PS#4: Competitive Balance (Chapter 5)1. Explain how the law of diminishing returns provides a natural tendency toward competitive balance.Answer: The law of diminishing marginal returns means that adding good players will eventually add less and less to a team’s performance on the field and at the box office. Adding an All Star goalie will not bring many more wins if a hockey team already has an All Star goalie. Thus, the marginal benefit of the second goalie is probably less than the marginal cost of hiring him. As a result, even the wealthiest teams do not have an incentive to sign all the best players. Diminishing returns thus add to competitive balance.2. Suppose in a six-team league, the winning percentages were as follows at the end of the season: Team A, .750; Team B, .600; Team C, .500; Team D, .500; Team E, .400; Team F, .250. Compute the standard deviation of win percentage.Answer: The standard deviation is given by the equationwhere WPCTi is team i’s winning percentage, and N is the number of teams in the league. In the case of sports leagues, the mean winning percentage is always 0.5 because one team’s win must be another team’s loss. Using the numbers given, the standard deviation is roughly 0.156.3. In question 5.3, suppose each team plays a 50-game schedule. Compute the “ideal” benchmark standard deviation of winning percentage based on equal playing strength, and the ratio of the actual to the ideal.Answer: The “ideal” standard deviation for a league in which all teams are equally likely to win is given by the equationwhere G is the number of games played. The ideal standard deviation in a league that plays 50 games is roughly 0.071. The ratio of the actual standard deviation to the actual standard deviation is4. Why do many economists believe that free agency has not affected competitive balance?Answer: According to the Coase Theorem, resources flow to their most valued use as long as there are well-defined property rights, regardless of who holds those rights. In the case of professional sports, free agency has transferred the rights to players’ services from the team owners to the players. The Coase Theorem predicts that the distribution of talent—and hence competitive balance—will be no different under free agency than it was when players were bound to teams. The change in property rights affects only who is compensated for the move. The Yankees paid the owner of the Red Sox to obtain Ruth’s services. To obtain a free agent like Jason Giambi, they pay only the player.5. Draw the Lorenz curves from Figure 5.3. Based on what you know from this chapter, add a Lorenz curve for the NFL and a Lorenz curve for the English Premier League. Why did you place them where you did?Answer: Figure 5.3 shows that the standard deviation of win percentage ratios for the EPL and NFL are 1.97 and 1.66 respectively while they are 1.50 and 3.07 for the NHL and NBA respectively. While there is not a perfect correlation between the standard deviation of win percentage ratio and the shape of a Lorenz curve, in general, the higher the standard deviation of win percentage ratio, the more curved the Lorenz curve should be. Thus, both the NFL and EPL curves are likely to lie between the NHL and NBA curves with the EPL curve being closer to the NBA curve than the NFL’s curve.6. Suppose over five seasons, the order of finish for five teams in the West League and the East League is as follows. Use the HHI to determine which league has better competitive balance across seasons.West League SeasonEast League Season1234512345AAAEEABCDEBBDDDEAAAACCCCCCBDEDDDBBBBDBBBEEEAADEECCAnswer: In the West League, Team A finished first 3 times, Team E finished first twice, and no other team finished first. In the East League, each team finished first once. The HHI index is given by the equationwhere c is the number of first-place finishes, and T is the number of seasons, N=number of teams. Here, ci is 3 for Team A and 2 for Team B in the West League and is 1 for each team in the East League. T equals 5 for both leagues. Inserting these values yields HHI  0.52 for the West League and HHI  0.2 for the East League. Because HHI is lower for leagues with greater competitive balance, the HHI shows that the East League is more balanced than the West League. These two leagues can be compared using the HHI only because they have the same number of teams. As noted previously, one would expect teams in the Big Ten (with its paradoxical eleven teams) to win championships more frequently than teams in the 32-team NFL.7. What is a shortcoming of the HHI measure?Answer: One flaw with the HHI measure is that as N (the number of teams increases) the value of HHI decreases even if the league has not become more competitive. Therefore, the HHI cannot generally be used to compare between-season competitive balance between two leagues of different sizes.8. Which move would be more effective for increasing the level of competitive balance in baseball, a hard salary cap or a 50-50 gate revenue-sharing plan?Answer: First of all, gate revenues are only a small portion of total revenues collected in professional sports, so even complete sharing of gate revenues will leave some teams with significantly higher revenues. Even if all revenues are shared equally, however, players still flow to the same teams they would have gone to if there were no revenue sharing. When teams evenly split revenue they effectively put all their revenue in a pile in the middle of the room and then divide the pile equally among themselves. If teams maximize profits, they want to make the pile as large as possible. If a star player is more valuable in Los Angeles than in Cleveland, then the pile in the middle of the room is larger—and the revenue for Los Angeles and Cleveland—will be greater if the player moves to Los Angeles. Thus, even a 50-50 revenue split may have little impact on the distribution of talent or competitive balance. A hard salary cap limits what a team can pay its players. Even if a player is worth more in Los Angeles then in Cleveland, Los Angeles may not be able to lure him away if it is already spending close to its limit on its other players. This limit evens out what teams can spend for players and has the potential for evening out the distribution of talent.PS#4: Competitive Balance (Chapter 5)1. Explain how the law of diminishing returns provides a natural tendency toward competitive