Axiomatic characterizations of the core and the Shapley value of the broadcasting game (cid:3)

We study the cooperative game associated with a broadcasting problem (the allocation of revenues raised from the collective sale of broadcasting rights for a sports tournament). We show that the set of core allocations can be characterized with three axioms: additivity , null team and monotonicity . We also show that the Shapley value can be characterized with additivity , equal treatment of equals and core selection


Introduction
Sports organizations (clubs) largely rely on the sale of broadcasting and media rights as their main source of revenue.Typically, such a sale is via collective bargaining (between the unionized clubs and the broadcasting companies).Thus, once an agreement is reached, the collected revenues have to be shared among the clubs.This is, by no means, an easy problem because the individual contributions (to those revenues) are not clearly identi…ed.
Bergantiños and Moreno-Ternero (2020a) introduce a formal model to analyze the problem of sharing the revenues from broadcasting sports leagues among participating clubs based on the audiences they generate.This model has already generated a sizable literature, dealing with several aspects of the problem. 1 A special emphasis within that literature has been made on the axiomatic approach.But a game-theoretical has also received attention.In particular, Bergantiños and Moreno-Ternero (2020a) also associate with each broadcasting problem a natural cooperative game and study several aspects of it.More precisely, they characterize the allocations in the core of such a game and prove that its Shapley value coincides with the so called equal-split rule (a rule highlighted in the axiomatic approach to solve broadcasting problems directly).
In this paper, we study further the core and the Shapley value of the broadcasting game mentioned above.More precisely, we prove that the set of core allocations can be characterized with three axioms.On the one hand, additivity and null team. 2 The former says that revenues should be additive on audiences.The latter says that clubs with null audience should receive a null allocation.On the other hand, one of the following three monotonicity axioms. 3Weak club monotonicity states that if the audiences of the games played by a certain club increase, and the rest of audiences remain the same, then this club cannot receive less.Overall monotonicity states that the rule should be monotonic on audiences.Finally, pairwise monotonicity states that if the aggregate audience of the games played by any pair of clubs increases, then no club can receive less. 1 See Bergantiños and Moreno-Ternero (2023e) for a recent survey of this literature. 2These are two standard axioms in cooperative game theory formalizing principles that can be traced back to Shapley (1953). 3Monotonicity axioms have a long tradition in axiomatic work.We then formalize as an axiom that the solution (to a broadcasting game) should only select allocations within the core.This axiom has obvious normative appeal, as it guarantees the stability of the league being considered, preventing participating clubs to secede.We refer to the axiom as core selection.We obtain a new characterization of the Shapley value when combining such an axiom with additivity (already mentioned above) and equal treatment of equals (the standard notion of impartiality, which states that clubs generating the same audiences should get the same allocation).
We conclude this introduction mentioning that our work is obviously related to two important strands of the game-theory literature.
On the one hand, the strand of that literature that addresses various sharing problems by associating a transferable utility game to the problem, and constructing sharing rules by means of the standard values in that game.Classical instances are the so-called airport problems (e.g., Littlechild and Owen, 1973; Littlechild, 1973), in which the cost of a runway has to be shared The rest of the paper is organized as follows.In Section 2, we introduce the model (of broadcasting problems and games) and the notation.In Section 3, we study the core (of broadcasting games).In Section 4, we study the Shapley value (of broadcasting games).In section 5, we conclude.

The model
We consider the model introduced in Bergantiños and Moreno-Ternero (2020a).Let N be a …nite set of clubs.Its cardinality is denoted by n.We assume n 3.For each pair of clubs i; j 2 N , we denote by a ij the broadcasting audience (number of viewers) for the game played by i and j at i's stadium.We use the notational convention that a ii = 0, for each i 2 N .
Let A 2 A n n denote the resulting matrix of broadcasting audiences generated in the whole tournament involving the clubs within N .
A problem is A 2 A n n with zero entries in the diagonal.Let P denote the set of all problems.
Let i (A) denote the total audience achieved by club i, i.e., When no confusion arises, we write i instead of i (A).
For each A 2 A n n , let jjAjj denote the total audience of the tournament.Namely, A (sharing) rule (R) is a mapping that associates with each problem the list of the amounts the clubs get from the total revenue.Without loss of generality, we normalize the revenue generated from each viewer to 1 (to be interpreted as the "pay per view"fee).Thus, formally, The equal-split rule, divides the audience of each game equally, among the two participating clubs.Formally, Equal-split rule, ES: for each A 2 P, and each i 2 N ,

Axioms
We now introduce the axioms we shall use in this paper.Notice that the axioms are de…ned on the set of broadcasting problems and not on the set of cooperative games.
The …rst axiom says that if two clubs have the same audiences, then they should receive the same amount.
Equal treatment of equals: For each A 2 P, and each pair i; j 2 N such that a ik = a jk , and a ki = a kj , for each k 2 N n fi; jg, The second axiom says that revenues should be additive on A: Formally, Additivity: For each pair A, A 0 2 P, The third axiom says that if a club has a null audience, then such a club gets no revenue.

Formally,
Null team: For each A 2 P, and each i 2 N , such that a ij = 0 = a ji for each j 2 N; R i (A) = 0: The The second one says that the rule should be monotonic on A: Formally, Overall monotonicity: For each pair A; A 0 2 P and each i 2 N , The third one says that if the aggregate audience of the games played by any pair of clubs increases, then no club can be worse o¤.Formally, Pairwise monotonicity: For each pair A; A 0 2 P and each i 2 N ,

Cooperative games
A cooperative game with transferable utility, brie ‡y a TU game, is a pair (N; v), where N denotes a set of agents and v : 2 N !R satis…es v (?) = 0: The core is de…ned as the set of feasible payo¤ vectors, upon which no coalition can improve.
As the population N will remain …xed, we avoid its use in the notation.Formally, The Shapley value (Shapley, 1953) is de…ned for each player as the average of his contributions across orders of agents.
Bergantiños and Moreno-Ternero (2020a) associate with each broadcasting problem A 2 P a T U game (N; v A ).To so so, they take an optimistic approach, noting that the highest possible revenue that a game between teams i and j in the former's stadium may generate is a ij .So, by breaking away from the league, the most optimistic scenario for any coalition of teams is to generate the same revenue they generated before leaving the league.Formally, given a broadcasting problem A 2 P, for each S N , v A (S) is de…ned as the total audience of the games played by the clubs in S. Namely, 3 The core of the broadcasting game We prove in this section that the allocations within the core can be obtained as the allocations induced by the set of rules satisfying additivity, null team and one of the three monotonicity axioms listed above.To do so, we …rst consider the following lemma, which provides the characterization of the core obtained in Bergantiños and Moreno-Ternero (2020a).
Lemma 1 Let A 2 P.Then, x = (x i ) i2N 2 Core(v A ) if and only if, for each i 2 N; there exist x j i j2N nfig satisfying three conditions: for each j 2 N n fig; (ii) P j2N nfig Let D be the set of rules satisfying additivity, null team and weak club monotonicity.Given a problem A, let D (A) be the set of allocations induced by the rules in D for that problem. Namely, We can now state our result.
Theorem 1 For each A 2 P; Proof.We …rst prove that Core(v A ) D (A) : Let A 2 P. Let x = (x i ) i2N 2 Core(v A ).By Lemma 1, for each i 2 N we can …nd x j i j2N nfig satisfying conditions (i) ; (ii) ; and (iii).Given A 0 2 P and i; j 2 N we de…ne Given A 0 2 P and i 2 N we now de…ne the rule R y as It is obvious that R y satis…es additivity, null team and weak club monotonicity.Now, for We now prove that Core(v A ) D (A) : Let R be a rule satisfying additivity, null team and weak club monotonicity.
For each pair i; j 2 N , with i 6 = j, let 1 ij denote the matrix with the following entries: Notice that 1 ij ji is the zero matrix (which we denote as 0), i:e:, the matrix with only zero entries.Let A 2 P and i 2 N .As R satis…es additivity, As R satis…es null team, For each pair i; j 2 N with i 6 = j we de…ne By Lemma 1, it is enough to show that x j i j2N nfig satis…es conditions (i) ; (ii) ; and (iii).By null team, R i (0) = 0.By weak club monotonicity, R i 1 ij R i (0) = 0 and R i 1 ji R i (0) = 0: Then, x j i 0: Thus, condition (i) holds.
By (2) ; R i (A) = P j2N nfig x j i : Thus, condition (ii) holds.Besides, for each pair i; j 2 N , with i 6 = j, which implies that condition (iii) also holds.
In the next proposition we obtain a similar characterization to Theorem 1 with overall monotonicity or pairwise monotonicity instead of weak club monotonicity.
Let D 1 (respectively D 2 ) be the set of rules satisfying additivity, null team and overall monotonicity (respectively pairwise monotonicity).Given a problem A, let D 1 (A) (respectively D 2 (A)) be the set of allocations induced by the rules of D 1 (respectively D 2 ) on A. Namely, Proposition 1 For each A 2 P; Proof.Bergantiños and Moreno-Ternero (2022b) prove that if a rule satis…es overall monotonicity or pairwise monotonicity, then it also satis…es weak club monotonicity.Thus, As for the converse inclusions, it su¢ ces to notice that the rule R y de…ned in the proof of Theorem 1 satis…es overall monotonicity and pairwise monotonicity.Thus, similarly to the proof of Theorem 1, we can prove that Core(v A ) D 1 (A) and Core(v A ) D 2 (A) : 4 The Shapley value of the broadcasting game Proof.We already know that the equal-split rule satis…es additivity and equal treatment of equals (e.g., Bergantiños and Moreno-Ternero, 2020a).By Lemma 1 we deduce that the equalsplit rule also satis…es core selection.
Conversely, let R be a rule satisfying the three axioms.Let A 2 P and i 2 N: By (1), As R satis…es additivity and equal treatment of equals, using similar arguments to those used in the proof of Theorem 1 in Bergantiños and Moreno-Ternero (2021), we can prove that there exists x 2 R such that for each fi; jg N; for each l 2 N n fi; jg: As R satis…es core selection, it follows from Lemma 1 that R l 1 ij = 0 for each l 2 N nfi; jg: Then, x = 1 2 and hence R 1 ij = ES 1 ij : By (1), we conclude that R (A) = ES (A).
Remark 1 The axioms used in Theorem 2 are independent.
Let R 1 be the rule in which, for each game (i; j) 2 N N the revenue goes to the club with the lowest number of the two.Namely, for each problem A 2 P, and each i 2 N , R 1 satis…es additivity and core selection, but not equal treatment of equals.
The uniform rule, which divides jjAjj equally among all clubs, satis…es additivity and equal treatment of equals, but not core selection.
Let R 2 be the rule in which, for each game (i; j) 2 N N the revenue goes to the club with the highest audience and ties are divided equally among both clubs.Namely, for each problem A 2 P and i 2 N; R 2 satis…es equal treatment of equals and core selection, but not additivity.
among di¤erent types of airplanes, bankruptcy problems from the Talmud (e.g.,Aumann and   Maschler, 1985; Aumann, 2010), minimum cost spanning tree problems (e.g.,Bergantiños and   Vidal-Puga, 2007) or telecommunications problems (e.g.,van den Nouweland et al., 1992; 1996).On the other hand, the strand of the literature that addresses various problems related to sports.Instances are the scoring or ranking of participants in tournaments or competitions (e.g., Slutzki and Volij, 2005; Kondratev et al., 2023), the prize allocation therein (e.g., Dietzenbacher and Kondratev, 2023; Alcalde-Unzu et al., 2023) or, more generally, the design of stable and fair competitions or memberships (e.g., Le Breton et al., 2013; Anbarci et al., 2021).Finally, Palacios-Huerta (2014) gathers numerous intriguing connections between game theory and the most popular sport worldwide, which provide interesting lessons for research in mainstream economics.
three axioms de…ned above were introduced in Bergantiños and Moreno-Ternero (2020a) and used later elsewhere (e.g., Bergantiños and Moreno-Ternero, 2021; 2022a; 2022c; 2023a; 2023c).Bergantiños and Moreno-Ternero (2022b) introduced several monotonicity axioms.We consider three of them here.The …rst one says that if the audiences of the games played by a certain club increase and the rest of audiences remain the same, then this club can not receive less.Formally, Weak club monotonicity: For each pair A; A 0 2 P and each i 2 N , a ij a 0 ij for all j 2 N n fig and a ji a 0 ji for all j 2 N n fig a jk = a 0 jk when i = 2 fj; kg Bergantiños and Moreno-Ternero (2020a)show that the equal-split rule coincides with the Shapley value of the associated cooperative game.That is, for each problem A 2 P, Theorem 2 A rule satis…es additivity, equal treatment of equals and core selection if and only if it is the equal-split rule.
Core selection: For each A 2 P, R(A) 2 Core(v A ):