Distributing the votes using an Italian Flag
As we said on the previous page using an Italian Flag quantitatively is similar to using standard probability but is just a little more complicated because we include the white interval of incompleteness.
In standard probability theory on the previous page we had just 4 areas over which to distribute the votes for Fred and Vera i.e. (A & B), (A & notB), (not A & B), (notA & notB). Now we have 9 areas as shown in the table below. Now we are allowing the voters to vote for (green) or against (red) or ‘don’t know’ (white).
The top row and the left column are the headers for the results of the voting. The areas to which the votes must be allocated according to the dependencies between them are the 9 areas labelled gg to rr.
Thus the area gg contains the number or proportion of green votes for A & B. The area gw contains the green votes for A & uncertain white votes for B. The other 7 areas are defined likewise.
The table is equivalent to the Venn diagram below again with areas proportional to the no. of votes.
In practice it becomes very difficult to allocate votes to each area – there are so many combinations to consider. So it is much more convenient to choose a dependency between A and B, to choose sufficiency and necessity values for a higher level proposition R as before and then to estimate the bounds on the green and red areas for R.
A full explanation of the mathematics behind the logical combination of Italian Flags is described by Blockley (2007). A The formulas used here were first developed by Marashi in his 2006 PhD thesis at the University of Bristol.
A short explanation is as follows.
Let’s capture the IF for A using interval bounds [g, (1-r)].
So as before g = the green part of the flag for A and r = the red part but now the white part w is not zero but = (1- g- r)
As before the Italian Flag A is evidence for R and we want to calculate the IF for R.
We judge that the bounds on the sufficiency of A for R i.e. p(R/A)) are [sl, su]. Likewise the bounds on the necessity of A for R i.e. p(notR/notA) are [nl, nu].
As before we use the total probability theorem which we stated as
p(R) = suff*g + (1 – nec)*(1 – r)
but now we have to find the bounds on the values of g, r and w for R.
The strategy is this: we find the largest and smallest amounts of evidence that could go from A to R, first for the green and then for the red, and then we allocate evidence to white in a least biased way. We do the same for the evidence from B to R and combine the answers.
So first we calculate the upper bound on the positive evidence g for R. The largest g for R can be is su*g. The smallest the red can be is nl*r and so the corresponding largest green is (1-nl)*r. The least biased allocation to white is the product of (1-su) and nl subtracted from 1.
Next we calculate the upper bound on notR and then subtract it from 1. The reason we do this is because we want to capture all of the negative combinations of sets.
The largest the red can be for notR is nu*r. The smallest the green can be is sl*g and so the the largest the corresponding red can be is (1-sl)*g. The least biased allocation to white is the product of (1-nl) and sl subtracted from 1.
Thus the green part of the IF for R is gR = su*g + (1-nl*r + w*f1
The red part of the IF for R is rR = (1-sl)*g + nuR + w*f2
where f1 = (1-(1-su)*nl) and f2 = 1-(1-nu)*sl
and so the bounds on R from the evidence A are [gR, (1-rR)]
Next we find the equivalent bounds on the evidence B for R.
Finally we combine the two results as the union based on the dependency between A and B. For more than two pieces of evidence we perform a pairwise set of comparisons and combine those.
The first software to include the quantitative use of Italian Flags is called Perimeta. It has been used in a a number of research projects but is not now available commercially as far as we are aware. Software called Tesla has been developed without any input from us apart from our published papers. However the software relies on a very early version of the thinking behind the Italian Flags and the Tesla handbook refers to nothing after 2004.
In order to help anyone interested in their learning journey towards using the Italian Flag quantitively a demo page has been developed by David Blockley to ‘play’ with the variables and understand better how the methodology might be developed for use in practice.
However you are strongly encouraged to read the guidance available here before using the tool.