Assignment 4
T.Joy
Section #: 01
Students must abide by UVic academic regulations and observe standards of scholarly integrity (i.e. no plagiarism or cheating). Therefore, this assignment must be taken individually and not with a friend, classmate, or group. You are also prohibited from sharing any information about the assignment with others. I affirm that I will not give or receive any aid on this assignment and that all work will be my own. T. Joy
Many aspects of the environment are public goods. Public goods are nonrivalrous and nonexcludable. Nonrivalrous means that you deriving benefit from the good does not diminish the amount of the good available for others. Nonexcludable means that there is no way to prevent people from deriving benefits from the good. Examples of environmental public goods include obvious things like the ozone layer, but can also be more subtle like the absence of litter in a park.
In experiment 4 we saw that the health of a society is also public good, where individuals can take costly actions to contribute. Experiment 4 put you in a situation where you faced a trade-off between a private good (having fun) and a public good (the health of your society). Your task was to choose a costly effort level that would impact the transmission of a virus. Effort involves things like wearing a mask, washing your hands and staying socially isolated. Obviously these things are un-fun: we assume that effort reduces fun 1 for 1. Of course the flip-side is that these actions are what keeps the virus under control. We assume that public health is 1.5 times the average effort in your group, and all members of your group benefited equally from public health. Thus, individual 1’s payoff was:
\(\pi_1=10-e_1+1.5\times\frac{e_1+...+e_n}{n}\)
where 10 is the maximum effort you can exert: you can think of this as your endowment.
The treatments determined how subjects were rematched between rounds. In the not treatment subjects played all 20 rounds with the same group. In the randomly treatment subjects were randomly rematched for each round. Finally in the assortatively treatment subjects played each round with players who had made similar contributions in the previous round. Higher payoffs can be sustained in a public goods game if behaviour can be coordinated (each member of the group contributing a similar amount.) These treatments were designed to influence how easy it would be to coordinate behaviour.
1 (10 marks)
If I increased your endowment from 10 to 11, but you kept your effort level constant, by how much would your payoff increase? If instead you responded by increasing your effort by one unit, by how much would your payoff increase (assume 3 people in your group)? If a player is maximizing their own payoff (as given by the equation above) how much effort should they exert? Does your answer depend on group size? Does your answer depend on round number? Does your answer depend on the effort level of others?
It would increase by one because my effort level is constant. say my effort is 5 and my groups effort is (5,6,7), average effort is 6 and my functions look like this: 11-5+1.5(6)=15 and previously: 10-5+1.5(6)=14. If i increase my effort level by one i should expect my payoff to increase by less than one due to the average effort increasing. i am going from: 10-5+1.5(6)=14 to 11-6+1.5(6.33)=14.5 assuming others are constant. Players should exert less effort, ideally 0, relative to the other players to maximize their utility this would be considered the dominant strategy as we do not know the other players effort until after. no group size doesn’t matter because when maximizing you are considering private benefit. you receive more happiness by keeping your effort low than by contributing to the social good, so group size doesn’t matter and if you were considering social effects, it takes more effort to get a large group of people to work together compared to a small number so effort may decrease. No my answer doesn’t depend on round number as my maximum utility is exerting 0 effort right off the bat, although if you were behaving in a socially responsible manner you would likely decrease your effort over the rounds. My answer doesn’t depend on the effort level of others because I don’t want other players exerting less effort than i am, so I am automatically keeping effort low. If we were colluding to receive maximum utility, then my answer does depend on others because if someone broke the alliance we would start exerting low effort.
2 (10 marks)
Suppose that each group had a “government”, which could mandate effort levels: (all members have to exert the same effort) Would this government intervention increase player payoffs relative to the equilibrium prediction? What would be the optimal mandated effort level? Yes it would increase players payoff relative to their payoff without intervention. Without mandated effort level, people will slowly decrease their effort to receive more utility relative to their peers. the optimal mandated effort level is 10 because everyone gets 15 units of happiness, any lower mandate lowers the happiness per person.
3 (5 marks)
Use dataframe mydf and functions ggplot(), geom_boxplot(), and geom_jitter() to visualize how effort choice differs by treatment rematching. Use labs() to give the plot an informative title.
4 (10 marks)
Use dataframe mydf and functions ggplot(), geom_boxplot(), and geom_jitter() to visualize how effort choice varies across round. Use function facet_wrap(~rematching) to create a different facet for each treatment. Use labs() to give the plot an informative title.
5 (10 marks)
What differences do you notice in the first round choices across the three treatments?
The variability in choices was the largest for the assorted groups. The variability was the smallest for the not group. effort was the highest for the random treatment and around 5 for both assorted and not. random treatment has no upper tail, likely because a lot of people put 10 as their effort for round 1. Many people decided to make a lesser effort right off the bat, without seeing what their peers did.
6 (10 marks)
In your .R file, create a new dataframe called within_group_variability by taking dataframe mydf THEN group_by variables whichgroup, round, rematching THEN summarise by calculating variability_contributions=sd(choice). Include a copy of your code in the chunk below (note eval=FALSE)
within_group_variability<-group_by(mydf, whichgroup, round, rematching)
within_group_variability<-summarise(within_group_variability, variability_contributions=sd(choice))7 (10 marks)
Use dataframe within_group_variability and functions ggplot(), geom_boxplot(), and geom_jitter() to visualize how variability_contributions differs by treatment rematching. Use labs() to give the plot an informative title. Make use of function fix_gg_labs() to fix the labels of third_plot.
8 (10 marks)
Randomization of which treatment each subject receives is meant to ensure that subjects are comparable. Above we saw some evidence that in the first round, prior to being re-matched the subjects in the randomly treatment made much larger choices than the other two treatments. This suggests that the subjects were not in fact comparable prior to receiving their treatments. One way we can address this is look at choices relative to the choice made in the first round. To this end, create a new dataframe called relative_to_r1 by taking dataframe mydf THEN group_by() variables oneid,rematching THEN nest() THEN mutate(new_df=map(data,calc_relative)) THEN select(-data) THEN unnest(cols = new_df) THEN filter(round!=1). Include a copy of your code in the chunk below (note eval=FALSE)
relative_to_r1<-mydf%>%group_by(oneid, rematching)%>%nest()%>%
mutate(new_df=map(data, calc_relative))%>%select(-data)%>%unnest(cols=new_df)%>%
filter(round!=1)9 (5 marks)
Explain what function calc_relative() does (it can be found in the .R file) choice-head(choice,n=1))
it mutates choice by taking the difference between someones choice in the current round minus their first round choice. This measures the differences in peoples choices overtime with a relative starting point. It likely shows how effort decresed over time. # (10 marks)
Use dataframe relative_to_r1 and functions ggplot(), geom_boxplot(), and geom_jitter() to visualize how choice_relative_to_first_round differs by treatment rematching. Use labs() to give the plot an informative title. Make use of function fix_gg_labs() to fix the labels of fourth_plot
10 (0 marks)
Use dataframe mydf and functions ggplot(), stat_density_ridges(), scale_fill_viridis_c() and facet_wrap() to create a density ridge plot for choice across rematching and round. Use labs() to give the plot an informative title. Use function fix_gg_labs() to fix the labels for this fifth_plot.
11 (10 marks)
Use dataframe relative_to_r1 and functions ggplot(), stat_density_ridges(), scale_fill_viridis_c() and facet_wrap() to create a density ridge plot for choice_relative_to_first_round across rematching and round. Add a geom_vline() with arguments xintercept = 0, col="white",alpha=.5,lwd=2 as a guide. Use labs() to give the plot an informative title. Use function fix_gg_labs() to fix the labels for this sixth_plot.
