Power index-Based Ranking–Semantics

Master degree in Computer Science

Candidate
Francesco Faloci

Advisors
Stefano Bistarelli
Francesco Santini

Power index-Based Ranking–Semantics

AAF

Voting Games

Ranking Semantics

Power index-Based Ranking–Semantics
  • Argumentation
  • Ranking Semantics
  • Power Indexes
  • PI based Ranking-Semantics
Power index-Based Ranking–Semantics
  • Argumentation
  • Ranking Semantincs
  • Power Indexes
  • PI based Ranking-Semantics
Argumentation
Argumentation
  • Bob: «I want to go soccer»
     
  • Alice: «I want to go to the theater»
     
  • Bob: «My ex-girlfriend is in the theater»
     
  • Alice: «You have not an ex-girlfriend»
Argumentation

«I want to go soccer»

«I want to go to the theater»

«My ex-girlfriend is in the theater»

«You have not an ex-girlfriend»

Argumentation

Abstract Argumentation Framework  [1]

A: { a, b, c, d }

R: { (a,b), (b,a), (c,b), (d,c) }

Argumentation

Abstract Argumentation Framework

A: { a, b, c, d }

R: { (a,b), (b,a), (c,b), (d,c) }

Argumentation: Sets
  • Conflict-free
    {} {a} {b} {c} {d} {a c} {a d} {b d}
     
  • Admissible
    {} {a} {d} {a d} {b d}

{} {a} {b} {c} {d} {a b} {a c} {a d} {b c} {b d} {c d}
{a b c} {a b d}​ {a c d}​ {b c d}​ {a b c d}​

Argumentation: Semantics
  • Complete
    {d} {a d} {b d}
     
  • Preferred
    {a d} {b d}
     
  • Stable
    {a d} {b d}
     
  • Grounded
    {d}

{} {a} {b} {c} {d} {a b} {a c} {a d} {b c} {b d} {c d}
{a b c} {a b d}​ {a c d}​ {b c d}​ {a b c d}​

Argumentation: Labelling
in(L) = \{a \in A \mid L(a) = in \}
out(L) = \{a \in A \mid L(a) = out \}
undec(L) = \{a \in A \mid L(a) = undec\}

Another Method to express acceptability of argument

  • ∀ a, b ∈ A, a ∈ in(L) only if  (b,a) ∈ R and b ∈ out(L)
     
  • ∀ a, b ∈ A, a ∈ out(L) only if (b,a) ∈ R and b ∈ in(L)
     
  • ∀ a ∈ A, if a ∉ in(L) and a ∉ out(L), then a ∈ undec(L)
Argumentation: Labelling
in(L) = \{a \in A \mid L(a) = in \}
out(L) = \{a \in A \mid L(a) = out \}
undec(L) = \{a \in A \mid L(a) = undec\}
Power index-Based Ranking–Semantics
  • Argumentation
  • Ranking Semantics
  • Power Indexes
  • PI based Ranking-Semantics
Ranking Semantics
Ranking Semantics

d ≻ a ≻ c ≻ b

Ranking Semantics

Bbs and Dbs, Categorize function, Tuples,  functions, Matt & Toni... [2]

  • Focus on attacks between arguments
  • Do not consider Dung's semantics
Power index-Based Ranking–Semantics
  • Argumentation
  • Ranking Semantics
  • Power Indexes
  • PI based Ranking-Semantics
Power indexes

Amaranth

Blue

Canary

36%

40%

17%

Voting Game!

Desert

11%

Power indexes

let's try the... Shapley Value [3]

SV (A) = 1,25

B ≻ A ≃ C ≻ D

  • if a coalition S reaches the 51% it values 1, 0 otherwise

  • we consider only minimal coalition: not the whole set

  • coalitions (A,B), (A,C), (B,C), (B,D) and (A,C,D) are minimal winning

A > C               A ≻ C

A: 36%

C: 17%

SV (B) = 1,5

SV (C) = 1,25

SV (D) = 0,75

Power indexes: Shapley Value

We used the characteristic function         defined on the coalition S that contains the agent i, as following:

... the Shapley Value formula for all the agents (i) as following:

\phi_i(v)= \sum_{S \subseteq N \setminus \{i\} }\frac{|S|! \; (|N| - |S| - 1)! }{|N|!} \;\; v_{S_i}
v_{S_i} = v(S \cup \{i\}) - v(S)
v_{S_i} 
Power indexes: MORE Indexes

Banzhaf Power Index [4]

Deegan-Pakel Index  [5]

Johnston Index  [5]

\beta_i(v) = \frac{1}{2^{|N|-1}} \sum_{S \subseteq N \setminus \{i\} } v_{S_i}
\rho_i(v) = \frac{1}{|M(v)|} \sum_{S \subseteq M_i(v) \setminus \{i\} \\ S\neq \emptyset} \frac{v_{S_i}}{|S|}
\gamma_i(v) = \sum_{S \subseteq N \setminus \{i\} \\ \varkappa(S) \geq 1} \frac{v_{S_i}}{\varkappa(S)}
Power index-Based Ranking–Semantics
  • Argumentation
  • Ranking Semantics
  • Power Indexes
  • PI based Ranking-Semantics
PI Ranking–Semantics

A ⇒ agents

  ⇒ coalitions

v_{S_i}

 {1} {2} {3} {4}

Conflict-free, Admissible, Complete,  Preferred, Stable,  Grounded

\sigma
PI Ranking–Semantics

Characteristic Function(s)

v^{I}_{\sigma,F}(S) = \begin{cases} 1, & \text{if } S \in \mathit{in}(L_\sigma) \\ 0, & \text{if } otherwise \end{cases}\hspace{4em}
v^{O}_{\sigma, F}(S) = \begin{cases} 1, & \text{if } S \in \mathit{out}(L_\sigma) \\ 0, & \text{if } otherwise \end{cases}

Given      a Dung semantics and         the set of all possible labellings on F satisfying     , we define:

\sigma
L_\sigma
\sigma
< v^{I}_{\sigma,F}(S), v^{O}_{\sigma,F}(S) > 
PI Ranking–Semantics

http://www.dmi.unipg.it/conarg/web_interface.php

PI Ranking–Semantics

Shapley Value
(admissible)

d ≻ a ≻ b ≻ c

d (in: 0.08333; out: -0.58333)
a (in: -0.08333; out: -0.41667)
b (in: -0.41667; out: -0.08333)
c (in: -0.58333; out: 0.08333)
 

Banzhaf Power Index
(complete)

d ≻ a ≃ b ≻ c

d (in: 0.37500; out: -0.37500)
a (in: -0.12500; out: -0.12500)
b (in: -0.12500; out: -0.12500)
c (in: -0.37500; out: 0.37500)

PI Ranking–Semantics

1. S. Bistarelli, F. Faloci, F. Santini, and C. Taticchi. A Tool For Ranking Arguments Through Voting-Games Power Indexes. In Proceedings 34th CILC, volume 2396 of CEUR Workshop Proceedings, pages 193–201. CEUR-WS.org, 2019.

RESULTS

2. S. Bistarelli, F. Faloci, and C. Taticchi. Implementing ranking-based semantics in conarg: a preliminary report. ICTAI, 2019.

.

CoRR , abs/1908.07784, 2019. URL http://arxiv.org/abs/1908.07784

.

PI Ranking–Semantics
  • Other Power Indexes
     
  • Deep study on local and global properties
     
  • Ranking function parametric to diferent subset

FUTURE WORKS

References

[2] Elise Bonzon, Jérôme Delobelle, Sébastien Konieczny, and Nicolas Maudet. A Comparative Study of Ranking-Based Semantics for Abstract Argumentation. In Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, February 12-17, 2016, Phoenix, Arizona, USA, pages 914–920. AAAI Press, 2016. ISBN 978-1-57735-760-5.

[1] Phan Minh Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif. Intell., 77(2):321–358, 1995. doi: 10.1016/0004-3702(94)00041-X.
URL https://doi.org/10.1016/0004-3702(94)00041-X.

[3] Lloyd S. Shapley. Contributions to the Theory of Games (AM-28), Volume II. Princeton University Press, 1953.

[4] John F. Banzhaf. Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review, 19(2):317–343, 1965.

[5] D. Keith. Encyclopedia of Power. SAGE Publications, Inc., 2011. ISBN 978-1-4129-2748-2.

PI Ranking–Semantics

</end>

Thanks for your attention

Made with Slides.com