Steve Watson

F(G): C → C

But to understand what it entails we will have to further modify the context function we have been using. Previously, we considered:

1. The function C(b, x) = c, for b a behaviour of x and c the social context parameters that feature in the partial satisfaction functions S for x.
2. The refinement C(B(x), x) = {C(b₁, x), …, C(b_m, x)} = {c₁, …, c_m}, where B(x) = {b₁, …, b_m} refers to the range of x’s possible behaviours

Now we can build up a further series of refinements;

3. Make the substitutions for the new notation to get C(A(x), x) = {C(a₁, x), …, C(a_m, x)} = {c₁, …, c_m}, where A(x) = {a₁, …, a_m} refers to the range of x’s possible actions
4. Note that we need to index the output contexts for the agent whose partial satisfactions they are taken to determine. Thus the function C(a_x, y) = c_y, for a_x a behaviour of x and c_y the context parameters that feature in the partial satisfaction functions S for y.
5. C(A(x), y) = {C(a_x,1, y), …, C(a_x,m, y)} = {c_y,1, …, c_y,m}, where A(x) = {a_x,1, …, a_x,m} refers to the range of x’s possible actions
6. C(A(x), c_x,0, y) = {C(a_x,1, c_x,0, y), …, C(a_x,m, c_x,0, y)} = {c_y,1, …, c_y,m}, where c_x,0 refers to the context in which x acts.
7. C(A(X), C_X,0, y) = {C(A(x), C_X,0, y): x ∈ X} = {c_y,1, …, c_y,n}, (note the index change from m to n,) where
   • X is a set of agents
   • A(X) is the range of possible actions for each x in X
   • C_X,0 refers to the context in which each x in X acts. It is the complete set of social parameters that feature in the partial satisfaction functions S for x in X
8. C(A(X), C_X,0, Y) = {C(A(x), C_X,0, Y): x ∈ X } = {C_Y,1, …, C_Y,n}, where C_Y,i are the complete sets of social parameters that feature in the partial satisfaction functions S for y in Y.

We can now observe that the group function described means that G is so organized that

F(G) (C_in) = C_out

Which, in fact, means that

C(A(|G|), C_|G|,in, |G|) = C_|G|,out