Partial ordering of OMPs

Based on the combinations of OMPs, a partial order $\preccurlyeq$ over the OMPs can be computed by exploiting the constituent BPQs of the OMPs considered. This partial order implies that a comparison of any pair of OMPs either returns equal preference ($=$), smaller preference ($\prec$), greater preference ($\succ$) or incomparable preference ($?$). This calculus is developed assuming the following:

• Prioritisation: A combination of BPQs is never an order of magnitude greater than its constituent BPQs. That is, given the set of BPQs belonging to the same order of magnitude $\{b_1,b_2,\ldots,b_n\}\subseteq O_1$ and a BPQ $b\in O_2$ belonging to a higher order of magnitude, i.e. $O_1\ll O_2$, then

  $$b_1\oplus b_2\oplus\ldots\oplus b_n\prec b$$

  
  
  With respect to the ongoing example, this means that any BPQ taken from the order of magnitude $O_1$ is preferred over any combination of BPQs taken from $O_2$. In other words, the choice of a model to describe a host-parasitoid phenomenon is considered more important than the choice of population growth model (see Figure 1).

  Prioritisation also means that distinctions at higher orders of magnitude are considered to be more significant than those at lower orders of magnitude. Consider a number of BPQs $b_1,\ldots, b_{m-1}, b_m,\ldots,b_n$ taken from one order of magnitude $O_1$ and a pair of BPQs $\{b,b'\}$ taken from an order of magnitude that is higher than $O_1$. If $b<b'$, then (irrespective of the ordering of the BPQs taken from $O_1$)

  $$b_1\oplus\ldots\oplus b_{m-1}\oplus b\prec b_m\oplus\ldots\oplus b_n\oplus b'$$

  
  
  
  
  
  
• Strict monotonicity: Even though distinctions at higher orders of magnitude are more significant, distinctions at lower orders of magnitude are not negligible. That is, given an OMP $P$ and two BPQs $b_1$ and $b_2$ taken from the same order of magnitude with $b_1<b_2$, then (irrespective of the orders of magnitude of the BPQs that constitute $P$)

  $$b_1\oplus P\prec b_2\oplus P$$

  
  
  For instance, the preference ordering depicted in Figure 1 shows that a scenario model with a Roger's host-parasitoid model and two logistic predation models is preferred over one with a Roger's host-parasitoid model and two exponential predation models:

  $$b_{11}\oplus b_{22}\oplus b_{22}\prec b_{11}\oplus b_{21}\oplus b_{21}$$

  
  

  Note that this is a departure from conventional order-of-magnitude reasoning. If the OMPs associated with two (partial) outcomes contain equal BPQs at a higher order of magnitude, it is usually desirable to compare both solutions further in terms of the (less important) constituent BPQs at lower orders of magnitude, as the example illustrated. However, conventional order-of-magnitude reasoning techniques can not handle this.

  
  
  
• Partial ordering maintenance: Conventional order-of-magnitude reasoning is motivated by the need for abstract descriptions of real-world behaviour, whereas the OMP calculus is motivated by incomplete knowledge for decision making. As opposed to conventional order-of-magnitude reasoning and real numbers, OMPs are not necessarily totally ordered. This implies that, when the user states, for example, that $b_1<b_2<b$ and that $b_3<b_4<b$, the explicit absence of ordering information between the BPQs in $\{b_1,b_2\}$ and those in $\{b_3,b_4\}$ means that the user is unable to compare them (e.g. because they are entirely different things). Consequently, $b_1\oplus b_2$ would be deemed incomparable to $b_3\oplus b_4$ (i.e. $b_1\oplus b_2 ? b_3\oplus b_4$), rather than roughly equivalent.

From the above, it can be derived that given two OMPs $P_1$ and $P_2$ and an order of magnitude $O$, $P_1$ is less or equally preferred to $P_2$ with respect to the order of magnitude $O$ (denoted $P_1\preccurlyeq_O P_2$) provided that

$$\forall b_i\in O, \big(f_{P_1}(b_i)+\sum_{b_j\in O, b_i<b_j}f_{P_1}(b_j)\big)\leq\big(f_{P_2}(b_i)+\sum_{b_j\in O, b_i<b_j}f_{P_2}(b_j)\big)$$

Thus, comparing two OMPs within an order of magnitude can yield four possible results:

• $P_1$ is less preferred than $P_2$ with respect to $O$ ( $P_1\prec_O P_2$) iff $(P_1\preccurlyeq_O P_2)\wedge\neg(P_2\preccurlyeq P_1)$,
• $P_1$ is more preferred than $P_2$ with respect to $O$ ( $P_1\succ_O P_2$) iff $\neg(P_1\preccurlyeq_O P_2)\wedge(P_2\preccurlyeq P_1)$,
• $P_1$ is equally preferred than $P_2$ with respect to $O$ ( $P_1=_O P_2$) iff $(P_1\preccurlyeq_O P_2)\wedge(P_2\preccurlyeq P_1)$, and
• $P_1$ is incomparable to $P_2$ with respect to $O$ ( $P_1?_O P_2$) iff $\neg(P_1\preccurlyeq_O P_2)\wedge\neg(P_2\preccurlyeq P_1)$.

In the ongoing example of Figure 1, for instance, the preference of a scenario model with a Roger's host-parasitoid model and a Holling predation model is $P_1=b_{11}\oplus b_{13}$ and the preference of a scenario model with a Roger's host-parasitoid model and a Lotka-Volterra predation model is $P_2=b_{11}\oplus b_{15}$. The latter model is less than or equally preferred to the former within the ``host-parasitoid'' order of magnitude ($O_1$), i.e. $P_2\preccurlyeq_{O_1} P_1$, because

$$f_{P_2}(b_{11})=1 \leq 1=f_{P_1}(b_{11}),$$

$$f_{P_2}(b_{11})\oplus f_{P_2}(b_{12})=1 \leq 1=f_{P_1}(b_{11})\oplus f_{P_1}(b_{12}),$$

$$f_{P_2}(b_{11})\oplus f_{P_2}(b_{13})=1 \leq 2=f_{P_1}(b_{11})\oplus f_{P_1}(b_{13}),$$

$$f_{P_2}(b_{11})\oplus f_{P_2}(b_{12})\oplus f_{P_2}(b_{14})=1 \leq 1=f_{P_1}(b_{11})\oplus f_{P_1}(b_{12})\oplus f_{P_1}(b_{14}),$$

$$f_{P_2}(b_{11})\oplus f_{P_2}(b_{12})\oplus f_{P_2}(b_{13})\oplus f_{P_2}(b_{14})=2 \leq 2=f_{P_1}(b_{11})\oplus f_{P_1}(b_{12})\oplus f_{P_1}(b_{13})\oplus f_{P_1}(b_{14}).$$

Similarly, it can be established that the reverse, i.e. $P_1\preccurlyeq_{O_1} P_2$, is not true. Therefore, the latter scenario model is less preferred than the former within $O_1$, i.e. $P_2\prec_{O_1} P_1$.

The above result can be further generalised such that given two OMPs $P_1$ and $P_2$, $P_1$ is less or equally preferred to $P_2$ (denoted $P_1\preccurlyeq P_2$) if

$$\forall O_i\in\mathbf{O},(P_1\preccurlyeq_{O_i} P_2)\vee(\exists O_j\in\mathbf{O}, O_i\ll O_j\wedge P_1\prec_{O_j} P_2)$$

More generally, the relations $\prec$, $\succ$, $=$ and $?$ can be derived in the same manner as with the relation $\preccurlyeq$ where $\prec_O$, $\succ_O$, $=_O$ and $?_O$ with $\preccurlyeq_O$.

To illustrate the utility of such orderings, consider the scenario of one predator population that feeds on two prey populations while the two prey populations compete for scarce resources. The following are two plausible scenario models for this scenario:

• Model 1 contains two Holling predation models and three logistic population growth models, and has preference $P_1=b_{13}\oplus b_{13}\oplus b_{21}\oplus b_{21}\oplus b_{21}$.
• Model 2 contains one competition model, two Holling predation models, two logistic population growth models and an exponential population growth model, and has preference $P_2=b_{13}\oplus b_{13}\oplus b_{21}\oplus b_{21}\oplus b_{22}\oplus b_{31}$.

As demonstrated earlier, it can be shown that $P_1=_{O_1}P_2$, $P_1\succ_{O_2}P_2$, and $P_1\prec_{O_3}P_2$. From these relations it follows that $P_1\preccurlyeq P_2$ because

• for $O_1$: $P_1\preccurlyeq_{O_1} P_2$ since $P_1=_{O_1}P_2$,
• for $O_2$: there exists an order of magnitude $O_3$ where $O_3\gg O_2$ and $P_1\prec_{O_3}P_2$,
• for $O_3$: $P_1\preccurlyeq_{O_3}P_2$ since $P_1\prec_{O_3}P_2$.

As the reverse is not true, it can be concluded that scenario model 2 is preferred over scenario model 1.