# Universally Balanced Combinatorial Optimization Games

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Definitions

#### 2.1. Examples

#### 2.1.1. Owen’s linear production model

#### 2.1.2. Linear cost sharing game

#### 2.1.3. Partitioning games

#### 2.2. Linear and Combinatorial Optimization Games

**Definition 1.**

#### 2.3. Universally Balanced Games

**Definition 2.**

#### 2.4. Computational Complexity

## 3. A General Result

**Theorem 1.**

## 4. Simple Combinatorial Optimization Games

#### 4.1. Description of the Simple Games

**Theorem 2.**

#### 4.2. Universally Balancedness in Simple Games

**Theorem 3.**

**Corollary 1.**

**Corollary 2.**

#### 4.3. Simple Games on Graphs

**Packing Games**Consider first the game with vertices as players and matrix A. Before describing the result, let us explain a little bit the game. We show that the problem faced by a coalition is to find a “maximum weighted matching”. Hence we call such a game a maximum weighted matching game. The class is obtained by keeping the graph fixed and varying v.

**Theorem 4.**

**Theorem 5.**

**Covering Games**The constraints for covering games are easier to understand than for packing games.

**Theorem 6.**

**Theorem 7.**

**Partition Games**Consider the partition game associated to the edge-vertex incident matrix B of the graph. First of all, the polyhedron $\{y\in {R}^{V}:By=1\phantom{\rule{-3.99994pt}{0ex}}{1}_{E},y\ge 0\}$ always has a feasible fractional solution: $y=\frac{1}{2}1\phantom{\rule{-3.99994pt}{0ex}}{1}_{V}$. Therefore, for any v the linear program relaxation, $max\{{v}^{t}y,By=1\phantom{\rule{-3.99994pt}{0ex}}{1}_{E},y\ge 0\}$, has a finite solution for any edge-vertex incident matrix B.

**Theorem 8.**

#### 4.4. Applications

## 5. Universally Balanced Property with respect to Resources Constraints

**Theorem 9.**

**Definition 3.**

**Theorem 10.**

## 6. Concluding Remarks

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^{1}In the absence of interactions between disjoint coalitions, super-additivity is not a restriction. If ν is not super-additive, the value of a coalition can be redefined as the maximum of the value of its partitions, as in a partitioning game defined below. The new game—called the super-additive cover—is super-additive and represents the same possibilities of cooperation as the initial game. For a discussion on cooperative games see Demange and Wooders [7], part 2 for instance.^{3}By convention, when a program is not feasible, the solution is defined to be $+\infty $ for minimization problems and $-\infty $ for maximization problems.^{4}A tricky part is that there are no time constraints to verify that the input belongs to the class of NO outcomes^{5}For an analysis of complexity issues in games without transferable utility, see Ballester [18].

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Demange, G.; Deng, X. Universally Balanced Combinatorial Optimization Games. *Games* **2010**, *1*, 299-316.
https://doi.org/10.3390/g1030299

**AMA Style**

Demange G, Deng X. Universally Balanced Combinatorial Optimization Games. *Games*. 2010; 1(3):299-316.
https://doi.org/10.3390/g1030299

**Chicago/Turabian Style**

Demange, Gabrielle, and Xiaotie Deng. 2010. "Universally Balanced Combinatorial Optimization Games" *Games* 1, no. 3: 299-316.
https://doi.org/10.3390/g1030299