Mutual agreement and procedural fairness

There are five treatments:
1) CONTROL: Subjects are randomly allocated to the biased or the unbiased spin.
2) MUTUAL: Both subjects in a pair have to agree to choose the biased spin. If one of them refuses, they both play the unbiased spin.

3) AGREEMENT: Sames as mutual except that subjects in a pair have to explicitely state that they agree to choose the biased spin.

4) Control0:
5) AGREEMENT0

2025-04-29

2025-11-28

Rejection rates of red (disadvantaged) players (when getting the low payoff in the pair) after paired spin

We assume that players who get the higher outcome of the pair will generally accept. All rejection rates below will therefore refer to rejection after receiving the lower payoff.
We conjecture that rejection rates after the paired lottery spin will be lower in Mutual than in Control. This should hold in particular for the red (disadvantaged) player.
To account for a selection problem, namely that we do not observe what players who opt out by choosing the individual spin would do in the paired spin, we see three options:
1) (very conservative) We assume that all subjects in Mutual who choose the individual lottery would reject in the paired lottery.
2) We make no assumption on unobserved behavior and simply compare rejection rates in Mutual and Control.
3) We assume that all subjects in Mutual who choose the individual lottery would accept in the paired lottery (and they choose the individual lottery because they do not want to be put in the situation that they have to accept in this a case).

To inform us which interpretation is most appropriate we will use the results from a non-incentivized (hypothetical) question asking whether subjects would have rejected the outcome of the paired lottery.

Hypothesis: percentage of rejections of disadvantaged players in Mutual < percentage of rejections of disadvantaged players in Control

Hypothesis New: percentage of rejections of disadvantaged players in AGREEMENT< percentage of rejections of disadvantaged players in CONTROL

The hypothesis for treatments CONTROL0 and AGREEMENT0 is as before (there was a typo in the old preregistration):

Hypothesis: percentage of rejections of disadvantaged players in Agreement0 < percentage of rejections of disadvantaged players in Control0

Other rejection rates (i.e. rejection rates when receiving the higher payoff and rejection rates when individual lottery is chosen)
Unincentivized fairness assessments of the various situations

Experimental Design Details
Our proposed experiment will run on the online platform Prolific using their subject pool, recruiting subjects from the UK.
After reading the instructions and completing control questions, each subject is assigned a color, either Red or Blue. Subjects know their color when the next steps happen.
Each subject is then either randomly assigned (treatment Control) or chooses (treatment Mutual) one of two options: (1) Play an individual lottery giving 3 pounds with probability 1/4 and 0 pounds else, or (2) playing a paired lottery with another participant of the opposite color. In the paired lottery, there is a 3/4 chance that the blue player earns 10 pounds and the red player zero, and a 1/4 chance that the red player earns 10 pounds and the blue player zero. In treatment Mutual, subjects who choose the paired lottery are subsequently matched to another subject who also chose this and is of the opposite color.
After the outcome of the paired lottery is determined, each subject in the pair can choose to accept or reject the outcome. If both accept, the payoffs from the lottery are paid. If either subject in a pair rejects, both are paid 1 pound instead.

UPDATED Preregistration:

After running treatments Control and Mutual we found no significant differences for the measures described in the preregistration from April 30, 2025.
Therefore, we decided to run a modified version of treatment Mutual, Agreement, which is identical in sample size and game form. The only difference is that subjects have to agree more explicitely that they want to play Spin the Wheel Version 2, e.g. by a pop-up windows that says: “I agree to play Version 2”.

2nd update:

After running treatment Agreement as preregistered on July 17, 2025, we again found no significant differences for the measures described in the preregistration.
We therefore decided to run another two treatments, Control0 and Agreement0, which differ from Control and Agreement, respectively, by what happens after the paired lottery is determined and at least one player rejects the outcome: rather than both receiving 1 pound, they receive 0 pound in the new treatments. This eliminates selfish reasons for rejecting an outcome and should make the power of the agreement stronger.

Done by Prolific

individual

Yes

500 pairs. This is under the assumption that 80% of pairs choose the paired lottery in Mutual. In case this percentage is substantially lower, we would need to increase the number of pairs in Mutual.

Update: plus 250 for new treatment AGREEMENT
2nd Update: plus 500 for new treatments CONTROL0 and AGREEMENT0

500 pairs + 250 pairs + 500 pairs

250 pairs per treatment

Assuming that 80% of pairs choose the paired lottery in Mutual, we would get 200 pairs in the paired lottery. We would then also send around 200 pair in Control in the paired lottery. Of those, 75% would get the zero payoff, which would give us 300 independent observations in total. Assuming a rejection rate of 80% in CONTROL, the minimum detectable effect size would be around 15 percentage points (i.e. a rejection rate of 65% or lower) with alpha=0.05 and a desired power of 80%.