Steve Watson

Let X be an ensemble.
We say x is a Conscious Member of X if x ∈ X & B_x[ x ∈ X ]
X is a Conscious Ensemble if (∀x ∈ X) B_x[ x ∈ X ]

• Note that it is widely accepted that groups are necessarily conscious ensembles; i.e.:
  γ(G) → (∀x ∈ G) B_x[ x ∈ G ]

The conscious memberships of an individual define its Identity wrt society. Thus for the individual x who is a conscious member of ensembles X₁, …, X_n, the identity of x is that collection, and we write:
ID(x) = {X₁, …, X_n} iff x ∈ X₁ ∩ … ∩ X_n & B_x[ x ∈ X₁ ∩ … ∩ X_n ]

Some sets of ensembles are mutually exclusive wrt membership. A collection of such ensembles δ = {X1, …, Xn}, that minimally covers another ensemble, X, is a Division of that ensemble and we write δ|X. Thus,
{X₁, …, X_n} is a division of X iff:

1. X₁, …, X_n are relatively disjoint, and
2. X ⊂ ∪_i=1,…,n X_i, and
3. For i = 1, …, n, X ∩ X_i ≠ ∅.

Where δ is a division of X and x ∈ X, ID(x|δ) is the element of δ of which x is a conscious member. Read it as the Identity of x wrt δ. Thus: For δ = {X₁, …, X_n}, δ|X, ID(x|δ) = X_j iff x ∈ X_j & B_x[ x ∈ X_j ]

• Where δ₁, …, δ_n are divisions of X, we can call A = {δ₁, …, δ_n} an Identity Analysis of X.
• Let A = {δ₁, …, δ_n} be an analysis of X. Then the identity of x wrt that analysis is:
  ID(x|A) = {ID(x|δ₁), …, ID(x|δ_n)}

This paper gives an introduction to stochastic actor-based models for dynamics of directed networks, using only a minimum of mathematics. The focus is on understanding the basic principles of the model, understanding the results, and on sensible rules for model selection.

TUTORIAL ON AGENT-BASED MODELING AND SIMULATION PART 2: HOW TO MODEL WITH AGENTS (pdf) by Charles M. Macal, Michael J. North

This tutorial describes the foundations of ABMS, identifies ABMS toolkits and development methods illustrated through a supply chain example, and provides thoughts on the appropriate contexts for ABMS versus conventional modeling techniques.

AGENT-BASED MODELS (pdf) by Nigel Gilbert

This ?rst chapter begins with a brief overview of agent-based modeling before contrasting it with other, perhaps more familiar forms of modeling and describing several examples of current agent-based modeling research.

Modeling Social Mechanisms: Mechanism-Based Explanations and Agent-Based Modeling in the Social Sciences (pdf) by Cyril Hédoin

This article evaluates the relevance of ABM to representing (social) mechanisms. It emphasizes the difficulties surrounding the representation of a particular form of social mechanisms I call “institutional mechanisms”, where the behavior of the social system and of its components (the social agents) are determined by institutional objects such as norms or conventions.

Some Links to Simulation Resources on the Web. Materials supporting sociology 242G by Robert Hanneman of the Department of Sociology at the University of California, Riverside.
From Factors to Actors: Computational Sociology and Agent-Based Modeling (pdf) by Michael W. Mac, Robert Willer

Like flocks of birds, human group processes are highly complex, nonlinear, path dependent, and self-organizing. We may be able to understand these dynamics much better not by trying to model them at the global level but instead as emergent properties of local interaction among adaptive agents who influence one another in response to the influence they receive.

Why Sociology Should Use Agent Based Modelling by Edmund Chattoe-Brown

Although Agent Based Models (hereafter ABM) are now regularly reported in sociology journals, explaining the approach, describing models and reporting results leaves little opportunity to examine wider implications of ABM for sociological practice. This article uses an established ABM (the Schelling model) for this.

From micro to macro and back again: Agent-based models for sociology (pdf) by Federico Bianchi

In this paper the importance of agent-based simulation is advocated for mechanism-based sociology [1]. The main epistemological reasons are to be found in the analogies between mechanism-based sociological theory [1.1], the study of complexity and emergent properties [1.2] and the generative explanatory power of computer simulations [1.3].

Finally, an Amazon wish: Agent-Based Computational Sociology [Hardcover] by Flaminio Squazzoni

This book looks at a new research stream that makes use of advanced computer simulation modelling techniques to spotlight agent interaction that allows us to explain the emergence of social patterns. It presents a method to pursue analytical sociology investigations that look at relevant social mechanisms in various empirical situations, such as markets, urban cities, and organisations.

A Norm is an action-directing rule, n, for which

(∃C_x,applies ⊂ C_x) (∃α_bad(x) ⊂ α(x)) (∀c_x,0 ∈ C_x,applies) (∀a_x ∈ α_bad(x)) [n & a_x |– ⊥ ]

• An equivalent condition might be written positively as
  (∃C_x,applies ⊂ C_x) (∃α_good(x) ⊂ α(x)) (∀c_x,0 ∈ C_x,applies) (∀a_x ∉ α_good(x)) [n & a_x |– ⊥ ]

• C_x,applies and C_x are sets of contexts for x. C_x is the set of all possible contexts for x.
• The condition that makes n normatively prescriptive is a logical condition: the forbidden action is logically inconsistent with the norm. Accurate logical reasoning is at least a part of the process by which x goes about determining his action in the circumstances, though we cannot depend upon it being a main part or that it will be very accurate.

An agent x is an Acceptant of the norm n iff x accepts the rule n. We will write

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

• In this case, there is a condition for accepting that B(x)[ n ] based on the notion that we can discover whether x really accepts a norm by observation of his behaviour. Thus
  B(x)[ n ] → (∃C_x,applies ⊂ C_x) (∃α_bad(x) ⊂ α(x)) (∀cx,applies ∈ C_x,applies) (∀a_x,bad ∈ α_bad(x))
  [n & a_x |– ⊥ → A(x, c_x,0) ≠ a_x]
  which is to say that, if x produces an action in a circumstance in which the norm determines that that action is impermissible, then x cannot be said to accept that norm.
• The reasons why x might fail this test are many. It might just be that x is not very clever and fails to reason logically accurately; perhaps the norm is too complex for any human; perhaps information is lacking; etc. We do not need to assume that x is actually dishonest.
• It would seem, on the face of it anyway, that it’s quite possible that ~B(x)[ n ] & B(x)[ B(x)[ n ]]; and other curiosities of epistemic logic may also apply. It will be necessary eventually to decide which axiomatization of epistemic logic is most appropriate for the operator B. We can ignore the question for now.
• The condition given for x being an acceptant of n may not adequately reflect the fuzziness of our belief states, or the degrees to which we believe something. Assuming that we wish to do so, and that this cannot be done by massaging the context variable of the action function as it applies to the conditions under which we will apply the norm, we may propose the following refinement:

  B(x)[ n ] → (∃C_x,applies ⊂ C_x) (∃α_bad(x) ⊂ α(x)) (∀cx,applies ∈ C_x,applies) (∀a_x,bad ∈ α_bad(x))
  [((n & a_x |– ⊥) &
  (&forall>a_x,alt ≠ ax)~B(x)[ (T(C(a_x,alt, c_x,0, x), I(x), x) ] >> T(C(a_x, c_x,0, x), I(x), x)) ]))
  → A(x, c_x,0) ≠ a_x]

  Which is simply stating that x will not contravene n unless he estimates that the total satisfaction to be gained by doing so is sufficiently greater than the satisfaction to gained by abiding by n.

• The degree of excess expressed by ‘>>’ is left deliberately vague – as in other uses of it. (To be set, perhaps, by empirical research.)

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 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) ]