Problem Set 3 Contests Please Provide Explanations And Cal

Problem Set 3contestsplease Provideexplanationsandcal

Suppose that Laverne plans to train 48 hours per week, and Shirley plans to train 16 hours. What is the probability that Shirley will be the county champ? What is Laverne’s payoff? How does Laverne's payoff change if she decreases her training to 32 hours or increases it to 64 hours? Is the chosen effort allocation an equilibrium or Nash equilibrium? How does changing the prize from 100 hours to 60 hours affect the equilibrium? What happens when a third athlete with the same ability, Edna, enters the race? How does effort in symmetric versus asymmetric contests vary according to contest theory, based on the data from the golf tournaments study? Additionally, a practical data collection project involves recording success or failure in a chosen activity during 100 attempts, and analyzing this data using statistical tests.

Paper For Above instruction

The analysis of effort and strategic behavior in contests is a fundamental aspect of behavioral and economic theory. In this problem set, the focus is on the classic Tullock contest model, which provides a probabilistic framework for understanding the likelihood of winning based on effort exerted by each contestant. The model assumes that the probability of an athlete winning is proportional to their effort relative to the total effort exerted by all competitors. This allows us to explore various strategic considerations, including equilibrium efforts, how changes in effort alter payoffs, and the stability of these efforts under different contest conditions.

Initially, consider two athletes, Laverne and Shirley, with effort levels of 48 and 16 hours per week, respectively. According to the Tullock model, the probability that Shirley wins can be computed as the ratio of Shirley’s effort to the total effort: P(Shirley wins) = effort of Shirley / (effort of Laverne + effort of Shirley). Substituting the given values gives P = 16 / (48 + 16) = 16 / 64 = 0.25, or 25%. This probability reflects Shirley’s chance of winning given her lower effort level.

The payoff for Laverne, measured in hours, combines the expected value of her victory minus her effort cost. Since the prize is worth 100 hours, her expected payoff is (probability of winning × prize) minus her own effort. Specifically, Laverne’s payoff = (0.75 × 100) - 48 = 75 - 48 = 27 hours. This calculation helps determine whether her effort level is optimal or if she could improve her payoff by adjusting her effort.

If Laverne decreases her effort to 32 hours, the probability that Shirley wins increases because her own effort decreases, making her less likely to win. Recomputing, P(Shirley wins) = 16 / (32 + 16) = 16 / 48 ≈ 0.333, or 33.3%. Correspondingly, Laverne’s expected payoff becomes (0.6667 × 100) - 32 ≈ 66.67 - 32 = 34.67 hours. Although her probability of winning drops, her expected payoff actually increases, indicating that reducing effort from 48 to 32 hours can be beneficial in this case.

Conversely, when Laverne increases effort to 64 hours, the probability of Shirley's victory decreases to 16 / (64 + 16) = 16 / 80 = 0.2, or 20%. Her expected payoff then is (0.8 × 100) - 64 = 80 - 64 = 16 hours, which is lower than when she exerted 48 hours. This demonstrates that over-investment in effort can reduce the expected payoff, emphasizing the importance of strategic effort levels.

Regarding the equilibrium, if Laverne trains 48 hours while Shirley trains 16, and neither athlete can improve their payoff by unilaterally changing their effort, this effort distribution is an equilibrium. A Nash equilibrium exists where neither participant can increase their expected payoff by deviating alone. For the given efforts, if Shirley considers increasing her effort to gain a higher winning probability, her payoff might increase or decrease depending on the trade-off between effort and winning probability, which should be checked explicitly.

When both athletes are training at 25 hours each, the effort levels are symmetric. To verify if this is a Nash equilibrium, we calculate Laverne’s payoff if she deviates slightly, say by increasing to 26 hours or decreasing to 24 hours. If her payoff at 26 hours exceeds 25 hours, then training at 25 is not an equilibrium, and similar logic applies for decreasing efforts. Repeating this for Shirley under symmetric effort levels verifies whether the effort levels constitute a stable equilibrium.

If the prize diminishes from 100 to 60 hours, the incentives shift since the potential reward shrinks. Repeating the payoff calculations at 25 hours effort levels, we find the incentive for deviation changes. It's observed that the payoff maximizing effort reduces, indicating that the 25-hour effort level might no longer be a Nash equilibrium under the lower prize. This example underscores how prize size influences strategic effort levels and equilibrium stability.

Introducing a third athlete, Edna, with identical ability and payoff prospects, complicates the strategic landscape. The effort levels that equilibrium efforts previously involved will change, as now three identical contestants compete simultaneously. If all three exert the same effort, the probability of each winning is 1/3, and each’s expected payoff must be recalculated. Under symmetric effort, equilibrium efforts are likely to adjust to ensure no player can improve by unilateral deviations, which can be identified by similar calculations as before.

From the perspective of contest theory, effort levels tend to be higher in symmetric settings—where abilities are equal—than in asymmetric contexts. The theory predicts that equality of ability encourages higher effort since each player's probability of winning is more balanced, creating stronger incentives to exert effort. Conversely, asymmetry tends to reduce effort from less capable players who perceive their chances as low or less worth exerting effort for. Empirical evidence from the golf tournaments study supports this: adjusted scores considering handicaps (leveling the playing field) are associated with more effort, aligning with the prediction that symmetric contests foster higher effort levels.

Finally, data collection projects involving success/failure in closed activities—such as free throws or goal kicks—offer valuable empirical insights. Recording outcomes over a series of attempts provides granular data for statistical analysis. Using this data, we can compute success probabilities and perform hypothesis testing to verify if observed success rates align with theoretical probabilities, assess consistency over time, or compare different activity settings. Such empirical work helps validate contest models and improve our understanding of effort and strategic behavior in competitive environments.

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