Qualitative vs Quantitative Reasoning.

In order to illustrate how quantitative elements change classical analysis, we will present an example borrowed from the theory of stochastic processes (see Example 2.1 of [38]), which is related to economics and in particular to risk management.

Example 1 (Random Walk)   A company starts with initial capital of ${\it Cap}_0$, at each time step its income is ${\it In}_i$ and its outlay to meet claims is ${\it Out}_i$; the sequence of incomes and outlays are modelled by mutually independent and identically distributed variables. The fortunes of the company are modelled by a simple random walk with an absorbing barrier at 0 and jumps ${\it Step}_n = {\it In}_n -{\it Out}_n$:

$${\it Cap}_n = \left \{ \begin{array}{ll} {\it Cap}_{n-1} ... ...\it Step}_n > 0\\ 0 & \mbox{otherwise} \end{array} \right.$$

Qualitatively, we can analyse the random walk and just conclude that ${\it Cap}$ ranges over the interval $(0,\infty)$; quantitatively, we can ask the more interesting question: What is the probability of bankruptcy for a given statistical behaviour of claims and income? Obviously, one can ask similar questions also with respect to computational processes which in one way or another use limited computational resources.

Another example, which is more related to security and coordination problems, concerns the exchange of resources.

Example 2 (Resource Theft)   Consider the case of an autonomous agent trying to find the optimal provider of some service. The agent might use some service directory to achieve this task and pay some ``fee'' for obtaining directory information. Supposing that the agent has only a limited ``budget'' to spend on obtaining information one can think of a malicious service directory which fraudulently overcharges the agent such that the search for services has to terminate as there are no more ``funds'' available. This not only leads to direct damages but also to indirect ones as the result of the search might lead to choosing an inappropriate service provider. The problem is therfore to analyse the possibilities and chances to protect against such resource thefts and to develop countermeasures to prevent them.