Steve Watson

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) ]}