1. In the first 5 rounds of the lab, you were randomly assigned into n=2 person groups. You incurred lobbying costs that affected the probability that you would win a prize of $64,000. Each unit of lobbying effort cost $1000. If both people cooperated and chose not to incur any lobbying effort, the prize would be awarded randomly and the group would receive the maximum payment of $64,000. Explain why the outcome just described is not a noncooperative Nash equilibrium of this lobbying game.
2. If both participants in a group incur $6000 in lobbying expenditures each, how much would the winning individual earn? How much would the losing individual earn? What is the expected payoff of each of the 2 participants, before the winner is randomly determined? (Each individual’s probability of winning is proportional to her lobbying expenditures, as explained in the Holt chapter.) If either of the participants deviated from this “$6000 each” lobbying expenditure to increase her expenditure to $7,000, would her expected payoff increase? Use your answer to explain why or why not this “$6000 each” lobbying expenditure is a Nash equilibrium.
3. Based on your careful reading of the Holt chapter on rent seeking, what do you think is the Nash equilibrium in lobbying expenditures? How much does each individual spend in Nash equilibrium, and how much do all of the individuals in each group spend in total? (Hint: you definitely should read the mathematical details appendix that starts on page 211 to answer this question.)
4. Using the formula in the Holt book for the equilibrium amount of lobbying effort (equation 17.2), determine whether the equilibrium effort (x*) increases or decreases in the number of lobbyists (N). Also determine whether the total equilibrium effort (Nx*) increases or decreases in the number of lobbyists (N). For simplicity you can use calculus even though technically N is not continuous.
5. In the second 5rounds of the lab session, nothing changed except the number of participants in each group increased to n=4. If all 4 participants in a group incur $6000 in lobbying expenditures each, how much would the winning individual earn? How much would the three losing individuals earn? What is the expected payoff of any of the 4 participants, before the winner is randomly determined? If one of the 4 participants deviated from this “$6000 each” lobbying expenditure to increase his expenditure to $7,000, would his expected payoff increase? Use your answer to explain why or why not this “$6000 each” lobbying expenditure is a Nash equilibrium.
6. What do you think is the Nash equilibrium in lobbying expenditures for this n=4 treatment? How much does each individual spend in Nash equilibrium, and how much do all of the individuals in each group spend in total? Do total expenditures on lobbying (incurred by the group in total) increase or decrease when the group size increases?
7. Use the lab data to compare the actuallobbying expenditures to the Nash equilibrium lobbying expenditures, separately for each of the two initial sets of 5rounds. Display the two time paths for the two treatments on the same graph. You might want to create separate graphs for average individual lobbying expenditures, and total lobbying expenditures, displayed for each round.
8. Is there a difference in the two treatments? Is this difference consistent with the “ordering” of contributions predicted by the Nash equilibria you described in questions 3 and 5? Are average contributions greater than or less than the relevant Nash equilibria for the treatments? Is this result consistent with what Holt reports for a similar experiment in Figure 17.1? Explain.
9. In the final 10rounds of the lab session we conducted an experiment on the Volunteer’s Dilemma described in Chapter 15 of the Holt textbook (pages 183-191). In all 10 rounds the payoff with a volunteer (V) was $10, the payoff without a volunteer (L) was $2, and the cost of volunteering (C) was $3. The first 5 of these 10 rounds employedn=2 person groups, and the final 5 rounds employed n=6person groups.Based on your careful reading of the Holt chapter, what is the equilibrium probability of volunteering in the mixed strategy equilibrium derived? How does it depend on the number of potential volunteers in the group?
10. Use the lab data to summarize the volunteering frequency, separately for each round, as a fraction of the number of potential volunteers. Is there much of a difference in the volunteering rate as n increases from 2 to 6? Is this difference consistent with the ordering of the equilibrium volunteering probability calculated in question 8? How closely does the observed volunteering rate correspond to the equilibrium volunteering probability? If there is a difference, why do you think this difference occurs?
Attachment:- Lab 5 data.xlsx