Second-order productivity, second-order payoffs, and the Owen value

We introduce the concepts of the components’ second-order productivities in cooperative games with transferable utility (TU games) with a coalition structure (CS games) and of the components’ second-order payoffs for one-point solutions for CS games as generalizations of the players’ second-order productivities in TU games and of the players’ second-order payoffs for one-point solutions for TU games (Casajus in Discrete Appl Math 304:212–219, 2021). The players’ second-order productivities are conceptualized as second-order marginal contributions, that is, how one player affects another player’s marginal contributions to coalitions containing neither of them by entering these coalitions. The players’ second-order payoffs are conceptualized as the effect of one player leaving the game on the payoff of another player. Analogously, the components’ second-order productivities are conceptualized as their second-order productivities in the game between components; the components’ second-order payoffs are conceptualized as their second-order payoffs in the game between components. We show that the Owen value is the unique efficient one-point solution for CS games that reflects the players’ and the components’ second-order productivities in terms of their second-order payoffs.


Introduction
A cooperative game with transferable utility for a …nite player set (TU game or simply game) is given by a coalition function that assigns a worth to any coalition (subset of the player set), where the empty coalition obtains zero. (One-point) solutions for TU games corresponding author Email addresses: mail@casajus.de (André Casajus), rodriguetido@yahoo.fr (Rodrigue Tido Takeng) URL: www.hhl.de/casajus (André Casajus) 1 In this paper, a player's productivity in a game refers to her in ‡uence on the generation of worth as expressed by her marginal contributions to coalitions not containing her, that is, the di¤erences between the worth generated after she entered such a coalition and the worth generated before she entered.
In this paper, we suggest a second-order version of Khmelnitskaya and Yanovskaya's (2007) characterization of the Owen value. In particular, we show that the Owen value is the unique CS solution that satis…es e¢ ciency, second-order marginality, and second-order versions of symmetry within components and symmetry across components. (Theorem 10). Second-order symmetry within components is just the restriction of second-order symmetry to players within the same component. Second-order symmetry across components: components that are equally second-order productive in the game between components obtain the same sum of second-order payo¤s of their members. This result is partly based on three facts. Second-order marginality implies marginality (Proposition 6). E¢ ciency and second-order symmetry within components imply symmetry within components (Proposition 7). E¢ ciency and second-order symmetry across components imply symmetry across components (Proposition 9).
The remainder of this paper is organized as follows. In Section 2, we provide basic de…nitions and notation. In Section 3, we survey the characterizations of the Shapley value by Young (1985) and by Casajus (2021). In Section 4, we …rst survey the characterization of the Owen value by Khmelnitskaya and Yanovskaya (2007). Then, we provide our secondorder approach to the Owen value. Some remarks conclude the paper.

Basic de…nitions and notation
Let the universe of players U be a countably in…nite set, and let N denote the set of all …nite subsets of U. The cardinalities of S; T; N 2 N are denoted by s; t; and n; respectively. A (…nite TU) game for the player set N 2 N is given by a coalition function v : 2 N ! R, v (;) = 0; where 2 N denotes the power set of N . Subsets of N are called coalitions; v (S) is called the worth of coalition S. The set of all games for N is denoted by V (N ); the set of all games is denoted by V := S N 2N V (N ) : For N 2 N , T N; and v 2 V (N ), the subgame vj T 2 V (T ) is given by vj T (S) = v (S) for all S T ; for i 2 N and S N; we occasionally write v i and v S instead of vj N nfig and vj N nS ; respectively. For N 2 N , v; w 2 V (N ) ; and 2 R, the coalition functions v + w 2 V (N ) and v 2 V (N ) are given by (v + w) (S) = v (S) + w (S) and ( v) (S) = v (S) for all S N: For T N; T 6 = ;; the game u N T 2 V given by u N T (S) = 1 if T S and u N T (S) = 0 otherwise is called a unanimity game. Any v 2 V (N ) ; N 2 N can be uniquely represented by unanimity games. In particular, we have v = X T N :T 6 =; where the coe¢ cients T (v) are known as the Harsanyi dividends (Harsanyi, 1959) and can be determined recursively by for all S N n fi; jg : A rank order of N 2 N is a bijection : N ! f1; 2; : : : ; jN jg with the interpretation that i is the (i)th player in ; the set of rank orders of N is denoted by R (N ) : The set of players before i in is denoted by B i ( ) := f`2 N : (`) < (i)g : The marginal contribution of i in and v 2 V (N ) is denoted by A solution for V is an operator that assigns to any N 2 N , v 2 V (N ) ; and i 2 N a payo¤ ' i (v) : The Shapley value (Shapley, 1953) for V, Sh, is given by for all N 2 N ; v 2 V (N ) ; and i 2 N: For N 2 N ; let P (N ) denote the set of all partitions (coalition structures) of N ; the component of player i 2 N in P 2 P (N ) is denoted by P (i) : For N 2 N ; P 2 P (N ) ; T N and i 2 N; let P (T ) P be given by P (T ) := fP 2 P j T \ P 6 = ;g, let Pj T 2 P (T ) be given by Pj T := fT \ P j P 2 P (T )g ; let P T 2 P (N n T ) be given by P T := Pj A (CS) solution for VP is an operator ' that assigns to any N 2 N , i 2 N , and (v; P) 2 VP (N ) a payo¤ ' i (v; P) ; for P 2 P; we set ' P (v; P) = P i2P ' i (v; P) : For N 2 N and P 2 P (N ) ; the set of all rank orders that respect P is denoted by R (N; P) := f 2 R (N ) j for all P 2 P and i; j 2 P : j (i) (j)j < jP jg ; that is, in any such rank order, the players from any component follow each other without players from other components between them. The Owen value (Owen, 1977) for VP, Ow, is the CS solution given by Fix an injection { : N ! U; N 7 ! { N for N 2 N : For any N 2 N , P 2 P (N ) ; and v 2 V (N ) ; set [P] := f{ P j P 2 Pg and let v P 2 V ([P]) be given by The TU game v P is called the game between components or intermediate game for the CS game (v; P). For N 2 N ; (v; P) 2 VP (N ), and P 2 P; we have 3  (4) indicates that the players' Shapley value payo¤s re ‡ects their productivities in games as expressed by their own marginal contributions. Young (1985) shows that the Shapley value is the unique e¢ cient such solution.
Theorem 1 (Young, 1985). The Shapley value is the unique solution for V that satis…es e¢ ciency (E), symmetry (S), and marginality (M). 5 Symmetry and marginality can be paraphrased as follows. Symmetry: players who are equally productive in a game should obtain the same payo¤. Marginality: a player who is equally productive in two games should obtain the same payo¤ in these games. Therefore, a solution that is intended to re ‡ect the players'productivities should satisfy these properties.
Later on, Casajus (2021) introduces the notions of the players'second-order productivities and second-order payo¤s. Second-order productivities are conceptualized as secondorder marginal contributions: the second-order marginal contributions of player i 2 N; N 2 N with respect to player j 2 N n fig in a game v 2 V (N ) are given as S N n fi; jg : These describe how player i a¤ects the productivity of player j: 6 The second-order payo¤ of player i 2 N; N 2 N with respect to player j 2 N n fig in a game v 2 V (N ) is given by It describes how player i a¤ects the payo¤ of player j. 7 Based on these notions, Casajus (2021) motivates natural second-order versions of symmetry and marginality. For all N 2 N ; v 2 V (N ) ; and i; j; k 2 N; i 6 = j 6 = k 6 = j; players i and j are called second-order symmetric with respect to player k if for all T N n fi; j; kg : Second-order symmetry, 2S. For all N 2 N ; v 2 V (N ) and i; j; k 2 N; i 6 = j 6 = k 6 = j such that players i and j are second-order symmetric with respect to player k; we have for all T N n fi; jg ; we have Second-order symmetry and second-order marginality can be paraphrased as follows. Second-order symmetry: players who are equally second-order productive with respect to a third player in a game should be assigned the same second-order payo¤ with respect to the latter. Second-order marginality: a player who is equally second-productive with respect to a another player in two games should be assigned the same second-order payo¤ with respect to the latter in these games. Therefore, it seems to be plausible that a solution the second-order payo¤s of which are intended to re ‡ect the players'second-order productivities satis…es these properties.
It turns out that the Shapley value re ‡ects the players' second-order productivities in terms of their second-order payo¤s in the same vein as it re ‡ects the players'(…rst-order) productivities in terms of their (…rst-order) payo¤s.
The proof of this theorem uses the fact that second-order marginality implies marginality, the proof of which is rather short.
Nevertheless, the proof of Theorem 2 is much more involved than the proof of Theorem 1 due to use of second-order symmetry instead of symmetry.
On the one hand, second-order symmetry does not inply symmetry (Casajus, 2021, Remark 3). On the other hand, the counterexamples in Casajus (2021, Remark 3) fail e¢ ciency. As our …rst result, we show that the proof of Theorem 2 can be simpli…ed substantially by providing a rather short proof that e¢ ciency and second-order symmetry imply symmetry.
that is, a player h is added to v such that (**) i and j remain symmetric in w; (***) h is symmetric to both i and j in w; and (****) w h = v: Since i and j are symmetric in w, they are second-order symmetric with respect to any k 2 M n fi; jg in w: Hence, we have for all k 2 M n fi; jg : Now, we obtain In view of (***), we analogously obtain Finally, we have which concludes the proof.

The Owen value
In this section, we …rst survey the characterization of the Owen value by Khmelnitskaya and Yanovskaya (2007). Then, we provide a second-order version of this characterization similar to the second-order characterization of the Shapley value by Casajus (2021) as surveyed in Section 3.

The (…rst-order) characterization by Khmelnitskaya and Yanovskaya (2007)
Khmelnitskaya and Yanovskaya (2007) generalize the characterization of the Shapley value due to Young (1985). This characterization indicates that the Owen value is the unique e¢ cient CS solution that re ‡ects both the players'and the components'(…rst-order) productivities in terms of the players'(…rst-order) payo¤s. 8 E¢ ciency, E. Symmetry within components, SwC. For all N 2 N , (v; P) 2 VP (N ) ; P 2 P, and i; j 2 P such that i and j are symmetric in v; we have ' i (v) = ' j (v) : For all N 2 N and (v; P) 2 VP (N ) ; the components P; Q 2 P are called symmetric in ! for all C P n fP; Qg ; that is, if and only if the representatives of P and Q are symmetric in the intermediate game v P : Symmetry across components, SaC. For all N 2 N , (v; P) 2 VP (N ), and P; Q 2 P such that P and Q are symmetric in (v; P) ; we have ' P (v; P) = ' Q (v; P) : 8 Recently, Hu (2021, Theorem 3.2) kind of rediscovered this characterization. Instead of marginality, he uses coalitional strategic equivalence (Chun, 1989). Nowadays, however, it is well understood that coalitional strategic equivalence is equivalent to marginality (see, for example, Casajus, 2011, Footnote 3). Coalitional strategic equivalence: For all N 2 N ; T N; T 6 = ;; i 2 N n T; 2 R; and v 2 V (N ) ; we have 8 E¢ ciency and marginality are just the CS versions of the properties for (TU) solutions with the same name and with the same interpretation. Symmetry within components is a natural relaxation of symmetry within the CS framework. Symmetry across components treats the components as players: equally productive components should obtain the same payo¤ as expressed by the sum of their members'payo¤s. Moreover, both symmetry within components and symmetry across component can be viewed as generalizations of symmetry. Whereas the former is equivalent to symmetry for the trivial coalition structure fN g ; the latter is so for the atomistic coalition structure ffig j i 2 N g : Theorem 5 (Khmelnitskaya and Yanovskaya, 2007). The Owen value is the unique CS solution for VP that satis…es e¢ ciency (E) symmetry within components (SwC), symmetry across components (SaC), and marginality (M).

A second-order characterization
In this subsection, we simultaneously translate the second-order characterization of the Shapley value to CS solutions and the (…rst-order) characterization of the Owen value to the second-order framework.
Second-order marginality, 2M. For all N 2 N ; (v; P) ; (w; P) 2 VP (N ) ; and i; j 2 N; for all T N n fi; jg ; we have In essence, this property is just a restatement of second-order marginality for TU games, where the coalition structure is …xed but can be ignored otherwise. Therefore, the proof of Proposition 3 runs through smoothly within the framework of CS games and we obtain Proposition 6. If a solution for VP satis…es second-order marginality (2M), then it satis-…es marginality (M).
Second-order symmetry within components, 2SwC. For all N 2 N ; (v; P) 2 VP (N ) ; P 2 P; i; j 2 P; and k 2 N n P such that i and j are second-order symmetric with respect to k in v; we have This property restricts second-order symmetry for TU games to players within the same component. Yet, the coalition structure can be ignored regarding the third player to whom the second-order marginal contributions and the second-order payo¤s are related. The proof of Proposition 4 essentially runs through smoothly with second-order symmetry within components instead of symmetry within components and symmetry within components instead of symmetry: one just has to put player h into the component containing players i and j: Hence, we obtain Proposition 7. If a solution for V satis…es strong second-order symmetry within components (2SwC) and e¢ ciency (E), then it satis…es symmetry within components (SwC).
In order to obtain a second-order version of symmetry across components, we …rst provide the notion of second-order symmetry of components. For all N 2 N ; (v; P) 2 VP (N ) ; and A; B; C 2 P pairwise di¤erent, components A and B are called second-order symmetric with respect to component C in for all D P n fA; B; Cg : Remark 8. Note that the components A and B are second-order symmetric with respect to component C in ( Second-order symmetric components are equally second-order productive with respect to a third component. Therefore, if a CS solution is intended to re ‡ect the components' second-order productivities in terms of their second-order payo¤s, it seems to be plausible that the second-order payo¤s of second-order symmetric components are the same. Second-order symmetry across components, 2SaC. For all N 2 N ; (v; P) 2 VP (N ) ; and A; B; C 2 P pairwise di¤erent such that A and B are second-order symmetric with respect to C in v; we have Using the general idea of the proof of Proposition 4 one shows that second-order symmetry across components and e¢ ciency imply symmetry across components.
Proposition 9. If a solution for VP satis…es second-order symmetry across components (2SaC) and e¢ ciency (E), then it satis…es symmetry across components (SaC).
Proof. Let the CS solution ' satisfy 2SaC and E. If jPj = 1; then nothing is to show. Let now N 2 N and (v; P) 2 VP (N ) be such that jPj > 1: Moreover, let (*) P; Q 2 P; P 6 = Q be symmetric in v: that is, fhg is added to (v; P) such that (**) P and Q remain symmetric in (w; Q) ; (***) fhg is symmetric to both P and Q in w; and (****) w fhg ; Q fhg = (v; P) : Since P and Q are symmetric in (w; Q), they are second-order symmetric with respect to any R 2 Q n fP; Qg in (w; Q) : Hence, we have for all R 2 Q n fP; Qg : Now, we obtain In view of (***), we analogously obtain and ' P w fhg ; Q fhg = ' fhg (w P ; Q P ) : Finally, we have = ' fhg (w P ; Q P ) which concludes the proof. Propositions 6, 7, and 9, allow us to "transfer"Theorem 5 to the second-order framework. We obtain Theorem 10. The Owen value is the unique CS solution for VP that satis…es e¢ ciency (E), second-order marginality (2M), second-order symmetry within components (2SwC), and second-order symmetry across components (2SaC).
Proof. It is well-known that the Owen value satis…es E. Straightforward but tedious calculations using (5b) show the following formulas for the second-order Owen value payo¤s in terms of second-order marginal contributions. Let N 2 N ; i; k 2 N; i 6 = k; and (v; P) 2 V (N ) : If k 2 P (i) ; then If k 2 N n P (i) ; then : (16) From (15) and (16) it is immediate that the Owen value satis…es 2M and 2SwC. By Remark ?? and in view of the well-known fact that the Owen values satis…es IG, it also satis…es 2SaC.
Remark 11. The characterization in Theorem 10 is non-redundant for jN j > 1. The zero CS solution, Z; given by Z i (v; P) := 0 for all N 2 N ; (v; P) 2 VP (N ) ; and i 2 N satis…es all properties but e¢ ciency. The component egalitarian CS solution, CE; given by CE i (v; P) := v (N ) jP (i)j jPj for all N 2 N ; (v; P) 2 VP (N ) ; and i 2 N satis…es all properties but marginality. Fix a bijection % : U ! N: For any N 2 N and P 2 P (N ) ; let R (N; P; %) := f 2 R (N; P) j for all P 2 P and i; j 2 P : (i) > (j) if and only if % (i) > % (j)g : The %-Owen value, Ow % ; given by M C v i ( ) for all N 2 N ; (v; P) 2 VP (N ) ; and i 2 N satis…es all properties but second-order symmetry within components. The Shapley value for CS games ignoring the coalition structure satis…es all properties but second-order symmetry across components.

Concluding remarks
In this paper, we suggest a characterization of the Owen value indicating that the latter is the unique e¢ cient CS solution that re ‡ects the players'and components'second-order productivities in terms of their second-order payo¤s. The natural question now arises whether this may hold true for higher-order productivities and higher-order payo¤s. In view of the results of Casajus (2020, Appendix A), the Owen value should satisfy the corresponding higher-order properties, whereas not being the unique e¢ cient CS solution to do so. Winter (1989) generalizes the Owen value to games enriched with a level structure, that is, a …nite sequence of coalition structures becoming successively …ner. Khmelnitskaya and Yanovskaya (2007, Theorem 2) indicate how their characterization can be extended to this level structure value. We leave it to the reader to provide the obvious extension of our characterization of the Owen value to that level structure value.

Acknowledgements
We are grateful to a number of referees who commented on Casajus (2021) and thereby also improved the exposion of the ideas in this paper. André Casajus: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -388390901.