It is worth to compare values obtained by the application of the MOSI with those expected intuitively. Let’s calculate, e. g., the value of the shared 2-4. place in a group of N = 10, then the same value for a population of N = 1000, comparing those values with those of both the preceding (1. place) and the following (5. place) position:
N = 10
N = 1000
shared 2-3-4. Place
The difference 3 – 2,40 – 2,30 what we get for the values in N = 1000 is much more moderate than the one 1 – 0,35 – 0,08 for the values in N = 10. And this is what would be expected by our intuition, the 5th place when N = 1000 being almost as distinguished a position as the shared 2-3-4th place, while when N = 10 the difference between the quite mediocre 5th place and the shared 2-3-4th one that is closer to the top must be more significant.
However, economic psychologists have known for quite a time that there may be also a divergence between what is implied by the economic rationality as calculated by a mathematical formula and the psychological intuition.
Such divergence has already been described by Bernoulli in the St. Petersburg paradox. Allais took this a step further by describing the paradox named after him. According to this, psychological intuition diverges not only from economic rationality, but also from a psychological rationality, which would mean that the divergence from a mathematically calculated result could itself be calculated mathematically.
The basis for the latter calculation would be the expectation that psychological intuition is consistent. In contrast, Allais found that the consistency assumed by Bernoulli and his followers does not exist; our intuition diverges from the rational differently in the direct vicinity of full certainty (where it prefers profit occurring at a greater level of probability even when the aggregate sum of all the positive cases is smaller) than in the domain that is far from certainty (where greater profits are preferred, despite the fact that the aggregate is decreased by the small probability of occurrence).
The measure of outstanding social identity to be discussed in this paper aims to give an approximation of such a divergence of second degree: from a degree rationally expected for a divergence from rational calculations. For this reason we attempt to trace deviant intuition with subsequent corrections.
A difficulty is, for example, that the differences resulting from the comparisons of the positions at the bottom end of a ranking within a group as calculated in the way given above is not in accord with the estimates stemming from our intuition. Namely, for such a calculation a population would be symmetrical, where differences between the rankings at the top of the scale should correspond to those at the bottom: in a group of N = 10, for instance, where the values of the first and second place equal, as we have seen it, 1 and 0,65, respectively, the values for the last and the second last place would similarly be calculated as -1 and -0,65 though for our intuition the difference between these places is smaller. In fact it is more so in a population of N = 1000 where for our intuition there is almost no difference between being placed as 999th or as 1000th.
Hence, a correction has to be done to the calculated symmetry, and the larger the population in question the most powerful must be that correction. For such a correction we divide the stigmatizing value by logN+1 (that is, by 2 if N=10, by 4 in the case N=1000 etc.). Thus, the corrected formula is:
logpa – logpb/(logN+1)
Unfortunately, this formula is more complicated than the simplified one we have employed up till now; on the other hand, it is worth looking at the following table and seeing the values obtained so far for the two populations examined.
N = 10
N = 1000
shared 1-2. place
shared 2-4. place
second last place
In the figures indicating values we may get rid of a good deal of unwieldy decimal fractions by multiplying all of them (arbitrarily but consistently) by 100: