# Characteristic Function of Maxmax Defensive-Equilibrium Representation for TU-Games with Strategies

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Cooperative Game with a Characteristic Function

**Definition**

**1.**

**Assumption**

**1.**

**Definition**

**2.**

**Example**

**1.**

**Property**

**1.**

**Proof.**

**Corollary**

**1.**

## 3. Solutions of Cooperative Games and Their Relations

**Example**

**2.**

**Property**

**2.**

**Proof.**

**Property**

**3.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Example**

**3.**

**Theorem**

**2.**

**Proof.**

**Example**

**4.**

**Example**

**5.**

- (1)
- We say that V is superadditive on $x\in X$ if $V\left(x\right)({S}_{1}\cup {S}_{2})\ge V\left(x\right)\left({S}_{1}\right)+V\left(x\right)\left({S}_{2}\right)$ and V is superadditive on X if it is superadditive for any $x\in X$;
- (2)
- We say that V is subadditive on $x\in X$ if $V\left(x\right)({S}_{1}\cup {S}_{2})\le V\left(x\right)\left({S}_{1}\right)+V\left(x\right)\left({S}_{2}\right)$ and V is subadditive on X if it is subadditive for any $x\in X$; and
- (3)
- We say that V is additive on $x\in X$ if $V\left(x\right)({S}_{1}\cup {S}_{2})=V\left(x\right)\left({S}_{1}\right)+V\left(x\right)\left({S}_{2}\right)$ and V is additive on X if it is additive for any $x\in X$.

**Example**

**6.**

## 4. Discussion

**Assumption**

**2.**

**Theorem**

**3.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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S | $\left\{1\right\}$ | $\left\{2\right\}$ | $\{1,2\}$ |
---|---|---|---|

$V(U,L)$ | 6 | 5 | 18 |

$V(U,R)$ | 0 | 0 | 1 |

$V(D,L)$ | 0 | 0 | 1 |

$V(D,R)$ | 4 | 7 | 15 |

S | $\left\{1\right\}$ | $\left\{2\right\}$ | $\{1,2\}$ |
---|---|---|---|

$\omega (X,V)$ | 4 | 5 | 18 |

${\psi}_{1}(X,V)$ | 0 | 0 | 18 |

${\psi}_{2}(X,V)$ | 4 | 5 | 18 |

S | $\left\{1\right\}$ | $\left\{2\right\}$ | $\left\{3\right\}$ | $\{1,2\}$ | $\{1,3\}$ | $\{2,3\}$ | $\{1,2,3\}$ |
---|---|---|---|---|---|---|---|

$V(0,0,0)$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

$V(0,1,0)$ | 0 | $-1$ | 0 | $-1$ | 0 | $-1$ | $-1$ |

$V(1,0,0)$ | 2 | 0 | 0 | 2 | 2 | 0 | 2 |

$V(1,1,0)$ | 2 | $-1$ | 0 | 4 | 2 | $-1$ | 4 |

$V(0,0,1)$ | 0 | 0 | $-3$ | 0 | $-3$ | $-3$ | $-3$ |

$V(0,1,1)$ | 0 | $-1$ | $-3$ | $-1$ | $-3$ | 2 | 2 |

$V(1,0,1)$ | 2 | 0 | $-3$ | 2 | 3 | $-3$ | 3 |

$V(1,1,1)$ | 2 | $-1$ | $-3$ | 4 | 3 | 2 | 4.4 |

S | $\left\{1\right\}$ | $\left\{2\right\}$ | $\left\{3\right\}$ | $\{1,2\}$ | $\{1,3\}$ | $\{2,3\}$ | $\{1,2,3\}$ |
---|---|---|---|---|---|---|---|

$V({a}_{1},{b}_{1},{c}_{1})$ | 1 | 2 | 3 | 3 | 5 | 5 | 12 |

$V({a}_{1},{b}_{2},{c}_{1})$ | 0 | 2 | 2 | 3 | 5 | 4 | 8 |

$V({a}_{2},{b}_{1},{c}_{1})$ | 1 | 1 | 3 | 3 | 4 | 5 | 10 |

$V({a}_{2},{b}_{2},{c}_{1})$ | 0 | 1 | 3 | 3 | 4 | 4 | 7 |

$V({a}_{1},{b}_{1},{c}_{2})$ | 1 | 2 | 3 | 3 | 3 | 4 | 8 |

$V({a}_{1},{b}_{2},{c}_{2})$ | 1 | 2 | 2 | 3 | 3 | 5 | 9 |

$V({a}_{2},{b}_{1},{c}_{2})$ | 1 | 2 | 3 | 3 | 5 | 4 | 7 |

$V({a}_{2},{b}_{2},{c}_{2})$ | 1 | 2 | 3 | 3 | 5 | 5 | 6 |

S | $\left\{1\right\}$ | $\left\{2\right\}$ | $\left\{3\right\}$ | $\{1,2\}$ | $\{1,3\}$ | $\{2,3\}$ | $\{1,2,3\}$ |
---|---|---|---|---|---|---|---|

$\omega (X,V)$ | 1 | 2 | 3 | 3 | 5 | 5 | 12 |

${\psi}_{1}(X,V)$ | 0 | 1 | 2 | 3 | 5 | 5 | 12 |

${\psi}_{2}(X,V)$ | 0 | 1 | 2 | 3 | 5 | 5 | 12 |

S | $\left\{1\right\}$ | $\left\{2\right\}$ | $\left\{3\right\}$ | $\{1,2\}$ | $\{1,3\}$ | $\{2,3\}$ | $\{1,2,3\}$ |
---|---|---|---|---|---|---|---|

$V(U,L,F)$ | 2 | 2 | 0 | 5 | 4 | 4 | 5 |

$V(U,R,F)$ | 3 | 0 | 0 | 3 | 3 | 3 | 11 |

$V(D,L,F)$ | 0 | 3 | 0 | 3 | 3 | 3 | 11 |

$V(D,R,F)$ | 2 | 2 | 0 | 8 | 3 | 3 | 12 |

S | $\left\{1\right\}$ | $\left\{2\right\}$ | $\left\{3\right\}$ | $\{1,2\}$ | $\{1,3\}$ | $\{2,3\}$ | $\{1,2,3\}$ |
---|---|---|---|---|---|---|---|

$\mu (X,V)=\omega (X,V)$ | 2 | 2 | 0 | 8 | 4 | 4 | 12 |

${\psi}_{1}(X,V)$ | 2 | 2 | 0 | 8 | 3 | 3 | 12 |

${\psi}_{2}(X,V)$ | 2 | 2 | 0 | 8 | 3 | 3 | 12 |

S | $\left\{1\right\}$ | $\left\{2\right\}$ | $\left\{3\right\}$ | $\{1,2\}$ | $\{1,3\}$ | $\{2,3\}$ | $\{1,2,3\}$ |
---|---|---|---|---|---|---|---|

${V}^{(U,L,F)}$ | 2 | 2 | 0 | 5 | 4 | 4 | 12 |

${V}^{(U,R,F)}$ | 2 | 0 | 0 | 3 | 4 | 3 | 12 |

${V}^{(D,L,F)}$ | 0 | 2 | 0 | 3 | 3 | 4 | 12 |

${V}^{(D,R,F)}$ | 0 | 0 | 0 | 8 | 3 | 3 | 12 |

S | $\left\{1\right\}$ | $\left\{2\right\}$ | $\left\{3\right\}$ | $\{1,2\}$ | $\{1,3\}$ | $\{2,3\}$ | $\{1,2,3\}$ |
---|---|---|---|---|---|---|---|

$V(U,L,F)$ | 1 | 8 | 2 | 8 | 2 | 8 | 8 |

$V(U,R,F)$ | 2 | 9 | 5 | 9 | 5 | 9 | 9 |

$V(D,L,F)$ | 5 | 10 | 7 | 10 | 7 | 10 | 10 |

$V(D,R,F)$ | 6 | 7 | 9 | 7 | 9 | 9 | 9 |

S | $\left\{1\right\}$ | $\left\{2\right\}$ | $\left\{3\right\}$ | $\{1,2\}$ | $\{1,3\}$ | $\{2,3\}$ | $\{1,2,3\}$ |
---|---|---|---|---|---|---|---|

$\mu (X,V)=\omega (X,V)$ | 1 | 8 | 9 | 7 | 2 | 8 | 8 |

${\psi}_{1}(X,V)$ | 2 | 9 | 9 | 7 | 5 | 9 | 8 |

${\psi}_{2}(X,V)$ | 2 | 8 | 9 | 7 | 5 | 9 | 8 |

S | $\left\{1\right\}$ | $\left\{2\right\}$ | $\left\{3\right\}$ | $\{1,2\}$ | $\{1,3\}$ | $\{2,3\}$ | $\{1,2,3\}$ |
---|---|---|---|---|---|---|---|

$V(0,2,2)$ | 0 | 8 | 8 | 8 | 8 | 16 | 16 |

$V(0,3,2)$ | 0 | 9 | 6 | 9 | 6 | 15 | 15 |

$V(1,2,2)$ | 3 | 6 | 6 | 9 | 9 | 12 | 15 |

$V(1,3,2)$ | 2 | 6 | 4 | 8 | 6 | 10 | 12 |

$V(0,2,3)$ | 0 | 6 | 9 | 6 | 9 | 15 | 15 |

$V(0,3,3)$ | 0 | 6 | 6 | 6 | 6 | 12 | 12 |

$V(1,2,3)$ | 2 | 4 | 6 | 6 | 8 | 10 | 12 |

$V(1,3,3)$ | 1 | 3 | 3 | 4 | 4 | 6 | 7 |

S | $\left\{1\right\}$ | $\left\{2\right\}$ | $\left\{3\right\}$ | $\{1,2\}$ | $\{1,3\}$ | $\{2,3\}$ | $\{1,2,3\}$ |
---|---|---|---|---|---|---|---|

$\omega (X,V)$ | 3 | 6 | 6 | 6 | 6 | 12 | 16 |

${\psi}_{1}(X,V)$ | 1 | 3 | 3 | 6 | 6 | 12 | 16 |

${\psi}_{2}(X,V)$ | 1 | 4 | 4 | 6 | 6 | 12 | 16 |

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**MDPI and ACS Style**

Liu, C.; Xiang, S.; Yang, Y.
Characteristic Function of Maxmax Defensive-Equilibrium Representation for TU-Games with Strategies. *Axioms* **2023**, *12*, 521.
https://doi.org/10.3390/axioms12060521

**AMA Style**

Liu C, Xiang S, Yang Y.
Characteristic Function of Maxmax Defensive-Equilibrium Representation for TU-Games with Strategies. *Axioms*. 2023; 12(6):521.
https://doi.org/10.3390/axioms12060521

**Chicago/Turabian Style**

Liu, Chenwei, Shuwen Xiang, and Yanlong Yang.
2023. "Characteristic Function of Maxmax Defensive-Equilibrium Representation for TU-Games with Strategies" *Axioms* 12, no. 6: 521.
https://doi.org/10.3390/axioms12060521