# A Multicriteria Standard to Rank Plea Bargain Proposals

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*Standards*)

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Composition of Probabilistic Preferences

_{1j},…, a

_{nj}) be the vector of numerical evaluations of n alternatives A

_{1},…, A

_{n}by the criterion C

_{j}. For each k, from 1 to n, and j, from 1 to m, the size of the set S of criteria, let X

_{kj}denote a random variable with the distribution of preference for alternative A

_{k}according to criterion C

_{j}. In the absence of more accurate information, X

_{kj}will have preference probabilities directly derived from the values of the trichotomic pairwise comparisons between the alternatives.

_{ij}the count of preferences for the i-th alternative according to C

_{j}and by A(j,i

_{1},i

_{2}) the result of the trichotomic comparison between alternatives i

_{1}and i

_{2}according to the j-th criterion. To A(j,i

_{1},i

_{2}) is assigned one of three possible values: 1 if i

_{1}is preferable to i

_{2}by C

_{j}, 0 if i

_{2}is preferable to i

_{1}, and ½ if there is indifference between i

_{1}and i

_{2}by C

_{j}.

_{ij}= ∑

_{i2}A(j,i

_{,}i

_{2})

_{2}ranging over all the n-1 alternatives i

_{2}that A

_{i}is compared with.

_{i}is therefore the sum of the number of pairwise comparisons where i is preferred with half the number of comparisons where A

_{i}is considered equivalent to another alternative. The estimate of the probability of preference for A

_{i}by C

_{j}is the quotient

_{ij}= C

_{ij}⁄(n(n − 1)/2)

_{ij}by the number of comparisons.

_{1},i

_{2}) = 1 − A(j,i

_{2},i

_{1}).

#### 2.2. The Treatment of Collinearity

_{iJ}for alternative A

_{i}according to J by adding the C

_{ij}along the criteria j in J. However, the simple addition of the counts of the preferences by the elements of J may be distorted by the presence of common factors implicit in different criteria, what would lead us to overvalue such criteria.

_{j1j2}the Spearman correlation coefficient between vectors of evaluations (a

_{1j1},…, a

_{nj1}) and (a

_{1j2},…, a

_{nj2}), for S

_{j1j2}> 0, a portion of the P

_{ij1}and P

_{ij2}proportional to S

_{j1j2}is subtracted from each of them. For j

_{1}and j

_{2}with positive S

_{j1j2}and without positive correlation with other criteria, this leads to a reduction of their individual importance by subtracting P

_{ij1}S

_{j1j2/2}/2 from P

_{ij1}and P

_{ij2}S

_{j1j2/2}/2 from P

_{ij2}.

_{ij}an amount proportional to the correlation with each positively correlated criterion. This results in determining the C

_{ij}, replacing the simple sum by a weighted average

_{iJ}= ∑

_{j∈J}W

_{j}P

_{ij},

_{j}= Π

_{j2∈S-{j}}(1 − (max(S

_{jj2},0)/2).

#### 2.3. The Treatment of Interaction

_{1},..., A

_{n}} is determined through the following steps:

_{iJ}derived from the treatment of collinearity:

_{J}= max

_{i∈{1,…, n}}C

_{iJ}.

_{j1},..., C

_{js}} will be

_{J}/M

_{S}.

_{1},…, C

_{m}} with respect to a capacity μ on S is

_{1}, C

_{2}, and C

_{3}, with evaluations 90, 70, and 10, respectively. Suppose individual capacities are μ(C

_{1}) = 0.3, μ(C

_{2}) = 0.1, and μ(C

_{3}) = 0.6. In the absence of interaction, the joint evaluation of the alternative by the three criteria would be given by the mean, equal to 40. Let us assume now a strong positive interaction between the criteria C

_{1}and C

_{2}leading to μ({C

_{1},C

_{2}}) = 0.9. Since these criteria offer high evaluations for the alternative, by the Choquet integral, the joint evaluation (10 + 0.9(70 − 10) + 0.3(90 − 70) = 70) is much higher than the mean. If, on the other hand, a strong negative interaction leads, for example, to μ({C

_{1},C

_{2}}) = 0.3, the joint evaluation falls to 10 + 0.3(70 − 10) + 0.3(90 − 70) = 34, smaller than the mean.

## 3. Results and Discussion

#### 3.1. Multicriteria Formulation of Negotiation Terms

#### 3.2. An Example

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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[Crit1] | [Crit2] | [Crit3] | [Crit4] | [Crit5] | |
---|---|---|---|---|---|

[Alt1] | 4 | 4 | 4 | 0 | 2 |

[Alt2] | 6 | 6 | 0 | 1 | 1 |

[Alt3] | 6 | 0 | 6 | 1 | 1 |

[Alt4] | 0 | 6 | 6 | 1 | 1 |

[Alt5] | 12 | 0 | 0 | 2 | 1 |

[Alt6] | 0 | 12 | 0 | 2 | 1 |

[Alt7] | 0 | 0 | 12 | 2 | 0 |

[Alt8] | 2 | 2 | 2 | 3 | 0 |

[Crit2] | [Crit3] | [Crit4] | [Crit5] | |
---|---|---|---|---|

[Crit1] | −0.424 | −0.424 | −0.206 | 0.340 |

[Crit2] | −0.424 | −0.206 | 0.340 | |

[Crit3] | −0.206 | −0.255 | ||

[Crit4] | −0.809 |

[Crit1] | [Crit2] | [Crit3] | [Crit4] | [Crit5] | |
---|---|---|---|---|---|

[Alt1] | 0.107 | 0.107 | 0.107 | 0.250 | 0.000 |

[Alt2] | 0.054 | 0.054 | 0.214 | 0.179 | 0.107 |

[Alt3] | 0.054 | 0.214 | 0.054 | 0.179 | 0.107 |

[Alt4] | 0.214 | 0.054 | 0.054 | 0.179 | 0.107 |

[Alt5] | 0.000 | 0.214 | 0.214 | 0.071 | 0.107 |

[Alt6] | 0.214 | 0.000 | 0.214 | 0.071 | 0.107 |

[Alt7] | 0.214 | 0.214 | 0.000 | 0.071 | 0.232 |

[Alt8] | 0.143 | 0.143 | 0.143 | 0.000 | 0.232 |

[Crit4] | [Crit5] | [1&5] | [2&5] | [3&5] | [4&5] | |
---|---|---|---|---|---|---|

[Alt1] | 0.250 | 0.000 | 0.089 | 0.089 | 0.107 | 0.250 |

[Alt2] | 0.179 | 0.074 | 0.118 | 0.118 | 0.288 | 0.252 |

[Alt3] | 0.179 | 0.074 | 0.118 | 0.118 | 0.127 | 0.252 |

[Alt4] | 0.179 | 0.074 | 0.252 | 0.252 | 0.127 | 0.252 |

[Alt5] | 0.071 | 0.074 | 0.074 | 0.074 | 0.288 | 0.145 |

[Alt6] | 0.071 | 0.074 | 0.252 | 0.252 | 0.288 | 0.145 |

[Alt7] | 0.071 | 0.160 | 0.338 | 0.338 | 0.16 | 0.231 |

[Alt8] | 0.000 | 0.160 | 0.278 | 0.278 | 0.303 | 0.160 |

Maximum | 0.250 | 0.160 | 0.338 | 0.338 | 0.303 | 0.252 |

Capacity | 0.426 | 0.272 | 0.575 | 0.575 | 0.516 | 0.430 |

[Crit1] | [Crit2] | [Crit3] | [Crit4] | [Crit5] | |
---|---|---|---|---|---|

Probability | 0.190 | 0.190 | 0.190 | 0.222 | 0.206 |

Probability & collinearity | 0.181 | 0.181 | 0.219 | 0.255 | 0.163 |

Choquet | 0.218 | 0.218 | 0.151 | 0.176 | 0.237 |

Choquet & collinearity | 0.201 | 0.201 | 0.204 | 0.217 | 0.176 |

Probabilistic | Probabilistic & Collinearity | Choquet | Choquet & Collinearity | |
---|---|---|---|---|

[Alt1] | 0.117 | 0.126 | 0.132 | 0.158 |

[Alt2] | 0.123 | 0.129 | 0.139 | 0.157 |

[Alt3] | 0.123 | 0.123 | 0.140 | 0.147 |

[Alt4] | 0.123 | 0.123 | 0.140 | 0.147 |

[Alt5] | 0.120 | 0.121 | 0.148 | 0.165 |

[Alt6] | 0.120 | 0.121 | 0.148 | 0.165 |

[Alt7] | 0.145 | 0.134 | 0.206 | 0.202 |

[Alt8] | 0.130 | 0.121 | 0.162 | 0.156 |

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**MDPI and ACS Style**

Sant’Anna, A.P.; Gavião, L.O.; Sant’Anna, T.L.
A Multicriteria Standard to Rank Plea Bargain Proposals. *Standards* **2023**, *3*, 198-209.
https://doi.org/10.3390/standards3020016

**AMA Style**

Sant’Anna AP, Gavião LO, Sant’Anna TL.
A Multicriteria Standard to Rank Plea Bargain Proposals. *Standards*. 2023; 3(2):198-209.
https://doi.org/10.3390/standards3020016

**Chicago/Turabian Style**

Sant’Anna, Annibal Parracho, Luiz Octávio Gavião, and Tiago Lezan Sant’Anna.
2023. "A Multicriteria Standard to Rank Plea Bargain Proposals" *Standards* 3, no. 2: 198-209.
https://doi.org/10.3390/standards3020016