We develop a purely ordinal model for aggregation functionals for lattice valued functions, comprising as special cases quantiles, the Ky Fan metric and the Sugeno integral. For modeling findings of psychological experiments like the reflection effect in decision behaviour under risk or uncertainty, we introduce reflection lattices. These are complete linear lattices endowed with an order reversing bijection like the reflection at 0 on the real interval $[-1,1]$. Mathematically we investigate the lattice of non-void intervals in a complete linear lattice, then the class of monotone interval-valued functions and their inner product.
Measuring and aggregation or integration techniques have a very long tradition. Here numbers play an important role. But how do humans perceive numbers? The numbers, say the set R of reals, support two basic structures, the algebraic structure defined by + and ×, and the ordinal structure given by ≤. There are many situations where only order is relevant, cardinals being used merely by tradition and convenience. During the last years the interest in ordinal aggregation has increased, see e.g. [4,5,6,14,19,20].
We are interested in the question if aggregation or integration can be done in purely ordinal terms and what results can be obtained. Of course many partial results are already available. Since often they are formulated in terms of numbers, we ask what can be sustained if one ignores the algebraic structure or what weaker additional structure has to be imposed on the linear ordinal scale in order to formulate some well known important issues. It turns out that enough structure is given by an order reversing bijection of the scale leaving one point fixed. Thus the scale decomposes into two symmetric parts. This can be interpreted as the first step to numbers. Since, repeating the procedure with each resulting part of the scale infinitely often, one ends up with the binary representation of the numbers in the unit interval [0, 1] ⊂ R or some superstructure.
There are several ordinal concepts for aggregating values with respect to a measure. The oldest and best known selects a certain quantile, say the median, of a sample as the aggregated value. Next Ky Fan’s [8] metric on the space L 0 (µ) of µ-measurable functions is essentially ordinal. More recently and independently Sugeno [23] developed his integral which employs the same idea as Ky Fan. One aim of the present paper is to develop a common purely ordinal model for these three examples. This is done with a complete linear lattice M as scale, comprising the classical case M = [0, 1] ⊂ R. For the Sugeno integral the scales used for functions and the measure are identical. But in general we allow separate scales for the functions and the measure and the two scales are related by a commensurability application, as we call it.
The structure of a linear lattice seems not to be sufficient to model elementary human behaviour in the presence of risk or uncertainty. There is some empirical psychological evidence (cf. reflection effect, inverse S-shaped decision weights, etc.) that in certain decision situations humans have a point O of reference (often the status quo) on their scale which allows to distinguish good and bad or gains and losses, i.e. values above, respectively below, the reference point [17,18]. Then the attainable gains and the attainable losses are aggregated separately and finally these two aggregated values are compared to reach the final decision. In the cardinal models this behaviour can be modeled with the symmetric Choquet integral. We define the symmetric Sugeno integral in order to model the essentials of this behaviour in purely ordinal terms.
We also define the analogue of the asymmetric Choquet integral in our context. This can be done in introducing two commensurability functions, one for the positive part of the scale, the other one for the negative part.
Finally we comment on the new technical tools and the organisation of the paper. In Section 2 we model the scale with neutral reference point as a complete linear reflection lattice R corresponding to [-1, 1] ⊂ R. On R we use the binary relations from [14] to get operations corresponding to addition and multiplication in R.
In Section 3 we develop a theory of increasing interval valued functions and their inverse. Introducing these tools is motivated as follows. Mathematically, the idea of Fan and Sugeno for the aggregated value of a random variable is very simple, just take the argument at which the decreasing distribution function intersects a preselected increasing function, the identity function in their case. As already the quantiles show, the aggregated values are intervals rather than points on the linear scale L. So we look for a suitable ordering on the family I L of nonempty intervals in L. The ordering which had been introduced by Topkis (see [24]) on the family of nonempty sublattices of an arbitrary lattice turns out to be the right one to handle monotonic functions (Proposition 3.1). This ordering, restricted to I L , is only a partial one, but we show that I L is a completely distributive lattice in our situation. This structure is needed for an ordinal analogue of the inner product of vectors, which is introduced in Section 4. This product will, in Section 7, formalise the idea of Fan and Sugeno in our general context. Still in Section 4 the product provides a convenient tool to fill the gaps in the domain of the inverse of a monotone function, we call this saturation.
In close analogy to probability theory we introduce lattice valued measures in Section 5
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