Selftest Problems And Solutions

ST11.1 Game Theory. One of the most dynamic changes taking place in our economy is the evolution of the personal computer from a document preparation and computing device to a communicating device. What we used to view as stand-alone personal computers, televisions, VCRs, telephones, fax and copy machines are all converging toward nimble communications devices with the ability to fulfill a number of tasks simultaneously. Nowhere is the influence of this trend more obvious than in the communications equipment industries. This is the sector within which manufacturers produce household audio and video equipment, prerecorded records and tapes, telephone and telegraph apparatus, and radio and television communications equipment. Because many commercial users have unique needs, equipment suppliers sometimes have significant ability to influence the price charged for what is often a bundle of specialized goods and services. As a result, game theory concepts prove useful to both buyers and sellers of communications devices.

To illustrate, suppose two local suppliers are seeking to win the right to upgrade the communications capability of the internal "intranets" that link a number of customers with their suppliers. The system quality decision facing each competitor, and potential profit payoffs, are illustrated in the table. The first number listed in each cell is the profit earned by U.S. Equipment Supply; the second number indicates the profit earned by Business Systems, Inc. For example, if both competitors, U.S. Equipment Supply and Business Systems, Inc., pursue a high-quality strategy, U.S. Equipment Supply will earn $25,000 and Business Systems, Inc., will earn $50,000. If U.S. Equipment Supply pursues a high-quality strategy while Business Systems, Inc., offers low-quality goods and services, U.S. Equipment Supply will earn $40,000; Business Systems, Inc., earns $22,000. If U.S. Equipment Supply offers low-quality goods while Business Systems, Inc., offers high-quality goods, U.S. Equipment Supply will suffer a net loss of $25,000, and Business Systems, Inc., will earn $20,000. Finally, if U.S. Equipment Supply offers low-quality goods while Business Systems, Inc., offers low-quality goods, both U.S. Equipment Supply and Business Systems, Inc., will earn $25,000.

Business Systems, Inc. Quality Strategy High Quality Low Quality s

<»!§ High Quality $25,000, $50,000 $40,000, $22,000

A. Does U.S. Equipment Supply and/or Business Systems, Inc., have a dominant strategy? If so, what is it?

B. Does U.S. Equipment Supply and/or Business Systems, Inc., have a secure strategy? If so, what is it?

C. What is the Nash equilibrium concept, and why is it useful? What is the Nash equilibrium for this problem?

ST11.1 Solution

A. The dominant strategy for U.S. Equipment Supply is to provide high-quality goods. Irrespective of the quality strategy chosen by Business Systems, Inc., U.S. Equipment Supply can do no better than to choose a high-quality strategy. To see this, note that if Business Systems, Inc., chooses to produce high-quality goods, the best choice for U.S. Equipment Supply is to also provide high-quality goods because the $25,000 profit then earned is better than the $25,000 loss that would be incurred if U.S. Equipment Supply chose a low-quality strategy. If Business Systems, Inc., chose a low-quality strategy, the best choice by U.S. Equipment Supply would again be to produce high-quality goods. U.S. Equipment Supply's high-quality strategy profit of $40,000 dominates the low-quality payoff for U.S. Equipment Supply of $25,000.

Business Systems, Inc., does not have a dominant strategy. To see this, note that if U.S. Equipment Supply chooses to produce high-quality goods, the best choice for Business Systems, Inc., is to also provide high-quality goods because the $50,000 profit then earned is better than the $22,000 profit if Business Systems, Inc., chose a low-quality strategy. If U.S. Equipment Supply chose a low-quality strategy, the best choice by Business Systems, Inc., would be to produce low-quality goods and earn $25,000 versus $20,000.

B. The secure strategy for U.S. Equipment Supply is to provide high-quality goods. By choosing to provide high-quality goods, U.S. Equipment Supply can be guaranteed a profit payoff of at least $25,000. By pursuing a high-quality strategy, U.S. Equipment Supply can eliminate the chance of losing $25,000, as would happen if U.S. Equipment Supply chose a low-quality strategy while Business Systems, Inc., chose to produce high-quality goods.

The secure strategy for Business Systems, Inc., is to provide low-quality goods. By choosing to provide high-quality goods, Business Systems, Inc., can guarantee a profit payoff of only $20,000. Business Systems, Inc., can be assured of earning at least $22,000 with a low-quality strategy. Thus, the secure strategy for Business Systems, Inc., is to provide low-quality goods.

C. A set of strategies constitutes a Nash equilibrium if, given the strategies of other players, no player can improve its payoff through a unilateral change in strategy. The concept of Nash equilibrium is very important because it represents a situation where every player is doing the best possible in light of what other players are doing.

Although useful, the notion of a secure strategy suffers from a serious shortcoming. In the present example, suppose Business Systems, Inc., reasoned as follows: "U.S. Equipment Supply will surely choose its high-quality dominant strategy. Therefore, I should not choose my secure low-quality strategy and earn $22,000. I should instead choose a high-quality strategy and earn $50,000." A natural way of formalizing the "end result" of such a thought process is captured in the definition of Nash equilibrium.

In the present example, if U.S. Equipment Supply chooses a high-quality strategy, the Nash equilibrium strategy is for Business Systems, Inc., to also choose a high-quality strategy. Similarly, if Business Systems, Inc., chooses a high-quality strategy, the Nash equilibrium strategy is for U.S. Equipment Supply to also choose a high-quality strategy. Thus, a Nash equilibrium is reached when both firms adopt high-quality strategies.

Although some problems have multiple Nash equilibriums, that is not true in this case. A combination of high-quality strategies for both firms is the only set of strategies where no player can improve its payoff through a unilateral change in strategy.

ST11.2 Columbia Drugstores, Inc., based in Seattle, Washington, operates a chain of 30 drugstores in the Pacific Northwest. During recent years, the company has become increasingly concerned with the long-run implications of competition from a new type of competitor, the so-called superstore.

To measure the effects of superstore competition on current profitability, Columbia asked management consultant Mindy McConnell to conduct a statistical analysis of the company's profitability in its various markets. To net out size-related influences, profitability was measured by Columbia's gross profit margin, or earnings before interest and taxes divided by sales. Columbia provided proprietary company profit, advertising, and sales data covering the last year for all 30 outlets, along with public trade association and Census Bureau data concerning the number and relative size distribution of competitors in each market, among other market characteristics.

As a first step in the study, McConnell decided to conduct a regression-based analysis of the various factors thought to affect Columbia's profitability. The first is the relative size of leading competitors in the relevant market, measured at the Standard Metropolitan Statistical Area (SMSA) level. Columbia's market share, MS, in each market area is expected to have a positive effect on profitability given the pricing, marketing, and average-cost advantages that accompany large relative size. The market concentration ratio, CR, measured as the combined market share of the four largest competitors in any given market, is expected to have a negative effect on Columbia's profitability given the stiff competition from large, well-financed rivals. Of course, the expected negative effect of high concentration on Columbia profitability contrasts with the positive influence of high concentration on industry profits that is sometimes observed.

Both capital intensity, K/S, measured by the ratio of the book value of assets to sales, and advertising intensity, A/S, measured by the advertising-to-sales ratio, are expected to exert positive influences on profitability. Given that profitability is measured by Columbia's gross profit margin, the coefficient on capital intensity measured Columbia's return on tangible investment. Similarly, the coefficient on the advertising variable measures the profit effects of advertising. Growth, GR, measured by the geometric mean rate of change in total disposable income in each market, is expected to have a positive influence on Columbia's profitability, because some disequilibrium in industry demand and supply conditions is often observed in rapidly growing areas.

Profit-Margin and Market-Structure Data for Columbia

Drugstores,

Inc.

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