# Contextual and Possibilistic Reasoning for Coalition Formation

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## Abstract

**:**

## 1. Introduction

- How to identify all possible coalitions that agents can form to fulfil their goals?
- How to evaluate the coalitions given a set of requirements?
- How to compute and evaluate coalitions taking also into account the uncertainty in the agents’ actions?

## 2. Background

#### 2.1. Dependence Networks and Coalition Formation

#### 2.2. Multi-Context Systems

#### 2.2.1. Formalization

**KB**${}_{L}$,

**BS**${}_{L}$,

**ACC**${}_{L}$) consists of the following components:

**KB**${}_{L}$ is the set of well-formed knowledge bases of L. Each element of**KB**${}_{L}$ is a set of formulae.**BS**${}_{L}$ is the set of possible belief sets, where the elements of a belief set are a set of formulae.**ACC**${}_{L}$:**KB**${}_{L}$→${2}^{{\mathrm{BS}}_{L}}$ is a function describing the semantics of the logic by assigning to each knowledge base a set of acceptable belief sets.

**KB**${}_{i}$,

**BS**${}_{i}$,

**ACC**${}_{i}$) is a logic, $k{b}_{i}\in {\mathbf{KB}}_{i}$ a knowledge base, and $b{r}_{i}$ a set of ${L}_{i}$-bridge rules over (L${}_{1}$, …, L${}_{n}$). For each $H\subseteq \{{h}_{b}\left(r\right)\mid r\in b{r}_{i}\}$ it holds that $k{b}_{i}\cup H\in {\mathbf{KB}}_{i}$, meaning that the head of each bridge rule used to import information to context ${c}_{i}$, must be compatible with the knowledge base of ${c}_{i}$.

**Example**

**1.**

#### 2.2.2. Computational Complexity

**KB**and $S\in $

**ACC**$\left(kb\right)$ a set $\kappa (kb,S)\subseteq S$ of size (written as a string) polynomial in the size of $kb$, called the kernel of S, such that there is a one-to-one correspondence f between the belief sets in

**ACC**$\left(kb\right)$ and their kernels, i.e., $S\rightleftharpoons f\left(\kappa \right(kb,S\left)\right)$. Examples of logics with poly-size kernels include propositional logic, default logic, auto-epistemic logic and non-monotonic logic programs. If furthermore, given any knowledge base $kb$, an element b, and a set of elements K, deciding whether (i) $K=\kappa (kb,S)$ for some $S\in $

**ACC**$\left(kb\right)$ and (ii) $b\in S$ is in ${\Delta}_{k}^{p}$, then we say that L has kernel reasoning in ${\Delta}_{k}^{p}$. For example, default logic and auto-epistemic logic have kernel reasoning in ${\Delta}_{2}^{p}$.

#### 2.3. Possibilistic Reasoning in MCS

#### 2.3.1. Possibilistic Logic Programs

**Definition**

**1.**

**Definition**

**2.**

#### 2.3.2. Possibilistic MCS

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

**Example**

**2.**

## 3. Main Example

## 4. Computing and Evaluating Coalitions in the Perfect World

#### 4.1. Modeling Dependencies

**Definition**

**11.**

**Definition**

**12.**

**KB**${}_{i}$,

**BS**${}_{i}$,

**ACC**${}_{i}$) is the logic of agent $a{g}_{i}\in \mathcal{A}$, $k{b}_{i}\in {\mathbf{KB}}_{i}$ is a knowledge base that includes the goals of $a{g}_{i}$ and the actions it can perform, and $b{r}_{i}$ is a set of bridge rules, which include the rules that represent the dependencies of $a{g}_{i}$ on other agents in $\mathcal{A}$ for all the goals ${g}_{k}$ of the agent and their associated plans ${p}_{l}$.

**Example**

**3.**

- provide the current location of the paper (${a}_{2s}$)
- provide the location that the pen needs to be delivered to (${a}_{1d}$)
- provide the location that the glue needs to be delivered to (${a}_{3d}$)
- carry the pen or the glue (${a}_{1c}\vee {a}_{3c}$)

#### 4.2. Computing Coalitions

- The MCS-IE system [43] implements a centralized reasoning approach that is based on the translation of MCS into HEX-programs [44] (an extension of answer set programs with external atoms), and on their execution in the dlv-hex system (http://www.kr.tuwien.ac.at/research/systems/dlvhex/).
- The algorithms proposed in [25] implement a distributed computation, which however assumes that all contexts are homogeneous with respect to the logic that they use (defeasible logic).
- The three algorithms proposed in [45] enable distributed computation of equilibria. DMCS assumes that each agent has minimal knowledge about the world, namely the agents that it is connected through the bridge rules, but does not have any further metadata, e.g., topological information, of the system. Its computational complexity is exponential to the number of literals used in the bridge rules. DMCS-OPT uses graph theory techniques to detect any cycle dependencies in the system and avoid them during the evaluation of the equilibria, improving the scalability of the evaluation. DMCS-STREAMING computes the equilibria gradually (k equilibria at a time), reducing the memory requirements for the agents. The three algorithms have been implemented in a system prototype (http://www.kr.tuwien.ac.at/research/systems/dmcs).

**Example**

**4.**

#### 4.3. Evaluating the Coalitions

**Example**

**5.**

**$\mathrm{coal}(a,b)$**for any distinct pair of agents $a,b\in \mathcal{A}$ is the number of cycles that contain the ordered pair ($a,b$) in $DN$, l is the cycle length, and $\Omega $ denotes the maximal number of pairs of agents in cycles.

**Example**

**6.**

## 5. Computing Coalitions under Uncertainty

#### 5.1. Modeling Uncertainty

**Definition**

**13.**

**Definition**

**14.**

**Definition**

**15.**

**Example**

**7.**

#### 5.2. Computing Coalitions under Uncertainty

**Example**

**8.**

#### 5.3. Evaluating Coalitions under Uncertainty

**Definition**

**16.**

- the Weighted Sum Method [66], according to which each criterion ${R}_{j}$ is given a weight ${w}_{j}$, so that the sum of all weights is 1 ($\sum _{j=1}^{n}{w}_{j}=1$), and the overall score of each alternative ${D}_{i}$ is the weighted sum of ${q}_{ij}$, i.e., the scores of ${D}_{i}$ for each criterion ${R}_{j}$:$$WS\left({D}_{i}\right)=\sum _{j=1}^{n}{w}_{j}{q}_{ij}$$The optimal decision alternative is the one with the highest overall score ($WS$). This method can only be used in single-dimensional cases, in which all the score units are the same.
- the Weighted Product Method [67], which aggregates the individual scores using their product instead of their sum. Specifically, each decision alternative is compared with the others by multiplying several ratios, one for each criterion. Each ratio is raised to the power equivalent to the relative weight of the corresponding criterion. In general, to compare alternatives ${D}_{1}$ and ${D}_{2}$, the following product must be calculated:$$Ratio({D}_{1}/{D}_{2})=\prod _{j=1}^{n}{({q}_{1j}/{q}_{2j})}^{{w}_{j}}$$
- TOPSIS (Technique for Order Preference by Similarity to Ideal Solution [68]), which is based on the concept that the chosen alternative should have the shortest geometric distance from the positive ideal solution and the longest geometric distance from the negative ideal solution. TOPSIS assumes that each alternative has a tendency of monotonically increasing or decreasing utility. Therefore, it is easy to locate the ideal and negative ideal solutions.

**Example**

**9.**

## 6. Related Work

## 7. Summary and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**Coalitions ${C}_{0}$ (

**a**), and ${C}_{1}$ (

**b**) in bold; remaining dependencies in dotted lines.

Robot | $\mathit{a}{\mathit{g}}_{\mathbf{1}}$ | $\mathit{a}{\mathit{g}}_{\mathbf{2}}$ | ||||||

Task | ${t}_{1}$ | ${t}_{2}$ | ${t}_{3}$ | ${t}_{4}$ | ${t}_{1}$ | ${t}_{2}$ | ${t}_{3}$ | ${t}_{4}$ |

Source | x | x | ||||||

Destination | x | x | x | |||||

Robot | $\mathit{a}{\mathit{g}}_{\mathbf{3}}$ | $\mathit{a}{\mathit{g}}_{\mathbf{4}}$ | ||||||

Task | ${t}_{1}$ | ${t}_{2}$ | ${t}_{3}$ | ${t}_{4}$ | ${t}_{1}$ | ${t}_{2}$ | ${t}_{3}$ | ${t}_{4}$ |

Source | x | x | ||||||

Destination | x |

Distances among Locations | ||||
---|---|---|---|---|

Robot | Pen | Paper | Glue | Cutter |

$a{g}_{1}$ | 10 | 15 | 9 | 12 |

$a{g}_{2}$ | 14 | 8 | 11 | 13 |

$a{g}_{3}$ | 12 | 14 | 10 | 7 |

$a{g}_{4}$ | 9 | 12 | 15 | 11 |

Destination | Pen | Paper | Glue | Cutter |

${D}_{a}$ | 11 | 16 | 9 | 8 |

${D}_{b}$ | 14 | 7 | 12 | 9 |

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**MDPI and ACS Style**

Bikakis, A.; Caire, P.
Contextual and Possibilistic Reasoning for Coalition Formation. *AI* **2020**, *1*, 389-417.
https://doi.org/10.3390/ai1030026

**AMA Style**

Bikakis A, Caire P.
Contextual and Possibilistic Reasoning for Coalition Formation. *AI*. 2020; 1(3):389-417.
https://doi.org/10.3390/ai1030026

**Chicago/Turabian Style**

Bikakis, Antonis, and Patrice Caire.
2020. "Contextual and Possibilistic Reasoning for Coalition Formation" *AI* 1, no. 3: 389-417.
https://doi.org/10.3390/ai1030026