This document discusses computationally viable handling of beliefs in arguments for persuasion. It introduces persuasion problems where an agent tries to persuade another by exchanging arguments. It proposes using a belief distribution to represent the opponent's belief in arguments, and updating the distribution as new arguments are added. It describes splitting the distribution using a metagraph to group related arguments and reduce computational requirements. Experiments show the approach scales to larger argument graphs by updating individual flocks of arguments rather than the full distribution.

1. Computationally Viable Handling of Beliefs in
Arguments for Persuasion
Emmanuel Hadoux and Anthony Hunter
November 6, 2016
University College London
EPSRC grant Framework for Computational Persuasion

2. Introduction

3. Persuasion problems
• One agent (the proponent) tries to persuade the other
(the opponent)
• e.g., doctor persuading a patient to quit smoking, a
salesman, a politician, ...
1

4. Persuasion problems
• One agent (the proponent) tries to persuade the other
(the opponent)
• e.g., doctor persuading a patient to quit smoking, a
salesman, a politician, ...
• the agents exchange arguments during a persuasion
dialogue
1

5. Persuasion problems
• One agent (the proponent) tries to persuade the other
(the opponent)
• e.g., doctor persuading a patient to quit smoking, a
salesman, a politician, ...
• the agents exchange arguments during a persuasion
dialogue
• These arguments are connected by an attack relation
1

6. Abstract argumentation framework
A1 A2 A3
Figure 1: Argument graph with 3 arguments
Based on Dung’s abstract argumentation framework [1]
Example (Figure 1)
A1 = “It will rain, take an umbrella”
A2 = “The sun will shine, no need for an umbrella”
A3 = “Weather forecasts say it will rain”
2

7. Purpose of the work
The objective for the proponent:
1. Have an argument or a set of arguments holding at the
end of the dialogue
3

8. Purpose of the work
The objective for the proponent:
1. Have an argument or a set of arguments holding at the
end of the dialogue
2. Have these arguments believed by the opponent
3

9. Purpose of the work
The objective for the proponent:
1. Have an argument or a set of arguments holding at the
end of the dialogue
2. Have these arguments believed by the opponent
Need to maintain and update a belief distribution
3

10. Purpose of the work
The objective for the proponent:
1. Have an argument or a set of arguments holding at the
end of the dialogue
2. Have these arguments believed by the opponent
Need to maintain and update a belief distribution → to posit
the right argument
3

11. Belief distribution
Epistemic approach to probabilistic argumentation (e.g., [2])
Definition
Let G = ⟨A, R⟩ be an argument graph.
Each X ⊆ A is called a model.
A belief distribution P over 2A is such that
∑
X⊆A P(X) = 1 and
P(X) ∈ [0, 1], ∀X ⊆ A.
The belief in an argument A is P(A) =
∑
X⊆A s.t. A∈X P(X).
4

12. Belief distribution
Epistemic approach to probabilistic argumentation (e.g., [2])
Definition
Let G = ⟨A, R⟩ be an argument graph.
Each X ⊆ A is called a model.
A belief distribution P over 2A is such that
∑
X⊆A P(X) = 1 and
P(X) ∈ [0, 1], ∀X ⊆ A.
The belief in an argument A is P(A) =
∑
X⊆A s.t. A∈X P(X).
If P(A) > 0.5, argument A is accepted.
4

13. Belief distribution
Epistemic approach to probabilistic argumentation (e.g., [2])
Definition
Let G = ⟨A, R⟩ be an argument graph.
Each X ⊆ A is called a model.
A belief distribution P over 2A is such that
∑
X⊆A P(X) = 1 and
P(X) ∈ [0, 1], ∀X ⊆ A.
The belief in an argument A is P(A) =
∑
X⊆A s.t. A∈X P(X).
If P(A) > 0.5, argument A is accepted.
Example (of a belief distribution)
Let A = {A, B} where P({A, B}) = 1/6, P({A}) = 2/3, and
P({B}) = 1/6 is a belief distribution.
Then, P(A) = 5/6 > 0.5 and P(B) = 2/6 < 0.5.
4

14. Refinement of a belief distribution
Each time a new argument is added to the dialogue, the
distribution needs to be updated.
A
B
Figure 2
AB P H1
A(P) H0.75
A (P)
11 0.6 0.7 0.675
10 0.2 0.3 0.275
01 0.1 0.0 0.025
00 0.1 0.0 0.025
Table 1: Examples of Belief Redistribution
We can modulate the update to take into account different
types users (skeptical, credulous, etc.)
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15. Splitting the distribution

16. Splitting the distribution
• 30 arguments → 230 = 1, 073, 741, 824 models → 8.6 gB if
treated as a double type
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17. Splitting the distribution
• 30 arguments → 230 = 1, 073, 741, 824 models → 8.6 gB if
treated as a double type
• Fortunately, they are not all directly linked to each other
6

18. Splitting the distribution
• 30 arguments → 230 = 1, 073, 741, 824 models → 8.6 gB if
treated as a double type
• Fortunately, they are not all directly linked to each other
• We can group related arguments into flocks which are
themselves linked to each other
6

19. Splitting the distribution
• 30 arguments → 230 = 1, 073, 741, 824 models → 8.6 gB if
treated as a double type
• Fortunately, they are not all directly linked to each other
• We can group related arguments into flocks which are
themselves linked to each other
• We create a split distribution from the metagraph, as
opposed to the joint distribution
6

20. Metagraphs
A1
A2
A3
A4
A5 A6
A7
A8 A9 A10
(a)
A1 A2, A3
A4
A5 A6
A7, A8, A9, A10
(b)
Figure 3: Argument graph and possible metagraph
7

21. Creating a split distribution
A1 A2, A3
A4
A5 A6
A7, A8, A9, A10
Figure 4: Metagraph
We define three assumptions for
the split to be clean:
1. Arguments from non directly
connected flocks are
conditionaly independent
2. Arguments in a flock are
considered connected
3. Arguments in a flock are
conditionally dependent
No bayesian networks because: not probabilities, users are not
rational, etc.
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22. Creating a split distribution
• We can define an optimal, irreducible, split w.r.t. the graph
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23. Creating a split distribution
• We can define an optimal, irreducible, split w.r.t. the graph
• However, an irreducible split may not be computable
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24. Creating a split distribution
• We can define an optimal, irreducible, split w.r.t. the graph
• However, an irreducible split may not be computable
• Only the irreducible split is unique, we therefore need to
rank the others.
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25. Ranking the splits
Definition (valuation of a split)
x =
∑
A∈A
∑
Pi∈S s.t. A∈E(P)
|Pi|
and P ≻ P′ iff x < x′.
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26. Ranking the splits
Definition (valuation of a split)
x =
∑
A∈A
∑
Pi∈S s.t. A∈E(P)
|Pi|
and P ≻ P′ iff x < x′.
Example (of valuation and ranking)
Let P be the joint distribution for Figure 3a. Value of
P : 10 × 210 = 10, 240
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27. Ranking the splits
Definition (valuation of a split)
x =
∑
A∈A
∑
Pi∈S s.t. A∈E(P)
|Pi|
and P ≻ P′ iff x < x′.
A1 A2, A3
A4
A5 A6
A7, A8, A9, A10
Figure 5: Metagraph
Example (of valuation and
ranking)
P1 = (P(A5), P(A6 | A5),
P(A4 | A5, A6), P(A2, A3 | A4, A7),
P(A1 | A2, A3), P(A7, A8, A9, A10)):
21+22+23+2×24+23+4×24 = 118
10

28. Ranking the splits
Definition (valuation of a split)
x =
∑
A∈A
∑
Pi∈S s.t. A∈E(P)
|Pi|
and P ≻ P′ iff x < x′.
Example (of valuation and ranking)
P2 = (P(A1, A2, A3, A4, A5, A6 | A7), P(A7, A8, A9, A10)):
6 × 27 + 4 × 24 = 832.
We then see that P1 ≻ P2 ≻ P.
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29. Experiments

30. Splitting the distribution
A1 A2, A3
A4
A5 A6
A7, A8, A9, A10
Figure 5: Metagraph
• Original graph: 10
arguments → 1,024 values
→ 8kB
• Metagraph: 10 arguments
in 6 flocks → 54 values →
432B
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31. Splitting the distribution
A1 A2, A3
A4
A5 A6
A7, A8, A9, A10
Figure 5: Metagraph
• Original graph: 10
arguments → 1,024 values
→ 8kB
• Metagraph: 10 arguments
in 6 flocks → 54 values →
432B
• And the time taken to
update.
11

32. Splitting the distribution
A1 A2, A3
A4
A5 A6
A7, A8, A9, A10
Figure 5: Metagraph
• Original graph: 10
arguments → 1,024 values
→ 8kB
• Metagraph: 10 arguments
in 6 flocks → 54 values →
432B
• And the time taken to
update.
• An argument can be
updated by updating only
its flock.
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33. Experiments with flocks of different sizes
# flocks # links 1 update 50 updates
2 flocks
10 links 2ms 107ms
30 links 6ms 236ms
4 flocks
10 links 1ms 45ms
30 links 3ms 114ms
10 flocks
10 links 0.03ms 1.6ms
30 links 0.06ms 2.5ms
Table 2: Computation Time for Updates in Different Graphs of 50
Arguments (in ms)
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34. Experiments with different numbers of arguments
# args Time for 20 updates Comparative %
25 497ns +0%
50 517ns +4%
75 519ns +4%
100 533ns +7%
Table 3: Computation Time for 20 Updates (in ns)
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35. Experiments
A new version of the library is currently begin developped in
C++ and is available at: https:
//github.com/ComputationalPersuasion/splittercell.
As a rule of thumb, we should keep flocks to less than 25
arguments each.
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36. Conclusion
We have presented:
1. A framework to represent the belief of the opponent in
the arguments
2. How to create a split distribution using a metagraph
3. How to rank the splits in order to choose the most
appropriate one w.r.t. the problem
4. Experiments showing the viability of the approach
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37. Conclusion
We have presented:
1. A framework to represent the belief of the opponent in
the arguments
2. How to create a split distribution using a metagraph
3. How to rank the splits in order to choose the most
appropriate one w.r.t. the problem
4. Experiments showing the viability of the approach
Next step: adapt this work to the whole project to scale.
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38. Thank you!
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39. Phan Minh Dung.
On the acceptability of arguments and its fundamental
role in nonmonotonic reasoning, logic programming, and
n-person games.
Artificial Intelligence, 77:321–357, 1995.
Anthony Hunter.
A probabilistic approach to modelling uncertain logical
arguments.
International Journal of Approximate Reasoning,
54(1):47–81, 2013.
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