotonicity and consistency coincide foreach competition with two competitors, then the two rules coincide for each competition withan arbitrary number of competitors.
How do prize allocation rules satisfying order preservation, endowment monotonicity, andconsistency look for competitions with two competitors? To this end, it helps to graph-ically illustrate possible prize allocation paths , i.e. draw the prize allocations assigned tocompetitions with two competitors when the prize endowment increases from zero.
Example 2
Let ( N, R , E ) with | N | = 2 be a competition with two competitors. Denote N = { , } suchthat R (1) = 1 and 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 ϕ from Example 1 are illustrated as follows. ϕ ϕ ED( N, R , E )WTA( N, R , E ) ϕ ( N, R , E ) The horizontal axis depicts the prize allocated to competitor one and the vertical axis depictsthe prize allocated to competitor two. The dotted lines indicate different levels of the prizeendowment. △ Each prize allocation rule satisfying order preservation, endowment monotonicity, andconsistency is somehow a combination of the ED rule, the WTA rule, and the type of rulein Examples 1 and 2. Each such prize allocation rule can be described in the following way.Let N ⊆ U be a finite set of competitors and consider a corresponding ranking and prizeendowment. Then there exist disjoint intervals ( a k , b k ) with a k ∈ R + and b k ∈ R + ∪ { + ∞} for each k such that the following holds. Each competitor is allocated a prize of a k whenthe prize endowment equals | N | a k . If the endowment is higher, first the prize allocated A similar result in the context of claims problems was obtained by Aumann and Maschler (1985). to the competitor with the highest position is increased to b k , then the prize allocated tothe competitor with the second highest position is increased to b k , and so on. This meansthat each competitor is allocated a prize of b k when the prize endowment equals | N | b k . Ifthe average prize endowment does not belong to one of these intervals, it is equally dividedamong the competitors. The formal description of all these prize allocation rules is capturedby the following theorem. Theorem 1
A prize allocation rule ϕ satisfies order preservation, endowment monotonicity, and consis-tency iff there exist disjoint intervals ( a , b ) , ( a , b ) , . . . with a , a , . . . ∈ R + and b , b , . . . ∈ R + ∪ { + ∞} such that for each competition ( N, R , E ) and each competitor i ∈ N , we have ϕ i ( N, R , E ) = a k if | N | a k ≤ E ≤ ( | N | − R ( i ) + 1) a k + ( R ( i ) − b k ; x if ( | N | − R ( i ) + 1) a k + ( R ( i ) − b k ≤ E ≤ ( | N | − R ( i )) a k + R ( i ) b k ; b k if ( | N | − R ( i )) a k + R ( i ) b k ≤ E ≤ | N | b k ; E | N | otherwise , where x = E − ( | N | − R ( i )) a k − ( R ( i ) − b k . Example 3
The ED rule fits Theorem 1 by setting a k = b k = 0 for each k . The WTA rule fits Theorem 1by setting a = 0 and b = + ∞ . The prize allocation rule from Examples 1 and 2 fitsTheorem 1 by setting a k = k − and b k = k for each k .Let ( N, R , E ) with | N | = 2 be a competition with two competitors. Denote N = { , } such that R (1) = 1 and R (2) = 2 . The prize allocation path of the rule ϕ satisfying orderpreservation, endowment monotonicity, and consistency which fits Theorem 1 with a = 1 , b = a = 2 , b = 3 , a = 3 , and b = + ∞ is illustrated as follows. ϕ ϕ ϕ ( N, R , E ) △ Strengthening the properties.
Theorem 1 shows that many rules satisfy order preser-vation, endowment monotonicity, and consistency. This means that there is room for im-posing additional requirements. In particular, it may be possible to strengthen one of theproperties. Unfortunately, order preservation, which requires that the prize of a competitoris not lower than the prize of a competitor with a lower position, cannot be strengthenedto strict order preservation , which requires that the prize of a competitor is higher thanthe prize of a competitor with a lower position. However, it can be strengthened to theweaker property winner-loser strict order preservation , which requires in addition to orderpreservation that the prize of the competitor with the highest position is higher than theprize of the competitor with the lowest position. Only the Winner-Takes-All rule satisfieswinner-loser strict order preservation, endowment monotonicity, and consistency.
Strict order preservation
For each competition ( N, R , E ) with E > , if competitor i ∈ N has a higher position in theranking R than competitor j ∈ N , we have ϕ i ( N, R , E ) > ϕ j ( N, R , E ) . Winner-loser strict order preservation
For each competition ( N, R , E ) with E > , if competitor i ∈ N has a higher position in theranking R than competitor j ∈ N , we have ϕ i ( N, R , E ) ≥ ϕ j ( N, R , E ) and if R ( i ) = 1 and R ( j ) = | N | , then ϕ i ( N, R , E ) > ϕ j ( N, R , E ) . Corollary 1 (i) The Winner-Takes-All rule is the unique prize allocation rule satisfying winner-loserstrict order preservation, endowment monotonicity, and consistency.(ii) No prize allocation rule satisfies strict order preservation, endowment monotonicity,and consistency.
Instead of strengthening order preservation, it is also possible to strengthen endowmentmonotonicity, which requires that no competitor is worse off when the prize endowmentincreases. For instance, strict endowment monotonicity requires that each competitor isbetter off when the prize endowment increases, or the weaker property winner strict en-dowment monotonicity requires in addition to endowment monotonicity that the winner isbetter off when the prize endowment increases. Only the Equal Division rule satisfies orderpreservation, strict endowment monotonicity, and consistency. A prize allocation rule sat-isfies order preservation, winner strict endowment monotonicity, and consistency iff it is a Winner-Takes-Surplus rule , i.e. it coincides with the ED rule up to some value of the prizeendowment, and it allocates the surplus according to the WTA rule.
Strict endowment monotonicity
For each pair of competitions ( N, R , E ) and ( N, R , E ′ ) with E < E ′ and each competitor i ∈ N , we have ϕ i ( N, R , E ) < ϕ i ( N, R , E ′ ) . Winner strict endowment monotonicity
For each pair of competitions ( N, R , E ) and ( N, R , E ′ ) with E < E ′ and each competitor i ∈ N , we have ϕ i ( N, R , E ) ≤ ϕ i ( N, R , E ′ ) and if R ( i ) = 1 , then ϕ i ( N, R , E ) < ϕ i ( N, R , E ′ ) . Corollary 2 (i) A prize allocation rule ϕ satisfies order preservation, winner strict endowment mono-tonicity, and consistency iff ϕ is a Winner-Takes-Surplus rule, i.e. there exists a ∈ R + ∪ { + ∞} such that for each competitor i ∈ N , we have ϕ i ( N, R , E ) = E − ( | N | − a if E ≥ | N | a and R ( i ) = 1; a if E ≥ | N | a and R ( i ) = 1; E | N | otherwise . (ii) The Equal Division rule is the unique prize allocation rule satisfying order preservation,strict endowment monotonicity, and consistency. Example 4
The ED rule fits Corollary 2(i) by setting a = + ∞ . The WTA rule fits Corollary 2(i) bysetting a = 0 . The prize allocation path of a Winner-Takes-Surplus rule ϕ can be illustratedas follows. ϕ aϕ a ϕ ( N, R , E ) △ Alternatively, endowment monotonicity can be strengthened to endowment additivity .Suppose that the prize endowment turns out to be higher than expected. Then there are twoways of proceeding. Either the initial allocation is cancelled and the rule is applied to thecompetition with the new prize endowment, or the rule is applied to the competition withthe increment as the prize endowment and the resulting allocation is added to the initialallocation. A prize allocation rule satisfies endowment additivity if both ways of proceedinglead to the same prize allocation. Endowment additivity is equivalent to proportionality ,which requires that each position in the ranking of a competition is assigned a fixed pro-portion 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 prize allocationrules satisfying order preservation, endowment monotonicity, and consistency, only the EDrule and the WTA rule satisfy endowment additivity.
Endowment additivity
For each pair of competitions ( N, R , E ) and ( N, R , E ′ ) and each competitor i ∈ N , we have ϕ i ( N, R , E + E ′ ) = ϕ i ( N, R , E ) + ϕ i ( N, R , E ′ ) . Proportionality
For each competition ( N, R , E ) and each competitor i ∈ N , we have ϕ i ( N, R , E ) = Eϕ i ( N, R , . Lemma 4
A prize allocation rule satisfies endowment additivity iff the rule satisfies proportionality.
Corollary 3
The Equal Division rule and the Winner-Takes-All rule are the only two prize allocationrules satisfying order preservation, endowment additivity (proportionality), and consistency. Locally consistent prize allocation rules
We know that the Equal Division rule and the Winner-Takes-All rule are two membersof a family of prize allocation rules satisfying anonymity, order preservation, endowmentmonotonicity, and consistency. Strengthening order preservation or endowment monotonicityleads to impossibilities, uniqueness, or restricted families. However, consistency may beconsidered too strong a requirement, since an arbitrary subranking may not reflect the resultsof the original competition well. Suppose for instance that two competitors of a competitionwith a large number of competitors reevaluate their allocated prizes on the basis of theirsubranking. Then the reduced competition tends to lose significant features of originalcompetition, 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 positionand the lowest position.In fact, it may be desirable to weaken consistency to local consistency , which requires onlyconsistent allocations for reduced competitions where for each of two competitors, each othercompetitor with an intermediate position is also involved. In other words, local consistencyrequires invariance under splitting up the full competition into leagues, where the competitorsin the top segment of the ranking are put in a separate league, the competitors in the secondsegment of the ranking are put in a second league, and so on. Local consistency
For each competition ( N, R , E ) , each subset of competitors S ⊆ N such that |R ( i ) − R ( j ) | ≤| S | − for each of two competitors i ∈ S and j ∈ S , and each competitor i ∈ S , we have ϕ i ( N, R , E ) = ϕ i S, R S , X j ∈ S ϕ j ( N, R , E ) ! . In contrast to consistency, the following example shows that a prize allocation rule satisfy-ing order preservation, endowment monotonicity, and local consistency does not necessarilysatisfy anonymity.
Example 5
Let i ∈ U and j ∈ U be two potential competitors. Let the prize allocation rule ϕ be definedin the following way. If i has the highest position and j has the second highest position in theranking, then ϕ divides the prize endowment equally among competitors i and j . Otherwise, ϕ divides the prize endowment according to the WTA rule. Formally, ϕ assigns to eachcompetition ( N, R , E ) the prize allocation such that if i, j ∈ N , R ( i ) = 1 , and R ( j ) = 2 ,then ϕ k ( N, R , E ) = E if k ∈ { i, j } ;0 otherwise , and otherwise ϕ k ( N, R , E ) = E if R ( k ) = 1;0 otherwise . Then ϕ satisfies order preservation, endowment monotonicity, and local consistency, but doesnot satisfy anonymity. By Lemma 2, ϕ does not satisfy consistency. △ Needless to say, all the prize allocation rules described in Theorem 1 satisfy anonymity,order preservation, endowment monotonicity, and local consistency, but these rules are notthe only ones. The following example provides another rule satisfying these properties. In 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. Example 6
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 isdivided equally among the competitors with the highest position and the competitor with thesecond highest position until they are allocated two and one dollar, respectively, or until theprize endowment is exhausted. Then the surplus is divided equally among the competitorswith the three highest positions until they are allocated three, two, and one dollar, respec-tively. This procedure continues until the competitor with the lowest position is allocatedone dollar. If there is still money left, this is equally divided among the competitors. Thisprize allocation rule also satisfies anonymity, order preservation, endowment monotonicity,and local consistency.Let ( N, R , E ) with | N | = 2 be a competition with two competitors. Denote N = { , } such that R (1) = 1 and R (2) = 2 . Then the prize allocation paths of the ED rule, the WTArule, and the prize allocation rule ϕ described above are illustrated as follows. ϕ ϕ ED( N, R , E )WTA( N, R , E ) ϕ ( N, R , E ) △ Note that the arithmetic type of rule in Example 6 satisfies winner strict endowment mono-tonicity, i.e. the competitor with the highest position is better off when the prize endowmentincreases. Each prize allocation rule satisfying anonymity, order preservation, winner strictendowment monotonicity, and local consistency admits a compact indirect description. Foreach such rule there exists a continuous and non-decreasing function f : R + → R + with ≤ f ( x ) ≤ x for each x such that the assigned allocation for each competition can bedescribed in the following way. The competitor with the highest position is allocated a prizeof x ∈ 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. Intotal, the full prize endowment is allocated among the competitors. The formal descriptionof all these prize allocation rules is captured by the following theorem. Theorem 2
A prize allocation rule ϕ satisfies anonymity, order preservation, winner strict endowmentmonotonicity, and local consistency iff there exists a continuous and non-decreasing function f : R + → R + with ≤ f ( x ) ≤ x for each x such that for each competition ( N, R , E ) andeach competitor i ∈ N , we have ϕ i ( N, R , E ) = x if R ( i ) = 1; f ( R ( i ) − ( x ) otherwise , where x ∈ R + is such that x + P | N | k =2 f ( k − ( x ) = E . (Here, we denote f ( k ) ( x ) = f ( f ( k − ( x )) .) Example 7
The ED rule fits Theorem 2 by setting f ( x ) = x for each x . The WTA rule fits Theorem 2by setting f ( x ) = 0 for each x . The arithmetic type of rule from Example 6 fits Theorem 2by setting f ( x ) = if ≤ x ≤ ; x − if x ≥ . △ Theorem 2 does not hold for all prize allocation rules satisfying anonymity, order preser-vation, endowment monotonicity, and local consistency. If such rule does not satisfy winnerstrict endowment monotonicity, the prize of the competitor with the second highest positioncannot 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 competitorsdoes not necessarily have a unique locally consistent extension to competitions with morecompetitors.
Example 8
Let a prize allocation rule be defined by allocating the prize endowment in the followingway. Up to the first dollar is allocated to the competitor with the highest position. Sub-sequently, one dollar is allocated to the competitor with the second highest position andthen another dollar is allocated to the competitor with the highest position. Thereafter, onedollar is allocated to the competitor with the third highest position, then another dollar tothe competitor with the second highest position, and then another dollar to the competitorwith the highest position. This procedure continues until the competitor with the highestposition is allocated an amount of dollars equal to the number of competitors. If there is stillmoney left, first another dollar is allocated to the competitor with the lowest position, thenanother dollar to the competitor with the second lowest position, and so on. This continuesuntil the full prize endowment is allocated among the competitors. This prize allocation rulesatisfies anonymity, order preservation, endowment monotonicity, and local consistency, but does not satisfy winner strict endowment monotonicity. Moreover, for competitions with twocompetitors, this rule coincides with the prize allocation rule from Examples 1 and 2. △ Theorem 2 shows that there are many other rules satisfying anonymity, order preservation,winner strict endowment monotonicity, and local consistency. Subfamilies are obtained iforder preservation is strengthened to strict order preservation or winner-loser strict orderpreservation, or winner strict endowment monotonicity is strengthened to strict endowmentmonotonicity. For each prize allocation rule satisfying anonymity, order preservation, winnerstrict endowment monotonicity, and local consistency, the rule satisfies winner-loser strictorder preservation iff it does not coincide with the ED rule for each positive prize endowment,and the rule satisfies strict order preservation iff it does not coincide with the ED rule norwith the WTA rule for each positive prize endowment. Moreover, a prize allocation rulesatisfying anonymity, order preservation, winner strict endowment monotonicity, and localconsistency satisfies strict endowment monotonicity iff it does not allocate any additionalprize endowment according to the WTA rule.
Corollary 4
Let ϕ be a prize allocation rule satisfying anonymity, order preservation, winner strict en-dowment monotonicity, and local consistency. Let f : R + → R + with ≤ f ( x ) ≤ x for each x be a continuous and non-decreasing function such that for each competition ( N, R , E ) andeach competitor i ∈ N , we have ϕ i ( N, R , E ) = x if R ( i ) = 1; f ( R ( i ) − ( x ) otherwise , where x ∈ R + is such that x + P | N | k =2 f ( k − ( x ) = E . Then the following statements hold.(i) ϕ satisfies winner-loser strict order preservation iff ≤ f ( x ) < x for each x > ;(ii) ϕ satisfies strict order preservation iff < f ( x ) < x for each x > ;(iii) ϕ satisfies strict endowment monotonicity iff f is increasing. If winner strict endowment monotonicity is strengthened to endowment additivity (pro-portionality), then an interesting family of geometric prize allocation rules is obtained. Foreach rule satisfying anonymity, order preservation, endowment additivity, and local consis-tency, there exists λ ∈ [0 , such that the assigned allocation for each competition can bedescribed in the following way. The competitor with the second highest position is allocateda prize of λ times the prize of the competitor with the highest position. The competitor withthe third highest position is allocated a prize of λ times the prize of the competitor with thesecond highest position, i.e. λ times the prize of the competitor with the highest position.The competitor with the fourth highest position is allocated a prize of λ times the prizeof the competitor with the highest position, and so on. In total, the full prize endowment is allocated. The formal description of all these prize allocation rules is captured by thefollowing corollary. Corollary 5
A prize allocation rule ϕ satisfies anonymity, order preservation, endowment additivity (pro-portionality), and local consistency iff ϕ is a geometric prize allocation rule, i.e. there exists λ ∈ [0 , such that for each competition ( N, R , E ) and each competitor i ∈ N , we have ϕ i ( N, R , E ) = λ R ( i ) − P | N |− k =0 λ k E. Example 9
The ED rule fits Corollary 5 with λ = 1 . The WTA rule fits Corollary 5 with λ = 0 . Theprize allocation paths of the ED rule, the WTA rule, and the geometric prize allocation rule ϕ with λ = can be illustrated as follows. ϕ ϕ ED( N, R , E )WTA( N, R , E ) ϕ ( N, R , E ) △ Concluding remarks
This paper takes an axiomatic approach to study fair and consistent prize allocation insingle rank-order competitions. We introduce a model in which the competitors, their finalranking, and the prize endowment are the primitives and we characterize two families ofprize allocation rules. In the next remark, we formally state that the axioms are logicallyindependent.
Remark
Theorem 1 and Theorem 2 are each based on a set of logically independent prop-erties. To see this, consider the following prize allocation rules. The rule which allocates allthe prize money to the competitor with the lowest position satisfies endowment monotonicityand consistency, but does not satisfy order preservation. The rule which coincides with theEqual Division rule for competitions with a prize endowment of, at most, one dollar, and Here, we define λ = 1 for each λ ∈ [0 , . 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 5 satisfies order preservation, winner strict endowment monotonicity,and local consistency, but does not satisfy anonymity and consistency. The geometric prizeallocation rule that fits Corollary 5 with λ = 2 satisfies anonymity, winner strict endowmentmonotonicity, and local consistency, but does not satisfy order preservation. The rule fromExamples 1 and 2 satisfies anonymity, order preservation, and local consistency, but does notsatisfy winner strict endowment monotonicity. The rule which coincides with the ED rulefor competitions with two competitors and coincides with the WTA rule for competitionswith more than two competitors satisfies anonymity, order preservation, and winner strictendowment monotonicity, but does not satisfy local consistency.The practical relevance of our paper is that if an organizer wants to choose a prize alloca-tion rule and accepts some set of axioms, then the problem is reduced to choosing a singlerule from the corresponding family. Since the families are broad enough, the organizer canchoose the rule that maximizes some desirable goal, such as the total effort of competitors orthe participation of talented competitors. This problem could be studied in future researchusing 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 situationswhere 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 othercompetition types, such as knockout tournaments, round-robin tournaments, and multiplerank-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 anda number of points in each single tournament. Then the total sum of points determines thetotal ranking and the bonuses for the entire series. A straightforward question is how tojointly choose a points system, a prize structure for each single tournament, and a bonusstructure for the entire series. In particular, since Corollary 5 calls for geometric prizesequences and Kondratev et al. (2019) justify geometric point sequences, should we choosethe 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) developeda 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 beinterpreted 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 undesirablein a budgeting context.’ We can conclude from these arguments that any application ofwell-known ranking, voting, or allocation rules must be re-motivated and re-justified in thecontext of competitions. Appendix
Lemma 1
If a prize allocation rule satisfies endowment monotonicity, then the rule satisfies endowmentcontinuity.Proof.
Let ϕ be a prize allocation rule satisfying endowment monotonicity. Let ( N, R , E ) and ( N, R , E ′ ) be two competitions. Let i ∈ N . By endowment monotonicity, | E − E ′ | = | X j ∈ N ϕ j ( N, R , E ) − X j ∈ N ϕ j ( N, R , E ′ ) | = | X j ∈ N ( ϕ j ( N, R , E ) − ϕ j ( N, R , E ′ )) | = X j ∈ N | ϕ j ( N, R , E ) − ϕ j ( N, R , E ′ ) |≥ | ϕ i ( N, R , E ) − ϕ i ( N, R , E ′ ) | . This means that ϕ i ( N, R , E ) → ϕ i ( N, R , E ′ ) if E → E ′ . Hence, ϕ satisfies endowmentcontinuity. (cid:3) Lemma 2
If a prize allocation rule satisfies order preservation, endowment monotonicity, and consis-tency, then the rule satisfies anonymity.Proof.
Let ϕ be a prize allocation rule satisfying order preservation, endowment monotonic-ity, and consistency. By Lemma 1, ϕ satisfies endowment continuity.Suppose for the sake of contradiction that ϕ does not satisfy anonymity. Then there existtwo competitions ( N, R , E ) and ( N ′ , R ′ , E ) with equal numbers of competitors | N | = | N ′ | ,and two competitors i ∈ N and j ∈ N ′ with equal positions R ( i ) = R ′ ( j ) such that ϕ i ( N, R , E ) < ϕ j ( N ′ , R ′ , E ) . Since X k ∈ N ϕ k ( N, R , E ) = E = X k ∈ N ′ ϕ k ( N ′ , R ′ , E ) , there exist two competitors i ∈ N and j ∈ N ′ with equal position R ( i ) = R ′ ( j ) such that ϕ i ( N, R , E ) > ϕ j ( N ′ , R ′ , E ) . Suppose without loss of generality that R ′ ( j ) = R ( i ) < R ( i ) = R ′ ( j ) . Denote x = ϕ ( N, R , E ) and y = ϕ ( N ′ , R ′ , E ) . By consistency, ( x i , x i ) = ϕ ( { i , i } , R { i ,i } , x i + x i ) and ( y j , y j ) = ϕ ( { j , j } , R ′{ j ,j } , y j + y j ) . This is illustrated in the following way.(1) N R ϕi x i i x i N R ϕj y j j y j By order preservation, y j < x i ≤ x i < y j . One of the following six cases hold. Case 1 i = j and i = j By (1), N R ϕi = j x i i = j x i N R ϕj y j j y j Then there exists another competitor k ∈ U \ { i , i } . By order preservation, endowmentcontinuity, and consistency, there exist prize endowments E ′ and E ′′ such that N R ϕi x i i x i k E ′ − x i − x i N R ϕi x i k E ′′ − x i − x i i x i By order preservation, E ′ − x i − x i ≤ x i ≤ E ′′ − x i − x i . By consistency, N R ϕi x i k E ′ − x i − x i N R ϕi x i k E ′′ − x i − x i By endowment monotonicity, N R ϕi x i k x i By order preservation, endowment continuity, consistency, and (1), there exist prize endow-ments E ′′′ and E ′′′′ such that N R ϕi x i i E ′′′ − x i − x i k x i N R ϕj y j j y j k E ′′′′ − y j − y j By order preservation, E ′′′ − x i − x i ≤ x i and E ′′′′ − y j − y j ≤ y j . By consistency, N R ϕi E ′′′ − x i − x i k x i N R ϕi = j y j k E ′′′′ − y j − y j Then E ′′′ − x i − x i ≤ x i < y j and x i > y j ≥ E ′′′′ − y j − y j . This contradicts endowmentmonotonicity. Case 2 i = j and i = j By order preservation, endowment continuity, consistency, and (1), there exists prize endow-ment E ′ such that N R ϕi x i j E ′ − x i − x i i x i By order preservation, E ′ − x i − x i ≥ x i . By consistency, N R ϕi = j x i j E ′ − x i − x i Then x i < y j and E ′ − x i − x i ≥ x i > y j . By (1), a contradiction follows from Case 1. Case 3 i = j and i = j By order preservation, endowment continuity, consistency, and (1), there exists a prize en-dowment E ′ such that N R ϕi x i j E ′ − x i − x i i x i By order preservation, E ′ − x i − x i ≤ x i . By consistency, N R ϕj E ′ − x i − x i i = j x i Then E ′ − x i − x i ≤ x i < y j and x i > y j . By (1), a contradiction follows from Case 1. Case 4 i = j and i = j By order preservation, endowment continuity, consistency, and (1), there exists a prize en-dowment E ′ such that N R ϕi x i j E ′ − x i − x i i x i By order preservation, E ′ − x i − x i ≥ x i . By consistency, N R ϕi = j x i j E ′ − x i − x i Then x i < y j and E ′ − x i − x i ≥ x i > y j . By (1), this contradicts endowment mono-tonicity. Case 5 i = j and i = j By order preservation, endowment continuity, consistency, and (1), there exists a prize en-dowment E ′ such that N R ϕi x i j E ′ − x i − x i i x i By order preservation, E ′ − x i − x i ≤ x i . By consistency, N R ϕj E ′ − x i − x i i = j x i Then E ′ − x i − x i ≤ x i < y j and x i > y j . By (1), this contradicts endowment mono-tonicity. Case 6 i = j = i and i = j = i By order preservation, endowment continuity, consistency, and (1), there exist z j and z j such that N R ϕi x i j z j j z j i x i By order preservation, z j ≤ x i and z j ≥ x i . By consistency, N R ϕj z j j z j Then z j ≤ x i < y j and z j ≥ x i > y j . By (1), this contradicts endowment monotonicity. (cid:3) Lemma 3
If two prize allocation rules satisfying endowment monotonicity and consistency coincide foreach competition with two competitors, then the two rules coincide for each competition withan arbitrary number of competitors.Proof.
Let ϕ and ϕ ′ be two prize allocation rules satisfying endowment monotonicity andconsistency such that ϕ ( N, R , E ) = ϕ ′ ( N, R , E ) for each competition with two competitors.Suppose for the sake of contradiction that there exists a competition ( N, R , E ) such that ϕ ( N, R , E ) = ϕ ′ ( N, R , E ) . Since X i ∈ N ϕ i ( N, R , E ) = E = X i ∈ N ϕ ′ i ( N, R , E ) , there exist two competitors i ∈ N and j ∈ N such that(2) ϕ i ( N, R , E ) < ϕ ′ i ( N, R , E ) and ϕ j ( N, R , E ) > ϕ ′ j ( N, R , E ) . By consistency, ( ϕ i ( N, R , E ) , ϕ j ( N, R , E )) = ϕ ( { i, j } , R { i,j } , ϕ i ( N, R , E ) + ϕ j ( N, R , E )) and ( ϕ ′ i ( N, R , E ) , ϕ ′ j ( N, R , E )) = ϕ ′ ( { i, j } , R { i,j } , ϕ ′ i ( N, R , E ) + ϕ ′ j ( N, R , E ))= ϕ ( { i, j } , R { i,j } , ϕ ′ i ( N, R , E ) + ϕ ′ j ( N, R , E )) . By endowment monotonicity, ϕ i ( N, R , E ) ≤ ϕ ′ i ( N, R , E ) and ϕ j ( N, R , E ) ≤ ϕ ′ j ( N, R , E ) , or ϕ i ( N, R , E ) ≥ ϕ ′ i ( N, R , E ) and ϕ j ( N, R , E ) ≥ ϕ ′ j ( N, R , E ) . This contradicts (2). Hence, ϕ ( N, R , E ) = ϕ ′ ( N, R , E ) for each competition ( N, R , E ) . (cid:3) Theorem 1
A prize allocation rule ϕ satisfies order preservation, endowment monotonicity, and consis-tency iff there exist disjoint intervals ( a , b ) , ( a , b ) , . . . with a , a , . . . ∈ R + and b , b , . . . ∈ R + ∪ { + ∞} such that for each competition ( N, R , E ) and each competitor i ∈ N , ϕ i ( N, R , E ) = a k if | N | a k ≤ E ≤ ( | N | − R ( i ) + 1) a k + ( R ( i ) − b k ; x if ( | N | − R ( i ) + 1) a k + ( R ( i ) − b k ≤ E ≤ ( | N | − R ( i )) a k + R ( i ) b k ; b k if ( | N | − R ( i )) a k + R ( i ) b k ≤ E ≤ | N | b k ; E | N | otherwise , where x = E − ( | N | − R ( i )) a k − ( R ( i ) − b k .Proof. It is readily checked that each such ϕ satisfies order preservation, endowment mono-tonicity, and consistency.Let ϕ be a prize allocation rule satisfying order preservation, endowment monotonicity,and consistency. By Lemma 1, ϕ satisfies endowment continuity. By Lemma 2, ϕ satisfiesanonymity. By Lemma 3, we only need to show that ϕ satisfies the description for eachcompetition with two competitors, since each such rule has a unique consistent extension tocompetitions with more competitors. Let N ⊆ U with | N | = 2 be two competitors and let R be a ranking. Denote N = { , } such that R (1) = 1 and R (2) = 2 . The proof consistsof two steps. Step 1.
For each prize endowment E , if E = x + x such that(3) ϕ ( N, R , x + x ) = ( x , x ) , then(4) ϕ ( N, R , x + x ) = ( x , x ) or ϕ ( N, R , x + x ) = ( x , x ) . Proof of Step 1.
Let E be a prize endowment and denote ϕ ( N, R , E ) = ( x , x ) . Then E = x + x and (3) holds. By order preservation, x ≥ x . If x = x , then (4) followsimmediately from (3).Suppose that x > x . Let N ′ = { , , } and let R ′ be a ranking of N ′ such that R ′ (1) = 1 , R ′ (2) = 2 , and R ′ (3) = 3 . By order preservation and endowment continuity, there existsa prize endowment E ′ such that ϕ ( N ′ , R ′ , E ′ ) + ϕ ( N ′ , R ′ , E ′ ) = x + x . By anonymity,consistency, and (3),(5) ϕ ( N ′ , R ′ , E ′ ) = ( x , E ′ − x − x , x ) . By order preservation, x + x + x ≤ E ′ ≤ x + x + x . By anonymity and consistency, ϕ ( N, R , E ′ − x ) = ( E ′ − x − x , x ) (6) and ϕ ( N, R , E ′ − x ) = ( x , E ′ − x − x ) . (7) If E ′ = x + x + x , then (4) follows immediately from (6). If E ′ = x + x + x , then (4)follows immediately from (7).Suppose that x + x + x < E ′ < x + x + x . Then E ′ − x < x + x < E ′ − x . Byendowment monotonicity, (3), (6), and (7),(8) ϕ ( N, R , E ′′ ) = ( E ′′ − x , x ) if E ′ − x ≤ E ′′ ≤ x + x ; ( x , E ′′ − x ) if x + x ≤ E ′′ ≤ E ′ − x .Denote ( y , y , y ) = ϕ ( N ′ , R ′ , x + x + x ) and ( z , z , z ) = ϕ ( N ′ , R ′ , x + x + x ) . Byendowment monotonicity and (5), y ≤ x ≤ z ,y ≤ E ′ − x − x ≤ z ,y ≤ x ≤ z . Then y + y ≤ E ′ − x and E ′ − x ≤ z + z . Since y ≤ x and y + y + y = x + x + x ,we have x + x ≤ x + x + x − y = y + y . Since x ≤ z and z + z + z = x + x + x ,we have z + z = x + x + x − z ≤ x + x . This means that x + x ≤ y + y ≤ E ′ − x , and E ′ − x ≤ z + z ≤ x + x . By anonymity, consistency, and (8), ϕ ( N, R , y + y ) = y = ϕ ( N, R , y + y ) = x , and ϕ ( N, R , z + z ) = z = ϕ ( N, R , z + z ) = x . Since y ≤ z and y ≤ z , we have y + y ≤ z + z . By endowment monotonicity and (8), foreach E ′′ such that ϕ ( N, R , E ′′ ) ≥ x , we have E ′′ ≥ x + x . Since ϕ ( N, R , y + y ) = x ,we have y + y ≥ x + x . By endowment monotonicity and (8), for each E ′′ such that ϕ ( N, R , E ′′ ) ≤ x , we have E ′′ ≤ x + x . Since ϕ ( N, R , z + z ) = x , we have z + z ≤ x + x . Hence, x + x ≤ y + y ≤ z + z ≤ x + x . Then y + y = x + x and z + z = x + x . Since y = x and z = x , we have y = x and z = x . This means that y = ( x , x , x ) and z = ( x , x , x ) . By anonymity andconsistency, ϕ ( N, R , x + x ) = ( x , x ) and ϕ ( N, R , x + x ) = ( x , x ) . Hence, (4) holds. Step 2.
There exist disjoint intervals ( a , b ) , ( a , b ) , . . . with a , a , . . . ∈ R + and b , b , . . . ∈ R + ∪ { + ∞} such that(9) ϕ ( N, R , E ) = ( E − a k , a k ) if a k + a k ≤ E ≤ b k + a k ;( b k , E − b k ) if b k + a k ≤ E ≤ b k + b k ; E otherwise . Proof of Step 2. If ϕ ( N, R , E ) = ϕ ( N, R , E ) for each prize endowment E , then (9) followsimmediately by defining a k = b k = 0 for each k .Let E be a prize endowment with ϕ ( N, R , E ) > ϕ ( N, R , E ) . Denote ϕ ( N, R , E ) =( x , x ) . Then E = x + x , x > x , and ϕ ( N, R , x + x ) = ( x , x ) . By Step 1, ϕ ( N, R , x + x ) = ( x , x ) or ϕ ( N, R , x + x ) = ( x , x ) .Suppose that ϕ ( N, R , x + x ) = ( x , x ) . Define(10) b E = x and a E = min E ′ ∈ R + { ϕ ( N, R , E ′ ) | ϕ ( N, R , E ′ ) = b E } . Then a E < b E since a E ≤ x < x = b E . Moreover, ϕ ( N, R , b E + a E ) = ( b E , a E ) and ϕ ( N, R , b E + b E ) = ( b E , b E ) . By endowment monotonicity,(11) ϕ ( N, R , E ′ ) = ( b E , E ′ − b E ) if b E + a E ≤ E ′ ≤ b E + b E .This also means that(12) b E + a E < E ′ < b E + b E if a E < ϕ ( N, R , E ′ ) < b E . Let E ′ be a prize endowment with a E + a E < E ′ < b E + a E . Denote ( y , y ) = ϕ ( N, R , E ′ ) .By endowment monotonicity and (10), y < b E and y ≤ a E . Then a E < y since a E ≤ a E + a E − y < E ′ − y = y . By Step 1, ϕ ( N, R , y + y ) = ( y , y ) or ϕ ( N, R , y + y ) = ( y , y ) .If ϕ ( N, R , y + y ) = ( y , y ) , then a E < y = ϕ ( N, R , y + y ) < b E , (12) implies b E + a E
A prize allocation rule satisfies endowment additivity iff the rule satisfies proportionality.Proof.
Let ϕ be a prize allocation rule satisfying proportionality. Let ( N, R , E ) and ( N, R , E ′ ) be two competitions and let i ∈ N be a competitor. Then ϕ i ( N, R , E + E ′ ) = ( E + E ′ ) ϕ i ( N, R , Eϕ i ( N, R ,
1) + E ′ ϕ i ( N, R , ϕ i ( N, R , E ) + ϕ i ( N, R , E ′ ) . Hence, ϕ satisfies endowment additivity. Now, let ϕ be a prize allocation rule satisfying endowment additivity. Then ϕ satisfiesendowment monotonicity. By Lemma 1, ϕ satisfies endowment continuity. Let ( N, R , E ) bea competition and let i ∈ N be a competitor. If E is a rational number, then there exist twonatural numbers p ∈ N and q ∈ N such that E = pq . By endowment additivity, ϕ i ( N, R , E ) = ϕ i ( N, R , pq )= pϕ i ( N, R , q )= pq qϕ i ( N, R , q )= pq ϕ i ( N, R , Eϕ i ( N, R , . By endowment continuity, ϕ i ( N, R , E ) = Eϕ i ( N, R , for each real number E . Hence, ϕ satisfies proportionality. (cid:3) Theorem 2
A prize allocation rule ϕ satisfies anonymity, order preservation, winner strict endowmentmonotonicity, and local consistency iff there exists a continuous and non-decreasing function f : R + → R + with ≤ f ( x ) ≤ x for each x such that for each competition ( N, R , E ) andeach competitor i ∈ N , we have ϕ i ( N, R , E ) = x if R ( i ) = 1; f ( R ( i ) − ( x ) otherwise , where x ∈ R + is such that x + P | N | k =2 f ( k − ( x ) = E .Proof. It is readily checked that each such ϕ satisfies anonymity, order preservation, winnerstrict endowment monotonicity, and local consistency.Let ϕ be a prize allocation rule satisfying anonymity, order preservation, winner strictendowment monotonicity, and local consistency. Then ϕ satisfies endowment monotonicity.By Lemma 1, ϕ satisfies endowment continuity. Let N ⊆ U with | N | = 2 be a set oftwo competitors and let R be the corresponding ranking. Denote N = { , } such that R (1) = 1 and R (2) = 2 . For each x ∈ R + , define f ( x ) = y iff ϕ ( N, R , x + y ) = x and ϕ ( N, R , x + y ) = y . By winner strict endowment monotonicity, endowment continuity,and order preservation, f : R + → R + is a well-defined, continuous and non-decreasingfunction with ≤ f ( x ) ≤ x for each x . By anonymity, ϕ fits the description with f for eachcompetition with two competitors. Let ϕ ′ be a prize allocation rule that fits the descriptionwith f for each competition with arbitrary number of competitors.Suppose for the sake of contradiction that there exists a competition ( N, R , E ) such that ϕ ′ ( N, R , E ) = ϕ ( N, R , E ) . Denote N = { , . . . , | N |} such that R ( k ) = k for each k ∈ N .Let i ∈ N be such that ϕ ′ i ( N, R , E ) = ϕ i ( N, R , E ) and ϕ ′ j ( N, R , E ) = ϕ j ( N, R , E ) for each j ∈ N with j < i . Suppose without loss of generality that ϕ ′ i ( N, R , E ) > ϕ i ( N, R , E ) . Bylocal consistency and anonymity, ϕ ′ i +1 ( N, R , E ) = ϕ ′ i +1 ( { i, i + 1 } , R { i,i +1 } , ϕ ′ i ( N, R , E ) + ϕ ′ i +1 ( N, R , E ))= f ( ϕ ′ i ( N, R , E )) ≥ f ( ϕ i ( N, R , E ))= ϕ i +1 ( { i, i + 1 } , R { i,i +1 } , ϕ i ( N, R , E ) + ϕ i +1 ( N, R , E ))= ϕ i +1 ( N, R , E ) . In a similar way, this implies that ϕ ′ i +2 ( N, R , E ) ≥ ϕ i +2 ( N, R , E ) . Continuing this reasoning, ϕ ′ j ( N, R , E ) ≥ ϕ j ( N, R , E ) for each j ∈ N with j > i . This means that X j ∈ N ϕ ′ j ( N, R , E ) > X j ∈ N ϕ j ( N, R , E ) = E. This is a contradiction. Hence, ϕ ′ ( N, R , E ) = ϕ ( N, R , E ) for each competition ( N, R , E ) . (cid:3) References
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