Main differences between ``local'' and ``global'' valuations

[7] gives a comparison of these approaches with some existing approaches ([9,13,4]), and also a comparison of the ``local'' approaches and the ``global'' approach. The improvement of the global approach proposed in this paper does not modify the main results of this comparison.

Let us recall here an example of the essential point which differentiates them (this example has already been presented at the beginning of Section 3):

In the local approach, $B'$ is better than $B$ (since $B'$ suffers one attack whereas $B$ suffers two attacks).

In the global approach, $B$ is better than $B'$ (since it has at least a defence whereas $B'$ has none). In this case, $C_1$ loses its negative status of attacker, since it is in fact ``carrying a defence'' for $B$.

The following table synthesises the results about the different proposed valuations:

global approach
arguments having only attack branches	$\preceq$	arguments having attack branches and defence branches	$\preceq$	arguments having only defence branches	$\preceq$	arguments never attacked

local approach

arguments having several unattacked direct attackers	$\preceq$	arguments having only one unattacked direct attacker	$\preceq$	arguments having only one attacked direct attacker (possibly defended)
arguments having several attacked direct attackers (possibly defended)

$\preceq$

arguments never attacked

The difference between the local approaches and the global approach is also illustrated by the following property:

Property 11 (Independence of branches in the global approach)  
Let $A$ be an argument having the following direct attackers:

• $A_1$ whose value is $v(A_1) = [(a^1_{p_1}, \ldots, a^1_{p_{m_1}}),(a^1_{i_1}, \ldots, a^1_{i_{m_1}})]$,
• ...,
• $A_n$ whose value is $v(A_n) = [(a^n_{p_1}, \ldots, a^n_{p_{m_n}}),(a^n_{i_1}, \ldots, a^n_{i_{m_n}})]$.

Let $A'$ be an argument having the following direct attackers:

• $A^1_{p_1}$ whose value is $v(A^1_{p_1}) = [(a^1_{p_1})()]$,
• ...,
• $A^1_{p_{m_1}}$ whose value is $v(A^1_{p_{m_1}}) = [(a^1_{p_{m_1}})()]$,
• $A^1_{i_1}$ whose value is $v(A^1_{i_1}) = [()(a^1_{i_1})]$,
• ...,
• $A^1_{i_{m_1}}$ whose value is $v(A^1_{i_{m_1}}) = [()(a^1_{i_{m_1}})]$,
• ...,
• $A^n_{p_1}$ whose value is $v(A^n_{p_1}) = [(a^n_{p_1})()]$,
• ...,
• $A^n_{p_{m_n}}$ whose value is $v(A^n_{p_{m_n}}) = [(a^n_{p_{m_n}})()]$,
• $A^n_{i_1}$ whose value is $v(A^n_{i_1}) = [()(a^n_{i_1})]$,
• ...,
• $A^n_{i_{m_n}}$ whose value is $v(A^n_{i_{m_n}}) = [()(a^n_{i_{m_n}})]$.

Then $v(A) = v(A')$.

This property illustrates the ``independence'' of branches during the computation of the values in the global approach, even when these branches are not graphically independent. On the following example, $A$ and $A'$ have the same value $[(2,2)()]$ though they are the root of different subgraphs:

This property is not satisfied by the local approach since, using the underlying principles of the local approach (see Property 4), the value of the argument $A$ must be at least as good as (and sometimes better than²³) the value of the argument $A'$ ($A$ having one direct attacker, and $A'$ having two direct attackers).