Introduction

The presented a rather standard view of quantitative modeling. When dealing with probabilities, we have often taken for granted a frequentist perspective; our approach to statistics, especially in terms of parameter estimation, has been an orthodox one. Actually, these are not the only possible viewpoints. In fact, probability and statistics are a branch of mathematics at the boundary with philosophy of science, and as such they are not free from heated controversy. This might sound like a matter of academic debate, but it is not. The “death of probability” was invoked in the wake of the 2008 financial turmoil, when the quantitative modeling approach in finance has been blamed as one of the root causes of the disaster. Of course, truth always lies somewhere between extremes, but this is reason enough to see the need for an eye opening illustrating alternative views that have been put forward, like subjective probabilities and Bayesian statistics. A similar consideration applies to the on decision models. There, we have also followed a standard route, implicitly assuming that decisions are made by one person keeping all problem dimensions under direct control. We have hinted at some difficulties in trading off multiple and conflicting objectives, when dealing with multiobjective optimization in Section 12.3.3. However, we did not fully address the thorny issues that are raised when multiple decision makers are involved. On the one hand, they can be interested in different objectives; on the other one, they may behave without coordinating their decisions with other actors. Decisions involving not necessarily cooperative actors are the subject of game theory. Rather surprising results are obtained when multiple players interact, possibly leading to suboptimal decisions; here, we do not mean suboptimal only for a single decision maker, but, for the whole set of them. Finally, standard models assume that uncertainty is exogenous, whereas there are many practical situations in which decisions do influence uncertainty, such as big trades on thin and illiquid financial markets or inventory management decisions affecting demand. When all of the above difficulties compound, multiple decision makers can influence one another through decisions, behavior, and information flows, possibly leading to instability due to vicious feedbacks. Such mechanisms have been put forward as an explanation of some major financial crashes.

We are certainly in no position to deal with such demanding topics extensively. They all would require their own (voluminous) book and the technicalities involved are far from trivial. However, I strongly believe that there must be room for a fostering critical thinking about quantitative models. This is not to say that what we have dealt with so far is useless. On the contrary, it must be taken with a grain of salt and properly applied, keeping in mind that we could be missing quite important points, rendering our analysis irrelevant or even counterproductive. The aim is not to provide readers with working knowledge and ready-to-use methods; rather, a sequence of simple and stylized examples will hopefully stimulate curiosity and further study.

In Section 14.1 we discuss general issues concerning the difference between decision making under risk, the topic and decision making under uncertainty, which is related to a more radical view. In Section 14.2 we begin formalizing decision problems characterized by the presence of multiple noncooperative decision makers, setting the stage for the next sections, where we discuss the effects of conflicting viewpoints and introduce game theory. In Section 14.3 we illustrate the effect of misaligned incentives in a stylized example involving two decision makers in a supply chain. The two stakeholders aim at maximizing their own profit, and this results in a solution that does not maximize the overall profit of the supply chain. Such noncooperative behavior is the subject of game theory, which is the topic of Section 14.4. We broaden our view by outlining fundamental concepts about equilibrium, as well as games with sequential or simultaneous moves. Very stylized examples will illustrate the ideas, but this section is a bit more abstract than usual. Therefore, in Section 14.5, we show a more practical example related to equilibrium in traffic networks; this example, known as Braess’ paradox, shows that quite counterintuitive outcomes may result from noncooperative decision making. In Section 14.6 we discuss how the dynamic interaction among multiple actors may lead to instability and, ultimately, to disaster, by analyzing a couple of real-life financial market crashes. Finally, we are providing the reader with a scent of Bayesian statistics in Section 14.7. Bayesian learning is related to parameter estimation issues in orthodox statistics; there, the basic concept is that parameters are unknown numbers; within this alternative framework, we may cope with probabilistic knowledge about parameters, possibly subjective in nature. As a concrete example, we outline the Black–Litterman portfolio optimization model, which can be interpreted as a Bayesian approach.