Introduction

When star players are free agents, several teams may bid for their services in a kind of auction. Barry Zito is a good example of this. Several Major League Baseball (MLB) teams were interested in signing Zito when he became a free agent after the 2006 season. In the end, the San Francisco Giants and the New York Mets battled down to the wire over Zito. The Giants ultimately won the battle and Zito's services by outbidding everyone else. Zito, of course, was the real winner: $126 million over seven years. In this instance, Zito's salary was determined through competitive bidding. In other cases, however, salaries are determined through bilateral bargaining between one team and one player (or his agent). Top NFL draft choices for example, bargain over the terms of their initial contracts with the teams that selected them in the draft. Some players who are currently under contract bargain over multiyear contract extensions. At the end of the 2010 season, for example, Derek Jeter and the New York Yankees engaged in a very public bargaining struggle. Jeter wanted more than his market value, and the Yankees wanted to pay less than his market value. In the end, they settled on a three-year contract for $51 million. Some National Football League (NFL) players bargain over restructuring their current contract because of salary cap constraints.

In this chapter, we develop the simple economics of bidding at auctions and how it applies to professional sports. We also develop the basics of bargaining. Several examples illustrate the applicability of bargaining theory. Finally, we examine the posting system that applies to players under contract with professional baseball teams in Japan. As it applies in the United States, the posting system combines bidding at one stage with bargaining at the next stage.

Introduction

Advertising is part of the landscape in both professional and amateur sports. It comes in all shapes and sizes. In 2006, Anheuser-Busch spent more than a quarter of a billion dollars on sports advertising. American Express was a corporate sponsor of the 2006 National Basketball Association (NBA) draft. Sony sponsored the Hawaiian Open golf tournament on the Professional Golf Association (PGA) Tour. The Pittsburgh Steelers play their home games at Heinz Field. We now have “Bears football presented by Bank One,” and the White Sox start their home games at 7:11pm because of 7-Eleven's support. Phil Mickelson and Annika Sorenstam tout Callaway Golf, and Tiger Woods is a Nike man. Gatorade is the “official” sports drink of the National Football League (NFL). All of these are examples of advertising in sports, which is a multibillion dollar business.

The reason for all this activity is simple: profit. Advertising (and other forms of promotion) is designed to increase demand for the advertiser's product. Some advertising does this by being informative, whereas other forms do this by being persuasive. In either event, the idea is to increase the quantity demanded at the same price or, viewed differently, to increase the price at the same quantity. Teams advertise their games to increase attendance, whereas leagues and organizations advertise the athletic competition that their members produce. Of course, producers of a wide array of goods and services advertise to sports fans on television and radio as well as at the venue. All of the billions of dollars that are spent on advertising are geared toward improving the advertiser's profit. To be successful, the ads must increase the advertiser's total revenue by more than they increase its total cost.

This textbook grew out of my dissatisfaction with other textbooks on sports economics that were available. As my own course evolved over time, these books became increasingly unsuitable for my class in part because of coverage and in part because of level and organization. This book reflects my interests and those of my students at the University of Florida. Unlike other books on this subject, this one includes extensive use of present values, choice under uncertainty, pricing models, and numerical examples. The content includes multiyear contracts, insurance, sports gambling, misconduct (and its discipline), the steroid scandal, and many other topics that are not standard fare in the existing sports economics textbooks.

An important pedagogical feature of this book is the involvement of students in the learning process through problem solving. At the end of each chapter, there are Problems and Questions that are intended to help students learn the concepts in the text. There are also some Research Questions that will help students learn how to find information and present it.

Introduction

Many cities have an unmet demand for a professional sports franchise in their midst. Whenever an existing franchise expresses an interest in relocating to greener pastures, there is no apparent shortage of willing hosts. The same is true when one of the major sports leagues contemplates expansion. For strategic reasons, the major sports leagues make sure that there are a few viable locations that go unserved. In this way, they ensure the presence of excess demand and thereby improve the credibility of threats by existing franchises to move. This excess demand also provides leverage when either a new or existing franchise is bargaining with a city for benefits.

Cities also compete for major sports events. Major cities around the world compete with one another to host the Olympics or the World Cup. On a somewhat smaller scale, Omaha pursued the College World Series. Some U.S. cities compete to host the NFL player draft, which has turned into a media event. Cities compete in the only way they can by providing benefits to the organizers. Usually these are legitimate, but there have been instances of corruption.

Introduction

Professional teams, university athletic departments, event sponsors, and sports facility owners all provide something of value to sports fans and participants. Through their pricing decisions, they extract some of that value for themselves. The more that they extract, the more profit they earn at the expense of the fans and participants. As a result, pricing decisions are critical to their financial success. These pricing decisions also have important welfare consequences because they may result in allocative inefficiency. In this chapter, we examine various pricing models that have been used to extract value from the consumer. We begin with competitive pricing to provide a benchmark for comparison. We then examine various ways to exploit monopoly power: simple monopoly pricing, price discrimination, peak load pricing, bundling, two-part pricing, and pricing complements. Finally, we examine ticket scalping, which is the practice of reselling tickets at prices above the face value.

Competitive Pricing

We begin by examining the results of competitive pricing. Suppose that the demand for tickets to a basketball game is represented by D in Figure 5.1. The marginal cost of each additional spectator is low and we assume that it is constant. When marginal cost is constant, average cost is also constant and equal to marginal cost. The constant marginal and average cost are shown as MC = AC in Figure 5.1. Now, in competitive markets, price is driven to marginal cost by competing sellers. If the organizer of the basketball game were to price at the competitive level, price would be P 1, which is equal to MC, and the number of spectators would be Q 1.

Introduction

Major sports events undeniably bring with them large crowds of fans and media. The Super Bowl, Major League Baseball's (MLB's) All-Star Game, the Masters Golf Tournament, and big home football games in College Station, Auburn, Tallahassee, and Clemson all attract hordes of people and an apparent burst of economic activity. Thousands of people flock to these events, and they spend plenty of money in the local area. In addition to their spending at the event (tickets, concessions, and souvenirs), these fans spend substantial sums at local hotels and motels, car rental agencies, bars and restaurants, gas stations, T-shirt shops, and so on. This first round of spending provides income to residents in the local community, which results in subsequent rounds of spending. There is a multiplied effect of the initial spending, which is the economic impact of the sports event. Sports leagues and organizations often grossly exaggerate this economic impact for their own purposes. The National Football League (NFL), for example, claims that the impact of the Super Bowl is hundreds of millions of dollars. As we will see, however, such claims are dubious.

In this chapter, we examine the fundamentals of economic impact analysis. We begin with a simple multiplier model and explain why the impact of an event is not as large as some leagues and organizations would have us believe. We also review and evaluate some empirical studies that have been conducted to test the reliability of some claimed impacts.

This chapter is primarily concerned with algorithms for efficient computation of the Discrete Fourier Transform (DFT). This is an important topic because the DFT plays an important role in the analysis, design, and implementation of many digital signal processing systems. Direct computation of the N-point DFT requires computational cost proportional to N2. The most important class of efficient DFT algorithms, known collectively as Fast Fourier Transform (FFT) algorithms, compute all DFT coefficients as a “block” with computational cost proportional to Nlog2N. However, when we only need a few DFT coefficients, a few samples of DTFT, or a few values of z-transform, it may be more efficient to use algorithms based on linear filtering operations, like the Goertzel algorithm or the chirp z-transform algorithm.

Although many computational environments provide FFT algorithms as built-in functions, the user should understand the fundamental principles of FFT algorithms to make effective use of these functions. The details of FFT algorithms are important to designers of real-time DSP systems in either software or hardware.

Study objectives

After studying this chapter you should be able to:

• Understand the derivation, operation, programming, and use of decimation-in-time and decimation-in-frequency radix-2 FFT algorithms.

• Understand the general principles underlying the development of FFT algorithms and use them to make effective use of existing functions, evaluate competing algorithms, or guide the selection of algorithms for a particular application or computer architecture.

In theory, all signal samples, filter coefficients, twiddle factors, other quantities, and the results of any computations, can assume any value, that is, they can be represented with infinite accuracy. However, in practice, any number must be represented in a digital computer or other digital hardware using a finite number of binary digits (bits), that is, with finite accuracy. In most applications, where we use personal computers or workstations with floating point arithmetic processing units, numerical precision is not an issue. However, in analog-to-digital converters, digital-to-analog converters, and digital signal processors that use fixed-point number representations, use of finite wordlength may introduce unacceptable errors. Finite wordlength effects are caused by nonlinear operations and are very complicated, if not impossible, to understand and analyze. Thus, the most effective approach to analyze finite wordlength effects is to simulate a specific filter and evaluate its performance. Another approach is to use statistical techniques to derive approximate results which can be used to make educated decisions in the design of A/D converters, D/A converters, and digital filters. In this chapter we discuss several topics related to the effects of finite wordlength in digital signal processing systems.

Study objectives

After studying this chapter you should be able to:

• Understand the implications of binary fixed-point and floating-point representation of numbers for signal representation and DSP arithmetic operations.

• Understand how to use a statistical quantization model to analyze the operation of A/D and D/A converters incorporating oversampling and noise shaping.

• […]

To use stochastic process models in practical signal processing applications, we need to estimate their parameters from data. In the first part of this chapter we introduce some basic concepts and techniques from estimation theory and then we use them to estimate the mean, variance, ACRS, and PSD of a stationary random process model. In the second part, we discuss the design of optimum filters for detection of signals with known shape in the presence of additive noise (matched filters), optimum filters for estimation of signals corrupted by additive noise (Wiener filters), and finite memory linear predictors for signal modeling and spectral estimation applications. We conclude with a discussion of the Karhunen–Loève transform, which is an optimum finite orthogonal transform for representation of random signals.

Study objectives

After studying this chapter you should be able to:

• Compute estimates of the mean, variance, and covariance of random variables from a finite number of observations (data) and assess their quality based on the bias and variance of the estimators used.

• Estimate the mean, variance, ACRS sequence, and PSD function of a stationary process from a finite data set by properly choosing the estimator parameters to achieve the desired quality in terms of bias–variance trade-offs.

• Design FIR matched filters for detection of known signals corrupted by additive random noise, FIR Wiener filters that minimize the mean squared error between the output signal and a desired response, and finite memory linear predictors that minimize the mean squared prediction error.

• […]

The term “filter” is used for LTI systems that alter their input signals in a prescribed way. Frequency-selective filters, the subject of this chapter, are designed to pass a set of desired frequency components from a mixture of desired and undesired components or to shape the spectrum of the input signal in a desired way. In this case, the filter design specifications are given in the frequency domain by a desired frequency response. The filter design problem consists of finding a practically realizable filter whose frequency response best approximates the desired ideal magnitude and phase responses within specified tolerances.

The design of FIR filters requires finding a polynomial frequency response function that best approximates the design specifications; in contrast, the design of IIR filters requires a rational approximating function. Thus, the algorithms used to design FIR filters are different from those used to design IIR filters. In this chapter we concentrate on FIR filter design techniques while in Chapter 11 we discuss IIR filter design techniques. The design of FIR filters is typically performed either directly in the discrete-time domain using the windowing method or in the frequency domain using the frequency sampling method and the optimum Chebyshev approximation method via the Parks–McClellan algorithm.

Study objectives

After studying this chapter you should be able to:

• Understand how to set up specifications for design of discrete-time filters.

• Understand the conditions required to ensure linear phase in FIR filters and how to use them to design FIR filters by specifying their magnitude response. […]

• […]

This chapter is primarily concerned with the definition, properties, and applications of the Discrete Fourier Transform (DFT). The DFT provides a unique representation using N coefficients for any sequence of N consecutive samples. The DFT coefficients are related to the DTFS coefficients or to equally spaced samples of the DTFT of the underlying sequences. As a result of these relationships and the existence of efficient algorithms for its computation, the DFT plays a central role in spectral analysis, the implementation of digital filters, and a variety of other signal processing applications.

Study objectives

After studying this chapter you should be able to:

• Understand the meaning and basic properties of DFT and how to use the DFT to compute the DTFS, DTFT, CTFS, and CTFT transforms.

• Understand how to obtain the DFT by sampling the DTFT and the implications of this operation on how accurately the DFT approximates the DTFT and other transforms.

• Understand the symmetry and operational properties of DFT and how to use the property of circular convolution for the computation of linear convolution.

• Understand how to use the DFT to compute the spectrum of continuous-time signals and how to compensate for the effects of windowing the signal to finite-length using the proper window.

Computational Fourier analysis

The basic premise of Fourier analysis is that any signal can be expressed as a linear superposition, that is, a sum or integral of sinusoidal signals.

As we discussed in Chapter 2, any LTI can be implemented using three basic computational elements: adders, multipliers, and unit delays. For LTI systems with a rational system function, the relation between the input and output sequences satisfies a linear constant-coefficient difference equation. Such systems are practically realizable because they require a finite number of computational elements. In this chapter, we show that there is a large collection of difference equations corresponding to the same system function. Each set of equations describes the same input-output relation and provides an algorithm or structure for the implementation of the system. Alternative structures for the same system differ in computational complexity, memory, and behavior when we use finite precision arithmetic. In this chapter, we discuss the most widely used discrete-time structures and their implementation using Matlab. These include direct-form, transposed-form, cascade, parallel, frequency sampling, and lattice structures.

Study objectives

After studying this chapter you should be able to:

• Develop and analyze practically useful structures for both FIR and IIR systems.

• Understand the advantages and disadvantages of different filter structures and convert from one structure to another.

• Implement a filter using a particular structure and understand how to simulate and verify the correct operation of that structure in Matlab.

Block diagrams and signal flow graphs

Every practically realizable LTI system can be described by a set of difference equations, which constitute a computational algorithm for its implementation.

This chapter is primarily concerned with the conversion of continuous-time signals into discrete-time signals using uniform or periodic sampling. The presented theory of sampling provides the conditions for signal sampling and reconstruction from a sequence of sample values. It turns out that a properly sampled bandlimited signal can be perfectly reconstructed from its samples. In practice, the numerical value of each sample is expressed by a finite number of bits, a process known as quantization. The error introduced by quantizing the sample values, known as quantization noise, is unavoidable. The major implication of sampling theory is that it makes possible the processing of continuous-time signals using discrete-time signal processing techniques.

Study objectives

After studying this chapter you should be able to:

• Determine the spectrum of a discrete-time signal from that of the original continuous-time signal, and understand the conditions that allow perfect reconstruction of a continuous-time signal from its samples.

• Understand how to process continuous-time signals by sampling, followed by discrete-time signal processing, and reconstruction of the resulting continuous-time signal.

• Understand how practical limitations affect the sampling and reconstruction of continuous-time signals.

• Apply the theory of sampling to continuous-time bandpass signals and two-dimensional image signals.

Ideal periodic sampling of continuous-time signals

In the most common form of sampling, known as periodic or uniform sampling, a sequence of samples x[n] is obtained from a continuous-time signal xc(t) by taking values at equally spaced points in time.

In Chapter 2 we discussed representation and analysis of LTI systems in the time-domain using the convolution summation and difference equations. In Chapter 3 we developed a representation and analysis of LTI systems using the z-transform. In this chapter, we use Fourier representation of signals in terms of complex exponentials and the polezero representation of the system function to characterize and analyze the effect of LTI systems on the input signals. The fundamental tool is the frequency response function of a system and the close relationship of its shape to the location of poles and zeros of the system function. Although the emphasis is on discrete-time systems, the last section explains how the same concepts can be used to analyze continuous-time LTI systems.

Study objectives

After studying this chapter you should be able to:

• Determine the steady-state response of LTI systems to sinusoidal, complex exponential, periodic, and aperiodic signals using the frequency response function.

• Understand the effects of ideal and practical LTI systems upon the input signal in terms of the shape of magnitude, phase, and group-delay responses.

• Understand how the locations of poles and zeros of the system function determine the shape of magnitude, phase, and group-delay responses of an LTI system.

• Develop and use algorithms for the computation of magnitude, phase, and group-delay responses of LTI systems described by linear constant-coefficient difference equations. […]