Category: Multiple Decision Makers, Subjective Probability, and Other Wild Beasts

We considered portfolio optimization in Example 12.5 and in Section 13.2.2. For the sake of convenience, let us reconsider the problem here. We must allocate our wealth among n risky assets and a risk-free one. The returns of the risky assets are a vector of random variables with expected value μ and covariance matrix Σ; let rf be the return of the risk-free asset. Let w0 be…

Consider the problem of estimating a parameter θ, characterizing the probability distribution of a random available X. We have some prior information about θ, that we would like to express in a sensible way. We might assume that the unknown parameter lies anywhere in the unit interval [0, 1], or we might assume that it is close to…

We took a rather standard view. On the one hand, we have introduced events and probabilities according to an axiomatic approach. On the other hand, when dealing with inferential statistics, we have followed the orthodox approach: Parameters are unknown numbers, that we try to estimate by squeezing information out of a random sample, in the…

Game theory in its simplest form does not consider dynamics, as it revolves around a static equilibrium concept: It posits a situation such that no player has an incentive to deviate. But how is that equilibrium reached dynamically? And what about the disorderly interaction of many stakeholders, maybe stockholders in financial markets? Addressing such issues is beyond…

The result of the collective interaction of noncooperative players may be occasionally quite surprising. We illustrate here a little example of the Braess’ paradox for traffic networks.18 Imagine a traffic network consisting of links such as road segments, bridges, and whatnot. Most of us had some pretty bad experiences with traffic jams. Intuition would suggest that…

In this section we consider Nash equilibrium for the case in which a continuum of infinite actions is available to each player. To be specific, we analyze the behavior of two firms competing with each other in terms of quantities produced. Both firms would like to maximize their profit, but they influence each other since…

The concepts that we have used so far make sense, but they are a bit too restrictive and limit the set of games for which we may make reasonable predictions. A better approach, in a sense that we should clarify, is Nash equilibrium. Before formalizing the concept, imagine a game in which there is one…

Sometimes, it is fairly easy to argue which outcome is to be expected. If we consider the strategies for firm A in Table 14.1, we see that: So, whatever firm B plays, firm A is better off by playing low. The symmetry of the game implies that the same consideration applies to firm B, which will also…

The standard example to illustrate the normal form representation of a simple game is the prisoner’s dilemma, which is arguably the prototypical example of a two-player game. The prisoner’s dilemma has been phrased in many different ways;12 in the next example we use what is closest to a business management setting. Example 14.4 (Prisoner’s dilemma) Consider two…

In the previous section we have considered a case in which two stakeholders, a producer and a distributor, make their decisions in a specific order. The producer (leader) determines product quality, as well as the probability distribution of demand as a consequence; the distributor (follower) chooses the order quantity. In other cases, however, decisions are…