On the axiomatic approach to sharing the revenues from broadcasting sports leagues

We take the axiomatic approach to uncover the structure of the revenue-sharing problem from broadcasting sports leagues. Our starting point is to explore the implications of three basic axioms: additivity, order preservation and weak upper bound. We show that the combination of these axioms characterizes a large family of rules, which is made of compromises between the uniform rule and concede-and-divide, such as the one represented by the equal-split rule. The members of the family are fully ranked according to the Lorenz dominance criterion, and the structure of the family guarantees the existence of a majority voting equilibrium. Strengthening some of the previous axioms, or adding new ones, we provide additional characterizations within the family. Weakening some of those axioms, we also characterize several families encompassing the original one. JEL numbers: D63, C71, Z20.


Introduction
In a recent paper (Bergantiños and Moreno-Ternero, 2020a), we have introduced a formal model to analyze the problem of sharing the revenues from broadcasting sports leagues among participating teams, based on the audiences they generate. In this paper, we uncover the structure of this stylized model further, thanks to the axiomatic approach.
We start considering three basic axioms: additivity, order preservation and weak upper bound. The …rst one says that awards are additive on audiences. The second one says that awards preserve the order of teams' audiences. The third one says that individual awards are bounded above by the overall revenues obtained from the whole tournament. The three axioms are satis…ed by three rules that stand out as focal to solve this problem (Bergantiños and Moreno-Ternero, 2020c). They are the uniform rule, which shares equally among all participating teams the overall revenues obtained in the whole tournament, the equal-split rule, which splits the revenue generated from each game equally among the participating teams, and concede-and-divide, which concedes each team the revenues generated from its fan base and divides equally the residual.
Our …rst result shows that the combination of the three axioms mentioned above actually characterizes the family of rules compromising between the uniform rule and concede-anddivide, which actually has the equal-split rule as a focal member. Each rule within the family is simply de…ned by a certain convex combination of the uniform rule and concede-and-divide.
We shall refer to this family as the U C-family of rules.
We also show that all rules within the family satisfy the so-called single-crossing property, which allows one to separate those teams who bene…t from one rule or the other, depending on the rank of their aggregate audiences. This has important implications. On the one hand, the existence of a majority voting equilibrium (e.g., Gans and Smart, 1996). That is, if we allow teams to vote for any rule within the family, then there exists a rule that cannot be overturned by any other rule within the family through majority rule. On the other hand, the rules within the family yield outcomes that are fully ranked according to the Lorenz dominance criterion (e.g., Hemming and Keen, 1983). More precisely, for each problem, and each pair of rules within the family, the outcome suggested by the rule associated with a higher parameter dominates (in the sense of Lorenz) the outcome suggested by the other rule, which is equivalent to saying that the former will be more egalitarian than the latter (e.g., Dasgupta et al., 1973).
We then proceed to consider additional axioms to the structure supporting the U C-family of rules. We start showing that if we add non-negativity (no team receives negative awards), then only a speci…c part of the family survives; namely the rules that are actually convex combinations of the uniform rule and the equal-split rule, which we shall dub the U E-family of rules. More interestingly, we can dismiss the weak upper bound axiom to characterize such a family. To wit, we show that a rule satis…es additivity, order preservation and non-negativity if and only if it is a member of the U E-family of rules. This was actually an open question in Bergantiños and Moreno-Ternero (2020c).
It turns out that the other half of the U C-family of rules; namely, the rules that are actually convex combinations of the equal-split rule and concede-and-divide, dubbed here the EC-family of rules, can also be singled out. To do so, one simply needs to strengthen the weak upper bound axiom to maximum aspirations, which says that no team can receive an amount higher than its claim (i.e., the overall revenues obtained from all the games in which the team was involved). As a matter of fact, order preservation is not required in its full force for this characterization, and the cleanest result states that additivity, equal treatment of equals and maximum aspirations characterize the EC-family of rules. This is almost equivalent to the characterization in Bergantiños and Moreno-Ternero (2020b). 1 We also provide additional characterization results for families encompassing the U C-family of rules, by weakening some of the original axioms considered for its characterization. More precisely, we characterize the rules satisfying additivity, equal treatment of equals, and either weak upper bound or non-negativity. We also characterize the rules satisfying additivity and order preservation and, …nally, the rules satisfying additivity and equal treatment of equals. In all cases, we obtain linear (albeit not convex) combinations of the focal rules mentioned above.
The rest of the paper is organized as follows. We introduce the model, axioms and rules in Section 2. In Section 3, we provide the characterization result leading towards the U C-family of rules and then explore other properties of it. In Section 4, we obtain further characterizations for speci…c members of the family. In Section 5 we characterize more general families encompassing the U C-family of rules. Finally, we conclude in Section 6.

The model
We consider the model introduced by Bergantiños and Moreno-Ternero (2020a). Let N describe a …nite set of teams. Its cardinality is denoted by n. We assume n 3. For each pair of teams i; j 2 N , we denote by a ij the broadcasting audience (number of viewers) for the game played by i and j at i's stadium. We use the notational convention that a ii = 0, for each i 2 N .
Let A 2 A n n denote the resulting matrix of broadcasting audiences generated in the whole tournament involving the teams within N . 2 Each matrix A 2 A n n with zero entries in the diagonal will thus represent a problem and we shall refer to the set of problems as P. 3 Let i (A) denote the total audience achieved by team i, i.e., Without loss of generality, we normalize the revenue generated from each viewer to 1 (to be interpreted as the "pay per view" fee). Thus, we sometimes refer to i (A) by the claim of team i. When no confusion arises, we write i instead of i (A). We de…ne as the average audience of all teams. Namely, = P i2N i n : For each A 2 A n n , let jjAjj denote the total audience of the tournament. Namely,

Rules
A (sharing) rule is a mapping that associates with each problem the list of the amounts the teams get from the total revenue. Without loss of generality, we normalize the revenue generated from each viewer to 1 (to be interpreted as the "pay per view"fee). Thus, formally, R : P ! R n 2 We are therefore assuming a round-robin tournament in which each team plays in turn against each other team twice: once home, another away. This is the usual format of the main European football leagues. Our model could also be extended to leagues in which some teams play other teams a di¤erent number of times and play-o¤s at the end of the regular season, which is the usual format of North American professional sports. In such a case, a ij is the broadcasting audience in all games played by i and j at i's stadium. 3 As the set N will be …xed throughout our analysis, we shall not explicitly consider it in the description of each problem.
is such that, for each A 2 P, The following three rules have been highlighted as focal for this problem (e.g., Bergantiños and Moreno-Ternero, 2020a; 2020b; 2020c).
The uniform rule divides equally among all teams the overall audience of the whole tournament. Formally, Uniform, U : for each A 2 P, and each i 2 N , The equal-split rule divides the audience of each game equally, among the two participating teams. Formally, Equal-split rule, ES: for each A 2 P, and each i 2 N , Concede-and-divide compares the performance of a team with the average performance of the other teams. Formally, Concede-and-divide, CD: for each A 2 P, and each i 2 N , n 2 = (n 1) i jjAjj n 2 = 2 (n 1) i n 2(n 2) : The following family of rules encompasses the above three rules.
UC-family of rules U C 2[0;1] : for each 2 [0; 1] ; each A 2 P, and each i 2 N , Equivalently, At the risk of stressing the obvious, note that, when = 0, U C coincides with the uniform rule, whereas, when = 1, U C coincides with concede-and-divide. That is, U C 0 U and U C 1 CD. Bergantiños and Moreno-Ternero (2020a) prove that for each A 2 P, ES(A) = n 2 (n 1) U (A) + n 2 2 (n 1)

CD(A):
That is, U C ES, where = n 2 2(n 1) . 4 Consequently, the UC-family of rules can be split in two.
On the one hand, the family of rules compromising between the uniform rule and the equalsplit rule. Formally, UE-family of rules U E 2[0;1] : for each 2 [0; 1] ; each A 2 P, and each i 2 N , On the other hand, the family of rules compromising between the equal-split rule and concede-and-divide. 5 Formally, EC-family of rules fEC g 2[0;1] : for each 2 [0; 1] ; each A 2 P, and each i 2 N , As Figure 1 illustrates, the family of U C rules is indeed the union of the family of U E rules and EC rules. Note that is the unique rule belonging to both families.

Axioms
We now introduce the axioms we consider in this paper.
The …rst axiom we consider says that if two teams have the same audiences, then they should receive the same amount.
Equal treatment of equals: For each A 2 P, and each pair i; j 2 N such that a ik = a jk , and a ki = a kj , for each k 2 N n fi; jg, 4 Note that approaches 0:5 (from below) for n arbitrarily large. 5 We studied this family independently in Bergantiños and Moreno-Ternero (2020b).
The following axiom strengthens the previous one by saying that if the audience of team i is, game by game, not smaller than the audience of team j, then that team i should not receive less than team j.
Order preservation: For each A 2 P and each pair i; j 2 N , such that, for each k 2 N n fi; jg, a ik a jk and a ki a kj we have that The next axiom says that each team should receive, at most, the total audience of the games played by the team.
Maximum aspirations: For each A 2 P and each i 2 N , Alternatively, one could consider a weaker upper bound with the total audience of all games in the tournament.
Weak upper bound: For each A 2 P and each i 2 N , The next axiom provides instead a lower bound as it says that no team should receive negative awards. Formally, Non-negativity. For each A 2 P and i 2 N; It is not di¢ cult to show that both maximum aspirations and non-negativity imply weak upper bound.
The next axiom says that revenues should be additive on A. Formally, The …nal two axioms refer to the performance of the rule with respect to somewhat pathological teams. Null team says that if a team has a null audience, then such a team gets no revenue. Essential team says that if only the games played by some team have positive audience, then such a team should receive all its audience. Formally, Null team: For each A 2 P, and each i 2 N , such that a ij = 0 = a ji , for each j 2 N , Essential team: For each A 2 P and each i 2 N such that a jk = 0 for each pair

The benchmark family
We start this section with a characterization result of the U C-family of rules, our benchmark family.

Theorem 1 A rule satis…es additivity, order preservation and weak upper bound if and only if
it is a member of the UC-family of rules.
Proof. It is not di¢ cult to show that both the uniform rule and concede-and-divide satisfy all the axioms in the statement. It follows from there that all the members of the UC-family of rules satisfy them too.
Conversely, let R be a rule satisfying the three axioms. Note that, then, R satis…es equal treatment of equals too. Let A 2 P. For each pair i; j 2 N , with i 6 = j, let 1 ij denote the matrix with the following entries: Notice that 1 ij ji = 0: Let k 2 N: By additivity, By equal treatment of equals, for each pair k; l 2 N n fi; jg we have that Let k 2 N n fi; jg. By additivity, R j 1 ij + 1 ik = x ij + z ik , and R k 1 ij + 1 ik = z ij + x ik .
By equal treatment of equals, R j 1 ij + 1 ik = R k 1 ij + 1 ik . Thus, Therefore, there exists x 2 R such that for each fi; jg N; for each l 2 N n fi; jg: By weak upper bound, Let k 2 N n fi; jg. By order preservation, which implies that x 1 n . Let = nx 1 n 1 : As 1 n x 1, it follows that 0 1: Then, Thus, U C (1 ij ) = R (1 ij ). As U C and R satisfy additivity, we deduce from here that Theorem 1 shows that the U C-family comprises all rules satisfying three basic and intuitive properties. We show next that the family exhibits additional interesting features. To begin with, all rules within the family satisfy the so-called single-crossing property. That is, for each pair of rules within the family, and each problem A 2 P, there exists a team i 2 N separating those teams bene…tting with one rule and those bene…tting with the other. It turns out that i is precisely the team whose overall audience is closest (from below) to the average overall audience. Formally, Proposition 1 Let 0 1 2 1, and A 2 P such that, without loss of generality, N = f1; : : : ; ng and 1 2 n . Then, there exists i 2 N such that: for each i = i + 1; :::; n.
Proof. Let 0 1 2 1, and A 2 P be such that N = f1; : : : ; ng and 1 2 n . Let i 2 N . We distinguish two cases: : Thus, i is the agent whose claim is closest to from below.
It is well known that the single-crossing property of preferences is a su¢ cient condition for the existence of a majority voting equilibrium (e.g., Gans and Smart, 1996). Thus, we have the following corollary from Proposition 1.
Corollary 1 There is a majority voting equilibrium for the U C-family of rules.
Corollary 1 states that if we let teams vote for a rule within the U C-family, then there will be a majority winner. The identity of this winner will be problem speci…c and, thus, it will depend on the characteristics of the problem at stake. For problems with a distribution of claims skewed to the left, only the uniform rule is a majority winner. For problems with a distribution of claims skewed to the right, only concede-and-divide is a majority winner. For the remainder of the problems, each U C rule is a majority winner. This is a consequence of the fact that, as it can be inferred from (1), U C i is increasing (decreasing) in for agents with claims above (below) the average claim.
Another well-known consequence of the single-crossing property is that it guarantees progressivity comparisons of schedules (e.g., Jakobsson, 1976;Hemming and Keen, 1983). Thus, we can also obtain an interesting corollary from Proposition 1 in our setting, referring to the distributive power of the rules within the U C-family.
Formally, given x; y 2 R n satisfying x 1 x 2 ::: x n , y 1 y 2 ::: y n , and for each k = 1; :::; n 1, with at least one strict inequality. When x is greater than y in the Lorenz ordering, one can state (see, for instance, Dasgupta et al., 1973) that x is unambiguously "more egalitarian" than y. In our setting, we say that a rule R Lorenz dominates another rule R 0 if for each A 2 P, R(A) is greater than R 0 (A) in the Lorenz ordering. As the Lorenz criterion is a partial ordering, one might not expect to perform many comparisons of vectors. It turns out that, here, all rules within the family are fully ranked according to this criterion.
Corollary 2 implies that the parameter de…ning the family can actually be interpreted as an index of the distributive power of the rules within the family.

Decomposing the benchmark family
In this section, we scrutinize the U C-family of rules further. We summarize …rst the performance of the rules within the family with respect to the other axioms introduced above 6 .

essential team if and only if it is concede-and-divide.
Combining Proposition 2 with Theorem 1, and noting that both non-negativity and maximum aspirations imply weak upper bound, additional characterizations are obtained as immediate corollaries.

Corollary 3
The following statements hold: 6 The proof of Proposition 2, and some other results of the paper, can be found in the Appendix.

1.
A rule satis…es additivity, order preservation and non-negativity if and only if it is a member of the UE-family of rules.

A rule satis…es additivity, order preservation and maximum aspirations if and only if it
is a member of the EC-family of rules. : In

Beyond the benchmark family
In this section, we consider some combinations of axioms leading towards rules that extend the benchmark family studied in the previous sections. In Theorem 2, we characterize the rules satisfying additivity, equal treatment of equals and weak upper bound. In Theorem 3, we characterize the rules satisfying additivity, equal treatment of equals and non-negativity.
In Theorem 4, we characterize the rules satisfying additivity and order preservation. In all cases, we obtain rules that are linear (but not necessarily convex) combinations of the uniform rule and concede-and-divide. For that reason, we conclude the section studying explicitly the performance of all the rules within the extended family f(1 )U + CD : 2 ( 1; +1)g with respect to all the axioms, depending on : The next result extends Theorem 1, weakening order preservation to equal treatment of equals. Proof. As mentioned above, the uniform rule and concede-and-divide satisfy additivity and equal treatment of equals. It follows from there any linear combination of the two rules sat-is…es the two axioms too. As for weak upper bound, one can also show (after some algebraic computations) that, for each 2 1 n 2 ; 1 , (1 )U + CD satis…es it too. 7 Conversely, let R be a rule satisfying the three axioms. Let i .
We now explore the implications of the combination of additivity and order preservation. Proof. As mentioned above, any linear combination of the uniform rule and concede-anddivide satis…es additivity. As concede-and-divide satis…es order preservation, and the uniform rule assigns the same amount to all teams, it follows that (1 )U + CD also satis…es order preservation for each 2 [0; 1).
Conversely, let R be a rule satisfying the two axioms. Let In  We conclude this section studying the performance of all rules within the general family with respect to the axioms considered in this paper.

Proposition 5
The following statements hold:   Common to all of our characterization results is the axiom of additivity. This is an invariance requirement with a long tradition in axiomatic work (e.g., Shapley, 1953)  One could also be interested into approaching our problems with a (cooperative) gametheoretical approach, a standard approach in many related models of resource allocation (e.g., To save space, we have included in this appendix, which is not necessarily intended for publication, some technical aspects of our analysis, as well as secondary proofs. For each A 2 P, and each i 2 N; we de…ne the rule R 1 as R 1 satis…es all axioms in the theorem but additivity.
Let R 2 be de…ned as follows. For each fi; jg 2 N and k 2 N we de…ne We extend R 2 to each problem A using additivity. Namely, satis…es all axioms in the theorem but order preservation.
Let R 3 be de…ned as follows. For each fi; jg 2 N and k 2 N we de…ne Now, if i , the above holds trivially. If, instead, i < , then the above is equivalent to n 2 2 (n 1) ( i ) ; as desired. Equivalently, : , the above is equivalent to n 2 2 (n 1) + n i 2 (n 1) ( i ) : If i < ; the above is equivalent to n 2 2 (n 1) Equivalently, We consider three cases.
3. i < : Then (3) is equivalent to (n 2) 2( i ) : As 1 n 2 ; it is enough to prove that Equivalently, We consider three cases.
3. i < : Then (4) is equivalent to (n 2) 2(n 1)( i ) : As n 2 2(n 1) , it is enough to prove that i . 9 It is also shown therein that for each 2 h 0; n 2 2(n 1) , the property is violated. Thus, it remains to show that the property is also violated for any = 2 [0; 1].
To do so, consider, again, the same problem as above. Then, as U (A) = (100; 100; 100) and 9 Note that those rules correspond precisely with the U E-family of rules. 10 Note that those rules correspond precisely with the EC-family of rules. Statements (g) and (h) are straightforward consequences of Proposition 2.

Extra material
We now study which speci…c rule within the U C-family could be a majority winner for each problem. We obtain three di¤erent scenarios, depending on the characteristics of the problem at stake. For some problems, only the uniform rule is a majority winner. For some other problems, only concede-and-divide is a majority winner. For the remainder of the problems, each rule within the family is a majority winner.
For each A 2 P, we consider the following partition of N , with respect to the average claim ( ): N l (A) = fi 2 N : i < g, N u (A) = fi 2 N : i > g, and N e (A) = fi 2 N : i = g. That is, taking the average claim (within the tournament) as the benchmark threshold, we consider three groups referring to individuals with claims below, above, or exactly at, the threshold. When no confusion arises, we simply write N l , N u ; and N e . Note that n = jN l j + jN u j + jN e j.
Proposition 6 Let A 2 P. The following statements hold: (i) If 2jN l j > n, then U (A) is the unique majority winner.
(ii) If 2jN u j > n, then CD (A) is the unique majority winner.
(iii) Otherwise, each U C (A) is a majority winner.
Proof. Let 0 1, and A 2 P. For each i 2 N , If i > , then U C i (A) is an increasing function of , thus maximized at = 1. This implies that, for each i 2 N u , CD i (A) is the most preferred outcome (among those provided by the family).
If i < , then U C i (A) is a decreasing function of , thus maximized at = 0. This implies that, for each i 2 N l , U i (A) is the most preferred outcome (among those provided by the family).
If i = ; then U C i (A) = jjAjj n for each 2 [0; 1] : This implies that, for each i 2 N e , all rules in the family yield the same outcome.
From the above, statements (i) and (ii) follow trivially. Assume, by contradiction, that statement (iii) does not hold. Then, there exists A 2 P and 2 [0; 1] such that U C is not a majority winner for A. Thus, we can …nd 0 2 [0; 1] such that U C 0 i (A) > U C i (A) holds for the majority of the teams. We then consider two cases: In this case, U C Proposition 6 implies that if the distribution of claims is skewed to the left (i.e., the median claim is below the mean claim), then the uniform allocation (the most equal allocation within the family) is the majority winner, whereas if it is skewed to the right (i.e., the median claim is above the mean claim), then the concede-and-divide allocation (the most unequal allocation within the family, as proved below) is the majority winner. If it is not skewed, then any compromise allocation can be a majority winner.
The single-crossing property also guarantees that the social preference relationship obtained under majority voting is transitive, and corresponds to the median voter's. In our setting, the median voter corresponds to the team with the median overall audience (claim). Thus, depending on whether this median overall audience is below or above the average audience, the median voter's preferred rule (and, thus, the majority winner) will either be the uniform rule or concede-and-divide. In other words, a tournament with a small number of very strong teams (i.e., with very high claims in relative terms) will proclaim the uniform allocation (the one favoring weaker teams more within the family) as the majority winner, whereas a tournament with a small number of very weak teams (i.e., with very small claims in relative terms) will proclaim the concede-and-divide allocation (the one favoring stronger teams more within the family).
Corollary 4 Let A 2 P be such that n is odd. The following statements hold: (i) If m < , then U (A) is the unique majority winner.
(ii) If m > , then CD(A) is the unique majority winner.
(iii) If m = , then any U C (A) is a majority winner.
Proof. If m < , then jN l j m. Hence 2jN l j > n. By Proposition 6, statement (i) holds.
If m > , then jN u j m. Hence 2jN u j > n By Proposition 6, statement (ii) holds.
If m = , then jN l j < m; jN u j < m; and jN e j > 0. Hence, we are in case (iii) of the statement of Proposition 6, which concludes the proof.
Corollary 5 Let A 2 P be such that n is even. The following statements hold: