The Effects of Excluding Coalitions

games

Article The Effects of Excluding Coalitions

Tobias Hiller 1,2

1 Institute for Theoretical Economics, University of Leipzig, D-04109 Leipzig, Germany; [email protected] 2 HR Department, TU Dresden, D-01062 Dresden, Germany

Received: 2 November 2017; Accepted: 18 December 2017; Published: 1 January 2018

Abstract: One problem in cooperative game theory is to model situations when two players refuse to cooperate (or the problem of quarreling members in coalitions). One example of such exclusions is the coalition statements of parliamentary parties. Other situations in which incompatible players affect the outcome are teams in ﬁrms and markets, for example. To model these exclusions in cooperative game theory, the excluded coalitions value (ϕE value) was introduced. This value is based on the Shapley value and takes into account that players exclude coalitions with other players. In this article, we deduce some properties of this new value. After some general results, we analyze the apex game that could be interpreted as a team situation and the glove game that models markets where sellers and buyers deal. For team situations, we show that all employees have a common interest for cooperation. On asymmetric markets, excluding coalitions affect the market players of the scarce side to a higher extent.

Keywords: excluded coalitions; quarreling; ϕE value; Shapley value; cooperative game theory

1. Introduction One problem in cooperative game theory is to model situations when two players refuse to cooperate (or the problem of quarreling members in coalitions). One example of such exclusions is the coalition statements of parliamentary parties. Other situations in which incompatible players affect the outcome are teams in firms and markets, for example. To model this, the ϕE value (excluded coalitions value) for cooperative games was introduced and axiomatized [1]. This value is based on the Shapley value [2] and takes into account that players exclude coalitions with other players. The ϕE value enhances the approaches developed by [3] to model that players prefer some other players for cooperation.1 Whereas the approach by [3,5] modifies the coalitional function of the game, the model by restricts the set of admissible permutations of players. For modeling the preferences of players, both models are insufficient. Assume a parliament with three parties. One party is located on the left, one party is on the right and one party is in the middle. The left and the right parties exclude coalitions with each other. The party from the middle admits cooperation with both parties. In the models by [3,5] the left and the right parties are connected via the middle party. Hence, all parties cooperate. Using the ϕE value, the excluded coalition between the left and the right players is considered in a way that precludes a coalition of all three players.

1 Another enhancement of the approach by [3] was introduced by [4]. In this model, incompatible players are linked by an arc of the graph. As well as in the models by [3,5] , every coalition (especially the grand coalition) obtains a worth—the maximum that could achieved by the compatible players of the coalition. In the model by [4], loops or parallel arcs are not intended. This limitation does not exist when calculating the ϕE value. In addition, a further analysis of stable coalition structures based on the excluded coalition partners [6,7] may be complicated if the worth of the grand coalition, for example, is a superadditive hull.

Games 2018, 9, 1; doi:10.3390/g9010001 www.mdpi.com/journal/games Games 2018, 9, 1 2 of 7

The articles by [8,9] analyzed situations where each player could decide to exclude coalitions to all other players. Using exclusions of coalitions strategically was analyzed by [10,11]. In these articles, a non-cooperative game models the strategic decision of players whereas the values of cooperative game theory determine the payoffs at the last stage. Since the ϕE value is a new value, there are many fields of research with fruitful questions. One major question is how excluding of coalitions influences the players’ payoffs. For weighted majority games, this question was addressed in [1]. In this article, we deduce the influence of excluding coalitions on the players’ ϕE payoffs. We start with some general games (monotone games) and, in the final stage, we analyze the apex game that could be interpreted as a team situation and glove games that models markets where sellers and buyers deal. For team situations, we show that all employees have a common interest for cooperation. On asymmetric markets, excluding coalitions affect the market players of the scarce side to a higher extent. The remainder of this article is structured as follows. Basic definitions of cooperative game theory are presented in Section2. In Section3, we present our results. Section4 concludes.

2. Basic Notation A game is a pair (N, v) and N = {1, 2, ..., n} is the player set. The coalitional function v assigns every subset K of N a certain worth v (K) , reﬂecting the economic abilities of K (i.e., v : 2N → R) such that v (∅) = 0. A game is called monotone if v (T) < v (S) , for all T ⊂ S, T, S ⊆ N. A game is convex if v (T ∪ {i}) − v (T) < v (S ∪ {i}) − v (S) , for all T ⊂ S, T, S ⊆ N and i ∈/ S. A game (N, v) is designated symmetric or anonymous, if a function f :N→ R exists such that v(K) = f (|K|) for all non-empty sets K ⊆ N. Hence, the number of players determines the worth of a coalition. The cardinality of N is called n or |N| . One important value for games (N, v) is the Shapley value. To calculate the player’s payoffs, rank orders ρ on N are used. They are written as (ρ1, ... , ρn) where ρ1 is the ﬁrst player in the order, ρ2 the second player, etc. The player at position t is noted by ρ(t). The set of these orders is denoted by RO (N); n! rank orders exist. The set of players before i in rank order ρ together with player i is called Pi(ρ). The Shapley payoff of a player i is the average of the marginal contributions of i taken over all rank orders of the players:

1 ϕ (N, v) = v (P (ρ)) − v (P (ρ) \ {i}) . (1) i n ∑ i i ! ρ∈RO(N)

The ϕE value is based on the Shapley value. For the calculation of the ϕE value, we take into account the statements of the players on excluded coalitions. The set of i’s excluded coalition partners is denoted by Ei. A player excludes only coalitions to single players; |K| = 1, K ∈ Ei. The set of coalitions that are not allowed based on Ei is called Xi, Xi := { K ⊆ N| K\ {i} ∈ Ei} with |K| = 2. If i does not cooperate with j, we have Xi ∩ Xj = {i, j} , i.e., if a player i does not cooperate with j, also j cannot cooperate with i. All inadmissible coalitions are denoted by Γ := K ⊆ N| ∃ S ∈ Xj, j ∈ N, with S ⊆ K . Thus, the admissible coalitions in the game (N, v) are Ω := { K ⊆ N| K ∈/ Γ} .The set of admissible coalitions containing player i is denoted by Ω (i) . U+2037 \backtrprime ⇩ U+21E9 \downwhitearrow The tuple (N, v, Γ) is a game with excluded cooperation partners. ‸ U+2038 \caretinsert ⇪ U+21EA \whitearrowupfrombar ‼ U+203C \Exclam ∀ U+2200 \forall = = = = ⁃ U+2043 \hyphenbullet∗ Example∁ 1.U+2201Let N \complement{1, 2, 3}, E1 {{2} , {3}}, E2 {{1}} and E3 {{1}} . From this, we obtain ⁇ U+2047 \Question the following∃ U+2203 sets Xi \exists: X1 = {{1, 2} , {1, 3}} , X2 = {{1, 2}} , X3 = {{1, 3}} . From these sets, we ´ U+2057 \qprime deduce∄ the setU+2204 of inadmissible \nexists coalitions Γ = {{1, 2} , {1, 3} , {1, 2, 3}} and the set of admissible coalitions ⃝ U+20DD \enclosecircle Ω = {∅, {1}U+2205, {2} , { \varnothing3} , {2, 3}} . ⃞ U+20DE \enclosesquare∗ ç U+2205 \emptyset ∗ ⃟ U+20DF \enclosediamond The∆ ϕEU+2206value is \increment one value for games with excluded cooperation partners. The primary idea of the ∗ ⃤ U+20E4 \enclosetriangle ϕE value∎ isU+220E that only \QED marginal contributions to admissible coalitions inﬂuence the players’ payoffs. ℇ U+2107 \Eulerconst All supersets∞ U+221E of excluded \infty coalitions are inadmissible. ` U+210F \hbar∗ ∟ U+221F \rightangle ℏ U+210F \hslash ∠ U+2220 \angle ℑ U+2111 \Im ∡ U+2221 \measuredangle l U+2113 \ell ∢ U+2222 \sphericalangle ℘ U+2118 \wp ∴ U+2234 \therefore ℜ U+211C \Re ∵ U+2235 \because ℧ U+2127 \mho ∿ U+223F \sinewave ℩ U+2129 \turnediota ⊤ U+22A4 \top Å U+212B \Angstrom ⊥ U+22A5 \bot Ⅎ U+2132 \Finv ⊹ U+22B9 \hermitmatrix ℵ U+2135 \aleph ⊾ U+22BE \measuredrightangle ℶ U+2136 \beth ⊿ U+22BF \varlrtriangle ℷ U+2137 \gimel ⋯ U+22EF \cdots ∗ ℸ U+2138 \daleth ⌀ U+2300 \diameter ⅁ U+2141 \Game∗ ⌂ U+2302 \house ⅂ U+2142 \sansLturned∗ ⌐ U+2310 \invnot ⅃ U+2143 \sansLmirrored∗ ⌑ U+2311 \sqlozenge∗ ⅄ U+2144 \Yup∗ ⌒ U+2312 \profline∗ ⅊ U+214A \PropertyLine∗ ⌓ U+2313 \profsurf∗ ↨ U+21A8 \updownarrowbar ⌗ U+2317 \viewdata∗ ↴ U+21B4 \linefeed ⌙ U+2319 \turnednot ↵ U+21B5 \carriagereturn ⌬ U+232C \varhexagonlrbonds∗ ↸ U+21B8 \barovernorthwestarrow ⌲ U+2332 \conictaper∗ ↹ U+21B9 \barleftarrowrightarrowbar ⌶ U+2336 \topbot ↺ U+21BA \acwopencirclearrow ⍀ U+2340 \APLnotbackslash∗ ↻ U+21BB \cwopencirclearrow ⍓ U+2353 \APLboxupcaret∗ ⇞ U+21DE \nHuparrow∗ ⍰ U+2370 \APLboxquestion∗ ⇟ U+21DF \nHdownarrow∗ ⍼ U+237C \rangledownzigzagarrow∗ ⇠ U+21E0 \leftdasharrow∗ ⎔ U+2394 \hexagon∗ ⇡ U+21E1 \updasharrow∗ ⎶ U+23B6 \bbrktbrk ⇢ U+21E2 \rightdasharrow∗ ⏎ U+23CE \varcarriagereturn∗ ⇣ U+21E3 \downdasharrow∗ ⏠ U+23E0 \obrbrak ⇦ U+21E6 \leftwhitearrow ⏡ U+23E1 \ubrbrak ⇧ U+21E7 \upwhitearrow ⏢ U+23E2 \trapezium∗ ⇨ U+21E8 \rightwhitearrow ⏣ U+23E3 \benzenr∗

5 Games 2018, 9, 1 3 of 7

U+2037 \backtrprime ⇩ U+21E9 \downwhitearrow E ‸ U+2038 \caretinsert Hence, the ϕ payoff⇪ forU+21EA player \whitearrowupfrombari in (N, v, Γ) is calculated by [1]: ‼ U+203C \Exclam ∀ U+2200 \forall 1 ⁃ U+2043 \hyphenbullet∗ ϕE∁(N, vU+2201, Γ) = \complement v (P (ρ)) − v (P (ρ) \ {i}) . (2) i n! ∑ i i ⁇ U+2047 \Question ∃ U+2203 \existsρ∈RO(N), Pi(ρ)∈Ω ´ U+2057 \qprime ∄ U+2204 \nexists E ⃝ U+20DD \enclosecircle In the case of Γ = ∅, weU+2205 have ϕi \varnothing(N, v, Γ) = ϕi (N, v) . ⃞ U+20DE \enclosesquare∗ ç U+2205 \emptyset ⃟ U+20DF \enclosediamond3. Results∗ ∆ U+2206 \increment ∗ ⃤ U+20E4 \enclosetriangle ∎ U+220E \QED In this section, first we present some results on how excluding of coalitions affects the players’ ℇ U+2107 \Eulerconst ∞ U+221E \infty payoffs. We start our analysis with general games like monotone games or symmetric games. After this, ` U+210F \hbar∗ ∟ U+221F \rightangle ℏ U+210F \hslash we analyze apex games∠ andU+2220 glove games. \angle Results on weighted majority games were drawn in [1]. ℑ U+2111 \Im For monotone games,∡ weU+2221 deduce \measuredangle from Equation (2): l U+2113 \ell ∢ U+2222 \sphericalangle ℘ U+2118 \wp Corollary 1. Let (N, v, Γ∴) beU+2234 a monotone \therefore game with excluded cooperation partners. For i, j ∈ N, i 6= j such ℜ U+211C \Re that neither i ∈ Ej nor j ∈∵ Ei, U+2235{i, j} ∈/ Γ \because, we have ℧ U+2127 \mho ∿ U+223F \sinewave ⊤ U+22A4 \topE E 0 ℩ U+2129 \turnediota ϕi (N, v, Γ) ≥ ϕi N, v, Γ Å U+212B \Angstrom ⊥ U+22A5 \bot Ⅎ U+2132 \Finv where {i, j} ∈ Γ0. ⊹ U+22B9 \hermitmatrix ℵ U+2135 \aleph ⊾ U+22BE \measuredrightangle ⊿ ℶ U+2136 \beth Hence, excluding coalitionsU+22BF reduces \varlrtrianglei’s ϕE payoff in monotone games. ℷ U+2137 \gimel ⋯ U+22EF \cdots In addition, we deduce from Equation (2) the∗ next Corollary: ℸ U+2138 \daleth ⌀ U+2300 \diameter ⅁ U+2141 \Game∗ ⌂ U+2302 \house Corollary 2. Let (N, v, Γ) be a convex game with excluded cooperation partners. For i, j, l ∈ N, i 6= j 6= l ⅂ U+2142 \sansLturned∗ ⌐ U+2310 \invnot ∈ ∈ ∈ ∗ ∈ ∈ ∈ ⅃ U+2143 \sansLmirroredsuch∗ that neither i Ej nor⌑ j U+2311Ei, {i, j} \sqlozenge/ Γ, and neither i El nor l Ei, {i, l} / Γ, we have ⅄ U+2144 \Yup∗ ⌒ U+2312 \profline∗ E E 0∗ E 0 E 00 ⅊ U+214A \PropertyLine∗ ϕ⌓i (N,U+2313v, Γ) − ϕ \profsurfi N, v, Γ > ϕi N, v, Γ − ϕi N, v, Γ ↨ U+21A8 \updownarrowbar ⌗ U+2317 \viewdata∗ 0 00 00 ↴ U+21B4 \linefeed where {i, l} ∈ Γ , {i, l} ∈⌙ Γ andU+2319{i, j} \turnednot∈ Γ . ↵ U+21B5 \carriagereturn ⌬ U+232C \varhexagonlrbonds∗ ↸ U+21B8 \barovernorthwestarrowHence, excluding⌲ the “first”U+2332 player \conictaper affects the∗ ϕE payoff of player i more than excluding one ↹ U+21B9 \barleftarrowrightarrowbarmore player. ⌶ U+2336 \topbot ∗ ↺ U+21BA \acwopencirclearrowWith an example we⍀ showU+2340 that superadditivity\APLnotbackslash2 is not sufficient. Assume a game with excluded ∗ ↻ U+21BB \cwopencirclearrowcooperation partners with⍓ U+2353N = {1, \APLboxupcaret 2, 3} , v ({1}) = 0, v ({2}) = 2, v ({3}) = 1, v ({1, 2}) = 2, ⇞ ∗ ⍰ ∗ U+21DE \nHuparrow v ({1, 3}) = 4, v ({2, 3}) = 5U+2370, v (N) = \APLboxquestion6 and Γ = . The game is superadditive but not convex. Table1 ⇟ U+21DF \nHdownarrow∗ ⍼ U+237C \rangledownzigzagarrow∗ shows the marginal contributions and the resulting ϕE payoff for player 1. In a first step, cooperation ⇠ U+21E0 \leftdasharrow∗ ⎔ U+2394 \hexagon∗ ⇡ U+21E1 \updasharrowwith∗ player 2 is excluded.⎶ InU+23B6 the next \bbrktbrk step (last column), cooperation with 3 is excluded additionally. ⇢ U+21E2 \rightdasharrow∗ ⏎ U+23CE \varcarriagereturn∗ ∗ ⇣ U+21E3 \downdasharrow ⏠ TableU+23E0 1. superadditive \obrbrak game with excluded coalitions. ⇦ U+21E6 \leftwhitearrow ⏡ U+23E1 \ubrbrak ∗ ⇧ U+21E7 \upwhitearrow ⏢ ROU+23E2(N) Γ \trapezium= Γ = {{1, 2}, N} Γ = {{1, 2}, {1, 3}, N} ⇨ U+21E8 \rightwhitearrow ⏣ U+23E3 \benzenr∗ 1, 2, 3 0 0 0 1, 3, 2 0 0 0 5 2, 1, 3 0 − − 2, 3, 1 1 − − 3, 1, 2 3 3 − 3, 2, 1 1 − − E 5 3 0 ϕ1 6 6 6

2 A game is superadditive if v (S) + v (T) ≤ v (S ∪ T) , T ∩ S = , T, S ⊆ N Games 2018, 9, 1 4 of 7

In the next step, we analyze symmetric games. From Theorem 16 in [1], we deduce the following Corollary:

Corollary 3. Let (N, v, Γ) be a symmetric game with excluded cooperation partners. For i, j ∈ N, i 6= j such that neither i ∈ Ej nor j ∈ Ei, {i, j} ∈/ Γ, we have

E E 0 E E 0 ϕi (N, v, Γ) − ϕi N, v, Γ = ϕj (N, v, Γ) − ϕj N, v, Γ . where {i, j} ∈ Γ0.

Both players’ ϕE payoffs are affected in the same way. Corollary 3 is similar to properties of other values like the balanced contributions axiom of the Shapley value [12], the property of fair gain from bilateral links [3] or the splitting axiom of the χ value [7]. Corollary 4 is deduced from Theorem 17 in [1]. It presents the impact of the number of excluded coalitions in a symmetric game.

Corollary 4. Let (N, v, Γ) be a monotone symmetric game with excluded cooperation partners with n > 2.

For i, j ∈ N, i 6= j, such that |Ei| > Ej we have

E E ϕj (N, v, Γ) > ϕi (N, v, Γ) .

Hence, players with a higher number of excluded coalitions have lower ϕE payoffs than do players with less number of excluded coalitions. The apex game was introduced by [13]. An overview about the existing literature on apex games is provided by [14]. The apex game is defined for n ≥ 2. There is one apex player io. The other players U+2037 \backtrprime ⇩ U+21E9 \downwhitearrow are minor ones. All coalitions which contain i and at least one minor player as well the coalition ‸ U+2038 \caretinserto ⇪ U+21EA \whitearrowupfrombar which contains all minor players‼ getU+203C the worth \Exclam of 1 while all other coalitions get zero.∀ ThisU+2200 game \forall could be interpreted as a team situation⁃ inU+2043 firms where \hyphenbulletio is the manager.∗ He needs at least∁ oneU+2201 team member \complement to create a worth. ⁇ U+2047 \Question ∃ U+2203 \exists ´ U+2057 \qprime ∄ U+2204 \nexists Theorem 1. Let (N, v, Γ) be an apex⃝ gameU+20DD with \enclosecircleexcluded cooperation partners and Ei = ∅ forU+2205 all i ∈ N. \varnothingFor all ∗ l, m ∈ N\ {io} , l 6= m, we have ⃞ U+20DE \enclosesquare ç U+2205 \emptyset ⃟ U+20DF \enclosediamond∗ ∆ U+2206 \increment E E 0 E E 0 ∎ U+220E \QED∗ ϕl (N, v,⃤ Γ) −U+20E4ϕl N, v \enclosetriangle, Γ = ϕm (N, v, Γ) − ϕm N, v, Γ ℇ U+2107 \Eulerconst ∞ U+221E \infty 0 ∗ where {i, j} ∈ Γ , i, j ∈ N\ {io} .` U+210F \hbar ∟ U+221F \rightangle ℏ U+210F \hslash ∠ U+2220 \angle Proof. Without excluded coalitions,ℑ U+2111 the players’ \Im ϕE payoffs are: ∡ U+2221 \measuredangle l U+2113 \ell ∢ U+2222 \sphericalangle ℘ U+2118 \wp n − 2 ∴ U+2234 \therefore ϕE (N, v, Γ) = ϕ (N, v) = (3) i0 ℜ U+211C \Rei0 n ∵ U+2235 \because ℧ U+2127 \mho n−2 ∿ U+223F \sinewave E 1 − n 2 ϕ (N, v, Γ) = ϕl (N, v) = = ⊤ U+22A4 \top l ℩ U+2129 \turnediotan − 1 n (n − 1) Å U+212B \Angstrom ⊥ U+22A5 \bot Ⅎ U+2132 \Finv ⊹ U+22B9 \hermitmatrix with l ∈ N\ {io} . The player i0 does not obtain the marginal contribution 1 in the rank orders ⊾ U+22BE \measuredrightangle at positions 1 and n. In n − 2ℵfromU+2135n possible \aleph positions, he obtains the marginal contribution 1. ℶ U+2136− \beth ⊿ U+22BF \varlrtriangle The remaining marginal contributions 1 − n 2 are divided equally to the minor players. Excluding a ℷ U+2137 \gimeln ⋯ U+22EF \cdots ∈ \ { } ∗ coalition between i, j N io preventsU+2138 the \daleth possibility for all minor players, to⌀ obtainU+2300 the marginal \diameter ℸ E contribution 1 at position n − 1.⅁SinceU+2141 all minor \Game players∗ are symmetric, their ϕ payoffs⌂ U+2302 are affected \house to the same extent. The possibility⅂ to obtainU+2142 the \sansLturned marginal contribution∗ 1 at the second⌐ position,U+2310 if \invnot player ∗ ∗ i0 is at first position, is unchanged⅃ forU+2143 the minor \sansLmirrored players. ⌑ U+2311 \sqlozenge ⅄ U+2144 \Yup∗ ⌒ U+2312 \profline∗ ⅊ U+214A \PropertyLine∗ ⌓ U+2313 \profsurf∗ ↨ U+21A8 \updownarrowbar ⌗ U+2317 \viewdata∗ ↴ U+21B4 \linefeed ⌙ U+2319 \turnednot ↵ U+21B5 \carriagereturn ⌬ U+232C \varhexagonlrbonds∗ ↸ U+21B8 \barovernorthwestarrow ⌲ U+2332 \conictaper∗ ↹ U+21B9 \barleftarrowrightarrowbar ⌶ U+2336 \topbot ↺ U+21BA \acwopencirclearrow ⍀ U+2340 \APLnotbackslash∗ ↻ U+21BB \cwopencirclearrow ⍓ U+2353 \APLboxupcaret∗ ⇞ U+21DE \nHuparrow∗ ⍰ U+2370 \APLboxquestion∗ ⇟ U+21DF \nHdownarrow∗ ⍼ U+237C \rangledownzigzagarrow∗ ⇠ U+21E0 \leftdasharrow∗ ⎔ U+2394 \hexagon∗ ⇡ U+21E1 \updasharrow∗ ⎶ U+23B6 \bbrktbrk ⇢ U+21E2 \rightdasharrow∗ ⏎ U+23CE \varcarriagereturn∗ ⇣ U+21E3 \downdasharrow∗ ⏠ U+23E0 \obrbrak ⇦ U+21E6 \leftwhitearrow ⏡ U+23E1 \ubrbrak ⇧ U+21E7 \upwhitearrow ⏢ U+23E2 \trapezium∗ ⇨ U+21E8 \rightwhitearrow ⏣ U+23E3 \benzenr∗

5 Games 2018, 9, 1 5 of 7

E Excluding coalitions between j and i affects the ϕ payoffs of all players l ∈ N\ {io} in the same way. Hence, all minor players have a common interest that all of them are willing to cooperate. The next Corollary follows directly:

Corollary 5. Let (N, v, Γ) be an apex game with excluded cooperation partners with i, j ∈ N\ {io} , i 6= j, such that i ∈ Ej (and/or j ∈ Ei). For all l ∈ N\ {io} we have

E E 0 ϕl (N, v, Γ) = ϕl N, v, Γ

0 where {i, m} ∈ Γ , m ∈ N\ {io, i, j} and i ∈ N\ {io}.

After excluding a coalition between i, j ∈ N\ {io} , the minor players obtain only a marginal contribution 1 at the second position, if player i0 is at ﬁrst position. With excluding further coalitions between minor players this possibility is not affected. Hence, if one coalition between minor players is excluded, further exclusions of coalitions between the minor players do not change their ϕE payoffs. The last game analyzed is the so-called glove game [15]. In this game, each player owns either one left glove or one right glove. A single glove has no worth; a pair of gloves has a worth of one. Hence, this game models markets where sellers and buyers deal. The coalitional function for this game is given by U+2037 \backtrprime ⇩ U+21E9 \downwhitearrow v (K) = min {|K ∩ L| , |K ∩‸R|}U+2038with N \caretinsert= R ∪ L, R ∩ L = ∅,⇪ (4) U+21EA \whitearrowupfrombar ‼ U+203C \Exclam ∀ U+2200 \forall where L (R) denotes the set of left (right) glove⁃ ownersU+2043. The \hyphenbulletworth of a coalition∗ equals the number∁ of U+2201 \complement matching pairs it contains. ⁇ U+2047 \Question ∃ U+2203 \exists ´ U+2057 \qprime ∄ U+2204 \nexists Corollary 6. Let (N, v, Γ) be a glove game with excluded⃝ U+20DD cooperation \enclosecircle partners with |R| = |L| and Ei = ∅ for U+2205 \varnothing ∗ all i ∈ N. For i, j ∈ N, i 6= j with i ∈ L, j ∈ R we⃞ have U+20DE \enclosesquare ç U+2205 \emptyset ⃟ U+20DF \enclosediamond∗ ∆ U+2206 \increment E E 0 E E 0 ∎ U+220E \QED∗ ϕi (N, v, Γ) − ϕi N, v, Γ⃤ =U+20E4ϕj (N, v \enclosetriangle, Γ) − ϕj N, v, Γ ℇ U+2107 \Eulerconst ∞ U+221E \infty where {i, j} ∈ Γ0. ` U+210F \hbar∗ ∟ U+221F \rightangle ℏ U+210F \hslash ∠ U+2220 \angle Without excluded coalitions, the players’ℑϕE payoffsU+2111 are: \Im ∡ U+2221 \measuredangle l U+2113 \ell ∢ U+2222 \sphericalangle ℘ U+2118 \wp 1 ∴ U+2234 \therefore ϕE (N, v, Γ) = ϕE (N, v, Γ) = ϕ (N, v) = . (5) j i ℜ U+211Ci \Re 2 ∵ U+2235 \because ℧ U+2127 \mho ∿ U+223F \sinewave Both types of players are symmetric. Excluding℩ U+2129 coalitions \turnediota between j and i, i ∈ L, j ∈ R, affects⊤ both U+22A4 \top players to the same extent; they stay symmetric.Å Hence,U+212B on symmetric \Angstrom markets with an equal number⊥ U+22A5 \bot of players on each market side, excluding a coalitionℲ U+2132 with a player \Finv from the opposite side of the market⊹ U+22B9 \hermitmatrix reduces the players’ ϕE payoffs in an equal way.ℵ U+2135 \aleph ⊾ U+22BE \measuredrightangle Now, we analyze asymmetric glove gamesℶ withU+2136|R| < | \bethL| : ⊿ U+22BF \varlrtriangle ℷ U+2137 \gimel ⋯ U+22EF \cdots ∗ ℸ U+2138 \daleth ⌀ U+2300 \diameter Theorem 2. Let (N, v, Γ) be a glove game with excluded cooperation partners with |R| < |L| and Ei = for ⅁ U+2141 \Game∗ ⌂ U+2302 \house all i ∈ N. For i, j ∈ N, i 6= j with i ∈ L, j ∈ R we have ⅂ U+2142 \sansLturned∗ ⌐ U+2310 \invnot ⅃ U+2143 \sansLmirrored∗ ⌑ U+2311 \sqlozenge∗ ϕE (N, v, Γ) − ϕE N, v, Γ0 > ϕE (N, v, Γ) − ϕE N, v, Γ0 j j ⅄ U+2144i \Yup∗ i ⌒ U+2312 \profline∗ ∗ ∗ 0 ⅊ U+214A \PropertyLine ⌓ U+2313 \profsurf where {i, j} ∈ Γ . ∗ ↨ U+21A8 \updownarrowbar ⌗ U+2317 \viewdata ↴ U+21B4 \linefeed ⌙ U+2319 \turnednot Proof. Without excluded coalitions, we have [↵15] U+21B5 \carriagereturn ⌬ U+232C \varhexagonlrbonds∗ ↸ U+21B8 \barovernorthwestarrow ⌲ U+2332 \conictaper∗ E E ϕj (N, v) = ϕj (N, v,↹Γ) >U+21B9ϕi (N, v \barleftarrowrightarrowbar, Γ) = ϕi (N, v) ⌶(6) U+2336 \topbot ↺ U+21BA \acwopencirclearrow ⍀ U+2340 \APLnotbackslash∗ ↻ U+21BB \cwopencirclearrow ⍓ U+2353 \APLboxupcaret∗ ⇞ U+21DE \nHuparrow∗ ⍰ U+2370 \APLboxquestion∗ ⇟ U+21DF \nHdownarrow∗ ⍼ U+237C \rangledownzigzagarrow∗ ⇠ U+21E0 \leftdasharrow∗ ⎔ U+2394 \hexagon∗ ⇡ U+21E1 \updasharrow∗ ⎶ U+23B6 \bbrktbrk ⇢ U+21E2 \rightdasharrow∗ ⏎ U+23CE \varcarriagereturn∗ ⇣ U+21E3 \downdasharrow∗ ⏠ U+23E0 \obrbrak ⇦ U+21E6 \leftwhitearrow ⏡ U+23E1 \ubrbrak ⇧ U+21E7 \upwhitearrow ⏢ U+23E2 \trapezium∗ ⇨ U+21E8 \rightwhitearrow ⏣ U+23E3 \benzenr∗

5 Games 2018, 9, 1 6 of 7 U+2037 \backtrprime ⇩ U+21E9 \downwhitearrow ‸ U+2038 \caretinsert ⇪ U+21EA \whitearrowupfrombar i ∈ L, j ∈ R, |R| < |L| . The players in R obtain the marginal contribution 1 in more rank orders then ‼ U+203C \Exclam ∀ U+2200 \forall ⁃ theU+2043 players \hyphenbullet in L. Excluding∗ coalitions between∁ i andU+2201j reduces \complement the number of admissible rank orders 0 0 ⁇ forU+2047 both players \Question to the same extent; i.e., we have∃ |ΩU+2203(i)| − | \existsΩ (i)| = |Ω (j)| − |Ω (j)| . The number of ´ rankU+2057 orders \qprime that evoke a marginal contribution∄ 1 isU+2204 reduced \nexists for both players in a proportional way ⃝ withU+20DD respect \enclosecircle to the initial situation with Γ = ∅. HenceU+2205 absolutely, \varnothing player j loses a higher number of ∗ ⃞ rankU+20DE orders \enclosesquare with marginal contribution 1 thenç playerU+2205i. \emptyset ⃟ U+20DF \enclosediamond∗ ∆ U+2206 \increment ∗ E ⃤ U+20E4Hence, \enclosetriangle excluding coalitions on asymmetric∎ marketsU+220E affects \QED the ϕ payoffs of the scarce side of the ℇ marketU+2107 to \Eulerconst a higher extend. ∞ U+221E \infty ` U+210F \hbar∗ ∟ U+221F \rightangle ℏ 4.U+210F Discussion \hslash ∠ U+2220 \angle ℑ ∡ U+2221 \measuredangle U+2111In this \Im article, we analyzed properties of the ϕE value for some classes of games. For further l U+2113 \ell ∢ U+2222 \sphericalangle research, the following theoretical lines of development could be interesting. Since a large body of ℘ U+2118 \wp ∴ U+2234 \therefore ℜ literatureU+211C \Re deals with axiomatizations of the Shapley∵ U+2235 value, \because one possible purpose is to develop new E ℧ axiomatizationsU+2127 \mho of the ϕ value and to compare∿ them.U+223F In addition, \sinewavethe effects of excluding coalitions ℩ couldU+2129 be analyzed \turnediota in the framework of other structures⊤ U+22A4 of cooperative \top game theory like partitions [16,17], Å levelsU+212B [18 \Angstrom,19], networks [3,20] or hierarchies⊥ [21–23U+22A5]. Another \bot theoretical development could be a Ⅎ modifiedU+2132 \Finv version of the ϕE value that is based⊹ onU+22B9 other value-like \hermitmatrix solution concepts of cooperative ℵ gameU+2135 theory \aleph [24]. Additionally, the issue of⊾ refusingU+22BE cooperation \measuredrightangle could be analyzed for value-like ⊿ U+22BF \varlrtriangle ℶ solutionU+2136 concepts \beth like the core. One last suggested theoretical development could be the application of ℷ U+2137 \gimel ⋯ U+22EF \cdots the concept of stability [6] to the set of excluded coalitions. ∗ ℸ U+2138 \daleth ⌀ U+2300 \diameter ⅁ U+2141Another \Game line∗ of research is the experimental⌂ one.U+2302 With this \house research one could check if the payoffs E E ⅂ determinedU+2142 \sansLturned by the ϕ value∗ are realistic and,⌐ hence,U+2310 if the \invnotϕ value is an appropriate way to model ⅃ economicU+2143 \sansLmirrored situations. ∗ ⌑ U+2311 \sqlozenge∗ ⅄ U+2144With \Yup respect∗ to strategic applications of⌒ excludedU+2312 coalitions \profline [10∗,11], a two step model could ∗ ⅊ beU+214A analyzed. \PropertyLine3 After the∗ decision on the excluded⌓ U+2313 coalitions \profsurf (the set of inadmissible coalitions Γ), ∗ ↨ theU+21A8 players’ \updownarrowbarϕE payoff result. For the games⌗ analyzedU+2317 in \viewdata Section3, it is a dominant strategy to ↴ U+21B4 \linefeed ⌙ U+2319 \turnednot state Ei = for all i ∈ N. Hence, we have an equilibrium with Γ = and the Shapley payoffs ↵ U+21B5 \carriagereturn ⌬ U+232C \varhexagonlrbonds∗ result. For further research, an interesting topic could be the analysis of cooperative games with less ↸ U+21B8 \barovernorthwestarrow ⌲ U+2332 \conictaper∗ symmetric assumptions (see [26], for example) and non-cooperative games with negotiations between ↹ U+21B9 \barleftarrowrightarrowbar ⌶ U+2336 \topbot ↺ playersU+21BA after \acwopencirclearrow determining their strategies (i.e.,⍀ sequentialU+2340 decisions \APLnotbackslash on Ei). In∗ particular, our results ∗ ↻ onU+21BB the apex \cwopencirclearrow game (all minor players are affect⍓ adverselyU+2353 if \APLboxupcaret one of them refuse cooperation with i0) ⇞ andU+21DE asymmetric \nHuparrow glove∗ markets (the shorter side⍰ ofU+2370 the market \APLboxquestion is more affected∗ than the longer side) ⇟ indicateU+21DF fruitful \nHdownarrow research.∗ ⍼ U+237C \rangledownzigzagarrow∗ ⇠ U+21E0 \leftdasharrow∗ ⎔ U+2394 \hexagon∗ ⇡ Acknowledgments:U+21E1 \updasharrowI am grateful∗ to two anonymous⎶ refereesU+23B6 for comments \bbrktbrk on this paper. I acknowledge support ⇢ byU+21E2 the German \rightdasharrow Research Foundation∗ and the Open⏎ AccessU+23CE Publication \varcarriagereturn Funds of the TU Dresden.∗ ∗ ⇣ ConflictsU+21E3 of \downdasharrow Interest: The author declares no conflict⏠ of interest.U+23E0 \obrbrak ⇦ U+21E6 \leftwhitearrow ⏡ U+23E1 \ubrbrak ∗ ⇧ ReferencesU+21E7 \upwhitearrow ⏢ U+23E2 \trapezium ⇨ U+21E8 \rightwhitearrow ⏣ U+23E3 \benzenr∗ 1. 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