Lab experiments sometimes pair subjects together and have them play against one another in games where each subject is rewarded financially according to the game's outcome. One such game involves making monetary contributions to a public good (e.g., preserving the environment); the game can be arranged such that each player gains financially if both of them make a contribution, but each player is better off still if they contribute nothing while their partner in the game makes a contribution. The treatment is whether the pair of players is allowed to communicate prior to deciding whether to contribute. Suppose that a lab experimenter recruits four subjects and assigns them randomly as pairs to play this game. The outcome is whether each player makes a contribution: is 1 if the player contributes and 0 otherwise. Each player has three potential outcomes: 0 if the outcome if players are prevented from communicating, 1 is the outcome if a player communicates with another player who is ``persuasive,'' and 2 if the outcome if a player communicates with another player who is ``unpersuasive.'' The table below shows the schedule of potential outcomes for four players, two of whom are persuasive and two of whom are unpersuasive.

Potential outcome schedule

Subject	Type	0Y0i	1Y1i	2Y2i
1	Persuasive	0	1	0
2	Persuasive	1	1	0
3	Unpersuasive	0	0	0
4	Unpersuasive	1	1	1

1. What is the probability that a persuasive subject is treated by communicating with an unpersuasive subject? What is the probability that an unpersuasive subject is treated by communicating with an unpersuasive subject?

2. Briefly summarize why a violation of the non-interference assumption leads to biased difference-in-means estimates in this example.

3. Would bias be eliminated if the experimenter replicated this study (with four subjects) each day and averaged the results over a series of 100 daily studies?

4. Would bias be eliminated if the experimenter assembled 400 subjects at the same time (imagine 100 subjects for each of the four potential outcomes profiles in the table) and assigned them to pairs? Hint: Answer the question based on the intuition suggested by part (d).