Theory for Treatment
Rados suggestions
About the theory. I considered many of alternatives now. As we spoke with Wendelin the other day what would be nice to have is something simple. There are two alternatives for simple:
- fairness norms + loss function
- simple social preferences.
Suppose we have three people: person 1 gets x, person 2 gets y and person 3 gets z. Suppose person 1 is in the minority.
The first approach would suggest a utility to look something like: u(x,y,z) = x - g(x - x_n1)^2 , where n is a my payoff under a norm n1 and g>0 is a scalar. So n is determined by the norm based on x,y, and z. Then you can compare several norms, such as, payoff equality norm and equality of opportunity norm. But you can imagine others... There are many examples of this in the literature. But, none of these norms that I considered captures the essential feature of our problem, namely, that my choice affects payoffs of others. Choices have externalities and therefore efficiency is a factor. I don't know I do not see a satisfactory path via norms....
Therefore, I favor the second approach via simple soc. preferences. Suppose we have a utility function:
u(x,y,z) = x + f(x+y+z), where f is a concave function (it could be that own payoff is also concave but let me keep it linear for now). It is easy to see how conformism comes about. In VETO tr. I have a choice of either u(1,0,0) or u(0,1,1) so 1 vs. f(2). If the latter is larger than the former, individual 1 conforms. Our tr. difference comes from combining the concavity of f with the ex-ante payoff comparisons. Obviously for ex-post payoff comparisons there is no tr. difference. So let us consider ex-ante comparisons. That is, before it is decided whether the majority choice is enforced or not and before the rand. dictator is chosen. If f is linear, then there is no predicted difference between our treatments. But f is concave, so I care less and less about each additional dollar that others get, then we get prediction that conformism should be stronger in the VETO than in the RD treatment. This is also what we observe. In the RD: the comparison is u(1/6,5/6,5/6) vs u(0,1,1) if I do not conform vs. conform. In VETO the comparison is u(1,0,0) vs u(0,1,1). The intuition is that in the RD, if I do not conform then I am gaining a little on my end and also shading a little from others (in expectations). But because my f is concave, shading from others costs me less than gaining a little of my payoff. So that is a good deal for me. I will therefore choose to not conform.
It's a simple story. I had this written up back in November before I went into the Cappelen et al. business but then I dismissed this because it wasn't exactly an off-the-shelf model + I was not entirely convinced by the intuition... But now this is really growing on me.
Wendelins Question
Are decision makers not expected utility maximizers? Or am I calculating something wrong.
u(x,y,z) = x + f(x+y+z), where f is a concave function (it could be that own payoff is also concave but let me keep it linear for now). It is easy to see how conformism comes about. In VETO tr. I have a choice of either u(1,0,0) or u(0,1,1) so 1 vs. f(2). If the latter is larger than the former, individual 1 conforms.
I fully agree with that part. But why is the RD case not evaluated as follows?
payoff when insisting 1/6 u(1,0,0) + 5/6 u(0,1,1) payoff when conforming 6/6 u(0,1,1)
Decide for insisting when u(1,0,0) > u(0,1,1) , so 1 vs. f(2), which is exactly the same as in the VETO treatment.
Where does the concavity kick in if I use the probabilities (where I think they belong): outside the utility function?
Am I overlooking something?