Anonymity in sharing the revenues from broadcasting sports leagues

Abstract

We study the problem of sharing the revenues from broadcasting sports leagues axiomatically. Our key axiom is anonymity, the classical impartiality axiom. Other impartiality axioms already studied in broadcasting problems are equal treatment of equals, weak equal treatment of equals and symmetry. We study the relationship between all impartiality axioms. Besides we combine anonymity with other axioms that have been considered in the literature. Some combinations give rise to new characterizations of well-known rules. The family of generalized split rules is characterized with anonymity, additivity and null team. The concede-and-divide rule is characterized with anonymity, additivity and essential team. Other and combinations characterize new rules that had not been considered before. We provide three characterizations in which three axioms are the same (anonymity, additivity, and order preservation) and the fourth one is different (maximum aspirations, weak upper bound, and non-negativity). Depending on the fourth axiom we obtain three different families of rules. In all of them concede-and-divide plays a central role.

1 Introduction

The number of people paying to watch professional sports on television has steadily increased in recent years. The amount of revenues raised from selling broadcasting rights has consequently increased, to the extent of becoming crucial for the management of sports organizations. Typically, the sale is carried out collectively and then revenues are shared among participating organizations. The revenue sharing rules differ worldwide. In North American leagues, uniform sharing mostly prevails. In European leagues, hybrid schemes involving lower bounds and performance-based rewards are widespread.

In Bergantiños and Moreno-Ternero (2020a) we introduced what we dub the broadcasting problem, a simple model in which the sharing process is based on the broadcasting audiences. The model considers a double round-robin tournament in which all games have a constant pay-per view fee. Thus, the prior of the model is just a square matrix, whose entries indicate the audience of the game involving the row team and the column team, at the former’s stadium. We have also studied this model theoretically in Bergantiños and Moreno-Ternero (2020b, 2021, 2022a, 2022b, 2022c, 2023a). Besides, we apply it to La Liga (the Spanish Football League) in Bergantiños and Moreno-Ternero (2020a, 2021, 2023b).

The broadcasting problem is an instance of resource allocation. Other well known instances to which the literature has paid attention are: airport problems (e.g., Littlechild and Owen, 1973), bankruptcy problems (e.g., O’Neill, 1982; Thomson, 2019a), telecommunications problems (e.g., van den Nouweland et al., 1996), minimum cost spanning tree problems (e.g., Bergantiños and Vidal-Puga, 2021), transport problems (e.g., Algaba et al. 2019; Estañ et al. 2021), inventory problems (e.g., Guardiola et al., 2021), liability problems with rooted tree networks (e.g. Oishi et al., 2023), knapsack problems (e.g., Arribillaga and Bergantiños 2023), pooling games (Schlicher et al. 2020), m-attribute games (Özen et al., 2022), urban consolidation centers (Hezarkhani et al., 2019), and scheduling problems with delays (Gonçalves-Dosantos et al., 2020).

Two classical notions within the literature of resource allocation are additivity and impartiality. The former is a kind of robustness axiom. Suppose that we can divide a problem in two small problems. Two views are possible: we solve the original problem, or we solve the small problems separately. Additivity requires that these two views produce the same outcome (e.g., Thomson, 2019b). In broadcasting problems, additivity says that revenues should be additive on the audiences. As for impartiality, it is a basic requirement of justice which excludes ethically irrelevant aspects from the allocation process. (e.g., Moreno-Ternero and Roemer, 2006). It is frequently formalized in two ways. On the one hand, with the axiom of equal treatment of equals, which says that if two agents are equal/symmetric, according to some input of the problem, then both agents should receive the same. On the other hand, with the axiom of anonymity, which says that a permutation of the set of agents equally permutes the allocation. This has an immediate implication: the name of the agents becomes irrelevant in the allocation process. A rule that is partial to some specific agent because of his name reflects an obvious unfairness in practice. Thus, anonymity is a normatively appealing axiom both in theory and practice.

Three different versions of equal treatment of equals have been used in broadcasting problems, depending on when two teams are considered equal/symmetric. In the first one, precisely named equal treatment of equals, two teams are considered equal if each time they play a third team the audiences are the same in both games (e.g., Bergantiños and Moreno-Ternero, 2020a). In the second one, named weak equal treatment of equals, an additional condition is imposed to consider two teams as equal: the audiences in the two games played by the two teams (home and away) also coincide (e.g., Bergantiños and Moreno-Ternero, 2022b). In the third one, named symmetry, two teams are considered equal simply when their aggregate audiences in the league coincide (e.g., Bergantiños and Moreno-Ternero, 2021). Thus, it is obviously the strongest of the three axioms as it is the one imposing the least demanding condition for teams to be equal/symmetric.¹

In this paper, we focus instead on anonymity in broadcasting problems. We first argue that anonymity is stronger than weak equal treatment of equals but it is neither related to equal treatment of equals nor to symmetry. We then consider its implications, when combined with other standard axioms in the literature. As we shall show, this allows us to uncover the structure of this rich model of broadcasting problems.

Our starting point is its combination with additivity. As we recently showed in Bergantiños and Moreno-Ternero (2023a), the combination of both axioms characterizes a family of general rules where the amount received by each team has three parts: one depending on the home audience of this team, another depending on the away audience of this team, and the third part depending on the total audience of the tournament.

Two focal rules in broadcasting problems are members of the family just described: the so-called equal-split rule and concede-and-divide. In Bergantiños and Moreno-Ternero (2020a) we argued that viewers of each game can essentially be divided in two categories: those watching the game because they are fans of one of the teams playing (called hard-core fans) and those watching the game because they think that the specific combination of teams renders the game interesting (called neutral fans). Besides, we argued that the revenue generated by hard-core fans should be assigned to the corresponding team and the revenue generated by neutral fans should be divided equally between both teams. The equal-split rule and concede-and-divide are extreme rules from the point of view of treating fans. The equal-split rule assumes that only neutral fans exist, whereas concede-and-divide assumes that there are as many hard-core fans as possible (compatible with the audiences). Then, equal-split divides the audience of each game equally among the two teams. Concede-and-divide concedes to each team (for each game) its number of hard-core fans and divides equally the rest. Both rules are characterized by additivity, equal treatment of equals and a third axiom: null team for the former, and essential team for the latter (e.g., Bergantiños and Moreno-Ternero, 2020a). We show in this paper that replacing equal treatment of equals by anonymity in those results is a move with striking consequences. More precisely, we show that the combination of anonymity with additivity and null team characterizes a family of rules where the audience of each game is divided in proportions $\lambda$ and $1-\lambda$ for the home and the away team respectively (the equal-split rule corresponds to the case where $\lambda =0.5).$ We show, on the other hand, that the combination of anonymity with additivity and essential team still characterizes concede-and-divide.

Another focal rule satisfying anonymity and additivity is the uniform rule (the audience of the tournament is divided equally among all teams). Convex combinations of each of the resulting pairs from the three focal rules have been considered. The so-called family of EC rules is made of convex combinations of the equal-split rule and concede-and-divide. The family of UE rules is made of convex combinations of the uniform rule and the equal-split rule and the family of UC rules is made of convex combinations of the uniform rule and concede-and-divide. The three families of rules have been characterized in Bergantiños and Moreno-Ternero (2021, 2022a) always involving one of the three axioms formalizing equal treatment of equals. We show in this paper that, if we consider anonymity instead, we derive three new families of rules, dubbed extended EC rules, extended UE rules, and extended UC rules, which contain, respectively, the three families mentioned above. To wit, extended EC rules are defined by two parameters $x^{\prime },y^{\prime }\in \left[ 0,1\right] .$ Each team i receives an allocation that is the sum of two parts. The first part is given by an EC rule where $\lambda$ depends on $x^{\prime }$ and $y^{\prime }$. The second part depends on $x^{\prime },$ $y^{\prime }$ and the allocation concede-and-divide yields for an associated broadcasting problem. When $x^{\prime }\ge y^{\prime }$ the associated problem is obtained by nullifying the home audiences of team i, whereas when $x^{\prime }\le y^{\prime }$ the associated problem is obtained by nullifying the away audiences of team i. When $x^{\prime }=y^{\prime }$ the rule just becomes an EC rule. The other two extended families can be described in similar ways. It turns out we obtain characterizations of the three families of extended rules when combining additivity and anonymity with other axioms from the literature.

We conclude the introduction acknowledging that our paper is related to some other papers in which families of rules are also characterized using similar axioms to those used in this paper. Some examples are the following. In cooperative games with transferable utility, Casajus and Huettner (2013), van den Brink et al. (2013), 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 UE rules. In minimum cost spanning tree problems, making use of additivity, Trudeau (2014) characterizes the convex combination of the so-called folk rule and the cycle-complete rule, whereas Bergantiños and Lorenzo (2021) characterize the family of Kruskal sharing rules. In bankruptcy problems, Thomson (2015a, 2015b) characterizes families of rules satisfying the standard non-negativity condition.

The rest of the paper is organized as follows. In Sect. 2, we introduce the model, rules and axioms. In Sect. 3, we obtain new characterizations of some well-known families of rules. In Sect. 4, we obtain characterizations of new families of rules. We conclude in Sect. 5.

2 The model

We consider the model introduced in Bergantiños and Moreno-Ternero (2020a). Let N be 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.² As the set N will be fixed throughout our analysis, we shall not explicitly consider it in the description of each problem. 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 $\mathcal {P}$.

Let $\alpha _{i}\left( A\right)$ denote the total audience achieved by team i, i.e.,

$$\begin{aligned} \alpha _{i}\left( A\right) =\sum _{j\in N}(a_{ij}+a_{ji}). \end{aligned}$$

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 $\alpha _{i}\left( A\right)$.

For each $A\in {\mathcal {A}}_{n\times n}$, let ||A|| denote the total audience of the tournament. Namely,

$$\begin{aligned} ||A||=\sum _{i,j\in N}a_{ij}=\frac{1}{2}\sum _{i\in N}\alpha _{i}. \end{aligned}$$

2.1 Rules

A (sharing) rule $\left( R\right)$ is a mapping that associates with each problem the list of the amounts teams get from the total revenue. As we have mentioned above we normalize the revenue generated from each viewer to 1. Formally, $R:\mathcal {P}\rightarrow \mathbb {R}^{N}$ is such that, for each $A\in \mathbf {\mathcal {P}}$,

$$\begin{aligned} \sum _{i\in N}R_{i}(A)=||A||. \end{aligned}$$

We first introduce three focal rules that have been studied in Bergantiños and Moreno-Ternero (2020a, 2020b, 2021, 2022a, 2022b, 2022c, 2023c).

The uniform rule divides equally among all teams the overall audience of the whole tournament. Formally,

Uniform rule, U: for each $A\in \mathbf { \mathcal {P}}$, and each $i\in N$,

$$\begin{aligned} U_{i}(A)=\frac{\left| \left| A\right| \right| }{n}. \end{aligned}$$

The equal-split rule divides the audience of each game equally, among the two participating teams. Formally,

Equal-split rule, ES: for each $A \in \mathbf { \mathcal {P}}$, and each $i\in N$,

$$\begin{aligned} ES_{i}(A) =\frac{\alpha _{i}}{2}. \end{aligned}$$

For each game, concede-and-divide concedes each team its number of hard-core fans and divides equally the rest. We introduce it in an equivalent and simpler way.³ Formally,

Concede-and-divide, CD: for each $A\in \mathbf { \mathcal {P}}$, and each $i\in N$,

$$\begin{aligned} CD_{i}(A)=\frac{\left( n-1\right) \alpha _{i}-\left| \left| A\right| \right| }{n-2}. \end{aligned}$$

We now introduce a family of rules studied in Bergantiños and Moreno-Ternero (2022a, 2022b). For each $\lambda \in \mathbb {R}$ and each game $\left( i,j\right) ,$ $S^{\lambda }$ divides the audience $a_{ij}$ among the teams i and j proportionally to $\left( 1-\lambda ,\lambda \right)$. Formally, for each $A\in \mathcal {P}$ and each $i\in N,$

$$\begin{aligned} S_{i}^{\lambda }\left( A\right) =\sum _{j\in N\backslash \left\{ i\right\} }\left( 1-\lambda \right) a_{ij}+\sum _{j\in N\backslash \left\{ i\right\} }\lambda a_{ji}. \end{aligned}$$

The equal-split rule corresponds to the case where $\lambda =0.5.$ When $\lambda =0$ all the audience is assigned to the home team and when $\lambda =1$ all the audience is assigned to the away team. We say that R is a split rule if $R\in \left\{ S^{\lambda }:\lambda \in \left[ 0,1 \right] \right\} .$ We say that R is a generalized split rule if $R\in \left\{ S^{\lambda }:\lambda \in \mathbb {R}\right\} .$

The next family of rules, studied in Bergantiños and Moreno-Ternero (2022c, 2023a), contains all the rules defined above. The amount received by each team i has three parts: one depending on the home audience of this team $\left( \sum \limits _{j\in N\backslash \left\{ i\right\} }a_{ij}\right) ,$ other depending on the away audience of this team $\left( \sum \limits _{j\in N\backslash \left\{ i\right\} }a_{ji}\right) ,$ and the third part depending on the total audience of the tournament $\left( \left| \left| A\right| \right| \right) .$ Formally,

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 $i\in N$,

$$\begin{aligned} G_{i}^{xyz}(A)=x\sum _{j\in N\backslash \left\{ i\right\} }a_{ij}+y\sum _{j\in N\backslash \left\{ i\right\} }a_{ji}+z\left| \left| A\right| \right| . \end{aligned}$$

The next table yields the value for the trio x, y, z, so that we obtain the three focal rules introduced above.

$$\begin{aligned} \begin{array}{cccc} &{} x &{} y &{} z \\ \text {Uniform rule} &{} 0 &{} 0 &{} \frac{1}{n} \\ \text {Equal-split rule} &{} \frac{1}{2} &{} \frac{1}{2} &{} 0 \\ \text {Concede-and-divide} &{} \frac{n-1}{n-2} &{} \frac{n-1}{n-2} &{} \frac{ -1}{n-2} \end{array} \end{aligned}$$

The next three families of rules, which have been studied in Bergantiños and Moreno-Ternero (2021, 2022a, 2022b) are also included in the family of general rules, as they are defined via convex combinations of two of the focal rules.

EC 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 $i\in N$,

$$\begin{aligned} EC_{i}^{\lambda }\left( A\right) =\lambda ES_{i}\left( A\right) +(1-\lambda )CD_{i}\left( A\right) . \end{aligned}$$

UC 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 $i\in N$,

$$\begin{aligned} UC_{i}^{\lambda }\left( A\right) =\lambda U_{i}\left( A\right) +(1-\lambda )CD_{i}\left( A\right) . \end{aligned}$$

UE 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 $i\in N$,

$$\begin{aligned} UE_{i}^{\lambda }\left( A\right) =\lambda U_{i}\left( A\right) +(1-\lambda )ES_{i}\left( A\right) . \end{aligned}$$

Bergantiños and Moreno-Ternero (2020a) prove that, for each $A\in \mathcal {P}$,

$$\begin{aligned} ES(A)=\frac{n}{2\left( n-1\right) }U(A)+\frac{n-2}{2\left( n-1\right) }CD(A). \end{aligned}$$

Thus,

$$\begin{aligned} \left\{ UC^{\lambda }\right\} _{\lambda \in \left[ 0,1\right] }= & {} \left\{ UE^{\lambda }\right\} _{\lambda \in \left[ 0,1\right] }\cup \left\{ EC^{\lambda }\right\} _{\lambda \in \left[ 0,1\right] },\text { and } \\ ES= & {} \left\{ UE^{\lambda }\right\} _{\lambda \in \left[ 0,1\right] }\cap \left\{ EC^{\lambda }\right\} _{\lambda \in \left[ 0,1\right] }. \end{aligned}$$

We now give an example to illustrate the various allocation rules.

Example 1

Let $A\in \mathbf {\mathcal {P}}$ be such that

$$\begin{aligned} A=\left( \begin{array}{ccc} 0 &{}\quad 1200 &{}\quad 1030 \\ 750 &{}\quad 0 &{}\quad 140 \\ 630 &{}\quad 210 &{}\quad 0 \end{array} \right) \end{aligned}$$

Then, $||A||=3960$ and $\left( \alpha _{i}\right) _{i\in N}=\left( 3610,2300,2010\right) .$

The three focal rules yield the following allocations in this example:

$$\begin{aligned} \begin{array}{cccc} \textrm{Rule} &{} \textrm{Team}\,1 &{} \textrm{Team}\,2 &{} \textrm{Team}\,3 \\ { U} &{} 1320 &{} 1320 &{} 1320 \\ { ES} &{} 1805 &{} 1150 &{} 1005 \\ { CD} &{} 3260 &{} 640 &{} 60 \end{array} \end{aligned}$$

We also compute the allocations from several generalized split rules $S^{\lambda }$:

$$\begin{aligned} \begin{array}{cccc} \lambda &{} \textrm{Team}\,1 &{} \textrm{Team}\,2 &{} \textrm{Team}\,3 \\ 0 &{}\quad 2230 &{}\quad 890 &{}\quad 840 \\ 0.2 &{}\quad 2060 &{}\quad 994 &{}\quad 906 \\ 1 &{}\quad 1380 &{}\quad 1410 &{}\quad 1170 \\ 4 &{}\quad -1170 &{}\quad 2970 &{}\quad 2160 \end{array} \end{aligned}$$

And the allocations from several general rules $G^{xyz}$:

$$\begin{aligned} \begin{array}{cccc} \left( x,y,z\right) &{} \textrm{Team}\,1 &{} \textrm{Team}\,2 &{} \textrm{Team}\,3 \\ \left( 0.5,0.2,0.1\right) &{} 1787 &{} 1123 &{} 1050 \\ \left( 1,3,-1\right) &{} 2410 &{} 1160 &{} 390 \end{array} \end{aligned}$$

Finally, we compute the allocations from $EC^{\lambda },$ $UC^{\lambda },$ and $UE^{\lambda }$ when $\lambda =0.5$:

$$\begin{aligned} \begin{array}{cccc} \textrm{Rule}\,&{} \textrm{Team}\,1 &{} \textrm{Team}\, 2 &{} \textrm{Team}\,3 \\ EC^{0.5} &{} 2532.5 &{} 895 &{} 532.5 \\ UC^{0.5} &{} 2290 &{} 980 &{} 690 \\ UE^{0.5} &{} 1562.5 &{} 1235 &{} 1162.5 \end{array} \end{aligned}$$

2.2 Axioms

We now introduce the axioms considered in this paper.

Our key axiom is anonymity, which says that a permutation of the set of teams equally permutes the allocation. Formally, let $\sigma$ be a permutation of the set of teams. Thus, $\sigma :N\rightarrow N$ such that $\sigma \left( i\right) \ne \sigma \left( j\right)$ when $i\ne j.$ Given a permutation $\sigma$ and $A\in \mathcal {P}$, we define the problem $A^{\sigma }$ where for each $i,j\in N,$ $a_{ij}^{\sigma }=a_{\sigma \left( i\right) \sigma \left( j\right) }.$

Anonymity $\left( AN\right)$: For each $A\in \mathcal {P}$, each permutation $\sigma ,$ and each $i\in N$,

$$\begin{aligned} R_{i}(A)=R_{\sigma \left( i\right) }(A^{\sigma }). \end{aligned}$$

The next axiom says that revenues should be additive on A. Formally,

Additivity $\left( AD\right)$: For each pair A and $A^{\prime }\in \mathcal {P}$,

$$\begin{aligned} R\left( A+A^{\prime }\right) =R(A)+R\left( A^{\prime }\right) . \end{aligned}$$

All the results presented in this paper involve the two axioms introduced above. We also consider other axioms. The first three axioms represent the three different ways of applying the principle of impartiality we mentioned at the Introduction.

Equal treatment of equals says that if two teams have the same audiences, when facing each of the other teams, then they should receive the same amount. This axiom has been used in Bergantiños and Moreno-Ternero (2020a, 2020b, 2021, 2022a, 2022b).

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 $k\in N{\setminus } \{i,j\}$,

$$\begin{aligned} R_{i}(A)=R_{j}(A). \end{aligned}$$

In Bergantiños and Moreno-Ternero (2022a, 2022b, 2022c) we also require something more to consider two teams as equals. Not only the two teams must have the same audiences when facing each of the other teams, but they also must have the same audiences when facing themselves at each stadium. Formally,

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 $k\in N{\setminus } \{i,j\}$,

$$\begin{aligned} R_{i}(A)=R_{j}(A). \end{aligned}$$

Finally, symmetry requires less to consider two teams as equal/symmetric. More precisely, the axiom says that if two teams have the same aggregate audience in the tournament, then they should receive the same amount. This axiom has been considered in Bergantiños and Moreno-Ternero (2021).

Symmetry $\left( SYM\right)$: For each $A\in \mathcal {P}$, and each pair $i,j\in N$, such that $\alpha _{i}\left( A\right) =\alpha _{j}\left( A\right)$,

$$\begin{aligned} R_{i}(A)=R_{j}(A). \end{aligned}$$

The following axioms strengthen the previous ones 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 $\left( OP\right)$: 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 $a_{ki}\ge a_{kj}$ we have that

$$\begin{aligned} R_{i}(A)\ge R_{j}(A). \end{aligned}$$

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 $a_{ij}\ge a_{ji}$ we have that

$$\begin{aligned} R_{i}(A)\ge R_{j}(A). \end{aligned}$$

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 $a_{ji}\ge a_{ij}$ we have that

$$\begin{aligned} R_{i}(A)\ge R_{j}(A). \end{aligned}$$

The next two axioms refer to the performance of the rule with respect to somewhat pathological teams. First, null team says that if a team has a null audience, then such a team gets no revenue. Second, 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 $\left( NT\right)$: For each $A\in \mathcal {P}$, and each $i\in N$, such that $a_{ij}=0=a_{ji}$, for each $j\in N$,

$$\begin{aligned} R_{i}(A)=0. \end{aligned}$$

Essential team $\left( ET\right)$: For each $A\in \mathcal {P}$ and each $i\in N$ such that $a_{jk}=0$ for each pair $\left\{ j,k\right\} \in N\backslash \left\{ i\right\}$,

$$\begin{aligned} R_{i}(A)=\alpha _{i}\left( A\right) . \end{aligned}$$

The next three axioms provide natural (lower/upper) bounds to the amount a team could receive.

The first one says that each team should receive, at most, the total audience of the games it played.

Maximum aspirations $\left( MA\right)$: For each $A\in \mathcal {P}$ and each $i\in N$,

$$\begin{aligned} R_{i}(A)\le \alpha _{i}\left( A\right) . \end{aligned}$$

The second one says that each team should receive, at most, the total audience of all games in the tournament.

Weak upper bound $\left( WUB\right)$: For each $A\in \mathcal {P}$ and each $i\in N$,

$$\begin{aligned} R_{i}(A)\le \left| \left| A\right| \right| . \end{aligned}$$

The third axiom says that no team should receive negative awards.

Non-negativity $\left( NN\right)$. For each $A\in \mathcal {P}$ and each $i\in N,$

$$\begin{aligned} R_{i}(A)\ge 0. \end{aligned}$$

The next result, whose straightforward proof we omit, summarizes the logical relations between the axioms introduced above.⁴

Proposition 1

The following statements hold:

$\left( 1\right)$ $AN\rightarrow WETE\leftarrow ETE\leftarrow SYM.$

$\left( 2\right)$ $ETE\leftarrow OP\rightarrow WETE.$

$\left( 3\right)$ $MA\rightarrow WUB\leftarrow NN.$

$\left( 4\right)$ $OP\rightarrow \left\{ HOP,AOP\right\} \rightarrow OP$

As for the impartiality axioms, we note first that there exist rules satisfying WETE but violating AN. For instance, the rule that divides the total audience equally among team 1 and all equally-deserving teams, according to WETE. Obviously this rule satisfies WETE. The next example shows that it does not satisfy AN.

Example 2

Let $A\in \mathbf {\mathcal {P}}$ be such that

$$\begin{aligned} A=\left( \begin{array}{ccc} 0 &{}\quad 6 &{}\quad 4 \\ 6 &{}\quad 0 &{}\quad 4 \\ 2 &{}\quad 2 &{}\quad 0 \end{array} \right) \end{aligned}$$

Then $R\left( A\right) =\left( 12,12,0\right) .$

Let $\sigma$ be such that $\sigma \left( 1\right) =2,$ $\sigma \left( 2\right) =3,$ and $\sigma \left( 3\right) =1.$ Now

$$\begin{aligned} A^{\sigma }=\left( \begin{array}{ccc} 0 &{}\quad 4 &{}\quad 6 \\ 2 &{}\quad 0 &{}\quad 2 \\ 6 &{}\quad 4 &{}\quad 0 \end{array} \right) \end{aligned}$$

Then $R\left( A^{\sigma }\right) =\left( 24,0,0\right) .$ As $R_{1}\left( A\right) \ne R_{2}\left( A^{\sigma }\right)$ we deduce that R does not satisfy AN.

Likewise, there exist rules satisfying ETE but violating AN. For instance, the rule that divides the total audience equally among team 1 and all equally-deserving teams, according to ETE. Obviously this rule satisfies ETE. The previous example shows that it does not satisfy AN (notice that teams 1 and 2 are equally-deserving teams, according to ETE).

Finally, there exist rules satisfying AN but violating ETE. For instance, the rule $S^{1}$ introduced above. By Theorem 1 in Bergantiños and Moreno-Ternero (2023a), $S^{1}$ satisfies AN. The next example shows that $S^{1}$ does not satisfy ETE, 

Example 3

Let $A\in \mathbf {\mathcal {P}}$ be such that

$$\begin{aligned} A=\left( \begin{array}{ccc} 0 &{}\quad 4 &{}\quad 6 \\ 2 &{}\quad 0 &{}\quad 6 \\ 3 &{}\quad 3 &{}\quad 0 \end{array} \right) \end{aligned}$$

Teams 1 and 2 satisfy the conditions of ETE. Nevertheless, $S^{1}\left( A\right) =\left( 5,7,12\right) .$

3 New characterizations of well-known families of rules

Our first result is a counterpart of Theorem 1 in Bergantiños and Moreno-Ternero (2020a), obtained by replacing equal treatment of equals therein by anonymity.

Theorem 1

The following statements hold:

$\left( 1\right)$ A rule satisfies additivity, anonymity, and null team if and only if it is a generalized split rule.

$\left( 2\right)$ A rule satisfies additivity, anonymity, and essential team if and only if it is concede-and divide.

Proof

We first discuss the main ideas of this proof (which will also be considered in the rest of the proofs). All our theorems characterize families of rules (in some cases, a single rule) with a combination of several axioms that include additivity and anonymity. The proof of the theorems has two parts. First, to prove that each rule of the family satisfies the combination of the axioms. This is made by checking that the rule satisfies the formula stated by the axiom. Second, to prove that if a rule satisfies the combination of axioms, then the rule belongs to the family. As additivity is always one of the axioms, the proof of this part is made in two steps. In the first step, we decompose a general problem in simple problems. We then characterize the rules satisfying the combination of our axioms for simple problems. In the second one, we extend the rules from simple problems to general problems using additivity. This two-step procedure is quite standard in the literature of resource allocation. The difference lies in the simple problems considered and the axioms used in their characterizations.

Theorem 1 in Bergantiños and Moreno-Ternero (2023a) states that a rule satisfies anonymity and additivity if and only if it is a general rule. Thus, it only remains to show that a general rule satisfies null team if and only if it is a generalized split rule, and that concede-and divide is the only general rule that satisfies essential team.

Let R be a general rule, i.e., $R=G^{xyz}$ for some $x,y,z\in R$ with $x+y+nz=1$. For each pair $i,j\in N$, with $i\ne j$, let $\varvec{1} ^{ij}\in \mathbf {\mathcal {P}}$ denote the matrix with the following entries:

$$\begin{aligned} \varvec{1}_{kl}^{ij}=\left\{ \begin{array}{ll} 1 &{} \textrm{if} \left( k,l\right) =\left( i,j\right) \\ 0 &{} \textrm{otherwise}. \end{array} \right. \end{aligned}$$

The proof of Theorem 1 in Bergantiños and Moreno-Ternero (2023a) shows that

$$\begin{aligned} z= & {} R_{k}\left( \varvec{1}^{ij}\right) \text { with }k\in N\backslash \left\{ i,j\right\} \nonumber \\ x= & {} R_{i}\left( \varvec{1}^{ij}\right) -z\text { and } \nonumber \\ y= & {} R_{j}\left( \varvec{1}^{ij}\right) -z. \end{aligned}$$

(1)

Besides, z,  x, and y do not depend on i, j, k.

Now, by null team, $z=0$ and hence $x=1-y.$ Then,

$$\begin{aligned} R_{i}(A)=G^{\left( 1-y\right) y0}=\left( 1-y\right) \sum _{j\in N\backslash \left\{ i\right\} }a_{ij}+y\sum _{j\in N\backslash \left\{ i\right\} }a_{ji}=S_{i}^{y}\left( A\right) . \end{aligned}$$

Hence, R is a generalized split rule, which concludes the proof of statement $\left( 1\right)$.

Now, by essential team, $R_{i}\left( \varvec{1}^{ij}\right) =R_{j}\left( \varvec{1}^{ij}\right) =1.$ Hence, $z=\frac{-1}{n-2}.$ Then, by additivity, $R\left( A\right) =CD\left( A\right)$, which concludes the proof of statement $\left( 2\right)$. $\square$

Remark 1

The axioms used in Theorem 1 are independent.

$\left( 1\right)$ The uniform rule satisfies AD and AN but fails NT.

Let $R^{1}$ be the rule in which the audience of each game goes to the team with the lowest number. Namely, for each $A\in \mathcal {P}$, and each $i\in N,$

$$\begin{aligned} R_{i}^{1}(N,A)=\sum _{j\in N:j>i}(a_{ij}+a_{ji}). \end{aligned}$$

Then, $R^{1}$ satisfies NT and AD but fails AN.

Let $R^{2}$ be the rule in which the audience of each game is divided among the teams playing the game proportionally to their audiences in the games played against the other teams. Namely, for each $A\in \mathcal {P}$, and each $i\in N,$

$$\begin{aligned} R_{i}^{2}(N,A)=\sum _{j\in N\backslash \left\{ i\right\} }\frac{ \sum \limits _{k\in N\backslash \left\{ i,j\right\} }\left( a_{ik}+a_{ki}\right) }{\sum \limits _{k\in N\backslash \left\{ i,j\right\} }\left( a_{ik}+a_{ki}\right) +\sum \limits _{k\in N\backslash \left\{ i,j\right\} }\left( a_{jk}+a_{kj}\right) }\left[ a_{ij}+a_{ji}\right] . \end{aligned}$$

Then, $R^{2}$ satisfies AN and NT but fails AD.

$\left( 2\right)$ The uniform rule satisfies AD and AN but fails ET.

Let $R^{3}$ be such that, for each pair $i,j\in N$, with $i\ne j$, each $\varvec{1}^{ij}\in \mathbf {\mathcal {P}}$ and each $k\in N,$

$$\begin{aligned} R_{k}^{3}\left( N,\varvec{1}^{ij}\right) =\left\{ \begin{array}{ll} 1 &{} \textrm{if}\,k\in \left\{ i,j\right\} \\ -1 &{} \textrm{if}\,k=\min \left\{ l:l\in N\backslash \left\{ i,j\right\} \right\} \\ 0 &{} \textrm{otherwise} \end{array} \right. \end{aligned}$$

We extend $R^{3}$ to all problems using additivity. Namely, $R^{3}\left( A\right) =\sum \limits _{i,j\in N:i\ne j}a_{ij}R^{3}\left( \varvec{1}^{ij}\right) .$

Then, $R^{3}$ satisfies AD and ET but fails AN.

Let $\mathbf {\mathcal {P}}^{\prime }$ be the set of problems having at least one essential team. Let $R^{4}$ be such that it coincides with concede-and-divide on $\mathbf {\mathcal {P}}^{\prime }$ and with the uniform rule on $\mathbf {\mathcal {P}}\backslash \mathbf {\mathcal {P}} ^{\prime }.$

Then, $R^{4}$ satisfies AN and ET but fails AD.

In many resource allocation problems, some relevant rules are also characterized with the help of an impartiality axiom. Usually, if we replace the impartiality axiom by anonymity in the characterizations, the result still holds.⁵ We stress that replacing equal treatment of equals by anonymity in Theorem 1 at Bergantiños and Moreno-Ternero (2020a) has different impacts. With null team, instead of characterizing the equal-split rule, we characterize a whole family of rules containing it. With essential team, we simply obtain an alternative characterization of concede-and-divide. We also note that Theorem 1 is the counterpart of Theorem 3.1 at Bergantiños and Moreno-Ternero (2022a), replacing weak equal treatment of equals by anonymity. Finally, if we add one of the bound axioms introduced above to the first statement of Theorem 1, we obtain a characterization of split rules instead of generalized split rules.

Corollary 1

A rule satisfies additivity, anonymity, null team and either maximum aspirations, weak upper bound or non-negativity if and only if it is a split rule.

Proof

Let x,  y, and z be defined as in the proof of Theorem 1.

Let $i,j\in N$, with $i\ne j.$ By maximum aspirations or weak upper bound, $x+z=R_{i}\left( \varvec{1}^{ij}\right) \le 1$ and $y+z=R_{j}\left( \varvec{1}^{ij}\right) \le 1.$ As $z=0,$ we deduce that $y\in \left[ 0,1\right]$ and $R=S^{y}.$

By non-negativity, $x+z=R_{i}\left( \varvec{1}^{ij}\right) \ge 0$ and $y+z=R_{j}\left( \varvec{1}^{ij}\right) \ge 0.$ As $z=0,$ we deduce that $y\in \left[ 0,1\right]$ and $R=S^{y}.$ $\square$

4 Characterizations of new families of rules

In this section we characterize new families of rules combining anonymity and additivity with the other axioms described above. All the families have two parts. The first part is obtained by applying a rule to the original problem. The second part is obtained by applying concede-and divide to an associated problem where some audiences of the original problem are nullified.

Formally, for each $A\in \mathcal {P}$ and each $i\in N$ let $A^{i0}$ be the matrix obtained from A by nullifying the audiences of all the games i played home. Namely,

$$\begin{aligned} a_{jk}^{i0}=\left\{ \begin{array}{ll} 0 &{} \textrm{if}\,j=i \\ a_{jk} &{} \textrm{otherwise}. \end{array} \right. \end{aligned}$$

Besides, let $A^{0i}$ the matrix obtained from A by nullifying the audiences of all the games i played away. Namely,

$$\begin{aligned} a_{jk}^{0i}=\left\{ \begin{array}{ll} 0 &{} \textrm{if}\,k=i \\ a_{jk} &{} \textrm{otherwise}. \end{array} \right. \end{aligned}$$

We say that R is an extended EC rule if there exist $x^{\prime },y^{\prime }\in \left[ 0,1\right]$ with $x^{\prime }+y^{\prime }\ge 1$ such that, for each $i\in N$,

$$\begin{aligned} R_{i}\left( A\right) =\left\{ \begin{array}{ll} EC_{i}^{\lambda }\left( A\right) -\left| x^{\prime }-y^{\prime }\right| CD_{i}\left( A^{i0}\right) &{} if x^{\prime }\ge y^{\prime } \\ EC_{i}^{\lambda }\left( A\right) -\left| x^{\prime }-y^{\prime }\right| CD_{i}\left( A^{0i}\right) &{} if x^{\prime }\le y^{\prime } \end{array} \right. \end{aligned}$$

where $\lambda =2-2\max \left\{ x^{\prime },y^{\prime }\right\} \in \left[ 0,1\right] .$

We denote by $EC^{x^{\prime }y^{\prime }}$ the extended EC rule associated to $x^{\prime }$ and $y^{\prime }$ as above. Notice that EC rules are extended EC rules (just take $x^{\prime }=y^{\prime }).$

In the next theorem, we characterize the extended EC rules.

Theorem 2

A rule satisfies additivity, anonymity and maximum aspirations if and only if it is an extended EC rule.

Proof

We have argued above that ES is a generalized split rule. Thus, by Theorem 1, both ES and CD satisfy additivity and anonymity. Therefore, so do all extended EC rules.

Let $EC^{x^{\prime }y^{\prime }}$ be an extended EC rule. We first show that $EC^{x^{\prime }y^{\prime }}$ satisfies maximum aspirations for each matrix $\varvec{1}^{ij}\in \mathbf { \mathcal {P}}$ with $i,j\in N$ such that $i\ne j.$

We consider several cases.

• 1. $x^{\prime }\ge y^{\prime }$.

  $$\begin{aligned} EC_{i}^{x^{\prime }y^{\prime }}\left( \varvec{1}^{ij}\right)= & {} \left( 2-2x^{\prime }\right) ES_{i}\left( \varvec{1}^{ij}\right) +\left( 2x^{\prime }-1\right) CD_{i}\left( \varvec{1}^{ij}\right) -\left| x^{\prime }-y^{\prime }\right| CD_{i}\left( \left( \varvec{1} ^{ij}\right) ^{i0}\right) \\= & {} \left( 2-2x^{\prime }\right) \frac{1}{2}+\left( 2x^{\prime }-1\right) =x^{\prime }\le 1. \end{aligned}$$

  Similarly, $EC_{j}^{x^{\prime }y^{\prime }}\left( \varvec{1}^{ij}\right) =x^{\prime }\le 1.$ For each $k\in N\backslash \left\{ i,j\right\}$,

  $$\begin{aligned} EC_{k}^{x^{\prime }y^{\prime }}\left( \varvec{1}^{ij}\right) =\left( 2x^{\prime }-1\right) \frac{-1}{n-2}=\frac{1-2x^{\prime }}{n-2}\le 0. \end{aligned}$$

• 2. $x^{\prime }\le y^{\prime }.$

  $$\begin{aligned} EC_{i}^{x^{\prime }y^{\prime }}\left( \varvec{1}^{ij}\right) =\left( 2-2y^{\prime }\right) \frac{1}{2}+\left( 2y^{\prime }-1\right) -\left( y^{\prime }-x^{\prime }\right) =x^{\prime }\le 1. \end{aligned}$$

  Similarly, $EC_{j}^{x^{\prime }y^{\prime }}\left( \varvec{1}^{ij}\right) =x^{\prime }\le 1.$ For each $k\in N\backslash \left\{ i,j\right\}$,

  $$\begin{aligned} EC_{k}^{x^{\prime }y^{\prime }}\left( \varvec{1}^{ij}\right) =\left( 2y^{\prime }-1\right) \frac{-1}{n-2}-\left( y^{\prime }-x^{\prime }\right) \frac{-1}{n-2}=\frac{1-x^{\prime }-y^{\prime }}{n-2}\le 0. \end{aligned}$$

Then, $EC^{x^{\prime }y^{\prime }}$ satisfies maximum aspirations for $\varvec{1}^{ij}\in \mathbf {\mathcal {P}}$. We now prove it in the general case. Let $A\in \mathcal {P}$ and $i\in N.$ As $EC^{x^{\prime }y^{\prime }}$ satisfies additivity,

$$\begin{aligned} EC_{i}^{x^{\prime }y^{\prime }}\left( A\right)= & {} \sum _{j,k\in N}a_{jk}EC_{i}^{x^{\prime }y^{\prime }}\left( \varvec{1}^{jk}\right) \\= & {} \sum _{j\in N\backslash \left\{ i\right\} }\left( a_{ij}EC_{i}^{x^{\prime }y^{\prime }}\left( \varvec{1}^{ij}\right) +a_{ji}EC_{i}^{x^{\prime }y^{\prime }}\left( \varvec{1}^{ji}\right) \right) +\sum _{j,k\in N\backslash \left\{ i\right\} }a_{jk}EC_{i}^{x^{\prime }y^{\prime }}\left( \varvec{1}^{jk}\right) \\\le & {} \sum _{j\in N\backslash \left\{ i\right\} }\left( a_{ij}+a_{ji}\right) +\sum _{j,k\in N\backslash \left\{ i\right\} }0 \\= & {} \alpha _{i}\left( A\right) . \end{aligned}$$

Conversely, let R be a rule satisfying the axioms from the statement. Let $A\in \mathcal {P}$. Given $i,j\in N,$ with $i\ne j$, let $x^{\prime }=R_{i}\left( \varvec{1}^{ij}\right)$ and $y^{\prime }=R_{j}\left( \varvec{1}^{ij}\right)$ and $z=R_{k}\left( \varvec{1}^{ij}\right)$ for $k\in N\backslash \left\{ i,j\right\} .$

By maximum aspirations, $x^{\prime }\le \alpha _{i}\left( \varvec{1}^{ij}\right) = 1,$ $y^{\prime }\le \alpha _{j}\left( \varvec{1}^{ij}\right) =1$ and $z\le \alpha _{k}\left( \varvec{1} ^{ij}\right) =0.$ As $z=\frac{1-x^{\prime }-y^{\prime }}{n-2}$, we deduce that $x^{\prime }+y^{\prime }\ge 1.$ Then, $x^{\prime }\ge 0$ and $y^{\prime }\ge 0.$

Let $i\in N.$ By additivity,

$$\begin{aligned} R_{i}(A)= & {} \sum _{j\in N}a_{ij}R_{i}\left( \varvec{1}^{ij}\right) +\sum _{j\in N}a_{ji}R_{i}\left( \varvec{1}^{ji}\right) +\sum _{j,k\in N\backslash \left\{ i\right\} }a_{jk}R_{i}\left( \varvec{1}^{jk}\right) \\= & {} x^{\prime }\sum _{j\in N}a_{ij}+y^{\prime }\sum _{j\in N}a_{ji}+\frac{ 1-x^{\prime }-y^{\prime }}{n-2}\sum _{j,k\in N\backslash \left\{ i\right\} }a_{jk}. \end{aligned}$$

We consider two cases.

Case 1. $\max \left\{ x^{\prime },y^{\prime }\right\} =x^{\prime }.$

Then,

$$\begin{aligned} R_{i}(A)=x^{\prime }\alpha _{i}\left( A\right) +\frac{1-2x^{\prime }}{n-2} \left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right) +\left( y^{\prime }-x^{\prime }\right) \sum _{j\in N}a_{ji}- \frac{y^{\prime }-x^{\prime }}{n-2}\left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right) . \end{aligned}$$

Thus,

$$\begin{aligned} x^{\prime }\alpha _{i}\left( A\right) +\frac{1-2x^{\prime }}{n-2}\left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right)= & {} x^{\prime }\alpha _{i}\left( A\right) +\left( 2x^{\prime }-1\right) \frac{\left( n-1\right) \alpha _{i}\left( A\right) -\left| \left| A\right| \right| }{n-2}\\{} & {} -\left( 2x^{\prime }-1\right) \alpha _{i}\left( A\right) \\= & {} \left( 2-2x^{\prime }\right) ES_{i}\left( A\right) +\left( 2x^{\prime }-1\right) CD_{i}\left( A\right) . \end{aligned}$$

As $x^{\prime },y^{\prime }\in \left[ 0,1\right] ,$ $x^{\prime }+y^{\prime }\ge 1$ and $x^{\prime }\ge y^{\prime }$ we have that $x^{\prime }\in \left[ \frac{1}{2},1\right] .$ Let $\lambda =2-2x^{\prime }.$ Then $\lambda \in \left[ 0,1\right]$.

Besides,

$$\begin{aligned} \left( y^{\prime }-x^{\prime }\right) \sum _{j\in N}a_{ji}-\frac{y^{\prime }-x^{\prime }}{n-2}\left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right)= & {} \left( y^{\prime }-x^{\prime }\right) \frac{\left( n-2\right) \sum \limits _{j\in N}a_{ji}+\alpha _{i}\left( A\right) -\left| \left| A\right| \right| }{n-2} \\= & {} \left( y^{\prime }-x^{\prime }\right) CD_{i}\left( A^{i0}\right) . \end{aligned}$$

Now,

$$\begin{aligned} R_{i}(A)=EC^{2-2\max \left\{ x^{\prime },y^{\prime }\right\} }\left( A\right) -\left| x^{\prime }-y^{\prime }\right| CD_{i}\left( A^{i0}\right) . \end{aligned}$$

Case 2. $\max \left\{ x^{\prime },y^{\prime }\right\} =y^{\prime }.$

Then,

$$\begin{aligned} R_{i}(A)=y^{\prime }\alpha _{i}\left( A\right) +\frac{1-2y^{\prime }}{n-2} \left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right) +\left( x^{\prime }-y^{\prime }\right) \sum _{j\in N}a_{ij}- \frac{x^{\prime }-y^{\prime }}{n-2}\left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right) . \end{aligned}$$

Similarly to Case 1, we deduce that

$$\begin{aligned} y^{\prime }\alpha _{i}\left( A\right) +\frac{1-2y^{\prime }}{n-2}\left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right) =EC^{2-2\max \left\{ x^{\prime },y^{\prime }\right\} }\left( A\right) . \end{aligned}$$

Besides,

$$\begin{aligned} \left( x^{\prime }-y^{\prime }\right) \sum _{j\in N}a_{ij}-\frac{x^{\prime }-y^{\prime }}{n-2}\left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right)= & {} \left( x^{\prime }-y^{\prime }\right) \frac{\left( n-2\right) \sum \limits _{j\in N}a_{ij}+\alpha _{i}\left( A\right) -\left| \left| A\right| \right| }{n-2} \\= & {} \left( x^{\prime }-y^{\prime }\right) CD_{i}\left( A^{0i}\right) . \end{aligned}$$

Thus,

$$\begin{aligned} R_{i}(A)=EC^{2-2\max \left\{ x^{\prime },y^{\prime }\right\} }\left( A\right) -\left| x^{\prime }-y^{\prime }\right| CD_{i}\left( A^{0i}\right) . \end{aligned}$$

$\square$

Theorem 2 highlights the central role of concede-and-divide as a rule to solve broadcasting problems. Extended EC rules have two parts: one depending on the family of EC rules $\left( EC_{i}^{\lambda }\left( A\right) \right)$ and another depending on concede-and-divide ($CD_{i}\left( A^{i0}\right)$ and $CD_{i}\left( A^{0i}\right) ).$

Remark 2

The axioms used in Theorem 2 are independent.

The uniform rule satisfies AD and AN but fails MA.

$R^{1}$, defined as in Remark 1, satisfies AD and MA but fails AN.

For each $A\in \mathcal {P}$, let

$$\begin{aligned} R^{5}\left( A\right) =\left\{ \begin{array}{ll} ES\left( A\right) &{} \textrm{when} \left| \left| A\right| \right| \le 10 \\ CD\left( A\right) &{} \textrm{otherwise}. \end{array} \right. \end{aligned}$$

Then, $R^{5}$ satisfies AN and MA but fails AD.

The EC rules are characterized by the combination of additivity, maximum aspirations and either symmetry or equal treatment of equals (e.g., Bergantiños and Moreno-Ternero, 2021; Bergantiños and Moreno-Ternero, 2022a). Theorem 2 shows that if we replace symmetry or equal treatment of equals by anonymity, more rules arise.

The next corollary states the effect of adding the three order preservation axioms to the statement of Theorem 2.

Corollary 2

The following statements hold:

$\left( 1\right)$ A rule satisfies additivity, anonymity, maximum aspirations and home order preservation if and only if it is an extended EC rule with $x^{\prime }\ge y^{\prime }.$

$\left( 2\right)$ A rule satisfies additivity, anonymity, maximum aspirations and away order preservation if and only if it is an extended EC rule with $x^{\prime }\le y^{\prime }.$

$\left( 3\right)$ A rule satisfies additivity, anonymity, maximum aspirations and order preservation if and only if it is an EC rule.

Proof

$\left( 1\right)$ Let $EC^{x^{\prime }y^{\prime }}$ be an extended EC rule with $x^{\prime }\ge y^{\prime }.$ By Theorem 2, $EC^{x^{\prime }y^{\prime }}$ satisfies additivity, anonymity and maximum aspirations. As $x^{\prime }\ge y^{\prime }$, and $\left\{ EC^{\lambda }:\lambda \in \left[ 0,1\right] \right\} \subset \left\{ UC^{\lambda }:\lambda \in \left[ 0,1\right] \right\}$, we deduce from Theorem 3 below that $EC^{x^{\prime }y^{\prime }}$ satisfies home order preservation.

Let R be a rule satisfying all the axioms from the statement. By the proof of Theorem 2, $R=EC^{x^{\prime }y^{\prime }}$ where $x^{\prime }=R_{i}\left( \varvec{1}^{ij}\right)$ and $y^{\prime }=R_{j}\left( \varvec{1}^{ij}\right)$ for each pair $i,j\in N$ with $i\ne j.$ By home order preservation we deduce that $x^{\prime }\ge y^{\prime }.$

$\left( 2\right)$ It is similar to $\left( 1\right) .$

$\left( 3\right)$ We have argued that each EC rule satisfies additivity, anonymity and maximum aspirations. It is obvious that they also satisfy order preservation.

Let R be a rule satisfying all the axioms from the statement. By the proof of Theorem 2, $R=EC^{x^{\prime }y^{\prime }}$ where $x^{\prime }=R_{i}\left( \varvec{1}^{ij}\right)$ and $y^{\prime }=R_{j}\left( \varvec{1}^{ij}\right)$ for each pair $i,j\in N$ with $i\ne j$. By Proposition 1, order preservation implies equal treatment of equals. Thus, $x^{\prime }=y^{\prime }$ and hence R is an EC rule. $\square$

We say that R is an extended UC rule if there exist $x^{\prime },y^{\prime }\in \mathbb {R}$ such that, for each $i\in N$,

$$\begin{aligned} R_{i}\left( A\right) =\left\{ \begin{array}{ll} UC_{i}^{\lambda }\left( A\right) -\left| x^{\prime }-y^{\prime }\right| CD_{i}\left( A^{i0}\right) &{} \textrm{if}\,x^{\prime }\ge y^{\prime } \\ UC_{i}^{\lambda }\left( A\right) -\left| x^{\prime }-y^{\prime }\right| CD_{i}\left( A^{0i}\right) &{} \textrm{if}\,x^{\prime }\le y^{\prime } \end{array} \right. \end{aligned}$$

where $\max \left\{ x^{\prime },y^{\prime }\right\} \in \left[ \frac{1}{n},1 \right] ,$ $\min \left\{ x^{\prime },y^{\prime }\right\} \in \left[ \frac{ 1-\max \left\{ x^{\prime },y^{\prime }\right\} }{n-1},\max \left\{ x^{\prime },y^{\prime }\right\} \right] ,$ and $\lambda =\frac{n\left( 1-\max \left\{ x^{\prime },y^{\prime }\right\} \right) }{n-1}\in \left[ 0,1\right]$.

We denote by $UC^{x^{\prime }y^{\prime }}$ the rule associated to the numbers $x^{\prime }$ and $y^{\prime }$, as above. Notice that UC rules are all extended UC rules (just take $x^{\prime }=y^{\prime }).$

In the next theorem we provide a characterization of extended UC rules.

Theorem 3

The following statements hold:

$\left( 1\right)$ A rule satisfies additivity, anonymity, home order preservation and weak upper bound if and only if it is an extended UC rule with $x^{\prime }\ge y^{\prime }.$

$\left( 2\right)$ A rule satisfies additivity, anonymity, away order preservation and weak upper bound if and only if it is an extended UC rule with $x^{\prime }\le y^{\prime }$.

Proof

$\left( 1\right)$ It is obvious that U satisfies additivity and anonymity. By Theorem 2, CD satisfies additivity and anonymity. As U and CD satisfy additivity and anonymity, so do all extended UC rules.

As for home order preservation, we focus first on each matrix $\varvec{1}^{jk}$ where $j,k\in N$ with $j\ne k$. Let $UC^{x^{\prime }y^{\prime }}$ be an extended UC rule with $x^{\prime }\ge y^{\prime }$. We compute $UC_{i}^{x^{\prime }y^{\prime }}\left( \varvec{1}^{jk}\right)$ for each $i\in N.$ We consider several cases.

• 1. $j=i.$ Then,

  $$\begin{aligned} UC_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{ik}\right)= & {} \frac{ n\left( 1-x^{\prime }\right) }{n-1}U_{i}\left( \textbf{1}^{ik}\right) +\frac{ nx^{\prime }-1}{n-1}CD_{i}\left( \textbf{1}^{ik}\right) -\left( x^{\prime }-y^{\prime }\right) CD_{i}\left( \left( \textbf{1}^{ik}\right) ^{i0}\right) \nonumber \\= & {} \frac{n\left( 1-x^{\prime }\right) }{n-1}\frac{1}{n}+\frac{nx^{\prime }-1 }{n-1}-0=x^{\prime }. \end{aligned}$$

  (2)

• 2. $k=i.$ Then,

  $$\begin{aligned} UC_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{ji}\right) =\frac{n\left( 1-x^{\prime }\right) }{n-1}\frac{1}{n}+\frac{nx^{\prime }-1}{n-1}-\left( x^{\prime }-y^{\prime }\right) =y^{\prime }. \end{aligned}$$

  (3)

• 3. $i\notin \left\{ j,k\right\} .$ Then,

  $$\begin{aligned} UC_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{jk}\right) =\frac{n\left( 1-x^{\prime }\right) }{n-1}\frac{1}{n}+\frac{nx^{\prime }-1}{n-1}\frac{-1}{ n-2}-\left( x^{\prime }-y^{\prime }\right) \frac{-1}{n-2}=1-x^{\prime }-y^{\prime }. \end{aligned}$$

  (4)

Let $R^{x^{\prime }y^{\prime }}$ be any extended UC rule with $x^{\prime }\ge y^{\prime }$, $A\in \mathcal {P},$ and i and j as in the definition of home order preservation. By additivity,

$$\begin{aligned} UC_{i}^{x^{\prime }y^{\prime }}\left( A\right)= & {} a_{ij}UC_{i}^{x^{\prime }y^{\prime }}\left( \varvec{\textbf{1}}^{ij}\right) +a_{ji}UC_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{ji}\right) +\sum _{k\in N\backslash \left\{ i,j\right\} }a_{ik}UC_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{ik}\right) \\{} & {} +\sum _{k\in N\backslash \left\{ i,j\right\} }a_{ki}UC_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{ki}\right) +\sum _{k,l\in N\backslash \left\{ i,j\right\} }a_{kl}UC_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1} ^{kl}\right) \text { and } \\ UC_{j}^{x^{\prime }y^{\prime }}\left( A\right)= & {} a_{ij}UC_{j}^{x^{\prime }y^{\prime }}\left( \varvec{\textbf{1}}^{ij}\right) +a_{ji}UC_{j}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{ji}\right) +\sum _{k\in N\backslash \left\{ i,j\right\} }a_{jk}UC_{j}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{jk}\right) \\{} & {} +\sum _{k\in N\backslash \left\{ i,j\right\} }a_{kj}UC_{j}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{kj}\right) +\sum _{k,l\in N\backslash \left\{ i,j\right\} }a_{kl}UC_{j}^{x^{\prime }y^{\prime }}\left( \textbf{1} ^{kl}\right) . \end{aligned}$$

By (2) and t(3),

$$\begin{aligned} a_{ij}UC_{i}^{x^{\prime }y^{\prime }}\left( \varvec{\textbf{1}} ^{ij}\right) +a_{ji}UC_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1} ^{ji}\right)\ge & {} a_{ij}UC_{j}^{x^{\prime }y^{\prime }}\left( \varvec{ \textbf{1}}^{ij}\right) +a_{ji}UC_{j}^{x^{\prime }y^{\prime }}\left( \textbf{ 1}^{ji}\right) \Leftrightarrow \\ a_{ij}x^{\prime }+a_{ji}y^{\prime }\ge & {} a_{ij}y^{\prime }+a_{ji}x^{\prime }\Leftrightarrow \\ x^{\prime }\left( a_{ij}-a_{ji}\right)\ge & {} y^{\prime }\left( a_{ij}-a_{ji}\right) , \end{aligned}$$

which holds because $x^{\prime }\ge y^{\prime }$ and $a_{ij}\ge a_{ji}.$

By (2), for each $k\in N\backslash \left\{ i,j\right\} ,$ $UC_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{ik}\right) =UC_{j}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{jk}\right) =x^{\prime }.$ As $a_{ik}\ge a_{jk}$, for each $k\in N\backslash \left\{ i,j\right\}$,

$$\begin{aligned} \sum _{k\in N\backslash \left\{ i,j\right\} }a_{ik}UC_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{ik}\right) \ge \sum _{k\in N\backslash \left\{ i,j\right\} }a_{jk}UC_{j}^{x^{\prime }y^{\prime }}\left( \textbf{1} ^{jk}\right) . \end{aligned}$$

By (3), for each $k\in N\backslash \left\{ i,j\right\} ,$ $UC_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{ki}\right) =UC_{j}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{kj}\right) =y^{\prime }.$ As for each $k\in N\backslash \left\{ i,j\right\}$ $a_{ki}\ge a_{kj}$,

$$\begin{aligned} \sum _{k\in N\backslash \left\{ i,j\right\} }a_{ki}UC_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{ki}\right) \ge \sum _{k\in N\backslash \left\{ i,j\right\} }a_{kj}UC_{j}^{x^{\prime }y^{\prime }}\left( \textbf{1} ^{kj}\right) . \end{aligned}$$

By (4), for each $k,l\in N\backslash \left\{ i,j\right\} ,$ $UC_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1} ^{kl}\right) =UC_{j}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{kl}\right) =1-x^{\prime }-y^{\prime }.$ Then,

$$\begin{aligned} \sum _{k,l\in N\backslash \left\{ i,j\right\} }a_{kl}UC_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{kl}\right) =\sum _{k,l\in N\backslash \left\{ i,j\right\} }a_{kl}UC_{j}^{x^{\prime }y^{\prime }}\left( \textbf{1} ^{kl}\right) . \end{aligned}$$

Then, $UC^{x^{\prime }y^{\prime }}$ satisfies home order preservation.

Finally, as for weak upper bound, and using arguments similar to those used in the proof of Theorem 2 for maximum aspirations, it suffices to show it for each matrix $\varvec{1}^{ij}\in \mathbf {\mathcal {P}}$, with $i,j\in N$, such that $i\ne j$. We consider three cases.

• 1. By (2), $UC_{i}^{x^{\prime }y^{\prime }}\left( \varvec{1}^{ij}\right) =x^{\prime }\le 1.$

• 2. By (3), $UC_{j}^{x^{\prime }y^{\prime }}\left( \varvec{1}^{ij}\right) =y^{\prime }\le x^{\prime }\le 1.$

• 3. By (4), for each $k\in N\backslash \left\{ i,j\right\} ,$ $UC_{k}^{x^{\prime }y^{\prime }}\left( \varvec{1} ^{ij}\right) =1-x^{\prime }-y^{\prime }.$ Now

  $$\begin{aligned} 1-x^{\prime }-y^{\prime }\le 1\Leftrightarrow x^{\prime }+y^{\prime }\ge 0 \end{aligned}$$

  which holds always because $x^{\prime }\in \left[ \frac{1}{n},1\right]$ and $y^{\prime }\in \left[ \frac{1-x^{\prime }}{n-1},x^{\prime }\right] .$

Then, $UC^{x^{\prime }y^{\prime }}$ satisfies weak upper bound.

Conversely, let R be a rule satisfying all the axioms from the statement. Similarly to the proof of Theorem 2, we can find $x^{\prime },y^{\prime }$ such that, for each pair $i,j\in N$ with $i\ne j,$ $x^{\prime }=R_{i}\left( \varvec{1}^{ij}\right) ,$ $y^{\prime }=R_{j}\left( \varvec{1}^{ij}\right) ,$ and for each $k\in N\backslash \left\{ i,j\right\} ,$ $\frac{1-x^{\prime }-y^{\prime }}{n-2}=R_{k}\left( \varvec{1}^{ij}\right)$.

By home order preservation,

$$\begin{aligned} \frac{1-x^{\prime }-y^{\prime }}{n-2}\le y^{\prime }\le x^{\prime }. \end{aligned}$$

Now,

• $x^{\prime }\ge \frac{1-x^{\prime }-y^{\prime }}{n-2}\ge \frac{ 1-2x^{\prime }}{n-2}.$ Notice that $x^{\prime }=y^{\prime }=\frac{1}{n}$ satisfies the previous conditions. Then, $x^{\prime }\ge \frac{1}{n}.$ By weak upper bound, $x^{\prime }\le 1.$ Thus, $\frac{1}{n}\le x^{\prime }\le 1.$ Hence, $\max \left\{ x^{\prime },y^{\prime }\right\} \in \left[ \frac{1}{n},1\right]$ holds.

• As $\frac{1-x^{\prime }-y^{\prime }}{n-2}\le y^{\prime },$ it follows that $\frac{1-x^{\prime }}{n-1}\le y^{\prime }.$ Hence, $\min \left\{ x^{\prime },y^{\prime }\right\} \in \left[ \frac{1-\max \left\{ x^{\prime },y^{\prime }\right\} }{n-1},\max \left\{ x^{\prime },y^{\prime }\right\} \right]$ holds.

Similarly to the proof of part $\left( 1\right)$ of Theorem 2, we can prove that for each $i\in N,$

$$\begin{aligned} R_{i}(A)=x^{\prime }\alpha _{i}\left( A\right) +\frac{1-2x^{\prime }}{n-2} \left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right) -\left( x^{\prime }-y^{\prime }\right) CD_{i}\left( A^{i0}\right) . \end{aligned}$$

Now,

$$\begin{aligned} x^{\prime }\alpha _{i}\left( A\right) +\frac{1-2x^{\prime }}{n-2}\left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right)= & {} \frac{\left( n-2\right) x^{\prime }\alpha _{i}\left( A\right) +\left( 1-2x^{\prime }\right) \left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right) }{n-2} \\= & {} \frac{\left( 1-2x^{\prime }\right) \left| \left| A\right| \right| }{n-2}+\frac{\left( nx^{\prime }-1\right) \alpha _{i}\left( A\right) }{n-2} \\= & {} \frac{n\left( 1-x^{\prime }\right) }{n-1}\frac{\left| \left| A\right| \right| }{n}+\left( \frac{1-nx^{\prime }}{\left( n-1\right) \left( n-2\right) }\right) \left| \left| A\right| \right| \\{} & {} + \frac{\left( nx^{\prime }-1\right) \alpha _{i}\left( A\right) }{n-2} \\= & {} \frac{n\left( 1-x^{\prime }\right) }{n-1}\frac{\left| \left| A\right| \right| }{n}+\frac{nx^{\prime }-1}{n-1}\left( \frac{\left( n-1\right) \alpha _{i}\left( A\right) -\left| \left| A\right| \right| }{n-2}\right) . \end{aligned}$$

Let

$$\begin{aligned} \lambda =\frac{n\left( 1-x^{\prime }\right) }{n-1}. \end{aligned}$$

As $\frac{1}{n}\le x^{\prime }\le 1$, it follows that $0\le \lambda \le 1.$ Thus,

$$\begin{aligned} x^{\prime }\alpha _{i}\left( A\right) +\frac{1-2x^{\prime }}{n-2}\left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right) =UC_{i}^{\lambda }\left( A\right) . \end{aligned}$$

$\left( 2\right)$ Using arguments similar to those used in $\left( 1\right)$ we can prove that all extended UC rules with $x^{\prime }\le y^{\prime }$ satisfy additivity, anonymity, weak upper bound and away order preservation. Conversely, let R be a rule satisfying all those axioms. Similarly to the proof of Theorem 2, we can find $x^{\prime },y^{\prime }$ such that, for each pair $i,j\in N$ with $i\ne j,$ $x^{\prime }=R_{i}\left( \varvec{1} ^{ij}\right) ,$ $y^{\prime }=R_{j}\left( \varvec{1}^{ij}\right) ,$ and for each $k\in N\backslash \left\{ i,j\right\} ,$ $\frac{1-x^{\prime }-y^{\prime }}{n-2}=R_{k}\left( \varvec{1}^{ij}\right)$.

Similarly to $\left( 1\right)$ above, we can prove that $y^{\prime }\in \left[ \frac{1}{n},1\right]$ and $x^{\prime }\in \left[ \frac{1-y^{\prime } }{n-1},y^{\prime }\right]$.

Similarly to the proof of part $\left( 2\right)$ of Theorem 2, we can prove that for each $i\in N,$

$$\begin{aligned} R_{i}(A)=y^{\prime }\alpha _{i}\left( A\right) +\frac{1-2y^{\prime }}{n-2} \left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right) -\left( y^{\prime }-x^{\prime }\right) CD_{i}\left( A^{0i}\right) . \end{aligned}$$

Similarly to Case 1, we can prove that

$$\begin{aligned} y^{\prime }\alpha _{i}\left( A\right) +\frac{1-2y^{\prime }}{n-2}\left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right) =UC_{i}^{\lambda }\left( A\right) , \end{aligned}$$

where

$$\begin{aligned} \lambda =\frac{n\left( 1-y^{\prime }\right) }{n-1}, \end{aligned}$$

as desired. $\square$

Theorem 3 also highlights the central role of concede-and-divide as a rule to solve broadcasting problems. Extended UC rules have two parts: one depending on the family of UC rules $\left( UC_{i}^{\lambda }\left( A\right) \right)$ and another depending on concede-and-divide ($CD_{i}\left( A^{i0}\right)$ and $CD_{i}\left( A^{0i}\right) ).$

Remark 3

The axioms used in Theorem 4 are independent.

$\left( 1\right)$ For each pair $\left\{ i,j\right\} \in N$, with $i\ne j$, and each $k\in N$, let

$$\begin{aligned} R_{k}^{6}\left( \varvec{1}^{ij}\right) =\left\{ \begin{array}{ll} 2 &{} \textrm{if}\,k\in \left\{ i,j\right\} \\ \frac{-3}{n-2} &{} \textrm{otherwise}. \end{array} \right. \end{aligned}$$

We extend $R^{6}$ to each problem A using additivity. Namely, $R^{6}\left( A\right) =\sum \limits _{\left\{ i,j\right\} \subset N}a_{ij}R^{3}\left( \varvec{1}^{ij}\right) .$

Then, $R^{6}$ satisfies AD,  AN and HOP but fails WUB .

For each pair $\left\{ i,j\right\} \in N$, with $i\ne j$, and each $k\in N$, let

$$\begin{aligned} R_{k}^{7}\left( \varvec{1}^{ij}\right) =\left\{ \begin{array}{ll} 0 &{} \textrm{if}\,k\in \left\{ i,j\right\} \\ \frac{1}{n-2} &{} \textrm{otherwise} \end{array} \right. \end{aligned}$$

We extend $R^{7}$ to each problem A using additivity. Namely, $R^{7}\left( A\right) =\sum \limits _{\left\{ i,j\right\} \subset N}a_{ij}R^{7}\left( \varvec{1}^{ij}\right) .$

Then, $R^{7}$ satisfies AD, AN, and WUB but fails HOP.

Let $N=\left\{ 1,2,3\right\}$ and $R^{8}$ be the separable rule (see Bergantiños and Moreno-Ternero, 2022c) where, for each $i,j\in N$,

$$\begin{aligned} \left( x^{ij}\right) _{i,j\in N}=\left( \begin{array}{ccc} &{}\quad 0.69 &{}\quad 0.83 \\ 0.63 &{}\quad &{}\quad 0.90 \\ 0.69 &{}\quad 0.83 &{}\quad \end{array} \right) \text { and }\left( y^{ij}\right) _{i,j\in N}=\left( \begin{array}{ccc} &{}\quad 0.56 &{}\quad 0.49 \\ 0.49 &{}\quad &{}\quad 0.35 \\ 0.35 &{}\quad 0.28 &{}\quad \end{array} \right) \end{aligned}$$

Then, $R^{8}$ satisfies AD,  HOP and WUB but fails AN.

For each $A\in \mathcal {P}$, and each $i\in N,$ let

$$\begin{aligned} R_{i}^{9}(A)=\left\{ \begin{array}{ll} U\left( A\right) &{} if \left| \left| A\right| \right| \le 10 \\ CD\left( A\right) &{} if \left| \left| A\right| \right| >10. \end{array} \right. \end{aligned}$$

Then, $R^{9}$ satisfies AN,  HOP,  and WUB but fails AD.

$\left( 2\right)$ $R^{6}$ satisfies AD,  AN,  AOP but fails WUB.

$R^{7}$ satisfies AD,  AN,  and WUB but fails AOP.

Let $N=\left\{ 1,2,3\right\}$ and $R^{10}$ be the separable rule (see Bergantiños and Moreno-Ternero, 2022c) where, for each $i,j\in N$,

$$\begin{aligned} \left( x^{ij}\right) _{i,j\in N}=\left( \begin{array}{ccc} &{}\quad 0.56 &{}\quad 0.49 \\ 0.49 &{}\quad &{}\quad 0.35 \\ 0.35 &{}\quad 0.28 &{}\quad \end{array} \right) \text { and }\left( y^{ij}\right) _{i,j\in N}=\left( \begin{array}{ccc} &{}\quad 0.69 &{}\quad 0.83 \\ 0.63 &{}\quad &{}\quad 0.90 \\ 0.69 &{}\quad 0.83 &{}\quad \end{array} \right) \end{aligned}$$

Then, $R^{10}$ satisfies AD,  AOP and WUB but fails AN.

$R^{9}$ satisfies AN,  AOP,  and WUB but fails AD.

A trivial consequence of Theorem 3 and part 4 of Proposition 1 is that a rule satisfies additivity, anonymity, order preservation and weak upper bound if and only if it is a UC rule. We do not stress this result because anonymity is actually redundant for this characterization, as shown in Bergantiños and Moreno-Ternero (2022a).

We say that R is an extended UE rule if there exist $x^{\prime },y^{\prime }\in \mathbb {R}$ such that for each $i\in N$

$$\begin{aligned} R_{i}\left( A\right) =\left\{ \begin{array}{ll} UE_{i}^{\lambda }\left( A\right) -\left| x^{\prime }-y^{\prime }\right| CD_{i}\left( A^{i0}\right) &{} if x^{\prime }\ge y^{\prime } \\ UE_{i}^{\lambda }\left( A\right) -\left| x^{\prime }-y^{\prime }\right| CD_{i}\left( A^{0i}\right) &{} if x^{\prime }\le y^{\prime } \end{array} \right. \end{aligned}$$

where $\max \left\{ x^{\prime },y^{\prime }\right\} \in \left[ \frac{1}{n},1 \right] ,$ $\min \left\{ x^{\prime },y^{\prime }\right\} \in \left[ 0,1-\max \left\{ x^{\prime },y^{\prime }\right\} \right] ,$ and $\lambda =\frac{ n\left( 1-2\max \left\{ x^{\prime },y^{\prime }\right\} \right) }{n-1}\in \left[ \frac{-n}{n-2},1\right]$.

We denote by $UE^{x^{\prime }y^{\prime }}$ the rule associated to the numbers $x^{\prime }$ and $y^{\prime }$, as above. Notice that all UE rules are extended UE rules too (just take $x^{\prime }=y^{\prime }$ and hence $x^{\prime }\in \left[ \frac{1}{n},0.5\right]$ and $\lambda \in \left[ 0,1\right] ).$

The next theorem characterizes the extended UE rules.

Theorem 4

The following statements hold:

$\left( 1\right)$ A rule satisfies additivity, anonymity, home order preservation and non-negativity if and only if it is an extended UE rule with $x^{\prime }\ge y^{\prime }.$

$\left( 2\right)$ A rule satisfies additivity, anonymity, away order preservation and non-negativity if and only if it is an extended UE rule with $x^{\prime }\le y^{\prime }.$

Proof

$\left( 1\right)$ As ES and U satisfy additivity and anonymity (Theorems 2 and 3), so do all extended UE rules.

Using similar arguments to those used in the proof of part $\left( 1\right)$ of Theorem 3, we can also prove that $UE^{x^{\prime }y^{\prime }}$ with $x^{\prime }\ge y^{\prime }$ satisfies home order preservation. Finally, as for non-negativity, and using arguments similar to those used in the proof of Theorem 2 for maximum aspirations, it suffices to prove that for each $j,k\in N$ with $j\ne k$ and each $i\in N,$ $UE_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{jk}\right) \ge 0.$ We consider several cases.

• 1. $j=i.$

  $$\begin{aligned} UE_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{ik}\right)= & {} \frac{ n-2nx^{\prime }}{n-2}U_{i}\left( \textbf{1}^{ik}\right) +\frac{2nx^{\prime }-2}{n-2}ES_{i}\left( \textbf{1}^{ik}\right) -\left( x^{\prime }-y^{\prime }\right) CD_{i}\left( \left( \textbf{1}^{ik}\right) ^{i0}\right) \\= & {} \frac{n-2nx^{\prime }}{n-2}\frac{1}{n}+\frac{2nx^{\prime }-2}{n-2}\frac{1 }{2}=x^{\prime }\ge 0. \end{aligned}$$

• 2. $k=i.$

  $$\begin{aligned} UE_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{ji}\right) =\frac{ n-2nx^{\prime }}{n-2}\frac{1}{n}+\frac{2nx^{\prime }-2}{n-2}\frac{1}{2} -\left( x^{\prime }-y^{\prime }\right) =y^{\prime }\ge 0. \end{aligned}$$

• 3. $i\in N\backslash \left\{ j,k\right\} .$

  $$\begin{aligned} UE_{i}^{x^{\prime }y^{\prime }}\left( \textbf{1}^{jk}\right) =\frac{ n-2nx^{\prime }}{n-2}\frac{1}{n}-\left( x^{\prime }-y^{\prime }\right) \frac{ -1}{n-2}=\frac{1-x^{\prime }-y^{\prime }}{n-2}\ge 0. \end{aligned}$$

Conversely, let R be a rule satisfying all the axioms from the statement. Similarly to the proof of Theorem 2, we can find $x^{\prime },y^{\prime }$ such that, for each pair $i,j\in N$ with $i\ne j,$ $x^{\prime }=R_{i}\left( \varvec{\textbf{1}}^{ij}\right) ,$ $y^{\prime }=R_{j}\left( \varvec{\textbf{1}}^{ij}\right) ,$ and for each $k\in N\backslash \left\{ i,j\right\} ,$ $\frac{1-x^{\prime }-y^{\prime }}{n-2} =R_{k}\left( \varvec{1}^{ij}\right)$.

By home order preservation,

$$\begin{aligned} \frac{1-x^{\prime }-y^{\prime }}{n-2}\le y^{\prime }\le x^{\prime }. \end{aligned}$$

Then,

$$\begin{aligned} x^{\prime }\ge \frac{1-x^{\prime }-y^{\prime }}{n-2}\ge \frac{1-2x^{\prime }}{n-2}. \end{aligned}$$

Notice that $x^{\prime }=y^{\prime }=\frac{1}{n}$ satisfies the previous conditions. Then, $x^{\prime }\ge \frac{1}{n}.$

By non-negativity, $x^{\prime }\ge 0,$ $y^{\prime }\ge 0$ and

$$\begin{aligned} \frac{1-x^{\prime }-y^{\prime }}{n-2}\ge 0\Leftrightarrow x^{\prime }+y^{\prime }\le 1. \end{aligned}$$

Then, $x^{\prime }\in \left[ \frac{1}{n},1\right]$ and $y^{\prime }\in \left[ 0,1-x^{\prime }\right] .$

Similarly to the proof of part $\left( 1\right)$ of Theorem 2, we can prove that, for each $i\in N$,

$$\begin{aligned} R_{i}(A)=x^{\prime }\alpha _{i}\left( A\right) +\frac{1-2x^{\prime }}{n-2} \left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right) -\left( x^{\prime }-y^{\prime }\right) CD_{i}\left( A^{i0}\right) . \end{aligned}$$

Now,

$$\begin{aligned} x^{\prime }\alpha _{i}\left( A\right) +\frac{1-2x^{\prime }}{n-2}\left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right) =\frac{n\left( 1-2x^{\prime }\right) }{n-2}\frac{\left| \left| A\right| \right| }{n}+\frac{2\left( nx^{\prime }-1\right) }{n-2}\frac{\alpha _{i}\left( A\right) }{2}. \end{aligned}$$

Let

$$\begin{aligned} \lambda =\frac{n\left( 1-2x^{\prime }\right) }{n-2}. \end{aligned}$$

As $\frac{1}{n}\le x^{\prime }\le 1$, it follows that $\frac{-n}{n-2}\le \lambda \le 1.$ Thus,

$$\begin{aligned} x^{\prime }\alpha _{i}\left( A\right) +\frac{1-2x^{\prime }}{n-2}\left( \left| \left| A\right| \right| -\alpha _{i}\left( A\right) \right) =UE_{i}^{\lambda }\left( A\right) . \end{aligned}$$

$\left( 2\right)$ Using similar arguments to those used in $\left( 1\right)$ we can prove that any extended UC rule with $x^{\prime }\le y^{\prime }$ satisfies additivity, anonymity, non-negativity and away order preservation. Conversely, let R be a rule satisfying all those axioms. Then, similarly to the proof of Theorem 2, we can find $x^{\prime },y^{\prime }$ such that, for each pair $i,j\in N$ with $i\ne j,$ $x^{\prime }=R_{i}\left( \varvec{1} ^{ij}\right) ,$ $y^{\prime }=R_{j}\left( \varvec{1}^{ij}\right) ,$ and for each $k\in N\backslash \left\{ i,j\right\} ,$ $\frac{1-x^{\prime }-y^{\prime }}{n-2}=R_{k}\left( \varvec{1}^{ij}\right)$. Similarly to $\left( 1\right)$, we can prove that $x^{\prime }\le y^{\prime },$ $y^{\prime }\in \left[ \frac{1}{n},1\right]$, $x^{\prime }\in \left[ 0,1-y^{\prime }\right] ,$ $\lambda =\frac{n\left( 1-2y^{\prime }\right) }{n-2 },$ and for each $i\in N$

$$\begin{aligned} R_{i}(A)=UE_{i}^{\lambda }\left( A\right) -\left( y^{\prime }-x^{\prime }\right) CD_{i}\left( A^{0i}\right) , \end{aligned}$$

as desired. $\square$

Theorem 4 also highlights the central role of concede-and-divide as a rule to solve broadcasting problems. Extended UE rules have two parts: one depending on the family of UE rules $\left( UE_{i}^{\lambda }\left( A\right) \right)$ and another depending on concede-and-divide ($CD_{i}\left( A^{i0}\right)$ and $CD_{i}\left( A^{0i}\right) ).$

Remark 4

$\left( 1\right)$ The axioms used in Theorem 4 are independent.

$R^{6},$ defined as in Remark 3, satisfies AD,  AN,  and HOP but fails NN.

$R^{7},$ defined as in Remark 3, satisfies AD,  AN,  and NN but fails HOP.

Let $N=\left\{ 1,2,3\right\}$ and $R^{11}$ be the separable rule (see Bergantiños and Moreno-Ternero, 2022c) where for each pair $i,j\in N$,

$$\begin{aligned} \left( x^{ij}\right) _{i,j\in N}=\left( \begin{array}{ccc} &{}\quad 0.50 &{}\quad 0.60 \\ 0.45 &{}\quad &{}\quad 0.65 \\ 0.50 &{}\quad 0.60 &{}\quad \end{array} \right) \text { and }\left( y^{ij}\right) _{i,j\in N}=\left( \begin{array}{ccc} &{}\quad 0.40 &{}\quad 0.35 \\ 0.35 &{}\quad &{}\quad 0.25 \\ 0.25 &{}\quad 0.20 &{}\quad \end{array} \right) \end{aligned}$$

Then, $R^{11}$ that satisfies AD,  HOP,  and NN but fails AN.

For each $A\in \mathcal {P}$, and each $i\in N,$ let as

$$\begin{aligned} R_{i}^{10}(A)=\left\{ \begin{array}{ll} U\left( A\right) &{} if \left| \left| A\right| \right| \le 10 \\ ES\left( A\right) &{} if \left| \left| A\right| \right| >10. \end{array} \right. \end{aligned}$$

Then, $R^{12}$ satisfies AN,  HOP,  and NN but fails AD.

$\left( 2\right)$ $R^{6},$ defined as in Remark 3, satisfies AD,  AN,  and AOP but fails NN.

$R^{7},$ defined as in Remark 3, satisfies AD,  AN,  and NN but fails AOP.

Let $N=\left\{ 1,2,3\right\}$ and $R^{13}$ be the separable rule (see Bergantiños and Moreno-Ternero, 2022c) where for each $i,j\in N$,

$$\begin{aligned} \left( x^{ij}\right) _{i,j\in N}=\left( \begin{array}{ccc} &{}\quad 0.40 &{}\quad 0.35 \\ 0.35 &{}\quad &{}\quad 0.25 \\ 0.25 &{}\quad 0.20 &{}\quad \end{array} \right) \text { and }\left( y^{ij}\right) _{i,j\in N}=\left( \begin{array}{ccc} &{}\quad 0.50 &{}\quad 0.60 \\ 0.45 &{}\quad &{}\quad 0.65 \\ 0.50 &{}\quad 0.60 &{}\quad \end{array} \right) \end{aligned}$$

Then, $R^{13}$ that satisfies AD,  AOP,  and NN but fails AN.

Finally, $R^{10}$, defined as in Remark 3, satisfies AN,  AOP,  and NN but fails AD.

A trivial consequence of Theorem 4 and part 4 of Proposition 1 is that a rule satisfies additivity, anonymity, order preservation and non-negativity if and only if it is a UE rule. We do not stress this result because anonymity is actually redundant for this characterization, as shown in Bergantiños and Moreno-Ternero (2022a).

5 Conclusion

We have studied in this paper the impact of anonymity as an axiom for broadcasting problems.

On the one hand, we have shown that, when combined with some axioms, it is possible to obtain new characterizations of rules and families of rules already studied in broadcasting problems. For instance, by adding essential team (as well as additivity), we characterize concede-and-divide. With null team instead of essential team, we characterize the family of generalized split rules, which generalize the focal equal split rule.

On the other hand, we have also shown that, when combined with some other axioms, it is possible to characterize new families of rules: the so called extended EC rules, extended UC rules, and extended UE rules. All these extended rules can be described as the sum of two components. In the first one, we apply to the original problem a rule within the corresponding family of rules already studied in broadcasting problems (EC rules, UC rules or UE rules). In the second component, we always apply the focal concede-and divide to the resulting problem after nullifying some of audiences in the original problem.