Portfolio Selection

This chapter is devoted to the application of the ideas described in Chapter 9 to the problem of sequential investment. Imagine a market of m assets (stocks) in which, in each trading period (day), the price of a stock may vary in an arbitrary way. An investor operates on this market for n days with the goal of maximizing his final wealth. At the beginning of each day, on the basis of the past behavior of the market, the investor redistributes his current wealth among the m assets. Following the approach developed in the previous chapters, we avoid any statistical assumptions about the nature of the stock market, and evaluate the investor's wealth relative to the performance achieved by the best strategy in a class of reference investment strategie (the “experts”).

In the idealized stock market we assume that there are no transaction costs and the amount of each stock that can be bought at any trading period is only limited by the investor's wealth at that time. Similarly, the investor can sell any quantity of the stocks he possesses at any time at the actual market price.

The model may be formalized as follows. A market vectorx = (x1, …, xm) for m assets is a vector of nonnegative real numbers representing price relatives for a given trading period. In other words, the quantity xi ≥ 0 denotes the ratio of closing to opening price of the ith asset for that period.

Games and Equilibria

The prediction problems studied in previous chapters have been often represented as repeated games between a forecaster and the environment. Our use of a game-theoretic formalism is not accidental: there exists an intimate connection between sequential prediction and some fundamental problems belonging to the theory of learning in games. We devote this chapter to the exploration of some of these connections.

Rather than giving an exhaustive account of the area of learning in games, we only focus on “regret-based” learning procedures (i.e., situations in which the players of the game base their strategies only on regrets they have suffered in the past) and our fundamental concern is whether such procedures lead to equilibria. We also limit our attention to finite strategic or normal form games.

In this introductory section we present the basic definitions of the games we consider describe some notions of equilibria, and introduce the model of playing repeated games that we investigate in the subsequent sections of this chapter.

K-Person Normal Form Games

A (finite) K-person game given in its strategic (or normal) form is defined as follows. Player k(k = 1, …, K) has Nk possible actions (or pure strategies) to choose from, where Nk is a positive integer.

Prediction of individual sequences, the main theme of this book, has been studied in various fields, such as statistical decision theory, information theory, game theory, machine learning, and mathematical finance. Early appearances of the problem go back as far as the 1950s, with the pioneering work of Blackwell, Hannan, and others. Even though the focus of investigation varied across these fields, some of the main principles have been discovered independently. Evolution of ideas remained parallel for quite some time. As each community developed its own vocabulary, communication became difficult. By the mid-1990s, however, it became clear that researchers of the different fields had a lot to teach each other.

When we decided to write this book, in 2001, one of our main purposes was to investigate these connections and help ideas circulate more fluently. In retrospect, we now realize that the interplay among these many fields is far richer than we suspected. For this reason, exploring this beautiful subject during the preparation of the book became a most exciting experience – we really hope to succeed in transmitting this excitement to the reader. Today, several hundreds of pages later, we still feel there remains a lot to discover. This book just shows the first steps of some largely unexplored paths. We invite the reader to join us in finding out where these paths lead and where they connect.

Prediction

Prediction, as we understand it in this book, is concerned with guessing the short-term evolution of certain phenomena. Examples of prediction problems are forecasting tomorrow's temperature at a given location or guessing which asset will achieve the best performance over the next month. Despite their different nature, these tasks look similar at an abstract level: one must predict the next element of an unknown sequence given some knowledge about the past elements and possibly other available information. In this book we develop a formal theory of this general prediction problem. To properly address the diversity of potential applications without sacrificing mathematical rigor, the theory will be able to accommodate different formalizations of the entities involved in a forecasting task, such as the elements forming the sequence, the criterion used to measure the quality of a forecast, the protocol specifying how the predictor receives feedback about the sequence, and any possible side information provided to the predictor.

In the most basic version of the sequential prediction problem, the predictor – or forecaster – observes one after another the elements of a sequence y1, y2,… of symbols. At each time t = 1, 2,…, before the tth symbol of the sequence is revealed, the forecaster guesses its value yt on the basis of the previous t – 1 observations.

In the classical statistical theory of sequential prediction, the sequence of elements, which we call outcomes, is assumed to be a realization of a stationary stochastic process.

Introduction

This chapter investigates several variants of the randomized prediction problem. These variants are more difficult than the basic version treated in Chapter 4 in that the forecaster has only limited information about the past outcomes of the sequence to be predicted. In particular, after making a prediction, the true outcome yt is not necessarily revealed to the forecaster, and a whole range of different problems can be defined depending on the type of information the forecaster has access to.

One of the main messages of this chapter is that Hannan consistency may be achieved under significantly more restricted circumstances, a surprising fact in some of the cases described later. The price paid for not having full information about the outcomes is reflected in the deterioration of the rate at which the per-round regret approaches 0.

In the first variant, investigated in Sections 6.2 and 6.3, only a small fraction of the outcomes is made available to the forecaster. Surprisingly, even in this “label efficient” version of the prediction game, Hannan consistency may be achieved under the only assumption that the number of outcomes revealed after n prediction rounds grows faster than log(n)log log(n).

Section 6.4 formulates prediction problems with limited information in a general framework. In the setup of prediction under partial monitoring, the forecaster, instead of his own loss, only receives a feedback signal. The difficulty of the problem depends on the relationship between losses and feedbacks.