{"id":62,"date":"2012-06-15T10:33:18","date_gmt":"2012-06-15T00:33:18","guid":{"rendered":"http:\/\/stevewatson.info\/blog\/2012\/06\/15\/modelling-a-social-agent-part-2-imperfect-rationality\/"},"modified":"2012-06-15T11:01:24","modified_gmt":"2012-06-15T01:01:24","slug":"modelling-a-social-agent-part-2-imperfect-rationality","status":"publish","type":"post","link":"https:\/\/stevewatson.info\/blog\/2012\/06\/15\/modelling-a-social-agent-part-2-imperfect-rationality\/","title":{"rendered":"Modelling a Social Agent: Part 2 (Imperfect Rationality)"},"content":{"rendered":"<p><font color=\"#000000\">The perfectly informed and perfectly rational agent (assumed <a href=\"http:\/\/stevewatson.info\/blog\/2012\/06\/14\/modelling-a-social-agent\/\">here<\/a>) is acceptable for a first approximation of agency; however, it is reasonably easy to modify the formulae above to take account of various forms of imperfection.<\/p>\n<p>Functions and variables, V say, that are estimated by x or are otherwise subjective wrt x shall be denoted <b>E<\/b>(x)[ V ].<\/p>\n<ul>\n<li>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. <\/li>\n<li>Until that complexity is introduced we shall simply write <b>E<\/b>[ V ]<\/li>\n<\/ul>\n<p>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 <b>E<\/b>[ S(c, j, x) ] which is a function that returns the <u>Expected Partial Satisfaction<\/u> of interest j for agent x in context c.<\/p>\n<ul>\n<li>We note that the expected partial satisfaction function may have little to no relationship to the partial satisfaction function.<\/li>\n<\/ul>\n<p>We do assume that the <u>Expected Total Satisfaction<\/u> function (<b>E<\/b>[ TS ]) is unchanged in form (modulo the partial satisfactions) from TS.<\/p>\n<ul>\n<li>The weight functions are operationally determined, so they are not imperfectly known: they are <i>essentially<\/i> subjective. (I expect controversy on that point from champions of false consciousness arguments.)<\/li>\n<li>It is unlikely that the agent considers all the interests that he may have. Let the salient interests be denoted <b>E<\/b>[ I(x) ]<\/li>\n<li>We can restrict the <b>E<\/b>[ TS ] sum to just the psychologically salient interests.<br \/>\n<blockquote><p>\n\t\t<b>E<\/b>[ TS(c, I(x), x) ] = &sum;<sub>j&isin;<b>E<\/b>[I(x)]<\/sub>w<sub>j<\/sub>(x)<b>E<\/b>[ S(C(b, x) , j, x) ]\n\t<\/p><\/blockquote>\n<\/li>\n<\/ul>\n<p>For each behaviour the agent will need to consider a range of <u>Subjectively Possible Outcomes<\/u>, rather than the single outcome that is actually determined by the laws of nature: let this be noted as <b>E<\/b>[ C(b, x) ] = {c<sub>1<\/sub>, \u2026, c<sub>m<\/sub>}<\/p>\n<ul>\n<li>Each subjectively possible outcome, c, has an associated <u>Subjective Estimate of Probability<\/u>, <b>E<\/b>[ P(c) ], where &sum;<sub>c&isin;<b>E<\/b>[C(b, x)]<\/sub> <b>E<\/b>[ P(c) ] = 1<\/li>\n<li>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 &isin; B(x), j &isin; I(x),<br \/>\n<blockquote><p>\n\t\t<b>E<\/b>[ S(C(b, x), j, x) ] = &sum;<sub>c&isin;<b>E<\/b>[C(b,x)]<\/sub> <b>E<\/b>[ P(c) ]<b>E<\/b>[ S(c, j, x) ]\n\t<\/p><\/blockquote>\n<\/li>\n<li>We thus require the further modification of the expected total satisfaction function for the action b of x:<br \/>\n<blockquote><p>\n\t\t<b>E<\/b>[ TS(C(b, x), I(x), x) ]  \t= &sum;<sub>j&isin;<b>E<\/b>[I(x)]<\/sub>w<sub>j<\/sub>(x)<b>E<\/b>[ S(C(b, x) , j, x) ]<br \/>\n\t\t\t\t= &sum;<sub>j&isin;<b>E<\/b>[I(x)]<\/sub>w<sub>j<\/sub>(x)&sum;<sub>c&isin;<b>E<\/b>[C(b, x)]<\/sub> <b>E<\/b>[ P(c) ]<b>E<\/b>[ S(c, j, x) ]\n\t<\/p><\/blockquote>\n<\/li>\n<\/ul>\n<p>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 <b>E<\/b>[ B(x) ]<\/p>\n<p>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<sub>out<\/sub>, such that:<\/p>\n<blockquote><p>\n\t<b>E<\/b>[ TS(C(bout, x), I(x), x) ]\t= max{<b>E<\/b>[ TS(C(b, x), I (x), x) ]: b &isin; <b>E<\/b>[ B(x) ]}<br \/>\n\t\t\t\t\t= max{&sum;<sub>j&isin;<b>E<\/b>[I(x)]<\/sub>w<sub>j<\/sub>(x)&sum;<sub>c&isin;<b>E<\/b>[C(b, x)]<\/sub> <b>E<\/b>[ P(c) ]<b>E<\/b>[ S(c, j, x) ]: b &isin; <b>E<\/b>[ B(x) ]}\n<\/p><\/blockquote>\n<p><\/font><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The perfectly informed and perfectly rational agent (assumed here) is acceptable for a first approximation of agency; however, it is reasonably easy to modify the formulae above to take account of various forms of imperfection. Functions and variables, V say, that are estimated by x or are otherwise subjective wrt x shall be denoted E(x)[ [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[28],"tags":[],"class_list":["post-62","post","type-post","status-publish","format-standard","hentry","category-sociology"],"_links":{"self":[{"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/posts\/62","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/comments?post=62"}],"version-history":[{"count":0,"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/posts\/62\/revisions"}],"wp:attachment":[{"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/media?parent=62"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/categories?post=62"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/tags?post=62"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}