# Cooperative Games Based on Coalition Functions in Biform Games

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## Abstract

**:**

## 1. Introduction

## 2. The Model

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

**Definition 1.**

- (1)
- V is the coalition function;
- (2)
- v is the characteristic function of minimax representation.

**Example 1.**

## 3. Main Results

#### 3.1. The Relations of V and v

**Theorem 1.**

**Proof.**

**Corollary 1.**

**Proof.**

**Remark 1.**

**Theorem 2.**

**Proof.**

**Theorem 3.**

**Proof.**

**Corollary 2.**

**Remark 2.**

#### 3.2. The Core and Shapley Value

- (1)
- efficiency if $\sum _{i=1}^{n}}{\phi}_{i}\left(v\right)=v\left(N\right)$;
- (2)
- dummy player property if ${\phi}_{i}\left(v\right)=v\left(\left\{i\right\}\right)$ for any dummy players $i\in N$, player i is a dummy player means that $v(S\cup \{i\left\}\right)=v\left(S\right)+v\left(\right\{i\left\}\right)$ for all $S\in {2}^{-i}$;
- (3)
- anonymity property if ${\phi}_{\sigma \left(i\right)}\left(\sigma v\right)={\phi}_{i}\left(v\right)$ for any $\sigma \in \pi \left(N\right)$, any $v\in {G}^{N}$, and all $i\in N$, where $\sigma v\left(\right\{\sigma \left(i\right),i\in S\left\}\right)=v\left(S\right)$ for all $S\in {2}^{N}\backslash \{\varnothing \}$;
- (4)
- additivity if ${\phi}_{i}(v+w)={\phi}_{i}\left(v\right)+{\phi}_{i}\left(w\right)$ for any $v,w\in {G}^{N}$.

- (1)
- An $n-$person cooperative game $(N,V,v)$ has a nonempty core $C\left(v\right)$ if and only if it is balanced. It can be proved by referring to Proposition 262.1 in Chapter 13 of Osborne and Rubinstein [22].
- (2)
- (3)
- If cooperative game $(N,V,v)$ is convex, then $C\left(v\right)$ is nonempty, by referencing Proposition 28.1 in Chapter 28 of Narahari [24].
- (4)
- We also obtain that if cooperative game $(N,V,v)$ is convex, then $\phi \left(v\right)\in C\left(v\right)$, by $C\left(v\right)$ is nonempty, according to Exercises 260.4 and 295.5 of Osborne and Rubinstein [22].

**Remark 3.**

**Theorem 4.**

- (1)
- $(N,V,v)$ is balanced;
- (2)
- $\phi \left(v\right)\in C\left(v\right)$.

**Proof.**

**Property 1.**

**Proof.**

**Remark 4.**

## 4. Comparison of the Solutions

**Example 2.**

**Example 3.**

## 5. Discussions

#### 5.1. CIS Value of the Cooperative Game $(N,V,v)$

**Example 4.**

#### 5.2. Two Stages of the Cooperative Game $(N,V,v)$

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Ui, T. A Shapley value representation of potential games. Game Econ. Behav.
**2000**, 31, 121–135. [Google Scholar] [CrossRef] - Brandenburger, A.; Stuart, H. Biform games. Manag. Sci.
**2007**, 53, 537–549. [Google Scholar] [CrossRef] - Fiestras–Janeiro, M.; Garca-Jurado, I.; Meca, A.; Mosquera, M. On benefits of cooperation under strategic power. Ann. Oper. Res.
**2020**, 288, 285–306. [Google Scholar] [CrossRef] - Gillies, D. Some Theorems on n-Person Games. Ph.D. Thesis, Princeton University, Princeton, NJ, USA, 1953. [Google Scholar]
- Shapley, L.S. A Value for n-Person Games. In Contributions to the Theory of Game, 2nd ed.; Princeton University Press: Princeton, NJ, USA, 1953. [Google Scholar] [CrossRef]
- Stuart, H. Cooperative games and business strategy. In Game Theory and Business Applications, 189–211; Chatterjee, K., Samuelson, W., Eds.; Kluwer Academic Publishers: Boston, MA, USA; Dordrecht, The Netherlands; London, UK, 2001. [Google Scholar] [CrossRef]
- Zhao, J.G. The hybrid solutions of an n-person game. Game. Econ. Behav.
**1992**, 4, 145–160. [Google Scholar] [CrossRef] - Grossman, S.; Hart, O. The costs and benefits of ownership: A theory of vertical and lateral integration. J. Polit. Econ.
**1986**, 94, 691–719. [Google Scholar] [CrossRef] - Hart, O.; Moore, J. Property rights and the nature of the firm. J. Polit. Econ.
**1990**, 98, 1119–1158. [Google Scholar] [CrossRef] - Summerfield, N.; Dror, M. Biform game: Reflection as a stochastic programming problem. Int. J. Prod. Econ.
**2013**, 142, 124–129. [Google Scholar] [CrossRef] - González, F.F.; van der Weijde, A.H.; Sauma, E. The promotion of community energy projects in chile and scotland: An economic approach using biform games. Energ. Econ.
**2020**, 86, 104677. [Google Scholar] [CrossRef] - Fox, B.; Grove, S.; Souder, D. When good deals need help getting done: Articulating side payment strategies. Long. Range. Plann.
**2021**, 3, 102072. [Google Scholar] [CrossRef] - Nash, J. Non-cooperative games. Ann. Math.
**1951**, 54, 286–295. [Google Scholar] [CrossRef] - Aumann, R. The core of a cooperative game without side payments. T. Am. Math. Sco.
**1961**, 98, 539–552. [Google Scholar] [CrossRef] - Hart, S.; Kurz, M. Endogenous formation of coalitions. Econometrica
**1983**, 51, 1047–1064. [Google Scholar] [CrossRef] - Von Neumann, J.; Morgenstern, O. Theory of Games and Economic Behavior; Princeton University Press: Princeton, NJ, USA, 2004. [Google Scholar] [CrossRef]
- Myerson, R. Game Theory: Analysis of Conflict; Harvard University Press: Cambridge, MA, USA, 1991. [Google Scholar] [CrossRef]
- Harsanyi, J. A simplified bargaining model for the n-person cooperative game. Int. Econ. Rev.
**1963**, 4, 194–220. [Google Scholar] [CrossRef] - Liu, C.W.; Xiang, S.W.; Yang, Y.L. A biform game model with the shapley allocation functions. Mathematics
**2021**, 9, 1872. [Google Scholar] [CrossRef] - Shapley, L.S. Cores of convex games. Int. J. Game. Theory.
**1971**, 1, 11–26. [Google Scholar] [CrossRef] - Ichiishi, T. Super-modularity: Applications to convex games and to the greedy algorithm for lp. J. Econ. Theory.
**1981**, 25, 283–286. [Google Scholar] [CrossRef] - Osborne, M.; Rubinstein, A. A Course in Game Theory; MIT Press: Cambridge, MA, USA, 1994. [Google Scholar] [CrossRef]
- Branzei, R.; Dimitrov, D.; Tijs, S. Models in Cooperative Game Theory, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar] [CrossRef]
- Narahari, Y. Game Theory and Mechanism Design; World Scientific Publishing: Princeton, NJ, USA, 1900. [Google Scholar] [CrossRef]
- Driessen, T.S.H.; Funaki, Y. Coincidence of and collinearity between game theoretic solutions. Oper. Res. Spektrum.
**1991**, 13, 15–30. [Google Scholar] [CrossRef] - Herrero, C.; Maschler, M.; Villar, A. Individual rights and collective responsibility: The rights–egalitarian solution. Math. Soc. Sci.
**1999**, 37, 59–77. [Google Scholar] [CrossRef] - Du, X.L.; Li, D.F.; Liang, K.R. A biform game approach to preventing block withholding attack of blockchain based on semi-cis value. Int. J. Comput. Int. Syst.
**2019**, 12, 1353–1360. [Google Scholar] [CrossRef]

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

$\left\{1\right\}$ | 0 | 0 | 2 | 2 | 0 | 0 | 2 | 2 |

$\left\{2\right\}$ | 0 | $-1$ | 0 | $-1$ | 0 | $-1$ | 0 | $-1$ |

$\left\{3\right\}$ | 0 | 0 | 0 | 0 | $-3$ | $-3$ | $-3$ | $-3$ |

$\{1,2\}$ | 0 | $-1$ | 2 | 4 | 0 | $-1$ | 2 | 4 |

$\{1,3\}$ | 0 | 0 | 2 | 2 | $-3$ | $-3$ | 3 | 3 |

$\{2,3\}$ | 0 | $-1$ | 0 | $-1$ | $-3$ | 2 | $-3$ | 2 |

$\{1,2,3\}$ | 0 | $-1$ | 2 | 4 | $-3$ | 2 | 3 | 4.4 |

(No, No, No) | (No, Yes, No) | (Yes, No, No) | (Yes, Yes, No) | |
---|---|---|---|---|

$\{1,2\}$ | 4 | 3 | 3 | 6 |

$\{1,3\}$ | 4 | 4 | 3 | 3 |

$\{2,3\}$ | 4 | 3 | 4 | 3 |

$\{1,2,3\}$ | 6 | 5 | 5 | 6 |

(No, No, Yes) | (No, Yes, Yes) | (Yes, No, Yes) | (Yes, Yes, Yes) | |

$\{1,2\}$ | 4 | 3 | 3 | 6 |

$\{1,3\}$ | 3 | 3 | 6 | 6 |

$\{2,3\}$ | 3 | 6 | 3 | 6 |

$\{1,2,3\}$ | 5 | 6 | 6 | 9 |

Player 2 × Player 3 | ||||||
---|---|---|---|---|---|---|

No, No | Yes, No | No, Yes | Yes, Yes | |||

Player 1 | No | 2, 2, 2 | 2, 1, 2 | 2, 2, 1 | 0, 3, 3 | |

Yes | 1, 2, 2 | 3, 3, 0 | 3, 0, 3 | 3, 3, 3 |

$({\mathit{a}}_{1},{\mathit{a}}_{2})$ | $({\mathit{b}}_{1},{\mathit{a}}_{2})$ | $({\mathit{a}}_{1},{\mathit{b}}_{2})$ | $({\mathit{b}}_{1},{\mathit{b}}_{2})$ | |
---|---|---|---|---|

$\left\{1\right\}$ | 2 | 1 | 1 | 3 |

$\left\{2\right\}$ | 1 | 1 | 1 | 2 |

$\{1,2\}$ | 2 | $3/2$ | $3/2$ | 4 |

Player 2 | ||||
---|---|---|---|---|

${a}_{2}$ | ${b}_{2}$ | |||

Player 1 | ${a}_{1}$ | $3/2,1/2$ | $3/4,3/4$ | |

${b}_{1}$ | $3/4,3/4$ | $5/2,3/2$ |

(0, 0, 0) | (0, 1, 0) | (1, 0, 0) | (1, 1, 0) | (0, 0, 1) | (0, 1, 1) | (1, 0, 1) | (1, 1, 1) | |
---|---|---|---|---|---|---|---|---|

$\left\{1\right\}$ | 0 | 0 | 0.0375 | 0.0375 | 0 | 0 | 0.0375 | 0.0375 |

$\left\{2\right\}$ | 0 | 0.1375 | 0.0019 | 0.1375 | 0.0019 | 0.1375 | 0.0019 | 0.1375 |

$\left\{3\right\}$ | 0 | 0 | 0 | 0 | 0.01 | 0.01 | 0.01 | 0.01 |

$\{1,2,3\}$ | 0 | 0.4225 | 0.3315 | 0.559 | 0.0145 | 0.5135 | 0.4225 | 0.65 |

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**MDPI and ACS Style**

Liu, C.; Xiang, S.; Yang, Y.; Luo, E.
Cooperative Games Based on Coalition Functions in Biform Games. *Axioms* **2023**, *12*, 296.
https://doi.org/10.3390/axioms12030296

**AMA Style**

Liu C, Xiang S, Yang Y, Luo E.
Cooperative Games Based on Coalition Functions in Biform Games. *Axioms*. 2023; 12(3):296.
https://doi.org/10.3390/axioms12030296

**Chicago/Turabian Style**

Liu, Chenwei, Shuwen Xiang, Yanlong Yang, and Enquan Luo.
2023. "Cooperative Games Based on Coalition Functions in Biform Games" *Axioms* 12, no. 3: 296.
https://doi.org/10.3390/axioms12030296