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On benefits of cooperation under strategic power

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Abstract

We introduce a new model involving TU-games and exogenous structures. Specifically, we consider that each player in a population can choose an element in a strategy set and that, for every possible strategy profile, a TU-game is associated with the population. This is what we call a TU-game with strategies. We propose and characterize the maxmin procedure to map every game with strategies to a TU-game. We also study whether or not the relevant properties of TU-games are transmitted by applying the maxmin procedure. Finally, we examine two relevant classes of TU-games with strategies: airport and simple games with strategies.

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Notes

  1. For an introduction to cooperative games and to the Shapley value, González-Díaz et al. (2010) can be consulted.

  2. In order not to lengthen with unnecessary details, we will only give a brief description of the political parties corresponding with the acronyms when relevant to the present example.

  3. It is the weighted majority game given by the players in N, the quota 176 and the weights (123, 66, 57, 42, 24, 15, 7, 6, 4, 2, 2, 1, 1).

  4. \((X^S,V^S)\) is given by \(X^S=X_{[S]}\times (\prod _{i\in N{\setminus } S}X_i)\) with \(X_{[S]}=\prod _{i\in S}X_i\), and for every \(x^S\in X^S\) and every \(T\subset N{\setminus } S\), \(V^S(x^S)(T)=V(x)(T)\), and \(V^S(x^S)(T\cup [S])=V(x)(T\cup S)\).

  5. This figure has been built with the toolbox TUGlab of \(\hbox {MATLAB}^{\circledR }\) (Mirás-Calvo and Sánchez-Rodríguez 2008). The web page of TUGlab can be found in http://mmiras.webs.uvigo.es/TUGlab/.

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Acknowledgements

The authors would like to thank two anonymous referees for their helpful suggestions to improve this article. This work has been supported by the Ministerio de Economía y Competitividad through Grants PGC2018-097965-B-100, MTM2017-87197-C3-1-P, MTM2017-87197-C3-2-P, MTM2014-53395-C3-1-P, MTM2014-53395-C3-3-P, MTM2014-54199-P, and by the Xunta de Galicia through the European Regional Development Fund (Grupos de Referencia Competitiva ED431C-2016-015 and ED431C-2016-040, and Centro Singular de Investigación de Galicia ED431G/01).

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Appendix

Appendix

Independence of the properties of Theorem 1.

Next we define some procedures in order to show the independence of the properties of Theorem 1.

Individual objectivity. Take the map \(\psi _0:SG\rightarrow G\) given, for all N and \((X,V)\in SG(N)\), by:

$$\begin{aligned} \psi _0 (X,V)(S)= 0 \end{aligned}$$

for all \(S\subset N\).

The procedure \(\psi _0\) satisfies all the properties but individual objectivity. Namely, it is clear that it does not satisfy individual objectivity. Now we will show that it satisfies the other properties:

  • Irrelevance of weakly dominated strategies: Let \((X,V)\in SG(N)\), \(i\in N\), and \(S\subset N\) with \(i\in S\). If strategy \(x_i\in X_i\) is weakly dominated in S, then

    $$\begin{aligned} \psi _0 (X,V)(S)=0=\psi _0 (X^{-x_i},V)(S). \end{aligned}$$
  • Irrelevance of weakly dominated threats: Let \((X,V)\in SG(N)\), \(j\in N\), and \(S\subset N{\setminus }\{ j\}\). If strategy \(x_j\in X_j\) is a weakly dominated threat to coalition S, then

    $$\begin{aligned} \psi _0 (X,V)(S)=0=\psi _0 (X^{-x_j},V)(S). \end{aligned}$$
  • Merge invariance: Let \((X,V)\in SG(N)\) and \(\emptyset \ne S\subset N\), then, for all \(T\subset N{\setminus } S\),

    $$\begin{aligned} \psi _0 (X,V)(T)=0=\psi _0 (X^S,V^S)(T) \end{aligned}$$

    and

    $$\begin{aligned} \psi _0 (X,V)(T\cup S)=0=\psi _0 (X^S,V^S)(T\cup \{[S]\}). \end{aligned}$$

Irrelevance of weakly dominated strategies. Take the minmin procedure\({\underline{\psi }}:SG\rightarrow G\) given, for all N and \((X,V)\in SG(N)\), by:

$$\begin{aligned} {\underline{\psi }} (X,V)(S)=\min _{x\in X}V(x)(S) \end{aligned}$$

for all \(S\subset N\).

The minmin procedure satisfies all the properties but irrelevance of weakly dominated strategies. It is clear that it does not satisfy irrelevance of weakly dominated strategies. Namely, take \((X,V)\in SG(N)\), \(i\in N\), \(S\subset N\) with \(i\in S\), and let \(x_i \in X_i\) be such that \(V({\bar{x}}_{-i},x'_i)(S)> V({\bar{x}}_{-i},x_i)(S)\) for all \({\bar{x}}_{-i}\in \prod _{j\in N{\setminus } \{i\}}X_j\) and \(x'_i\in X_i\) with \(x_i\ne x'_i\). Then, \(x_i\) is weakly dominated in S but

$$\begin{aligned} {\underline{\psi }} (X,V)(S)=\min _{x\in X}V(x)(S)<\min _{x\in X^{-x_i}}V(x)(S)={\underline{\psi }} (X^{-x_i},V)(S). \end{aligned}$$

Now we will show that it satisfies the other properties:

  • Individual objectivity: Let \((X,V)\in SG(N)\) and a player \(i\in N\) be such that \(V(x)(i)=c\), for all \(x\in X\), then

    $$\begin{aligned} {\underline{\psi }} (X,V) (i)=\min _{x\in X} c = c \end{aligned}$$
  • Irrelevance of weakly dominated threats: Let \((X,V)\in SG(N)\), \(j\in N\), and \(S\subset N{\setminus }\{ j\}\). If strategy \(x_j\in X_j\) is a weakly dominated threat to coalition S, then

    $$\begin{aligned} {\underline{\psi }} (X,V)(S)=\min _{x\in X}V(x)(S)=\min _{x\in X^{-x_j}}V(x)(S)={\underline{\psi }} (X^{-x_j},V)(S). \end{aligned}$$
  • Merge invariance: Let \((X,V)\in SG(N)\) and \(\emptyset \ne S\subset N\), then, for all \(T\subset N{\setminus } S\),

    $$\begin{aligned} {\underline{\psi }} (X,V)(T)=\min _{x\in X}V(x)(T)=\min _{x^S\in X^S}V^S(x^S)(T)={\underline{\psi }} (X^S,V^S)(T) \end{aligned}$$

    and

    $$\begin{aligned} {\underline{\psi }} (X,V)(T\cup S)=\min _{x\in X}V(x)(T\cup S)=\min _{x^S\in X^S}V^S(x^S)(T\cup \{[S]\})={\underline{\psi }} (X^S,V^S)(T\cup \{[S]\}). \end{aligned}$$

Irrelevance of weakly dominated threats. Take the maxmax procedure\({\overline{\psi }}:SG\rightarrow G\) given, for all N and \((X,V)\in SG(N)\), by:

$$\begin{aligned} {\overline{\psi }} (X,V)(S)=\max _{x\in X}V(x)(S) \end{aligned}$$

for all \(S\subset N\).

The maxmax procedure satisfies all the properties but irrelevance of weakly dominated threats. It is clear that it does not satisfy irrelevance of weakly dominated threats. Namely, take \((X,V)\in SG(N)\), \(j\in N\), \(S\subset N{\setminus }\{ j\}\), and let \(x_j \in X_j\) be such that \(V({\bar{x}}_{-j},x'_j)(S)< V({\bar{x}}_{-j},x_j)(S)\) for all \({\bar{x}}_{-j}\in \prod _{k\in N{\setminus } \{j\} }X_k\) and \(x'_j\in X_j\) with \(x_j\ne x'_j\). Then, \(x_j\) is a weakly dominated threat to coalition S but

$$\begin{aligned} {\overline{\psi }} (X,V)(S)=\max _{x\in X}V(x)(S)>\max _{x\in X^{-x_j}}V(x)(S)={\overline{\psi }} (X^{-x_j},V)(S). \end{aligned}$$

Now we will show that it satisfies the other properties:

  • Individual objectivity: Let \((X,V)\in SG(N)\) and a player \(i\in N\) be such that \(V(x)(i)=c\), for all \(x\in X\), then

    $$\begin{aligned} {\overline{\psi }} (X,V) (i)=\max _{x\in X} c = c \end{aligned}$$
  • Irrelevance of weakly dominated strategies: Let \((X,V)\in SG(N)\), \(i\in N\), and \(S\subset N\) with \(i\in S\). If strategy \(x_i\in X_i\) is weakly dominated in S, then

    $$\begin{aligned} {\overline{\psi }} (X,V)(S)=\max _{x\in X}V(x)(S)=\max _{x\in X^{-x_j}}V(x)(S)={\overline{\psi }} (X^{-x_j},V)(S). \end{aligned}$$
  • Merge invariance: Let \((X,V)\in SG(N)\) and \(\emptyset \ne S\subset N\), then, for all \(T\subset N{\setminus } S\),

    $$\begin{aligned} {\overline{\psi }} (X,V)(T)=\max _{x\in X}V(x)(T)=\max _{x^S\in X^S}V^S(x^S)(T)={\overline{\psi }} (X^S,V^S)(T) \end{aligned}$$

    and

    $$\begin{aligned} {\overline{\psi }} (X,V)(T\cup S)=\max _{x\in X}V(x)(T\cup S)=\max _{x^S\in X^S}V^S(x^S)(T\cup \{[S]\})={\overline{\psi }} (X^S,V^S)(T\cup \{[S]\}). \end{aligned}$$

Merge invariance. Take the map \(\psi _1:SG\rightarrow G\) given, for all N and \((X,V)\in SG(N)\), by:

$$\begin{aligned} \psi _1 (X,V)(S)=\psi (X,V)(S) + (|S|-1) \end{aligned}$$

for all \(\emptyset \ne S\subset N\).

The procedure \(\psi _1\) satisfies all the properties but merge invariance. Namely, it is clear that it does not satisfy merge invariance since if we take \((X,V)\in SG(N)\) and \(\emptyset \ne S\subset N\) with \(|S|> 1\), then for all \(T\subset N{\setminus } S\),

$$\begin{aligned} \psi _1 (X,V)(T\cup S)= & {} \psi (X,V)(T\cup S)+(|T|+ |S|-1)=\psi (X^S,V^S)(T\cup \{[S]\})+(|T|+ |S|-1)\\> & {} \psi (X^S,V^S)(T\cup \{[S]\})+(|T|+ |\{[S]\}|-1)=\psi _1 (X^S,V^S)(T\cup \{[S]\}) \end{aligned}$$

where the inequality follows from \(|S|> 1= |\{[S]\}|\).

Now we will show that it satisfies the other properties:

  • Individual objectivity: Let \((X,V)\in SG(N)\) and a player \(i\in N\) be such that \(V(x)(i)=c\), for all \(x\in X\), then

    $$\begin{aligned} \psi _1 (X,V) (i)=\psi (X,V) (i) = c \end{aligned}$$

    since \(\psi \) satisfies individual objectivity.

  • Irrelevance of weakly dominated strategies: Let \((X,V)\in SG(N)\), \(i\in N\), and \(S\subset N\) with \(i\in S\). If strategy \(x_i\in X_i\) is weakly dominated in S, then

    $$\begin{aligned} \psi _1 (X,V)(S)=\psi (X,V)(S) + (|S|-1)=\psi (X^{-x_i},V)(S)+(|S|-1)=\psi _1 (X^{-x_i},V)(S) \end{aligned}$$

    since \(\psi \) satisfies irrelevance of weakly dominated strategies.

  • Irrelevance of weakly dominated threats: Let \((X,V)\in SG(N)\), \(j\in N\), and \(S\subset N{\setminus }\{ j\}\). If strategy \(x_j\in X_j\) is a weakly dominated threat to coalition S, then

    $$\begin{aligned} \psi _1 (X,V)(S)=\psi (X,V)(S) + (|S|-1)=\psi (X^{-x_j},V)(S)+(|S|-1)=\psi _1 (X^{-x_j},V)(S) \end{aligned}$$

    since \(\psi \) satisfies irrelevance of weakly dominated threats.

Next table shows a summary of the procedures and properties.

 

\(\psi \)

\(\psi _0\)

\({\underline{\psi }}\)

\({\overline{\psi }}\)

\(\psi _1\)

Individual objectivity

Yes

No

Yes

Yes

Yes

Irrelevance of weakly dominated strategies

Yes

Yes

No

Yes

Yes

Irrelevance of weakly dominated threats

Yes

Yes

Yes

No

Yes

Merge invariance

Yes

Yes

Yes

Yes

No

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Fiestras-Janeiro, M.G., García-Jurado, I., Meca, A. et al. On benefits of cooperation under strategic power. Ann Oper Res 288, 285–306 (2020). https://doi.org/10.1007/s10479-019-03495-6

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