Suppose I want to divide a pie between you and another person.  It is known that the other person would get value $p$ from a fraction $p$ of the pie (that is, each “unit” of pie is worth 1 to him), but your value is known only to you.  You value a fraction $(1-p)$ of the pie at $\theta (1-p)$ dollars but nobody but you knows what $\theta$ is.

My goal is to allocate the pie efficiently.  If both of you are selfish, then this means that I would like to give all the pie to him if $\theta < 1$ and all the pie to you otherwise.  And if you are selfish then I can’t get you to tell me the truth about $\theta$.  You will always say it is larger than 1 in order to get the whole pie.

But what if you are inequity averse? Inequity aversion is a behavioral theory of preferences which is often used to explain non-selfish behavior that we see in experiments.  If you are inequity averse your utility has a kink at the point where your total pie value equals his.  When you have less than him you always like more pie both because you like pie and because you dislike the inequality.  When you have more than him you are conflicted because you like more pie but you dislike having even more than he has.

In that case, my objective is more interesting than when you are selfish.  If $\theta$ is not too much larger than 1, then both you and he want perfect equity.  So that’s the efficient allocation.  And to achieve that, I should give you less pie than he because you get more value per unit.  And now as we consider variations in $\theta$, increases in $\theta$ mean you should get even less!  This continues until $\theta$ is so much larger than 1 that your value for more pie outweighs your aversion to inequity, and now you want the whole pie (although he still wants equity.)

And its now much easier to get you to tell me the truth.  You will always tell me the truth when your value of $\theta$ is in the range where perfect equity is the unique efficient outcome because that way you will get exactly what you want.  Beyond that range you will again have an incentive to lie about $\theta$ to get as much pie as possible.

So inequity aversion has a very clear implication for an experiment like this.  If the experimenter is promising always to divide the pie equitably and is asking the subject to report his value of $\theta$, then inequity averse subjects will do only two possible things:  tell the truth, or exaggerate their value as much as possible.  They will never understate their value.

I would be curious to see if there are any experiments like this.