03092021  Original Paper Open Access
On the axiomatic approach to sharing the revenues from broadcasting sports leagues
 Journal:
 Social Choice and Welfare
Important notes
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
1 Introduction
In a recent paper (Bergantiños and MorenoTernero
2020a), we have introduced a formal model to analyze the problem of sharing the revenues from broadcasting sports leagues among participating teams (clubs), based on the audiences they generate. It is a stylized and simple model, yet rich enough to obtain numerous interesting insights. The model’s input is simply a square matrix (
A) whose entries (
\(a_{ij}\)) indicate the audience of the game the two corresponding teams play, with the convention that the row team (
i) plays home and the column team (
j) plays away.
^{1} The goal is to (axiomatically) derive rules that associate for each matrix an allocation (among participating teams) of the overall audience in the tournament (i.e., the aggregation of all entries in the matrix).
In this paper, we uncover the structure of this stylized model further, thanks to the axiomatic approach. Our starting point is the principle of
impartiality, with a long tradition in the theory of justice (e.g., MorenoTernero and Roemer
2006). The basic formulation of such a principle is typically the axiom of
equal treatment of equals. In our setting, this axiom can take two forms, depending on how we define the concept of
equals. One says that, to consider two teams as equals, they should generate the same audience each time they play a third team. Another says that, moreover, the (two) games played by those two teams also generate the same audience. We shall refer to both axioms as
equal treatment of equals and
weak equal treatment of equals, respectively. Depending on the stance one takes with respect to the audience in the games involving each pair of teams, one might consider one or the other axiom more appropriate. To wit, if we consider each pair of teams accountable for the total audience they generate at the two games involving them,
equal treatment of equals seems to be the right one. If, instead, teams should be held accountable for the audience generated at their own stadiums, so that
\(a_{ij}\ne a_{ji}\) justifies to treat
i and
j differently, then
weak equal treatment of equals seems to be the right one.
Advertisement
As the names suggest,
equal treatment of equals implies
weak equal treatment of equals. We also formalize an axiom that fills the gap between both axioms:
pairwise reallocation proofness. This axiom says that a redistribution between the two audiences of the games involving a pair of teams does not affect the amounts obtained by the teams in the pair.
We explore the implications of the previous axioms, when combined with some other basic axioms:
additivity,
maximum aspirations,
nonnegativity and
weak upper bound. The first one, which is standard in axiomatic work and can be traced back to Shapley (
1953), says that awards are additive on audiences. The other three formalize reasonable (lower or upper) bounds, which are principles with a long tradition in the literature on fair allocation (e.g., Thomson
2011,
2019). More precisely,
maximum aspirations says that no team can receive an amount higher than its
claim (i.e., the aggregate audience at all the games in which the team was involved).
Nonnegativity says that no team receives negative awards. Finally,
weak upper bound (which is weaker than the previous two axioms) says that individual awards are bounded above by the aggregate audience obtained at the whole tournament.
We show that
equal treatment of equals,
additivity and
maximum aspirations characterize the socalled
ECfamily of rules, which is made of compromises between two rules that stand out as focal to solve this problem.
^{2} They are the socalled
equalsplit rule, which splits the revenue generated from each game equally among the participating teams, and
concedeanddivide, which concedes each team the revenues generated from its fan base and divides equally the residual. Each rule within the
ECfamily of rules is defined by a parameter that establishes the specific convex combination between the solutions the
equalsplit rule and
concedeanddivide yield (for each problem). The
equalsplit rule can be seen itself as a convex combination between the socalled
uniform rule (another focal rule in this model, which allocates an equal portion of the whole audience to each participating team) and
concedeanddivide. We shall refer to the resulting family of rules compromising between the
uniform rule and
concedeanddivide via convex combinations as the
UCfamily of rules. Alternatively, one could consider linear (but not necessarily convex) combinations between those rules. And that would give rise to the
generalized
UCfamily. It turns out that if we replace
maximum aspirations by
nonnegativity or
weak upper bound then we characterize two other subfamilies of the
generalized
UCfamily of rules.
The three characterization results mentioned above can be formulated alternatively upon replacing
equal treatment of equals by the pair of axioms made of
weak equal treatment of equals and
pairwise reallocation proofness. More interestingly, it turns out that replacing the latter by a somewhat similar axiom yields completely different outcomes. More precisely, we consider the axiom
standalone pair, which states that in situations where only the games involving a pair of teams have a positive audience, the total audience should be allocated to such a pair of teams. The combination of this axiom with
additivity,
weak equal treatment of equals and any axiom from the group made of
maximum aspirations,
nonnegativity and
weak upper bound characterizes a new family of rules. We call them
split rules, as they generalize the
equalsplit rule to allow for unequal (but fixed) splits of the audience of each game between the two playing teams.
Advertisement
Somewhat related to the above, we also consider the axiom of
homogeneous effect of standalone pair. This axiom says that, in situations where only games played by a pair of teams have a positive audience, the remaining teams obtain a homogeneous amount. To complement the results mentioned above, we characterize the family of rules satisfying
homogeneous effect of standalone pair,
additivity, and
weak equal treatment of equals. In such a family, the amount each team receives depends on three numbers: the overall home audience of the team, the overall away audience of the team, and the total audience of the whole tournament. There is a weighted aggregation of these three numbers and the weights are the same for each team.
Finally, our axioms of equal treatment of equals can naturally be strengthened to axioms of
order preservation. If we do so, we can obtain additional characterization results. To wit, we show that
order preservation,
additivity and
maximum aspirations also characterize the
ECfamily of rules (thus, being an “inferior” result to the one mentioned above with
equal treatment of equals). On the other hand, we show that
order preservation,
additivity and
nonnegativity characterize the socalled
UEfamily of rules, which is made of compromises (in the form of convex combinations) between the
uniform rule and the
equalsplit rule. And
order preservation,
additivity and
weak upper bound characterize the
UCfamily of rules mentioned above, which is precisely the union of the previous two families.
The last three characterization results can also be formulated alternatively upon replacing
order preservation by the pair of axioms made of
home (or
away)
order preservation and
pairwise reallocation proofness. And replacing the latter by
standalone pair , we obtain completely different outcomes too. More precisely, the combination of
additivity,
home (respectively,
away)
order preservation,
standalone pair and any of the three bounds axioms, characterizes one half of the family of
split rules: those that impose a fixed split of the audience of each game between the two playing teams, but guaranteeing at least (respectively, at most) one half to the local team. We conclude our analysis dismissing the bounds axioms in these last results. That is, we characterize the rules that satisfy
additivity,
home (or
away) order preservation or
weak equal treatment of equals , and
pairwise reallocation proofness or
standalone pair.
2 The model
We consider the model introduced by Bergantiños and MorenoTernero (
2020a). Let
N describe a finite set of teams. Its cardinality is denoted by
n. We assume
\(n\ge 3\). For each pair of teams
\(i,j\in N\), 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
\(i\in N\). Let
\(A\in {{\mathcal {A}}}_{n\times n}\) denote the resulting matrix of broadcasting audiences generated in the whole tournament involving the teams within
N.
^{3} Each matrix
\(A\in {{\mathcal {A}}}_{n\times n}\) with zero entries in the diagonal will thus represent a
problem and we shall refer to the set of problems as
\({\mathcal {P}}\).
^{4}
Let
\(\alpha _{i}\left( A\right) \) 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
\(\alpha _{i}\left( A\right) \) by the
claim of team
i. When no confusion arises, we write
\(\alpha _{i}\) instead of
\(\alpha _{i}\left( A\right) \). We define
\({\overline{\alpha }}\) as the average audience of all teams. Namely,
For each
\(A\in {{\mathcal {A}}}_{n\times n}\), let 
A denote the total audience of the tournament. Namely,
$$\begin{aligned} \alpha _{i}\left( A\right) =\sum _{j\in N}(a_{ij}+a_{ji}). \end{aligned}$$
$$\begin{aligned} {\overline{\alpha }}=\frac{\sum \nolimits _{i\in N}\alpha _{i}}{n}. \end{aligned}$$
$$\begin{aligned} A=\sum _{i,j\in N}a_{ij}=\frac{1}{2}\sum _{i\in N}\alpha _{i}=\frac{n {\overline{\alpha }}}{2}. \end{aligned}$$
2.1 Rules
A (sharing)
rule is a mapping that associates with each problem the list of the amounts the teams get from the total revenue, which we have normalized to 1 per viewer. Thus, formally,
\(R:{\mathcal {P}}\rightarrow {\mathbb {R}}^{n}\) is such that, for each
\(A\in \mathbf {{\mathcal {P}}}\),
The following three rules have been highlighted as focal for this problem (e.g., Bergantiños and MorenoTernero
2020a,
b,
2021a,
b). The
uniform rule divides equally among all teams the overall audience of the whole tournament. The
equalsplit rule divides the audience of each game equally, among the two participating teams.
Concedeanddivide, which can be rationalized from a simple form of statistical estimation (e.g., Bergantiños and MorenoTernero
2020a), compares the performance of a team with the average performance of the other teams.
^{5} Formally,
$$\begin{aligned} \sum _{i\in N}R_{i}(A)=A. \end{aligned}$$
Uniform,
U: for each
\(A \in \mathbf {{\mathcal {P}} }\), and each
\(i\in N\),
Equalsplit rule,
ES: for each
\(A \in \mathbf { {\mathcal {P}}}\), and each
\(i\in N\),
Concedeanddivide,
CD: for each
\(A \in \mathbf {{\mathcal {P}}}\), and each
\(i\in N\),
The following family of rules encompasses the above three rules.
$$\begin{aligned} U_{i}(A) =\frac{\left \left A\right \right }{n}=\frac{ {\overline{\alpha }}}{2}. \end{aligned}$$
$$\begin{aligned} ES_{i}(A) =\frac{\alpha _{i}}{2}. \end{aligned}$$
$$\begin{aligned} CD_{i}(A) =\alpha _{i}\frac{\sum \nolimits _{j,k\in N\backslash \left\{ i\right\} }\left( a_{jk}+a_{kj}\right) }{n2}=\frac{\left( n1\right) \alpha _{i}\left \left A\right \right }{n2}=\frac{2\left( n1\right) \alpha _{i}n{\overline{\alpha }} }{2(n2)}. \end{aligned}$$
UCfamily of rules
\(\left\{ UC^{\lambda }\right\} _{\lambda \in \left[ 0,1\right] }\): for each
\(\lambda \in \left[ 0,1\right] ,\) each
\(A\in \mathbf {{\mathcal {P}}}\), and each
\(i\in N\),
Equivalently,
At the risk of stressing the obvious, note that, when
\(\lambda =0\),
\( UC^{\lambda }\) coincides with the
uniform rule, whereas, when
\( \lambda =1\),
\(UC^{\lambda }\) coincides with
concedeanddivide. That is,
\(UC^{0}\equiv U\) and
\(UC^{1}\equiv CD\). Bergantiños and MorenoTernero (
2020a) prove that, for each
\(A\in {\mathcal {P}}\),
That is,
\(UC^{\lambda }\equiv ES\), where
\(\lambda =\frac{n2}{2\left( n1\right) }\).
^{6}
$$\begin{aligned} UC_{i}^{\lambda }(A)=(1\lambda )U_{i}(A)+\lambda CD_{i}(A). \end{aligned}$$
$$\begin{aligned} UC_{i}^{\lambda }(A)=(1\lambda )\frac{\left \left A\right \right }{n}+\lambda \frac{\left( n1\right) \alpha _{i}\left \left A\right \right }{n2}=\frac{{\overline{\alpha }}}{2} +\lambda \frac{n1}{n2}\left( \alpha _{i}{\overline{\alpha }}\right) . \end{aligned}$$
$$\begin{aligned} ES(A)=\frac{n}{2\left( n1\right) }U(A)+\frac{n2}{2\left( n1\right) }CD(A). \end{aligned}$$
Consequently, the
UCfamily 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,
UEfamily of rules
\(\left\{ UE^{\lambda }\right\} _{\lambda \in \left[ 0,1\right] }\): for each
\(\lambda \in \left[ 0,1\right] ,\) each
\(A \in \mathbf {{\mathcal {P}}}\), and each
\(i\in N\),
On the other hand, the family of rules compromising between the
equalsplit rule and concedeanddivide.
^{7} Formally,
$$\begin{aligned} UE_{i}^{\lambda }(A) =(1\lambda ) U_{i}(A)+\lambda ES_{i}(A) =\frac{ {\overline{\alpha }}}{2}+\frac{\lambda }{2}(\alpha _{i}{\overline{\alpha }}). \end{aligned}$$
ECfamily of rules
\(\left\{ EC^{\lambda }\right\} _{\lambda \in \left[ 0,1\right] }\): for each
\(\lambda \in \left[ 0,1\right] ,\) each
\(A \in \mathbf {{\mathcal {P}}}\), and each
\(i\in N\),
As Fig.
1 illustrates, the family of
UC rules is indeed the union of the family of
UE rules and
EC rules. Note that
\(UE^{0}\equiv UC^{0}\equiv U,\),
\(EC^{1}\equiv UC^{1}\equiv CD\), whereas
\(ES\equiv UE^{1}\equiv EC^{0}\equiv UC^{\frac{n2}{2\left( n1\right) }}\) is the unique rule belonging to both families.
$$\begin{aligned} EC_{i}^{\lambda }(A) =(1\lambda ) ES_{i}(A) +\lambda CD_{i}(A)=\frac{\alpha _{i}}{2}+\lambda \frac{ n}{2\left( n2\right) }\left( \alpha _{i}\overline{ \alpha }\right) . \end{aligned}$$
×
We could generalize the previous families by considering any linear (but not necessarily convex) combination between
U and
CD, which we shall sometimes refer as
generalized compromise rules. Formally,
GUCfamily of rules
\(\left\{ GUC^{\lambda }\right\} _{\lambda \in {\mathbb {R}}}\): for each
\( \lambda \in {\mathbb {R}},\) each
\(A\in \mathbf {{\mathcal {P}}}\), and each
\(i\in N\) ,
We also consider another category of rules arising from generalizing the
equalsplit rule to allow for unequal (but fixed) splits of the audience of each game between the two playing teams. Formally,
$$\begin{aligned} GUC_{i}^{\lambda }\left( A\right) =(1\lambda ) U_{i}\left( A\right) +\lambda CD_{i}\left( A\right) . \end{aligned}$$
Split rules
\(\left\{ S^{\lambda }\right\} _{\lambda \in \left[ 0,1\right] }\): for each
\(\lambda \in \left[ 0,1\right] , \) each
\(A \in \mathbf {{\mathcal {P}}}\), and each
\(i\in N\),
The
equalsplit rule corresponds to the case where
\(\lambda =0.5.\) The case
\(\lambda =0\) corresponds to the rule assigning to each team its home audience and the case
\(\lambda =1\) corresponds to the rule assigning to each team its away audience. Thus, we shall refer to each half of the family as
homebiased split rules (
\(\left\{ S^{\lambda }\right\} _{\lambda \in \left[ 0,\frac{1}{2} \right] }\)) and
awaybiased split rules (
\(\left\{ S^{\lambda }\right\} _{\lambda \in \left[ \frac{1}{2},1\right] }\)), respectively. As before, we could also consider a generalization of the
split rules.
$$\begin{aligned} S_{i}^{\lambda }(A) =\left( 1\lambda \right) \sum _{j\in N\backslash \left\{ i\right\} }a_{ij}+\lambda \sum _{j\in N\backslash \left\{ i\right\} } a_{ji}. \end{aligned}$$
Generalized split rules
\(\left\{ GS^{\lambda }\right\} _{\lambda \in {\mathbb {R}} }\): for each
\(\lambda \in {\mathbb {R}} ,\) each
\(A \in \mathbf {{\mathcal {P}}}\), and each
\(i\in N,\)
To conclude with this inventory, we consider the most general family of rules encompassing all the rules introduced above. The allocation received by each team
i depends on three data: its home audience
\(\left( \sum _{j\in N\backslash \left\{ i\right\} }a_{ij}\right) ,\) its away audience
\(\left( \sum _{j\in N\backslash \left\{ i\right\} }a_{ji}\right) ,\) and the total audience of the tournament
\(\left( \left \left A\right \right \right) .\) The relative importance of each data is the same for each team. Formally,
$$\begin{aligned} GS_{i}^{\lambda }(A) =\left( 1\lambda \right) \sum _{j\in N\backslash \left\{ i\right\} }a_{ij}+\lambda \sum _{j\in N\backslash \left\{ i\right\} } a_{ji}. \end{aligned}$$
General rules
\(\left\{ G^{xyz}\right\} _{x+y+nz=1}\). For each trio
\(x,y,z\in {\mathbb {R}}\) with
\(x+y+nz=1,\) each
\( A\in \mathbf {{\mathcal {P}}}\), and each
\(i\in N\),
Note that the
uniform rule corresponds to the case in which
\(x=0,\)
\( y=0\) and
\(z=\frac{1}{n}.\) The
equalsplit rule corresponds to the case in which
\(x=\frac{1}{2},\)
\(y=\frac{1}{2}\) and
\(z=0.\)
Concedeanddivide corresponds to the case in which
\(x=\frac{n1}{n2},\)
\(y= \frac{n1}{n2}\) and
\(z=\frac{1}{n2}.\) Thus, all
GUC rules belong to this family. Besides, generalized split rules also belong to this family (they are obtained when
\(x=1\lambda ,\)
\(y=\lambda \) and
\(z=0).\)
$$\begin{aligned} G_{i}^{xyz}(A)=x\sum _{j\in N\backslash \left\{ i\right\} }a_{ij}+y\sum _{j\in N\backslash \left\{ i\right\} }a_{ji}+z\left \left A\right \right . \end{aligned}$$
The family of
general rules also contains rules outside the previous families. An interesting example is the following:
which corresponds to
\(\left( x,y,z\right) =\left( 1,1,\frac{1}{n}\right) .\) This rule is a kind of
dual rule of the
uniform rule.
$$\begin{aligned} R_{i}(A)=\alpha _{i}\frac{\left \left A\right \right }{n }, \end{aligned}$$
Another less intuitive rule is the following:
which corresponds to
\(\left( x,y,z\right) =\left( \frac{n+1}{2},\frac{n+1}{ 2},1\right) \). With this rule, the larger the audience of a team, the smaller the revenue it receives. In particular, a team with null audience receives all the revenue.
$$\begin{aligned} R_{i}(A)=\left \left A\right \right \frac{n1}{2}\alpha _{i}, \end{aligned}$$
2.2 Axioms
We now introduce the axioms we consider in this paper.
The first axiom says that if two teams have the same audiences, when facing each of the other teams, then they should receive the same amount.
Equal treatment of equals
\(\left( ETE\right) \): For each
\(A\in {\mathcal {P}}\), and each pair
\(i,j\in N\) such that
\(a_{ik}=a_{jk}\), and
\( a_{ki}=a_{kj}\), for each
\(k\in N{\setminus} \{i,j\}\),
Now, we may want to require something more to consider two teams as truly equals. To wit, the two teams have the same audiences, not only when facing each of the other teams, but also when facing themselves at each stadium. The next axiom only guarantees that they receive the same amount when this extra condition holds.
$$\begin{aligned} R_{i}(A)=R_{j}(A). \end{aligned}$$
Weak equal treatment of equals
\(\left( WETE\right) \): For each
\( A\in {\mathcal {P}}\), and each pair
\(i,j\in N\) such that
\(a_{ij}=a_{ji},\)
\( a_{ik}=a_{jk}\), and
\(a_{ki}=a_{kj}\), for each
\(k\in N{{\setminus} } \{i,j\}\),
Obviously, this axiom is weaker than the previous one. The following axiom fills the gap. It says that a redistribution between the audiences of the two games involving a pair of teams does not affect the revenues obtained by the teams in the pair.
$$\begin{aligned} R_{i}(A)=R_{j}(A). \end{aligned}$$
Pairwise reallocation proofness
\(\left( PRP\right) \): For each pair
\(A,A^{\prime }\in {\mathcal {P}}\), and each pair
\(i_{0},j_{0}\in N\), such that
\( a_{ij}=a_{ij}^{\prime }\), for each pair
\(\left\{ i,j\right\} \ne \{i_{0},j_{0}\}\), and
\(a_{i_{0}j_{0}}+a_{j_{0}i_{0}}=a_{i_{0}j_{0}}^{\prime }+a_{j_{0}i_{0}}^{\prime }\),
A somewhat related axiom refers to situations where only games played by a pair of teams have a positive audience. It says that the total audience should be allocated to such two teams.
$$\begin{aligned} R_{k}(A)=R_{k}(A^{\prime })\text { for each }k=i_{0},j_{0}. \end{aligned}$$
Standalone pair
\(\left( SAP\right) \): For each
\(A\in {\mathcal {P}}\) and each pair
\(i,j\in N\) such that
\(a_{kl}=0\) for each pair
\(\left\{ k,l\right\} \in N\) with
\((k,l)\ne (i,j)\) and
\((k,l)\ne (j,i)\),
We could instead consider that, for those situations, the remaining teams obtain a homogeneous amount.
$$\begin{aligned} R_{i}(A)+R_{j}(A)=\left \left A\right \right . \end{aligned}$$
Homogeneous effect of standalone pair
\(\left( HSAP\right) \): For each
\(A\in {\mathcal {P}}\) and each pair
\(i,j\in N\) such that
\(a_{kl}=0\) for each pair
\(\left\{ k,l\right\} \in N\), such that
\((k,l)\ne (i,j)\) and
\( (k,l)\ne (j,i)\),
A strengthening of
standalone pair says that if a team has null audience, then it gets no revenue. Formally,
$$\begin{aligned} R_{k}(A)=R_{l}(A)\text { for each }k,l\in N\backslash \left\{ i,j\right\} . \end{aligned}$$
Null team
\(\left( NT\right) \): For each
\(A\in {\mathcal {P}}\), and each
\(i\in N\), such that for each
\(j\in N\),
\(a_{ij}=0=a_{ji}\),
The following axiom strengthens
equal treatment of equals 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.
$$\begin{aligned} R_{i}(A)=0. \end{aligned}$$
Order preservation
\(\left( OP\right) \): For each
\(A\in {\mathcal {P}}\) and each pair
\(i,j\in N\), such that, for each
\(k\in N\backslash \left\{ i,j\right\} \),
\(a_{ik}\ge a_{jk}\) and
\(a_{ki}\ge a_{kj}\),
Alternatively, we can consider the natural weaker versions, aligned with
weak equal treatment of equals, when also paying attention to the two games involving the pair.
$$\begin{aligned} R_{i}(A)\ge R_{j}(A). \end{aligned}$$
Home order preservation
\(\left( HOP\right) \): For each
\(A\in {\mathcal {P}}\) and each pair
\(i,j\in N\), such that, for each
\(k\in N\backslash \left\{ i,j\right\} \),
\(a_{ik}\ge a_{jk}\),
\(a_{ki}\ge a_{kj}\), and
\( a_{ij}\ge a_{ji}\),
$$\begin{aligned} R_{i}(A)\ge R_{j}(A). \end{aligned}$$
Away order preservation
\(\left( AOP\right) \): For each
\(A\in {\mathcal {P}}\) and each pair
\(i,j\in N\), such that, for each
\(k\in N\backslash \left\{ i,j\right\} \),
\(a_{ik}\ge a_{jk}\),
\(a_{ki}\ge a_{kj}\), and
\( a_{ji}\ge a_{ij}\),
The next axiom provides a natural lower bound as it says that no team should receive negative awards. Formally,
$$\begin{aligned} R_{i}(A)\ge R_{j}(A). \end{aligned}$$
Nonnegativity
\(\left( NN\right) \): For each
\(A\in {\mathcal {P}}\) and
\(i\in N,\)
A natural upper bound would say that each team should receive, at most, the total audience of the games played by the team. Formally,
$$\begin{aligned} R_{i}(A)\ge 0. \end{aligned}$$
Maximum aspirations
\(\left( MA\right) \): For each
\(A\in {\mathcal {P}}\) and each
\(i\in N\),
Alternatively, one could consider a weaker upper bound set by the total audience of all games in the tournament.
$$\begin{aligned} R_{i}(A)\le \alpha _{i}. \end{aligned}$$
Weak upper bound
\(\left( WUB\right) \): For each
\(A\in {\mathcal {P}}\) and each
\(i\in N\),
Finally, we consider the axiom saying that revenues should be additive on
A. Formally,
$$\begin{aligned} R_{i}(A)\le \left \left A\right \right . \end{aligned}$$
Additivity
\(\left( ADD\right) \): For each pair
A and
\(A^{\prime }\in {\mathcal {P}}\)
The next proposition, whose straightforward proof we omit, summarizes the relations between the axioms introduced above.
$$\begin{aligned} R\left( A+A^{\prime }\right) =R(A)+R\left( A^{\prime }\right) . \end{aligned}$$
Proposition 1
The following implications among axioms hold:
1.
Equal treatment of equals implies weak equal treatment of equals.
2.
The combination of weak equal treatment of equals and pairwise reallocation proofness implies equal treatment of equals.
3.
Order preservation implies home order preservation and away order preservation.
4.
The combination of home order preservation (or away order preservation) and pairwise reallocation proofness implies order preservation.
5.
Order preservation implies equal treatment of equals.
6.
Home order preservation (or away order preservation) implies weak equal treatment of equals.
7.
Null team implies standalone pair.
8.
Non negativity implies weak upper bound.
9.
Maximum aspirations implies weak upper bound.
3 Characterization results
We consider two subsections. In the first one, we combine
weak equal treatment of equals with some other axioms. In the second one, we perform a similar analysis replacing
weak equal treatment of equals with
weak order preservation.
3.1 With weak equal treatment of equals
Our first result provides several characterizations of the
split rules. More precisely, Theorem
1 states that those rules are characterized combining
additivity,
weak equal treatment of equals and
standalone pair with one of the three bounds axioms:
maximum aspirations,
nonnegativity or
weak upper bound.
Theorem 1
A rule satisfies
additivity,
weak equal treatment of equals,
standalone pair,
and either
maximum aspirations,
nonnegativity
or
weak upper bound
if and only if it is a split rule.
As all
split rules satisfy
null team, and
null team implies
standalone pair (the seventh statement of Proposition
1), we obtain the following corollary from Theorem
1.
Corollary 1
A rule satisfies
additivity,
weak equal treatment of equals,
null team,
and either
maximum aspirations,
nonnegativity
or
weak upper bound
if and only if it is a split rule.
As the next result states, to replace
standalone pair by
pairwise reallocation proofness in Theorem
1 yields a completely different outcome. Instead of
split rules, we obtain
generalized compromise rules. As a matter of fact, each of the bounds axioms leads to a subfamily of
\(GUC^{\lambda }\), arising from considering that
\(\lambda \) ranges within a certain interval. More precisely,
maximum aspirations leads to the interval
\(\left[ \frac{ n2}{2\left( n1\right) },1\right] \),
nonnegativity to the interval
\(\left[ \frac{1}{n1},\frac{n2}{ 2\left( n1\right) }\right] \) and
weak upper bound to the interval
\( \left[ 1\frac{n}{2},1\right] \).
Theorem 2
The following statements hold:
1.
A rule satisfies
additivity,
weak equal treatment of equals,
pairwise reallocation proofness
and
maximum aspirations
if and only if it belongs to the
ECfamily of rules.
2.
A rule satisfies
additivity,
weak equal treatment of equals,
pairwise reallocation proofness
and
nonnegativity
if and only if it belongs to the
GUCfamily of rules
for
\(\lambda \in \left[ \frac{1}{n1},\frac{n2}{2\left( n1\right) }\right] \).
3.
A rule satisfies
additivity,
weak equal treatment of equals,
pairwise reallocation proofness
and
weak upper bound
if and only if it belongs to the
GUCfamily of rules
for
\( \lambda \in \left[ 1\frac{n}{2},1\right] \).
It turns out that dismissing all bounds axioms (
maximum aspirations,
nonnegativity or
weak upper bound) in the above results leads to the characterization of the two complete families of
generalized (
split or
compromise) rules.
Theorem 3
The following statements hold:
1.
A rule satisfies
additivity,
weak equal treatment of equals
and either
standalone pair or null team
if and only if it belongs to the family of generalized split rules.
2.
A rule satisfies
additivity,
weak equal treatment of equals
and
pairwise reallocation proofness
if and only if it belongs to the
GUCfamily of rules.
The
equalsplit rule is the unique rule satisfying
additivity,
equal treatment of equals and
null team (e.g., Bergantiños and MorenoTernero
2020a). The first statement of Theorem
3 says that if we replace
equal treatment of equals by
weak equal treatment of equals in that result we obtain the family of
split rules instead of the
equalsplit rule. It also follows from the second statement of Theorem
3 that the only rule satisfying
null team within the
GUCfamily of rules is precisely the
equalsplit rule.
The next proposition is obtained by replacing
weak equal treatment of equals and
pairwise reallocation proofness by
equal treatment of equals, in Theorem
2 and the second statement of Theorem
3.
Proposition 2
The following statements hold:
1.
A rule satisfies
additivity,
equal treatment of equals
and
maximum aspirations
if and only if it belongs to the
ECfamily of rules.
2.
A rule satisfies
additivity,
equal treatment of equals
and
nonnegativity
if and only if it belongs to the
GUCfamily of rules
for
\(\lambda \in \left[ \frac{1}{n1},\frac{n2}{ 2\left( n1\right) }\right] \).
3.
A rule satisfies
additivity,
equal treatment of equals
and
weak upper bound
if and only if it belongs to the
GUCfamily of rules
for
\(\lambda \in \left[ 1\frac{n}{2},1\right] \).
4.
A rule satisfies
additivity
and
equal treatment of equals
if and only if it belongs to the
GUCfamily of rules.
The first statement of Proposition
2 is a refinement of Theorem 1 in Bergantiños and MorenoTernero (
2021a), obtained by replacing
equal treatment of equals with an axiom dubbed
symmetry, which says that if two teams have the same audiences they receive the same amounts.
In the next theorem we focus on
homogenous effect of standalone pair. We replace in the statement of Theorem
1
standalone pair by this new axiom, and we dismiss the bounds axioms. We obtain a characterization of the general rules.
Theorem 4
A rule satisfies
additivity,
weak equal treatment of equals
and
homogenous effect of standalone pair
if and only if it is a general rule.
In Theorem
1, we obtain the same family of rules when adding either one of the three bounds axioms. This does not happen in Theorem
4. For instance, the rule corresponding to
\(\left( x,y,z\right) =\left( 1,1,\frac{1 }{n}\right) \) satisfies maximum aspirations (and hence weak upper bound) but violates nonnegativity. The rule corresponding to
\(\left( x,y,z\right) =\left( \frac{1}{4},\frac{1}{4},\frac{1}{2n}\right) ,\) which is
\(\frac{1}{2}U+\frac{1 }{2}ES,\) satisfies non negativity (and hence weak upper bound) but violates maximum aspirations. The rule corresponding to
\(\left( \frac{n+1}{2},\frac{ n+1}{2},1\right) \) satisfies weak upper bound but violates maximum aspirations and nonnegativity.
We conclude providing two tables summarizing the results obtained in this section. First, in Table
1, we summarize the results obtained for
generalized split rules. We use capital letters for necessary axioms in the characterizations and small letters for axioms that can be exchanged in the characterization (only one of them is necessary). Second, in Table
2, we summarize the results obtained for
generalized compromise rules. When we write
\(\left[ a,b\right] \) we refer to the family of rules
\(\left\{ GUC^{\lambda }:\lambda \in \left[ a,b\right] \right\} \).
Table 1
Characterizations of generalized split rules
Rules/axioms

ADD

WETE

SAP

NT

MA

NN

WUB


\(\left\{ S^{\lambda }\right\} _{\lambda \in \left[ 0,1\right] }\) (Theorem
1)

X

X

X

x

x

x


\(\left\{ S^{\lambda }\right\} _{\lambda \in \left[ 0,1\right] }\) (Corollary
1)

X

X

X

x

x

x


\(\left\{ GS^{\lambda }\right\} _{\lambda \in {\mathbb {R}} }\) (Theorem
3.1)

X

X

x

x

Table 2
Characterizations of generalized compromise rules
Rules/axioms

ADD

ETE

WETE

PRP

MA

NN

WUB


\(\left[ \frac{n2}{2\left( n1\right) },1\right] \) (Theorem
2.1)

X

X

X

X


\(\left[ \frac{1}{n1},\frac{n2}{2\left( n1\right) }\right] \) (Theorem
2.2)

X

X

X

X


\(\left[ 1\frac{n}{2},1\right] \) (Theorem
2.3)

X

X

X

X


\((\infty ,+\infty )\) (Theorem
3.2)

X

X

X


\(\left[ \frac{n2}{2\left( n1\right) },1\right] \) (Proposition
2.1)

X

X

X


\(\left[ \frac{1}{n1},\frac{n2}{2\left( n1\right) }\right] \) (Proposition
2.2)

X

X

X


\(\left[ 1\frac{n}{2},1\right] \) (Proposition
2.3)

X

X

X


\((\infty ,+\infty )\) (Proposition
2.4)

X

X

3.2 With weak order preservation
We now explore the implications of strengthening
(weak) equal treatment of equals to become
(weak) order preservation in the above results. On the one hand, we observe that the three characterizations of the
split rules become now characterizations of one half of the family; depending on whether they bias the splitting in favor of the home team or the away team.
Theorem 5
The following statements hold:
1.
A rule satisfies
additivity,
home order preservation,
standalone pair,
and either
maximum aspirations,
nonnegativity,
or
weak upper bound
if and only if it is a homebiased split rule.
2.
A rule satisfies
additivity,
away order preservation,
standalone pair,
and either
maximum aspirations,
nonnegativity,
or
weak upper bound
if and only if it is an awaybiased split rule.
As all
split rules satisfy
null team, and
null team implies
standalone pair (the seventh statement of Proposition
1), we obtain the following corollary from Theorem
5.
Corollary 2
The following statements hold:
1.
A rule satisfies
additivity,
home order preservation,
null team,
and either
maximum aspirations,
nonnegativity
or
weak upper bound
if and only if it is a homebiased split rule.
2.
A rule satisfies
additivity,
away order preservation,
null team,
and either
maximum aspirations,
nonnegativity
or
weak upper bound
if and only if it is an awaybiased split rule.
On the other hand, we obtain characterizations of each of the three families involving convex combinations between the three focal rules; namely, the
uniform rule,
equalsplit rule, and
concedeanddivide.
Theorem 6
The following statements hold:
1.
A rule satisfies
additivity,
home order preservation
or
away order preservation,
pairwise reallocation proofness
and
maximum aspirations
if and only if it belongs to the
ECfamily of rules.
2.
A rule satisfies
additivity,
home order preservation
or
away order preservation,
pairwise reallocation proofness
and
nonnegativity
if and only if it belongs to the
UEfamily of rules.
3.
A rule satisfies
additivity,
home order preservation
or
away order preservation,
pairwise reallocation proofness
and
weak upper bound
if and only if it belongs to the
UCfamily of rules
As the next result states, dismissing all bounds axioms (
maximum aspirations,
nonnegativity or
weak upper bound) in the above results leads to the characterization of half of each of the two complete families of
generalized (
split or
compromise) rules.
Theorem 7
The following statements hold:
1.
A rule satisfies
additivity,
home order preservation
and either
standalone pair or
null team
if and only if it belongs to the
GSfamily of rules
for
\(\lambda \le \frac{1}{2}\).
2.
A rule satisfies
additivity,
away order preservation
and either
standalone pair
or
null team
if and only if it belongs to the
GSfamily of rules
for
\(\lambda \ge \frac{1}{2}\).
3.
A rule satisfies
additivity,
home order preservation
or
away order preservation
and
pairwise reallocation proofness
if and only if it belongs to the
GUCfamily of rules
for
\(\lambda \ge 0\).
The next proposition is obtained by replacing in Theorem
6, and the last statement of Theorem
7, the combination of
home/away order preservation and
pairwise reallocation proofness by
order preservation.
Proposition 3
The following statements hold:
1.
A rule satisfies
additivity,
order preservation
and
maximum aspirations
if and only if it belongs to the
ECfamily of rules.
2.
A rule satisfies
additivity,
order preservation
and
nonnegativity
if and only if it belongs to the
UEfamily of rules.
3.
A rule satisfies
additivity,
order preservation
and
weak upper bound
if and only if it belongs to the
UCfamily of rules.
4.
A rule satisfies
additivity
and
order preservation
if and only if it belongs to the
GUCfamily of rules
for
\(\lambda \ge 0\).
We conclude providing two tables summarizing the results obtained in this section. First, in Table
3, we summarize the results obtained for
generalized split rules. When we write
\(\left[ a,b\right] \) we refer to the family of rules
\(\left\{ GS^{\lambda }:\lambda \in \left[ a,b \right] \right\} \). We use capital letters for necessary axioms in the characterizations and small letters for axioms that can be exchanged in the characterization (only one of them is necessary). Second, in Table
4, we summarize the results obtained for
generalized compromise rules. When we write
\(\left[ a,b\right] \) we refer to the family of rules
\(\left\{ GUC^{\lambda }:\lambda \in \left[ a,b\right] \right\} \).
Table 3
Further characterizations of generalized split rules
Rules/axioms

ADD

HOP

AOP

SAP

NT

MA

NN

WUB


\(\left[ 0,\frac{1}{2}\right] \) (Theorem
5.1)

X

X

X

x

x

x


\(\left[ \frac{1}{2},1\right] \) (Theorem
5.2)

X

X

X

x

x

x


\(\left[ 0,\frac{1}{2}\right] \) (Corollary
2.1)

X

X

X

x

x

x


\(\left[ \frac{1}{2},1\right] \) (Corollary
2.2)

X

X

X

x

x

x


\(\left( \infty ,\frac{1}{2}\right] \) (Theorem
7.1)

X

X

x

x


\(\left[ \frac{1}{2}, +\infty \right) \) (Theorem
7.2 )

X

X

x

x

Table 4
Further characterizations of generalized compromise rules
Rules/axioms

ADD

OP

WOP

PRP

MA

NN

WUB


\(\left[ \frac{n2}{2\left( n1\right) },1\right] \) (Theorem
6.1 )

X

X

X

X


\(\left[ 0,\frac{n2}{2\left( n1\right) }\right] \) (Theorem
6.2 )

X

X

X

X


\(\left[ 0,1\right] \) (Theorem
6.3)

X

X

X

X


\(\left[ 0,+\infty \right) \) (Theorem
7.3 )

X

X

X


\(\left[ \frac{n2}{2\left( n1\right) },1\right] \) (Proposition
3.1)

X

X

X


\(\left[ 0,\frac{n2}{2\left( n1\right) }\right] \) (Proposition
3.2)

X

X

X


\(\left[ 0,1\right] \) (Proposition
3.3)

X

X

X


\(\left[ 0,+\infty \right) \) (Proposition
3.4)

X

X

4 Discussion
We have explored in this paper the axiomatic approach to the problem of sharing the revenues raised from the collective sale of broadcasting rights in sports leagues. We have mostly focussed on two alternative axioms formalizing the notion of
impartiality:
equal treatment of equals and
weak equal treatment of equals. These two axioms, when combined with
additivity and some other basic axioms, lead towards two alternative categories of rules. To wit, the former leads to several families of rules compromising between focal existing rules: the
uniform rule, the
equalsplit rule and
concedeanddivide. The latter instead leads to a unique family generalizing the
equalsplit rule to allow for unequal (but fixed) sharing of the audience of each game between the home team and the away team. The choice between
equal treatment of equals and
weak equal treatment of equals is ultimately a choice between disregarding the effect of
\(a_{ij}\ne a_{ji} \) or not and, as such, it is reflected in the characterization results just summarized.
We also characterize a more general family of rules (encompassing all of the above) by combining
additivity,
weak equal treatment of equals and a third axiom dubbed
homogeneous effect of standalone pair, which states that in situations where only games played by a pair of teams have a positive audience, the remaining teams obtain a homogeneous amount. In the resulting family, the amount each team receives depends on three numbers: the overall home audience of the team, the overall away audience of the team, and the total audience of the whole tournament. There is a weighted aggregation of these three numbers and the weights are the same for each team.
We have also obtained further characterizations upon strengthening the axioms of
equal treatment of equals to become axioms of
order preservation in the previous results. In one case, that leads precisely towards convex combinations of the three focal rules. In the other case, to homebiased (or awaybiased) split rules, which prioritize home teams (or away teams) in the splitting process within each pair of teams for the games they play.
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) but also considered strong under some circumstances. For results without
additivity in this model, the reader is referred to Bergantiños and MorenoTernero (
2020b).
As mentioned above, our analysis is restricted to a fixedpopulation setting. A generalization to a variablepopulation setting is left for further research. In such a setting, one would expect that focal axioms such as
consistency, or
population monotonicity, would play a major role to obtain new characterization results.
To conclude, we mention that 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., Littlechild and Owen
1973; van den Nouweland et al.
1996; Ginsburgh and Zang
2003; Bergantiños and MorenoTernero
2015). In Bergantiños and MorenoTernero (
2020a), we associate to our problems a natural
optimistic cooperative TU game in which, for each subset of teams, we define its worth as the total audience of the games played by the teams in that subset.
^{8} For such a resulting game, only twoplayer coalitions have nonzero dividends. Thus, it follows that the Shapley value coincides with the
\(\tau \)value and the nucleolus (e.g., Deng and Papadimitriou
1994; van den Nouweland et al.
1996). All those values yield the same solutions as the
equalsplit rule for the original problem.
^{9} The egalitarian value (e.g., van den Brink
2007) of that game yields the same solutions as the
uniform rule. Casajus and Huettner (
2013), van den Brink et al. (
2013) and Casajus and Yokote (
2019) characterize the family of values arising from the convex combination of the Shapley value and the egalitarian value. In our setting, this would correspond to the family of rules compromising (via convex combinations) between the
equalsplit rule and the
uniform rule. Thus, the second statement of Proposition
3 in our paper could be considered as a parallel result to some of the results in that literature. One could conceivably go beyond convex combinations to consider linear combinations of the Shapley value and the egalitarian value. If so, the resulting generalized egalitarian Shapley values could be associated to our entire family of
generalized compromise rules (including
concedeanddivide as a member). Finally, the socalled egalitarian nonseparable contribution value (e.g., Driessen and Funaki
1991) of that game yields the same solutions as the
dual of the
uniform rule, i.e., the rule that imposes to each team a uniform loss (where loss is understood as the difference between claim and obtained amount).
^{10} Thus, the family of values arising from the convex combination of the egalitarian value and the egalitarian nonseparable contribution value (e.g., van den Brink et al.
2016) would correspond in our setting to the family of rules compromising (via convex combinations) between the
uniform rule and its
dual.
Nevertheless, the gametheoretical approach, although both natural and tractable, is not without loss of generality. For example, one may consider that being a member of a league also involves significant externalities, so that a subgroup of teams may not necessarily achieve the same audiences when part of a smaller league. This, together with the fact that there is not a unique way to associate a TUgame to our problems, leads us to prefer the axiomatic approach to the gametheoretical approach to analyze our problems. That is why we have endorsed the axiomatic approach in this paper.
Acknowledgements
We thank François Maniquet (Managing Editor of this journal), two anonymous referees, and participants at the International Seminar on Social Choice, ICAE Webinars, UPOECON Webinars, Korea University Micro Seminar Series, CIO Webinars, University of Arizona FCTalks (in Memory of Patrick Harless), Riccardo Faini CEIS Webinar, the 2021 SAET Conference and the 2021 International Conference on Social Choice and Voting Theory for helpful comments and suggestions. Financial support from the Spanish Ministry of Economics and Competitiveness, through the research projects ECO201782241R, ECO201783069P, and PID2020115011GBI00, Xunta de Galicia through Grant ED431B 2019/34 and Junta de Andalucía through Grant P18FR2933 is gratefully acknowledged.
Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit
http://creativecommons.org/licenses/by/4.0/.
Appendix: Proofs of the results
For ease of exposition, we present the proofs of the results in a different order. In most of the proofs, we shall make use of the following notation. For each pair
\(i,j\in N\), with
\(i\ne j\), let
\({\varvec{1}}^{ij}\) denote the matrix with the following entries:
Notice that
\({\varvec{1}}_{ji}^{ij}=0.\)
$$\begin{aligned} {\varvec{1}}_{kl}^{ij}=\left\{ \begin{array}{cc} 1 &{}\hbox { if }\left( k,l\right) =\left( i,j\right) \\ 0 &{}\hbox { otherwise.} \end{array} \right. \end{aligned}$$
Proof of the first statement of Theorem 3
It is straightforward to show that each
generalized split rule satisfies
additivity,
weak equal treatment of equals,
standalone pair and
null team. Conversely, let
R be a rule satisfying
additivity,
weak equal treatment of equals and either
standalone pair or
null team. Let
\(i,j\in N\) with
\(i\ne j\). By
weak equal treatment of equals, there exists
\(z^{ij}\in {\mathbb {R}}\) such that
\(R_{k}\left( {\varvec{1}} ^{ij}\right) =z^{ij}\) for all
\(k\in N\backslash \left\{ i,j\right\} .\) Let
\( R_{i}\left( {\varvec{1}}^{ij}\right) =x^{ij}\) and
\(R_{j}\left( \varvec{ 1}^{ij}\right) =y^{ij}\). Then,
\(x^{ij}+y^{ij}+(n2)z^{ij}={\varvec{1}} ^{ij}=1\).
By
standalone pair or
null team,
\(z^{ij}=0\) and
\(y^{ij}=1x^{ij}\).
Let
\(k\in N{\setminus} \{i,j\}\). By
additivity,
By
weak equal treatment of equals,
\(1x^{ij}=1x^{ik}\). Hence,
\( x^{ij}=x^{ik}\).
$$\begin{aligned} R_{j}\left( {\varvec{1}}^{ij}+{\varvec{1}}^{ik}\right)= & {} R_{j}\left( {\varvec{1}}^{ij}\right) +R_{j}\left( {\varvec{1}}^{ik}\right) =1x^{ij}, \text { and} \\ R_{k}\left( {\varvec{1}}^{ij}+{\varvec{1}}^{ik}\right)= & {} R_{k}\left( {\varvec{1}}^{ij}\right) +R_{k}\left( {\varvec{1}}^{ik}\right) =1x^{ik}. \end{aligned}$$
Therefore, for each
\(i\in N\) there exists
\(x_{i}\in {\mathbb {R}}\) such that
\( R_{i}\left( {\varvec{1}}^{ij}\right) =x_{i}\), for each
\(j\in N\). Thus,
By
weak equal treatment of equals,
\(x_{i}+1x_{j}=1x_{i}+x_{j}\). Hence,
\(x_{i}=x_{j}\). Thus,
\(R_{i}\left( {\varvec{1}}^{ij}\right) =x\), for each pair
\(i,j\in N\).
$$\begin{aligned} R_{i}\left( {\varvec{1}}^{ij}+{\varvec{1}}^{ji}\right)= & {} x_{i}+1x_{j} \text { and} \\ R_{j}\left( {\varvec{1}}^{ij}+{\varvec{1}}^{ji}\right)= & {} 1x_{i}+x_{j} \end{aligned}$$
Finally, let
\(A\in {\mathcal {P}}\) and
\(i\in N.\) By
additivity,
Let
\(\lambda =1x.\) Then,
\(R\left( A\right) =GS^{\lambda }\left( A\right) \), as desired.
$$\begin{aligned} R_{i}(A)=\sum _{j,k\in N}a_{jk}R_{i}\left( {\varvec{1}}^{jk}\right)= & {} \sum _{j\in N\backslash \left\{ i\right\} }a_{ij}R_{i}\left( {\varvec{1}} ^{ij}\right) +\sum _{j\in N\backslash \left\{ i\right\} }a_{ji}R_{i}\left( {\varvec{1}}^{ji}\right) \\= & {} x\sum _{j\in N\backslash \left\{ i\right\} }a_{ij}+\left( 1x\right) \sum _{j\in N\backslash \left\{ i\right\} }a_{ji}. \end{aligned}$$
Proof of Theorem 1
It is straightforward to show that each
split rule satisfies all the axioms in the statement. Conversely, let
R be a rule that satisfies
additivity,
weak equal treatment of equals,
standalone pair, and either
maximum aspirations,
nonnegativity or
weak upper bound. By the first statement of Theorem
3, there exists
\(\lambda \in {\mathbb {R}}\) such that
\(R(A)=GS^{\lambda }\left( A\right) .\)
Let
\(i,j\in N\) with
\(i\ne j.\)
Thus, either way,
\(\lambda \in \left[ 0,1\right] \), which concludes the proof.

By maximum aspirations, \(1\lambda =R_{i}\left( {\varvec{1}} ^{ij}\right) \le \alpha _{i}\left( {\varvec{1}}^{ij}\right) =1,\) and \( \lambda =R_{j}\left( {\varvec{1}}^{ij}\right) \le \alpha _{j}\left( {\varvec{1}}^{ij}\right) =1.\)

By nonnegativity, \(1\lambda =R_{i}\left( {\varvec{1}} ^{ij}\right) \ge 0,\) and \(\lambda =R_{j}\left( {\varvec{1}}^{ij}\right) \ge 0.\)

By weak upper bound, \(1\lambda =R_{i}\left( {\varvec{1}} ^{ij}\right) \le \left \left {\varvec{1}}^{ij}\right \right =1,\) and \(\lambda =R_{j}\left( {\varvec{1}}^{ij}\right) \le \left \left {\varvec{1}}^{ij}\right \right =1.\)
Proof of the fourth statement of Proposition 2
It is obvious that
U and
CD satisfy
additivity and
equal treatment of equals. Then, each rule within the
GUCfamily also satisfies the two axioms. Conversely, let
R be a rule satisfying
additivity and
equal treatment of equals. Let
\(i,j\in N\) with
\(i\ne j.\) By
equal treatment of equals, there exist
\(x^{ij},z^{ij}\in {\mathbb {R}}\) such that
\(R_{i}\left( \varvec{ 1}^{ij}\right) =R_{j}\left( {\varvec{1}}^{ij}\right) =x^{ij}\) and
\( R_{k}\left( {\varvec{1}}^{ij}\right) =z^{ij}=\frac{12x^{ij}}{n2}\) for all
\(k\in N\backslash \left\{ i,j\right\} \).
Let
\(k\in N{\setminus} \{i,j\}\). By
additivity,
By
equal treatment of equals,
\(R_{j}\left( {\varvec{1}}^{ij}+ {\varvec{1}}^{ik}\right) =R_{k}\left( {\varvec{1}}^{ij}+{\varvec{1}} ^{ik}\right) \). Thus,
\(x^{ij}+\frac{12x^{ik}}{n2}=x^{ik}+\frac{12x^{ij}}{ n2}\), which implies that
\(x^{ij}=x^{ik}\). Therefore, there exists
\(x\in {\mathbb {R}}\) such that for each
\(\left\{ i,j\right\} \subset N,\)
Let
\(\lambda =\frac{nx1}{n1}\). Then,
Thus,
\(R\left( {\varvec{1}}^{ij}\right) =GUC^{\lambda }\left( \varvec{1 }^{ij}\right) \). As both rules satisfy
additivity, we deduce from here that
\(R\left( A\right) =GUC^{\lambda }(A)\), for each
\(A\in {\mathcal {P}}\) , as desired.
$$\begin{aligned} R_{j}\left( {\varvec{1}}^{ij}+{\varvec{1}}^{ik}\right)= & {} x^{ij}+z^{ik} \text { and } \\ R_{k}\left( {\varvec{1}}^{ij}+{\varvec{1}}^{ik}\right)= & {} z^{ij}+x^{ik}. \end{aligned}$$
$$\begin{aligned} R_{i}\left( {\varvec{1}}^{ij}\right)= & {} R_{j}\left( {\varvec{1}} ^{ij}\right) =x\text {, and} \\ R_{l}\left( {\varvec{1}}^{ij}\right)= & {} \frac{12x}{n2}\text { for each } l\in N{\setminus} \{i,j\}. \end{aligned}$$
$$\begin{aligned} (1\lambda )U_{k}\left( {\varvec{1}}^{ij}\right) +\lambda CD_{k}\left( {\varvec{1}}^{ij}\right) =\left\{ \begin{array}{cc} (1\lambda )\frac{1}{n}+\lambda =x &{}\hbox { if }k=i,j \\ (1\lambda )\frac{1}{n}\lambda \frac{1}{n2}=\frac{12x}{n2} &{} \hbox { otherwise.} \end{array} \right. \end{aligned}$$
Proof of the first statement of Proposition 2
By the fourth statement of Proposition
2, each rule within the
ECfamily satisfies
additivity and
equal treatment of equals. It is straightforward to show that they also satisfy
maximum aspirations. Conversely, let
R be a rule satisfying those three axioms. By the fourth statement of Proposition
2, there exists
\(\lambda \in {\mathbb {R}}\) such that
\(R=GUC^{\lambda }\).
Let
\(i,j\in N\) with
\(i\ne j.\) By
maximum aspirations,
and for each
\(k\in N\backslash \left\{ i,j\right\} ,\)
It then follows that
\(\lambda \) ranges from
\(\frac{n2}{2(n1)}\) to 1. As
\( UC^{\lambda }\equiv ES\), when
\(\lambda =\frac{n2}{2\left( n1\right) }\), this concludes the proof of this statement.
$$\begin{aligned} R_{i}\left( {\varvec{1}}^{ij}\right) =(1\lambda )\frac{1}{n}+\lambda \le \alpha _{i}\left( {\varvec{1}}^{ij}\right) =1, \end{aligned}$$
$$\begin{aligned} R_{k}\left( {\varvec{1}}^{ij}\right) =(1\lambda )\frac{1}{n}\lambda \frac{1}{n2}\le \alpha _{k}\left( {\varvec{1}}^{ij}\right) =0. \end{aligned}$$
Proof of the second statement of Proposition 2
By the fourth statement of Proposition
2, each rule from the statement satisfies
additivity and
equal treatment of equals. As for
nonnegativity, one can also show (after some algebraic computations) that, for each
\(\lambda \in \left[ \frac{1}{n1},\frac{n2}{ 2\left( n1\right) }\right] \),
\(GUC^{\lambda }\) satisfies it too. Conversely, let
R be a rule satisfying those three axioms. By the fourth statement of Proposition
2, there exists
\(\lambda \in {\mathbb {R}} \) such that
\(R=GUC^{\lambda }\).
Let
\(i,j\in N\) with
\(i\ne j.\) By
nonnegativity,
and for each
\(k\in N\backslash \left\{ i,j\right\} ,\)
It then follows that
\(\lambda \) ranges from
\(\frac{1}{n1}\) to
\(\frac{n2}{ 2\left( n1\right) }\) to 1, which concludes the proof of this statement.
$$\begin{aligned} R_{i}\left( {\varvec{1}}^{ij}\right) =(1\lambda )\frac{1}{n}+\lambda \ge 0, \end{aligned}$$
$$\begin{aligned} R_{k}\left( {\varvec{1}}^{ij}\right) =(1\lambda )\frac{1}{n}\lambda \frac{1}{n2}\ge 0. \end{aligned}$$
Proof of the third statement of Proposition 2
By the fourth statement of Proposition
2, each rule from the statement satisfies
additivity and
equal treatment of equals. As for
weak upper bound, one can also show (after some algebraic computations) that, for each
\(\lambda \in \left[ 1\frac{n}{2},1\right] \),
\( GUC^{\lambda }\) satisfies it too. Conversely, let
R be a rule satisfying those three axioms. By the fourth statement of Proposition
2, there exists
\(\lambda \in {\mathbb {R}} \) such that
\(R=GUC^{\lambda }\).
Let
\(i,j\in N\) with
\(i\ne j.\) By
weak upper bound,
and for each
\(k\in N\backslash \left\{ i,j\right\} ,\)
It then follows that
\(\lambda \) ranges from
\(1\frac{n}{2}\) to 1, which concludes the proof of this statement.
$$\begin{aligned} R_{i}\left( {\varvec{1}}^{ij}\right) =(1\lambda )\frac{1}{n}+\lambda \le \left \left {\varvec{1}}^{ij}\right \right =1, \end{aligned}$$
$$\begin{aligned} R_{k}\left( {\varvec{1}}^{ij}\right) =(1\lambda )\frac{1}{n}\lambda \frac{1}{n2}\le \left \left {\varvec{1}}^{ij}\right \right =1. \end{aligned}$$
Proof of Theorem 2
Proof of the second statement of Theorem 3
Proof of Theorem 4
It is straightforward to show that each general rule satisfies the three axioms in the statement. Conversely, let
R be a rule satisfying the three axioms. Let
\(i,j\in N\) with
\(i\ne j\). By
weak equal treatment of equals, there exists
\(z^{ij}\in {\mathbb {R}}\) such that
\(R_{k}\left( {\varvec{1}}^{ij}\right) =z^{ij}\) for all
\(k\in N\backslash \left\{ i,j\right\} .\) Let
\(R_{i}\left( {\varvec{1}}^{ij}\right) =x^{ij}\) and
\( R_{j}\left( {\varvec{1}}^{ij}\right) =y^{ij}\). Then,
\( x^{ij}+y^{ij}+(n2)z^{ij}={\varvec{1}}^{ij}=1\).
Let
\(k\in N{\setminus} \{i,j\}\). By
additivity,
By
weak equal treatment of equals,
\(y^{ij}+z^{ik}=z^{ij}+y^{ik}\). Hence,
\(y^{ij}z^{ij}=y^{ik}z^{ik}.\) Thus,
\(y^{i}=y^{ij}z^{ij}\) is well defined (does not depend on
j). Now, for each
\(i,j\in N,\)
\( y^{ij}=z^{ij}+y^{i}.\)
$$\begin{aligned} R_{j}\left( {\varvec{1}}^{ij}+{\varvec{1}}^{ik}\right)= & {} R_{j}\left( {\varvec{1}}^{ij}\right) +R_{j}\left( {\varvec{1}}^{ik}\right) =y^{ij}+z^{ik},\text { and} \\ R_{k}\left( {\varvec{1}}^{ij}+{\varvec{1}}^{ik}\right)= & {} R_{k}\left( {\varvec{1}}^{ij}\right) +R_{k}\left( {\varvec{1}}^{ik}\right) =z^{ij}+y^{ik} \end{aligned}$$
By
additivity,
By
weak equal treatment of equals,
\(x^{ji}+z^{ki}=z^{ji}+x^{ki}\). Hence,
\(x^{ji}z^{ji}=x^{ki}z^{ki}.\) Thus,
\(x^{i}=x^{ij}z^{ij}\) is well defined (does not depend on
j). Now, for each
\(i,j\in N,\)
\( x^{ij}=z^{ij}+x^{i}.\)
$$\begin{aligned} R_{j}\left( {\varvec{1}}^{ji}+{\varvec{1}}^{ki}\right)= & {} R_{j}\left( {\varvec{1}}^{ji}\right) +R_{j}\left( {\varvec{1}}^{ki}\right) =x^{ji}+z^{ki},\text { and} \\ R_{k}\left( {\varvec{1}}^{ji}+{\varvec{1}}^{ki}\right)= & {} R_{k}\left( {\varvec{1}}^{ji}\right) +R_{k}\left( {\varvec{1}}^{ki}\right) =z^{ji}+x^{ki}. \end{aligned}$$
Let
\(i,j,k\in N\) with
\(i\ne j\) and
\(i\ne k.\) Then,
Hence,
\(z^{ij}=z^{ik}\) for each pair
\(j,k\in N\backslash \left\{ i\right\} .\)
$$\begin{aligned} 1= & {} {\varvec{1}}^{ij}=\sum _{l\in N}R_{l}\left( {\varvec{1}} ^{ij}\right) =nz^{ij}+x^{i}+y^{i}\text { and } \\ 1= & {} {\varvec{1}}^{ik}=\sum _{l\in N}R_{l}\left( {\varvec{1}} ^{ik}\right) =nz^{ik}+x^{i}+y^{i}. \end{aligned}$$
By
homogeneous effect of standalone pair, for each
\(i,j\in N,\)
\( z^{ij}=z^{ji}.\)
Let
\(i,j,k,l\in N\) with
\(i\ne j\) and
\(k\ne l.\) Then,
Thus, there exists
z such that
\(z^{ij}=z\) for all
\(i,j\in N.\)
$$\begin{aligned} z^{ij}=z^{il}=z^{li}=z^{lk}=z^{kl}. \end{aligned}$$
By
weak equal treatment of equals,
\(R_{i}\left( {\varvec{1}}^{ij}+ {\varvec{1}}^{ji}\right) =R_{j}\left( {\varvec{1}}^{ij}+{\varvec{1}} ^{ji}\right) .\) As
we conclude that
As
\(1={\varvec{1}}^{ij}=nz+x^{i}+y^{i}\), and
\(1={\varvec{1}} ^{ji}=nz+x^{j}+y^{j}\), we deduce that
Now,
Thus,
\(y^{i}=y^{j}\), and hence
\(x^{i}=x^{j}\). It follows that
\(x=x^{i}\) and
\(y=y^{i}\) are well defined (they do not depend on
\(i\in N\)).
$$\begin{aligned} R_{i}\left( {\varvec{1}}^{ij}+{\varvec{1}}^{ji}\right)= & {} R_{i}\left( {\varvec{1}}^{ij}\right) +R_{i}\left( {\varvec{1}}^{ji}\right) =z^{ij}+x^{i}+z^{ji}+y^{j}=2z+x^{i}+y^{j}\text { and } \\ R_{j}\left( {\varvec{1}}^{ij}+{\varvec{1}}^{ji}\right)= & {} R_{j}\left( {\varvec{1}}^{ij}\right) +R_{j}\left( {\varvec{1}}^{ji}\right) =z^{ij}+y^{i}+z^{ji}+x^{j}=2z+y^{i}+x^{j}, \end{aligned}$$
$$\begin{aligned} x^{i}+y^{j}=y^{i}+x^{j}. \end{aligned}$$
$$\begin{aligned} x^{i}+y^{i}=x^{j}+y^{j}. \end{aligned}$$
$$\begin{aligned} x^{i}=y^{i}+x^{j}y^{j}\text { and }x^{i}=x^{j}+y^{j}y^{i}. \end{aligned}$$
The above implies that
R is characterized by three numbers
x,
y, and
z, satisfying that
\(nz+x+y=1\), and that, for each
\(i,j\in N,\)
Finally, by
additivity, for each
\(i\in N\), and
\(A\in {\mathcal {P}},\)
as desired.
$$\begin{aligned} R_{k}\left( {\varvec{1}}^{ij}\right) =\left\{ \begin{array}{cc} z+x &{}\hbox { if }k=i \\ z+y &{}\hbox { if }k=j \\ z &{}\hbox { otherwise} \end{array} \right. \end{aligned}$$
$$\begin{aligned} R_{i}\left( A\right)= & {} \sum _{j\in N\backslash \left\{ i\right\} }a_{ij}R_{i}\left( {\varvec{1}}^{ij}\right) +\sum _{j\in N\backslash \left\{ i\right\} }a_{ji}R_{i}\left( {\varvec{1}}^{ji}\right) +\sum _{j,k\in N\backslash \left\{ i\right\} }a_{jk}R_{i}\left( {\varvec{1}} ^{jk}\right) \\= & {} \left( z+x\right) \sum _{j\in N\backslash \left\{ i\right\} }a_{ij}+\left( z+y\right) \sum _{j\in N\backslash \left\{ i\right\} }a_{ji}+z\sum _{j,k\in N\backslash \left\{ i\right\} }a_{jk} \\= & {} x\sum _{j\in N\backslash \left\{ i\right\} }a_{ij}+y\sum _{j\in N\backslash \left\{ i\right\} }a_{ji}+z\left \left A\right \right , \end{aligned}$$
Proof of the first two statements of Theorem 7
By the first statement of Theorem
3, each
generalized split rule satisfies
additivity,
standalone pair and
null team. As for
home order preservation, let
\( A\in {\mathcal {P}}\), and
\(i,j\in N\) as in the definition of the axiom. Note that
and
Now, for each
\(k\in N\backslash \left\{ i,j\right\} \),
\(GS_{i}^{\lambda }\left( {\varvec{1}}^{ik}\right) =GS_{j}^{\lambda }\left( {\varvec{1}} ^{jk}\right) =1\lambda \) and
\(GS_{i}^{\lambda }\left( {\varvec{1}} ^{ki}\right) =GS_{j}^{\lambda }\left( {\varvec{1}}^{kj}\right) =\lambda \). Similarly,
\(GS_{i}^{\lambda }\left( {\varvec{1}}^{kl}\right) =GS_{j}^{\lambda }\left( {\varvec{1}}^{kl}\right) =0\), for each pair
\( k,l\in N\backslash \left\{ i,j\right\} \). Finally,
\(a_{ij}GS_{i}^{\lambda }\left( {\varvec{1}}^{ij}\right) +a_{ji}GS_{i}^{\lambda }\left( {\varvec{1}}^{ji}\right) \ge a_{ij}GS_{j}^{\lambda }\left( {\varvec{1}} ^{ij}\right) +a_{ji}GS_{j}^{\lambda }\left( {\varvec{1}}^{ji}\right) \) if and only if
\(\left( a_{ij}a_{ji}\right) \left( 12\lambda \right) \ge 0\). As
\(\lambda \le \)
\(\frac{1}{2}\),
\(GS_{i}^{\lambda }\left( A\right) \ge GS_{j}^{\lambda }\left( A\right) \), as desired. Likewise, for each
\(\lambda \ge \)
\(\frac{1}{2}\),
\(GS^{\lambda }\) satisfies
away order preservation.
$$\begin{aligned} GS_{i}^{\lambda }\left( A\right)= & {} a_{ij}GS_{i}^{\lambda }\left( {\varvec{1}}^{ij}\right) +a_{ji}GS_{i}^{\lambda }\left( {\varvec{1}} ^{ji}\right) +\sum _{k\in N\backslash \left\{ i,j\right\} }a_{ik}GS_{i}^{\lambda }\left( {\varvec{1}}^{ik}\right) \\&+\sum _{k\in N\backslash \left\{ i,j\right\} }a_{ki}GS_{i}^{\lambda }\left( {\varvec{1}}^{ki}\right) +\sum _{k,l\in N\backslash \left\{ i,j\right\} }a_{kl}GS_{i}^{\lambda }\left( {\varvec{1}}^{kl}\right) , \end{aligned}$$
$$\begin{aligned} GS_{j}^{\lambda }\left( A\right)= & {} a_{ij}GS_{j}^{\lambda }\left( {\varvec{1}}^{ij}\right) +a_{ji}GS_{j}^{\lambda }\left( {\varvec{1}} ^{ji}\right) +\sum _{k\in N\backslash \left\{ i,j\right\} }a_{jk}GS_{j}^{\lambda }\left( {\varvec{1}}^{jk}\right) \\&+\sum _{k\in N\backslash \left\{ i,j\right\} }a_{kj}GS_{j}^{\lambda }\left( {\varvec{1}}^{kj}\right) +\sum _{k,l\in N\backslash \left\{ i,j\right\} }a_{kl}GS_{j}^{\lambda }\left( {\varvec{1}}^{kl}\right) . \end{aligned}$$
Conversely, let
R be a rule satisfying
additivity, either
home order preservation or
away order preservation, and either
standalone pair or
null team. By the sixth statement of Proposition
1,
R also satisfies
weak equal treatment of equals. Thus, by the first statement of Theorem
3, there exists
\(\lambda \in {\mathbb {R}}\) such that
\( R=GS^{\lambda }.\) Then, for each pair
\(i,j\in N\) such that
\(i\ne j\),
\( R_{i}\left( {\varvec{1}}^{ij}\right) =1\lambda \) and
\(R_{j}\left( {\varvec{1}}^{ij}\right) =\lambda \). Now, if
R satisfies
home order preservation,
\(1\lambda =R_{i}\left( {\varvec{1}}^{ij}\right) \ge R_{j}\left( {\varvec{1}}^{ij}\right) =\lambda \). Thus,
\(\lambda \in \left( \infty ,\frac{1}{2}\right] \), which concludes the proof of the first statement. Likewise, if
R satisfies
away order preservation,
\( 1\lambda =R_{i}\left( {\varvec{1}}^{ij}\right) \le R_{j}\left( {\varvec{1}}^{ij}\right) =\lambda \). Thus,
\(\lambda \in \left[ \frac{1}{2}, +\infty \right) \), which concludes the proof of the second statement.
Proof of Theorem 5
Except for the
weak order preservation axioms, we already know from Theorem
1 that each
split rule satisfies all the axioms in the statement. By the first two statements of Theorem
7, each
homebiased split rule satisfies
home order preservation and each
awaybiased split rule satisfies
away order preservation. Conversely, let
R be a rule satisfying
additivity, either
home order preservation or
away order preservation,
standalone pair, and either
maximum aspirations,
nonnegativity, or
weak upper bound. By the sixth statement of Proposition
1,
R also satisfies
weak equal treatment of equals. Thus, by Theorem
1, there exists
\(\lambda \in [0,1]\) such that
\( R=S^{\lambda }.\) In particular, for each pair
\(i,j\in N\) such that
\(i\ne j\) ,
\(R_{i}\left( {\varvec{1}}^{ij}\right) =1\lambda \) and
\(R_{j}\left( {\varvec{1}}^{ij}\right) =\lambda \). Now, if
R satisfies
home order preservation,
\(1\lambda =R_{i}\left( {\varvec{1}}^{ij}\right) \ge R_{j}\left( {\varvec{1}}^{ij}\right) =\lambda \). Thus,
\(\lambda \in \left[ 0,\frac{1}{2}\right] \), which concludes the proof of the first statement. Likewise, if
R satisfies
away order preservation,
\(1\lambda =R_{i}\left( {\varvec{1}}^{ij}\right) \le R_{j}\left( {\varvec{1}} ^{ij}\right) =\lambda \). Thus,
\(\lambda \in \left[ \frac{1}{2},1\right] \), which concludes the proof of the second statement.
Proof of the fourth statement of Proposition 3
We know each rule within the
GUCfamily satisfies
additivity. As
concedeanddivide satisfies
order preservation, and the
uniform rule assigns the same amount to all teams, it follows that
\((1\lambda )U+\lambda CD\) also satisfies
order preservation for each
\(\lambda \in \left[ 0,\infty \right) \). Conversely, let
R be a rule satisfying
additivity and
order preservation. By the fifth statement of Proposition
1,
R also satisfies
equal treatment of equals. Thus, by the fourth statement of Proposition
2, there exists
\(\lambda \in {\mathbb {R}}\) such that
\( R=GUC^{\lambda }.\) Besides, given
\(i,j\in N\) with
\(i\ne j,\)
By
order preservation,
\(x\ge \frac{12x}{n2}\) which implies that
\( x\ge \frac{1}{n}\). As
\(\lambda =\frac{nx1}{n1}\), it follows that
\(\lambda \in \left[ 0,+\infty \right) .\)
$$\begin{aligned} R_{k}\left( {\varvec{1}}^{ij}\right) =\left\{ \begin{array}{cc} x &{}\quad \hbox { if }k=i,j \\ \frac{12x}{n2} &{}\quad \hbox { otherwise.} \end{array} \right. \end{aligned}$$
Proof of the first statement of Proposition 3
As mentioned above, each rule within the
ECfamily satisfies
additivity and
maximum aspirations. It is obvious that
ES and
CD satisfy
order preservation. Then, each rule within the
ECfamily satisfies
order preservation. Conversely, let
R be a rule satisfying those three axioms. By the fourth statement of Proposition
3, there exists
\(\lambda \in \left[ 0,+\infty \right) \) such that
\( R=GUC^{\lambda }\). By
maximum aspirations, similarly to the proof of the first statement of Proposition
2, we can conclude that
\( \lambda \)
\(\in \left[ \frac{n2}{2(n1)},1\right] .\) Namely,
R belongs to the
ECfamily, as desired.
Proof of the second statement of Proposition 3
As mentioned above, each rule within the
UEfamily satisfies
additivity and
nonnegativity. As
U and
ES satisfy
order preservation, each rule within the
UEfamily satisfies it too. Conversely, let
R be a rule satisfying those three axioms. By the fourth statement of Proposition
3, there exists
\( \lambda \in \left[ 0,+\infty \right) \) such that
\(R=GUC^{\lambda }\). By
non negativity, similarly to the proof of the second statement of Proposition
2, we can conclude that
\(\lambda\,\in \left[ 0, \frac{n2}{2(n1)}\right].\) Namely,
R belongs to the
UEfamily, as desired.
Proof of the third statement of Proposition 3
As mentioned above, each rule within the
UCfamily satisfies
additivity and
weak upper bound. It is obvious that
U and
CD satisfy
order preservation. Then, each rule within the
UCfamily satisfies
order preservation. Conversely, let
R be a rule satisfying those three axioms. By the fourth statement of Proposition
3, there exists
\(\lambda \in \left[ 0,+\infty \right) \) such that
\(R=GUC^{\lambda }\). By
weak upper bound, similarly to the proof of the third statement of Proposition
2, we can conclude that
\( \lambda \in \left[ 0,1\right] \). Namely,
R belongs to the
UCfamily, as desired.
Proof of Theorem 6
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Footnotes
1
Thus, we assume a double round robin tournament. The model could be easily extended to accommodate other tournament forms.
2
This result is almost equivalent to the characterization in Bergantiños and MorenoTernero (
2021a). Therein, we use a stronger notion than
equal treatment of equals indicating that two teams with the same claims receive the same awards.
3
We are therefore assuming a roundrobin 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 different number of times and playoffs 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.
4
As the set
N will be fixed throughout our analysis, we shall not explicitly consider it in the description of each problem.
8
Alternative natural options could also be considered in order to construct a characteristic function leading to a TUgame associated to our problems. To do so, one has to take a stance on what revenue a coalition can generate on its own. It is not obvious that any coalition of teams can generate the broadcasting revenues its members generated while being members of the league (the grand coalition). The above is an “optimistic” option assuming that each coalition of teams can generate the broadcasting revenues its members generated while being members of the league.
9
The property that only twoplayer coalitions have nonzero dividends is used by Maniquet (
2003), who shows that the socalled minimal transfer rule for queueing problems coincides with the Shapley value of the associated TUgame. As mentioned above, it follows from the property that coincidence with the nucleolus of such a game also holds, which is shown directly by Chun and Hokari (
2007).
10
This is actually a specific convex combination between the
equalsplit rule and
concedeanddivide, thus satisfying
additivity,
equal treatment of equals (as a matter of fact,
order preservation) and
maximum aspirations. As the number of teams becomes large, the convex combination approaches
concedeanddivide.