Steve Watson

• an agent x is an Acceptant of the norm n iff x accepts the rule n.
  

  K(n) = {x: B(x)[ n ]}

• We can discover whether x accepts a norm by observation of his behaviour

A Norm Formation is a set of norms, N = {n₁, …, n_n}.

• an agent x is an acceptant of the norm formation N iff x accepts the rules in n.
  

  K(N) = {x: B(x)[ N ]}

• Standard classes of norm formations are conventions, traditions, etc. (which may be analysed later.)

An Institution may be formalized as a norm formation which imposes a group structure on K(N)

• This definition results in a particular selection of social structures being called institutions which may differ from the selections made by other theoretical views. This is no great difficulty, since we have the theoretical machinery to deal with all reasonable extensions or restrictions of this concept.

1. An institution imposes an (Institutional) Organization on K(N)
   • N → (∃Q ={r₁, …, r_n} ⊂ 2^K(N))
     

     [((∀x ∈ K(N))(∃r_i ∈ Q) [x ∈ r_i]) &
     ((∀x ∈ K(N))(∀r_i, r_j ∈ Q) [x ∈ r_i & x ∈ r_j → r_i = r_j])]

   • We call the elements of Q the (Institutional) Roles of the organization of N
2. An institution imposes a relation of (Institutional) Dominance on K(N)
   • x institutionally dominates y in N iff
     

     (∃α_response(x) ∈ B(y)[ α(x) ])
     (∃α_neutral(x) ∈ B(y)[ α(x) ] – α_response(x))
     (∃α_challenge(y) ∈ B(y)[ α(y) ])
     the following conditions are satisfied:

     1. (∀a_challenge ∈ α_challenge(y)) (∀a_response ∈ α_response(x)) B(y)[ P(α_neutral(x) | A(y)=a_challenge & A(x)≠a_response ] ≈ 1
     2. (∀a_challenge ∈ α_challenge(y)) (∃a_response ∈ α_response(x)) (∃N’ ⊂ N) [(B(y)[ N’ → (A(y)=a_challenge → A(x)=a_response )]]
     3. (∀a_response ∈ α_response(x)) (∀a_challenge ∈ α_challenge(y)) (∃N’ ⊂ N)[(B(y)[ (A(x)=a_response → A(y)=a_challenge) → N’]
     4. (∀c_response ∈ B(y)[ C(α_response, c_x,0, x) ]) (∀c_neutral ∈ B(y)[ C(α_neutral, c_x,0, x) ])[B(y)[ T(c_response, I(y), y) ] << B(y)[ T(c_neutral, I(y), y) ]

   • This condition differs from the condition for (non-institutional) dominance in parts 2 and 3, which express y’s belief that his action together with the rules of the institution will produce x’s response, and that without those rules there would be no such response. This is, therefore, a special case of the previous definition.
   • Note that since N creates the conditions under which T is degraded for noncompliant agents, N provides the mechanism of enforcing compliance.
   • Express x institutionally dominates y as x/_Ny
3. Let the Group determined by the institution N be denoted G_N = < K(N), R >, where R contains at least
   • O(G_N), the institutional organization as defined on |G_N|,
   • ADD(G_N) = {< x, y >: x, y ∈ |G_N|, x/_Ny}, the (Institutional) Agent Dominance Diagram of G_N, and
   • RDD(G_N) = {< r_i, r_j >: r_i, r_j ∈ O(G_N), r_i/_Nr_j}, the (Institutional) Role Dominance Diagram of G_N, where, for disjoint sets X and Y, Y institutionally dominates X iff (∀x ∈ X) (∀y ∈ Y) [y /_N x].

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

B(x)[ argument ] is the ‘belief’ of x about ‘argument’, which replaces E(x)[ argument ]
D(x)[ argument ] is the ‘desire’ of x about ‘argument’, which replaces D(x)
A(x) is the ‘action’ by x, which replaces A(x) and B(x). It yields the range of possible actions for x.

Previous terminology using ‘intensional’ will now use ‘believed’

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) ]]

For x ∈ |G|,
Let B_stat(x) ⊂ B(x) be all actions that constitute remaining in the group
Let B_exit(x) ⊂ B(x) be all actions that constitute leaving the group

(∀x ∈ |G|)[ E[ TS(C(B_stat(x), x), I(x), x) ] ≥ E[ TS(C(B_exit(x), x), I(x), x) ] → A(x)∈B_stat(x) ]

Where [ E[ TS(C(B_stat(x), x), I(x), x) ]
= ∑_{b∈E[ Bstat(x) ]} E[ P(b) ] (∑_j∈E[I(x)]w_j(x) ∑_{c∈E[C(b, x)]} E[ P(c) ]E[ S(c, j, x) ])
and [ E[ TS(C(B_exit(x), x), I(x), x) ]
= ∑_{b∈E[ Bexit(x) ]} E[ P(b) ] (∑_j∈E[I(x)]w_j(x) ∑_{c∈E[C(b, x)]} E[ P(c) ]E[ S(c, j, x) ])

We may similarly define a Recruitment Condition that needs to be satisfied in order that x should become a member of G:

For x ∈ |G|,
Let B_stat(x) ⊂ B(x) be all actions that constitute remaining outside the group
Let B_intrat(x) ⊂ B(x) be all actions that constitute joining the group

(∀x ∈ |G|)[ E[ TS(C(B_intrat(x), x), I(x), x) ] > E[ TS(C(B_stat(x), x), I(x), x) ] → A(x)∈B_intrat(x) ]

Where [ E[ TS(C(B_stat(x), x), I(x), x) ]
= ∑_{b∈E[ Bstat(x) ]} E[ P(b) ] (∑_j∈E[I(x)]w_j(x) ∑_{c∈E[C(b, x)]} E[ P(c) ]E[ S(c, j, x) ])
and [ E[ TS(C(B_intrat(x), x), I(x), x) ]
= ∑_{b∈E[ Bintrat(x) ]} E[ P(b) ] (∑_j∈E[I(x)]w_j(x) ∑_{c∈E[C(b, x)]} E[ P(c) ]E[ S(c, j, x) ])

Just for the sake of completeness, we’ll note two other conditions, which are perfectly obvious given the preceding. First, the Retirement Condition:

(∀x ∈ |G|)[ E[ TS(C(B_stat(x), x), I(x), x) ] < E[ TS(C(B_exit(x), x), I(x), x) ] → A(x)∈B_exit(x) ]

And, second, the Resistance Condition:

(∀x ∈ |G|)[ E[ TS(C(B_intrat(x), x), I(x), x) ] ≤ E[ TS(C(B_stat(x), x), I(x), x) ] → A(x)∈B_stat(x) ]

• The condition of equality between the two estimated total satisfactions will be taken to favour continuation of the status quo. It is assumed that inertia is preferred where we find ourselves in the position of Buridan’s Ass.
• It is not here assumed that (∀x ∈ |G|) [B_stat(x) = B_intrat(x)^c] or that (∀x ∈ |G|) [B_stat(x) = B_exit(x)^c], but that does seem like a reasonable assumption to make.

O(G) = {r₁, …, r_n} ∈ 2^|G| is the Organization of G, where r_i are the roles in G, satisfying

1. (∀x ∈ |G|)(∃r ∈ O(G)) [x ∈ r]
2. (∀x ∈ |G|)(∀r_i, r_j ∈ O(G)) [x ∈ r_i & x ∈ r_j → r_i = r_j]

Of course, merely defining a subset of subsets does not suffice to characterize an organization or the structure of a group.

The point of a group, we noted before, was to assist in the achievement of the interests of its members. Socially relevant groups, which are actors in the social realm, will have a structure that coordinates the actions of its members towards some end. Such coordination is achieved through the imposition of a system of dominance relations. The point of the roles is in large part that the dominance relations that are relevant to the function of the group are defined wrt the roles. We wish, therefore, to note the existence of a characteristic set of dominance relations amongst the roles in the organization, which can be done by referencing the dominance relationship defined between agents. First define a dominance relationship amongst sets of agents as follows:

Let X, Y be disjoint sets of agents. Y dominates X iff (∀x ∈ X) (∃y ∈ Y) [y/x].

We now apply this relation to the roles of a group, and claim that all the roles are involved in dominance relationships. We also need to specify some roles as being special in the sense that though they dominate others they are not themselves necessarily dominated by any other roles in the group. These are the Leadership Roles in an organization. In fact, we shall assume there is just one leadership role. This is certainly the case in most forms of organization, and certainly in the standard hierarchical organization. Let the leadership role be denoted r_L. Then we say that

3. (∀r_i ∈ O(G)-r_L) (∃r_j ∈ O(G)) [r_j / r_i]

• We should also distinguish leadership roles from Strict Leadership Roles. Strict leadership roles are undominated leadership roles. Not all LR are SLR. Every group has a LR but not every group has an SLR – in fact it’s likely that there are very few of the latter. The existence of an SLR is described thus:
  

  (∃r_SL ∈ O(G)) (∀r ∈ O(G)-r_SL) [~(r/r_SL)]

Setting that possibility to one side for now, there are many different structures possible under this minimal definition. It’s possible, for example, to have the leadership role entirely detached from the rest of the group roles. This is unacceptable – meaning just that it is not characteristic of significant social formations. We need to expand the definition of the leadership role to eliminate this possibility.

We say that there is a Domination Sequence (or a chain of command) in O(G) subordinating role p to role q iff (∃ < r_i: i=1, …,n > ∈ O(G)ⁿ) [r₁/p & r₂/r₁ & … & q/r_n], and we write q//p and read that as q Superdominates p

A reasonable suggestion, then, would be that for any other role in the group, the leadership role superdominates that role; thus:

4. (∀r ∈ O(G)-r_L) [r_L//r_n]

As an example of the kind of definition that is proposed to cover the latter conception they offer the following: “A has power over B when A can change B’s actions, or sanction B, and so forth.” This definition is only offered as an indicator by those authors, and it is, of course, unsatisfactory as it stands. For example, it does not distinguish between power and mere influence. To distinguish ‘power’ in the sense that it is usually used – and in which sense it has a special significance in informal sociological explanations – from ‘power’ as the mere existence of the effect of one agent on another, we need to concentrate on the implied coercive capabilities of A over B. Thus we would need to include the fact that B changes his behaviours because he believes that A has the ability to punish him if he fails to comply. Moreover, he believes that A actually will punish him if he fails to comply.

We can attempt a definition using the established formalism as follows:

x has power over y iff (∃b_response ∈ E(y))[ B(x) ] which satisfies the conditions

1. E(y)[ C(b_response, x) ] = c_response
2. E(y)[ TS(c_response, I(y), y) ] << E(y)[ TS(c_now, I(y), y) ]
3. (∃b ∈ E(y)[ B(y) ]) [(E(y)[ A(y)=b → A(x)=b_response ] & (E(y)[ (A(y)≠b → A(x)≠b_response) ]]

• Whenever x and y are in a relationship such as this we say that x Dominates y and write x/y
• As presented the relationship of dominance appears to have no essential structure. It is not necessarily transitive, and it is not necessarily non-symmetric.
• Degrees of power can be traced back to the degrees by which the total satisfaction functions are altered in 2.
• Note that E(y)[ statement ] in condition 3 is to be read as the subjective estimation of the fuzzy truth value of ‘statement,’ or the subjective degree of belief of y in ‘statement.’
• Complexity can be added by noting that in most cases b_response is merely one of a class of behaviours that are available to x which can punish y in various degrees, that b in condition 3 is merely one of a class of behaviours that are available to y which can prompt the response from x.
• c_now in condition 2 is the context at the time of consideration, but it should really refer to the context that exists when x does not respond punitively to the b in condition 3.
• Note that this is power through punishment, but there is a similar definition available for power through reward: simply replace condition 2 with the condition
  

  E(y)[ TS(c_response, I(y), y) ] >> E(y)[ TS(c_now, I(y), y) ]

In the light of the notes above, refine the original definition thus:

x dominates y iff
(∃B_response(x) ∈ E(y)[ B(x) ])
(∃B_neutral(x) ∈ E(y)[ B(x) ]-B_response(x))
(∃B_stimulus(y) ∈ E(y)[ B(y) ])
the following conditions are satisfied:

1. (∀b_stimulus ∈ B_stimulus(y)) (∀b_response ∈ B_response(x))
   E(y)[ P(B_neutral, x) | A(y)=b_stimulus & A(x)≠b_response ] ~ 1
2. ∀b_stimulus ∈ B_stimulus(y)) (∃b_response ∈ B_response(x))
   [(E(y)[ A(y)=b_stimulus → A(x)=b_response ]
3. ∀b_response ∈ B_response(x)) (∃b_stimulus ∈ B_stimulus(y))
   [(E(y)[ A(x)=b_response → A(y)=b_stimulus ]
4. ∀c_response ∈ E(y)[ C(B_response, x) ]) (∀c_neutral ∈ E(y)[ C(B_neutral, x) ])
   [E(y)[ TS(c_response, I(y), y) ] << E(y)[ TS(c_neutral, I(y), y) ]

There are a number of actions in the social world that can be associated with groups as actors.

The following are actions that are associated with single groups.

1. Formation: A group is formed when several individuals come together.
• A book club is formed to pursue a common interest in literature.
2. Expansion: More members attach themselves to the original group.
• Others who share the literary interest join the club when they observe that the club has been successful for the existing members.
3. Contraction: Membership declines
• Members leave and others do not join when it is seen that the organization of the club is poor, and does not function well.
4. Persistence: Individuals join and leave
• Some move away and others join, but the function of the club is unchanged.
5. Transformation: The nature of the group alters
• As membership changes the function of the club also drifts, becoming a political lobby group.
6. Dissolution: The group ceases to operate or to exist as a group.
• All the once-members join other clubs which suit them better and no others join, or the book club simply stops meeting through lack of interest.
7. Division: The group divides into separate groups.
• Those who are interested in genre fiction form a separate book club more in keeping with their interests. The two clubs have little to do with each other.
8. Subdivision: Subgroups form within the group.
• Those who prefer one literary style hold their own meetings, but without resigning or ceasing to participate in the general club activities.

The following are interactions that may be observed between groups

1. Absorption: A group attracts the membership of another group. (Set union.)
• The book club joins a larger literary society
2. Insertion: A group provides members to another group. (Set intersection.)
• Members of the university Trotskyist club infiltrate the book club (with a view to eventually taking it over and altering its function!)
3. Disruption: A group interrupts the functioning of another group
• The book club divides along political lines and it’s difficult to get any reading done with all the infighting happening. The Trotskyists insist that all book discussions are political meetings.
4. Concentration:A group causes the intensification of the membership relationships in another group
• The film club begins a campaign against the ‘old-fashioned’ book club, possibly threatening its unhampered continued operation. The book club members are motivated to defend their interests.

Functions and variables, V say, that are estimated by x or are otherwise subjective wrt x shall be denoted E(x)[ V ].

• The point of including x in that notation is that later we will want to be able to account for subjective judgements by x of subjective judgements by y, etc.
• Until that complexity is introduced we shall simply write E[ V ]

The partial satisfaction functions may be imperfectly known or imperfectly applied. Instead of the function S(c, j, x) we need to apply the function E[ S(c, j, x) ] which is a function that returns the Expected Partial Satisfaction of interest j for agent x in context c.

• We note that the expected partial satisfaction function may have little to no relationship to the partial satisfaction function.

We do assume that the Expected Total Satisfaction function (E[ TS ]) is unchanged in form (modulo the partial satisfactions) from TS.

• The weight functions are operationally determined, so they are not imperfectly known: they are essentially subjective. (I expect controversy on that point from champions of false consciousness arguments.)
• It is unlikely that the agent considers all the interests that he may have. Let the salient interests be denoted E[ I(x) ]
• We can restrict the E[ TS ] sum to just the psychologically salient interests.
  

  E[ TS(c, I(x), x) ] = ∑_j∈E[I(x)]w_j(x)E[ S(C(b, x) , j, x) ]

For each behaviour the agent will need to consider a range of Subjectively Possible Outcomes, rather than the single outcome that is actually determined by the laws of nature: let this be noted as E[ C(b, x) ] = {c₁, …, c_m}

• Each subjectively possible outcome, c, has an associated Subjective Estimate of Probability, E[ P(c) ], where ∑_{c∈E[C(b, x)]} E[ P(c) ] = 1
• To determine the satisfaction potential of an action b the agent x will consider the likely satisfactions to be had from each subjectively possible outcome of the action and weight it by the subjective estimate of probability of that outcome. Thus, for b ∈ B(x), j ∈ I(x),
  

  E[ S(C(b, x), j, x) ] = ∑_{c∈E[C(b,x)]} E[ P(c) ]E[ S(c, j, x) ]

• We thus require the further modification of the expected total satisfaction function for the action b of x:
  

  E[ TS(C(b, x), I(x), x) ] = ∑_j∈E[I(x)]w_j(x)E[ S(C(b, x) , j, x) ]
  = ∑_j∈E[I(x)]w_j(x)∑_{c∈E[C(b, x)]} E[ P(c) ]E[ S(c, j, x) ]

It is certain that the agent will not consider all possible behaviours. The set of Subjectively Possible Behaviours that the agent considers live options will be denoted E[ B(x) ]

Granted these forms of limited rationality, the behaviour produced by the agent will maximize the expected total satisfaction of all subjectively possible behaviours, yielding an output behaviour b_out, such that:

E[ TS(C(bout, x), I(x), x) ] = max{E[ TS(C(b, x), I (x), x) ]: b ∈ E[ B(x) ]}
= max{∑_j∈E[I(x)]w_j(x)∑_{c∈E[C(b, x)]} E[ P(c) ]E[ S(c, j, x) ]: b ∈ E[ B(x) ]}

Preliminaries

We have already defined I(x) = {i₁, …, i_n} to be the Interests of the individual x.
We have noted that there are scalars to account for the Weight of Interest for each of x’s interests:
W(x) = {W_j(x): j ∈ I(x)}
At any time there are a range of possible behaviours of x:
B(x = {b₁, …, b_m}
Each possible behavior b of x would result in a different social situation for x, which we call the Outcome of behaviour b of x. We write this as

C(b, x) = c,

• c is the Context of x. It is the set of relevant parameters describing the social context of x. The relevant parameters are only those that feature in the satisfaction function S, and this will be a very restricted set of all possible parameters.
• The same notation can be used unambiguously to refer to the outcomes of a range of x’s possible behaviours; thus:
  

  C(B(x), x) = {C(b₁, x), …, C(b_m, x)} = {c₁, …, c_m}

The different possible outcomes of x’s behaviour are responsible for differential advancement of x’s interests.

• They indicate/consist of changes to the relevant parameters of the satisfaction function described above.
• To record the fact that the satisfaction function is dependent upon the outcomes of particular behaviours of x, we include those outcomes as one of the arguments.

The outcomes of the behaviour b of x, yield the following satisfactions:

{S(C(b, x), j, x): b ∈ B(x), j ∈ I(x)}, or
{S(c, j, x): c ∈ C(B(x), x), j ∈ I(x)}

The perfectly informed actor will attempt to maximize the total satisfaction function by producing the specific behaviour b_out, such that:

TS(C(b_out, x), I(x), x) = max{TS(C(b, x), I(x), x): b ∈ B(x)}
= max{∑_j∈I(x)w_j(x)S(C(b, x), j, x): b ∈ B(x)}