Steve Watson

A significant part of the standard conception of a group is that group membership is valued because it is estimated that it will advance particular interests that are salient and weighty for the group member. We previously said that where a group is generally believed to particularly advance the interest j for its members; then j will be considered an intentional interest of the group, and we proposed the condition

(∀x) [S(c|_x∈|G|, j, x) > S(c|_x∉|G|, j, x)]

However, this condition is unsatisfactory on (at least) two counts. First, it is actually a condition for the extensional interest, since it doesn’t refer to the subjective evaluation of these variables or functions; and, second, it is actually claiming that the extensional interests of a group are just those interests that it is uniquely best placed to advance. If the same interest, j, could be served just as well by moving from G₁ to G₂, then j is not in the extensional interest of either group. This is not what is intended. We approach the problem instead from a different direction.

Define the Extensional Interest Groups for the interest j as:

G(j) = {G: (∀x ∈ |G|) [S(C(B_stat(x), x), j, x) >> 0]}

Then define the Extensional Interests of G as:

X(G)) = {j: G ∈ G(j)}

Or, more directly:

X(G)) = {j: (∀x ∈ |G|) [S(C(B_stat(x), x), j, x) >> 0]}

Extensional interests are objective facts about a group. Intentional interests, on the other hand, have to be defined with respect to the agents or collections of agents whose estimations are being considered. Therefore we need to define the Intensional Interest Groups for the interest j for the agent y as:

E(y)[ G(j) ] = {G: (∀x ∈ |G|) [E(y)[ S(C(B_stat(x), x), j, x) ] >> 0]}

Then define the Intensional Interests of G for the agent y as:

E(y)[ X(G)) ] = {j: G ∈ E(y)[ G(j) ]}

Or, more directly:

E(y)[ X(G)) ] = {j: (∀x ∈ |G|) [E(y)[ S(C(B_stat(x), x), j, x) ] >> 0]}

We can further define the Desired Interest Groups for the interest j for the agent y as:

D(y)[ G(j) ] = {G: (∀x ∈ |G|) [D(y)[ S(C(B_stat(x), x), j, x) ] >> 0]}

Then define the Desired Interests of G for the agent y as:

D(y)[ X(G)) ] = {j: G ∈ D(y)[ G(j) ]}

Or, more directly:

D(y)[ X(G)) ] = {j: (∀x ∈ |G|) [D(y)[ S(C(B_stat(x), x), j, x) ] >> 0]}

We can do the same sort of thing for the interests of roles within groups too. But that is a merely mechanical application of the principles here.

Unstructured Sets

Groups have a considerable amount of structure. Unstructured sets of agents may be defined wrt their awareness of their interests.

Define an Extensional Interest Set for the interest j as:

S(j) = {x: j ∈ I(x)}

Such unstructured sets are significant in several ways.

1. S(j) may provide a pool of agents ripe for recruitment into groups in G(j) that advance their interests.
2. As latent groups they may give rise to structure that converts some subset of S(j) into a group member of G(j)
3. They may consist of agents who recognize an interest that other agents might have in common.

Define an Intentional Interest Set for the interest j for the agent y as

E(y)[ S(j) ] = {x: j ∈ E(y)[ I(x) ]}

Agent y may see himself as a member of several intentional interest sets.

(∀y)(∀j ∈ E(y)[ I(y) ])[ y ∈ E(y)[ S(j) ]]