# The Integer Nucleolus of Directed Simple Games: A Characterization and an Algorithm

## Abstract

**:**

## 1. Introduction

## 2. A Characterization of $\mathbf{\nu}\left({\mathbf{\Gamma}}_{\mathit{t}}\right)$

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Remark**

**1.**

**Lemma**

**3.**

**Proof.**

**Remark**

**2.**

**Corollary**

**1.**

**Example**

**1.**

$\nu \left({\Gamma}_{t}\right)=\{$ | $(7,7,4,4,3,2,2,1,0),(7,7,4,4,3,2,2,0,1),(7,7,4,4,2,3,2,1,0),$ |

$(7,7,4,4,2,3,2,0,1),(7,7,4,4,2,2,3,1,0),(7,7,4,4,2,2,3,0,1)\}$. |

**Example**

**2.**

**Example**

**3.**

**Remark**

**3.**

## 3. An Algorithm to Compute $\nu \left({\Gamma}_{t}\right)$

**Lemma**

**4.**

**Proof.**

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6.**

**Remark**

**7.**

**Example**

**4.**

$\mathcal{W}\left({\Gamma}_{t}\right)\backslash N=\{$ | $\{1,2\},\{1,2,3\},\{1,2,4\},\{1,2,5\},\{1,3,4\},\{1,3,5\},\{2,3,4\},\{1,2,3,4\},$ |

$\{1,2,3,5\},\{1,2,4,5\},\{1,3,4,5\},\{2,3,4,5\}\}$. |

$P\left({\Gamma}_{t}\right)=\{$ | $(6,0,0,0,0),(5,1,0,0,0),(4,2,0,0,0),(4,1,1,0,0),(3,3,0,0,0),(3,2,1,0,0),$ |

$(3,1,1,1,0),(2,2,2,0,0),(2,2,1,1,0),(2,1,1,1,1)\}$. |

$\mathbf{x}$ | $\genfrac{}{}{0pt}{}{\mathit{last}\text{}\Theta \mathit{in}}{\mathit{inner}\text{}\mathit{loop}}$ | ${\mathbf{x}}^{*}\left(1\right)$ | ${\Theta}^{*}$ | l | $\genfrac{}{}{0pt}{}{\mathit{last}\text{}S\mathit{in}}{\mathit{inner}\text{}\mathit{loop}}$ |

— | — | $(2,1,1,1,1)$ | $(0,0,0,2,6,4,0)$ | 4 | — |

$(6,0,0,0,0)$ | $(1,0,0,0,0,0,6)$ | $(2,1,1,1,1)$ | $(0,0,0,2,6,4,0)$ | 4 | $\{2,3,4\}$ |

$(5,1,0,0,0)$ | $(0,1,0,0,0,2,4)$ | $(2,1,1,1,1)$ | $(0,0,0,2,6,4,0)$ | 4 | $\{2,3,4\}$ |

$(4,2,0,0,0)$ | $(0,0,1,0,2,0,4)$ | $(2,1,1,1,1)$ | $(0,0,0,2,6,4,0)$ | 4 | $\{2,3,4\}$ |

$(4,1,1,0,0)$ | $(0,0,1,0,0,5,1)$ | $(2,1,1,1,1)$ | $(0,0,0,2,6,4,0)$ | 4 | $\{2,3,4\}$ |

$(3,3,0,0,0)$ | $(0,0,0,3,0,0,4)$ | $(2,1,1,1,1)$ | $(0,0,0,2,6,4,0)$ | 4 | $\{2,3,4\}$ |

$(3,2,1,0,0)$ | $(0,0,0,2,3,4,3)$ | $(3,2,1,0,0)$ | $(0,0,0,2,3,4,3)$ | 4 | $\{2,3,4,5\}$ |

$(3,1,1,1,0)$ | $(0,0,0,2,3,6,1)$ | $(3,2,1,0,0)$ | $(0,0,0,2,3,4,3)$ | 4 | $\{2,3,4,5\}$ |

$(2,2,2,0,0)$ | $(0,0,0,0,9,0,3)$ | $(2,2,2,0,0)$ | $(0,0,0,0,9,0,3)$ | 5 | $\{2,3,4,5\}$ |

$(2,2,1,1,0)$ | $(0,0,0,1,3,2,0)$ | $(2,2,2,0,0)$ | $(0,0,0,0,9,0,3)$ | 5 | $\{1,3,5\}$ |

**Lemma**

**5.**

**Proof.**

## 4. Performance Analysis

## 5. Extension to the Integer Prenucleolus

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Lemma**

**8.**

**Remark**

**8.**

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

**Remark**

**9.**

**Remark**

**10.**

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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^{1}This conclusion was brought to the author’s attention through a valuable comment by J. Derks.^{2}The author thankfully acknowledges a helpful conversation with J. Derks who pointed out to him that, given a simple game ${\Gamma}_{t}$, the set $\left(\right)$ contains at most one element $\mathbf{x}$ with ${x}_{1}\ge \cdots \ge {x}_{n}$, whenever $N\backslash \left\{i\right\}\in \mathcal{W}\left({\Gamma}_{t}\right)$ for all $i\in N$. We add that it also suffices to impose the weaker condition that $r\le 1$.^{3}As to the inner`WHILE/DO`loop in Table 2, we assume short-circuit evaluation of relational expressions. This means that the expression ’$i>1$ and ${c}_{i-1}={c}_{j}$’ is marked to be false as soon as $i=1$. Hence, no attempt will be made to access a nonexistent ${c}_{0}$.

**Table 1.**Lexicographic minimization of $\theta \left(\mathbf{x}\right)$ over $P\left({\Gamma}_{t}\right)$.

Input: Directed simple game ${\Gamma}_{t}$ (assuming $I\left({\Gamma}_{t}\right)\ne \varnothing $), p Output: $(\mathbf{c},\mathbf{y})$ |

PROG LEXMIN |

GLOBAL $\mathcal{W}\left({\Gamma}_{t}\right)\backslash N,P\left({\Gamma}_{t}\right),{\mathbf{x}}^{*}\left(\right),p,t$ ; GLOBAL ${\mathbf{a}}_{S},{\mathbf{b}}_{S}\phantom{\rule{4pt}{0ex}}\mathrm{for}\text{}\mathrm{all}\phantom{\rule{4pt}{0ex}}S\in \mathcal{W}\left({\Gamma}_{t}\right)\backslash N$ |

$n\leftarrow \left|N\right|$ ; $t\leftarrow v\left(N\right)$ |

FOR $i=1$ TO n DO $\{{c}_{i}\leftarrow |\{S\subseteq \mathcal{W}\left({\Gamma}_{t}\right):i\in S\}\left|\right\}$ |

$P\left({\Gamma}_{t}\right)\leftarrow \left(\right)open="\{"\; close="\}">\mathbf{x}\in I\left({\Gamma}_{t}\right):{x}_{1}\ge \cdots \ge {x}_{n}\wedge |{x}_{i}-{x}_{j}|\in \{0,1\}\phantom{\rule{4pt}{0ex}}\mathrm{for}\text{}\mathrm{all}\phantom{\rule{4pt}{0ex}}i\ne j,\phantom{\rule{0.166667em}{0ex}}{c}_{i}={c}_{j}$ |

FOR each $S\in \mathcal{W}\left({\Gamma}_{t}\right)\backslash N$ DO {compute $({\mathbf{a}}_{S},{\mathbf{b}}_{S})$ as in (4) and (5)} |

DIM ${\mathbf{x}}^{*}(1\phantom{\rule{-0.166667em}{0ex}}:\phantom{\rule{-0.166667em}{0ex}}p)$ |

START THREAD LEXMIN1 ; … ; START THREAD LEXMINp |

DO {wait for threads $\mathrm{LEXMIN}1,\phantom{\rule{0.166667em}{0ex}}\dots ,\phantom{\rule{0.166667em}{0ex}}\mathrm{LEXMIN}p$ to complete} |

$\mathbf{y}\leftarrow {\mathbf{x}}^{*}\left(1\right)$ ; ${\Theta}^{*}\leftarrow \Theta \left(\mathbf{y}\right)$ |

FOR $i=2$ TO p DO { IF $\Theta \left({\mathbf{x}}^{*}\left(i\right)\right){<}_{lex}{\Theta}^{*}$ THEN $\mathbf{y}\leftarrow {\mathbf{x}}^{*}\left(i\right)$ ; ${\Theta}^{*}\leftarrow \Theta \left(\mathbf{y}\right)\}$ |

END PROG |

THREAD FUNCTION LEXMIN$\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{0.222222em}{0ex}}i$ |

${\mathbf{x}}^{*}\left(i\right)\leftarrow $ last $\mathbf{x}\in P\left({\Gamma}_{t}\right)$ ; ${\Theta}^{*}\leftarrow \Theta \left({\mathbf{x}}^{*}\left(i\right)\right)$ ; $l\leftarrow $ smallest index j for which ${\Theta}_{j}^{*}>0$ |

FOR the ith to the 2nd last $\mathbf{x}\in P\left({\Gamma}_{t}\right)$ STEP p DO { |

$k\leftarrow $ number of parts of t in $\mathbf{x}$ ; $\Theta \leftarrow (0,\phantom{\rule{0.166667em}{0ex}}\dots ,\phantom{\rule{0.166667em}{0ex}}0)$ |

FOR each $S\in \mathcal{W}\left({\Gamma}_{t}\right)\backslash N$ DO { |

$h\leftarrow t-{\sum}_{j\le {a}_{Sk}}{x}_{{b}_{Sj}}$ |

IF $t-h+1\ge l\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}$ THEN ${\Theta}_{t-h+1}\leftarrow {\Theta}_{t-h+1}+1$ ELSE EXIT FOR |

IF ${\Theta}_{l}>{\Theta}_{l}^{*}$ THEN EXIT FOR} |

IF $\Theta {<}_{lex}{\Theta}^{*}$ THEN ${\mathbf{x}}^{*}\left(i\right)\leftarrow \mathbf{x}$ ; ${\Theta}^{*}\leftarrow \Theta $ ; $l\leftarrow $ smallest index j for which ${\Theta}_{j}^{*}>0\}$ |

END FUNCTION |

Input: Directed simple game ${\Gamma}_{t}$ (assuming $I\left({\Gamma}_{t}\right)\ne \varnothing $) and associated pair $(\mathbf{c},\mathbf{y})$ Output: $\nu \left({\Gamma}_{t}\right)$ |

PROG COMPL |

$n\leftarrow \left|N\right|$ ; ${\mathbf{y}}^{1}\leftarrow \mathbf{y}$ ; $\nu \left({\Gamma}_{t}\right)\leftarrow \left\{{\mathbf{y}}^{1}\right\}$ |

$i\leftarrow n+1$ ; $m\leftarrow 1$ |

WHILE $i>2$ DO { |

$j\leftarrow i-1$ ; $i\leftarrow j$ |

WHILE $i>1$ and ${c}_{i-1}={c}_{j}$ DO $\{i\leftarrow i-1\}$ |

IF ${y}_{i}>{y}_{j}$ THEN { |

$l\leftarrow 0$ |

FOR each $\mathbf{b}\in B(i,j)$ DO { |

FOR $k=1$ TO m DO { |

$l\leftarrow l+1$ ; ${\mathbf{y}}^{l}\leftarrow ({y}_{1}^{k},\phantom{\rule{0.166667em}{0ex}}\dots ,\phantom{\rule{0.166667em}{0ex}}{y}_{i-1}^{k},\mathbf{b},{y}_{j+1}^{k},\phantom{\rule{0.166667em}{0ex}}\dots ,\phantom{\rule{0.166667em}{0ex}}{y}_{n}^{k})\}\}$ |

$m\leftarrow l$ ; $\nu \left({\Gamma}_{t}\right)\leftarrow \{{\mathbf{y}}^{1},\phantom{\rule{0.166667em}{0ex}}\dots ,\phantom{\rule{0.166667em}{0ex}}{\mathbf{y}}^{m}\}\left\}\right\}$ |

END PROG |

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Wolff, R.
The Integer Nucleolus of Directed Simple Games: A Characterization and an Algorithm. *Games* **2017**, *8*, 16.
https://doi.org/10.3390/g8010016

**AMA Style**

Wolff R.
The Integer Nucleolus of Directed Simple Games: A Characterization and an Algorithm. *Games*. 2017; 8(1):16.
https://doi.org/10.3390/g8010016

**Chicago/Turabian Style**

Wolff, Reiner.
2017. "The Integer Nucleolus of Directed Simple Games: A Characterization and an Algorithm" *Games* 8, no. 1: 16.
https://doi.org/10.3390/g8010016