Graduality in interaction-based valuations

We consider two different valuation methods for taking into account the quality of attackers and defenders of an argument in order to define the value of an argument using only the interaction between arguments¹³:

• In the first approach, the value of an argument only depends on the values of the direct attackers of this argument. Therefore, defenders are taken into account through the attackers. This approach is called local.

• In the second approach, the value of an argument represents the set of all the attack and the defence branches for this argument. This approach is called global.

The main difference between these two approaches is illustrated by the following example:

In the local approach, $B$ has two direct attackers ($C_2$ and $C_1$) whereas $B'$ has only one ($C'$). Thus $B'$ is better than $B$ (since $B'$ suffers one attack whereas $B$ suffers two attacks).

In the global approach, two branches (one of attack and one of defence) lead to $B$ whereas only one branch of attack leads to $B'$. Thus $B$ is better than $B'$ (since it has at least one 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 $B$.