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future_fabulators:formalised_decision_making [2013-10-17 19:20] – timbofuture_fabulators:formalised_decision_making [2013-10-17 19:44] – timbo
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 Some of the techniques that we have seen used that will be relevant: Some of the techniques that we have seen used that will be relevant:
- * dots: each participant get a number of dots to allocate to factors. More dots should indicate more of whatever it is that participants seek, whether that be relevance, importance, etc. +  * dots: each participant get a number of dots to allocate to factors. More dots should indicate more of whatever it is that participants seek, whether that be relevance, importance, etc. 
- * numbers: giving factors a numerical value, whether from 1 (uninteresting) to 5 (thrilling) or with a middle level from which two extremes vary i.e. plus and minus points, or having e.g. 5 as neutral, 10 as love and 0 as hate. +  * numbers: giving factors a numerical value, whether from 1 (uninteresting) to 5 (thrilling) or with a middle level from which two extremes vary i.e. plus and minus points, or having e.g. 5 as neutral, 10 as love and 0 as hate. 
- * ordering: selecting the factors in a list from highest to lowest in the evaluation.+  * ordering: selecting the factors in a list from highest to lowest in the evaluation.
  
 ===Partially ordered sets and Coverings== ===Partially ordered sets and Coverings==
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 A partial order, on the other hand, does not have the requirement that every pair of elements has an ordering. We require only the following axioms (note that we write A<B for "A is less than or equal to B" because we cannot find the symbol right now): A partial order, on the other hand, does not have the requirement that every pair of elements has an ordering. We require only the following axioms (note that we write A<B for "A is less than or equal to B" because we cannot find the symbol right now):
- * Transitivity: if A<B and B<C then A<C. If love is more important than sex and sex is more important than food, then love is more important than food.  +  * Transitivity: if A<B and B<C then A<C. If love is more important than sex and sex is more important than food, then love is more important than food.  
- * Antisymmetry: if A<B and B<A then A=B: If time is more important than money and money is more important than time, then time is money. +  * Antisymmetry: if A<B and B<A then A=B: If time is more important than money and money is more important than time, then time is money. 
- * Reflexivity: A <A, a sunset is at least as good as a sunset.+  * Reflexivity: A <A, a sunset is at least as good as a sunset.
  
 Given two total orderings, we can define a new partial order from them by saying that A>B if A is above B in **both** orders. This is called the **product** of the two orders. Given two total orderings, we can define a new partial order from them by saying that A>B if A is above B in **both** orders. This is called the **product** of the two orders.
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 So a possible selection technique is to take a set of factors S so that the number of factors __not__ in the downset of S is as small as possible.  So a possible selection technique is to take a set of factors S so that the number of factors __not__ in the downset of S is as small as possible. 
  
-=Some calculations=+==Some calculations==
 In Scenario planning, we want to select 2 (or perhaps 3) factors that are highest in our ordering of importance and uncertainty. So we have two orders, giving a product partial order, from which we want to select 2 factors. A mathematical question arises: how often will we have the situation that this is (not) possible? In Scenario planning, we want to select 2 (or perhaps 3) factors that are highest in our ordering of importance and uncertainty. So we have two orders, giving a product partial order, from which we want to select 2 factors. A mathematical question arises: how often will we have the situation that this is (not) possible?
  
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 To Do: work out the corect formula for counting 3-selections. These would be xxxnyyyYzzzZ where Y is the largest factor not in xxxn and Z is the largest factor not in xxxnyyyY. My current conjecture is: To Do: work out the corect formula for counting 3-selections. These would be xxxnyyyYzzzZ where Y is the largest factor not in xxxn and Z is the largest factor not in xxxnyyyY. My current conjecture is:
  
-(n-1)! sum(i=1,n-1)((1/i) sum(j=1,i-1)(1/j))+(n-1)! sum(i=1,n-1) ( (1/i) sum(j=1,i-1) (1/j))
  
 but this needs some work to check it. but this needs some work to check it.
  
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