Ben-Porath E, Gilboa I, Schmeidler D.

On the Measurement of Inequality under Uncertainty. Journal of Economic Theory. 1997;75 :194-204.

Abstract**On the Measurement of Inequality under Uncertainty**

with Itzhak Gilboa and David Schmeidler

To take into account both ex ante and ex post inequality considerations, one has to deal with inequality and uncertainty simultaneously. Under certainty, much of the literature has focused on "comonotonically linear" indices: functionals that are linear on cones of income profiles that agree on the social ranking of the individuals. This family generalizes both the Gini index and the egalitarian index(minimal income). However, it does not include functionals such as the average of expected-Gini and Gini-of-expectation. In contrast, the family of min-of-means functionals is rich enough for this purpose.

Ben-Porath E.

Rationality, Nash Equilibrium and Backward Induction in Perfect Information Games. Review of Economic Studies. 1997;64 :23-46.

Abstract**Rationality, Nash Equilibrium and Backward Induction in Perfect Information Games**

We say that a player is certain of an event *A* if she assigns probability 1 to *A*. There is common certainty *(CC)* of *A* if the event *A* occurred, each player is certain of *A*, each player is certain that every other player is certain of *A*, and so forth. It is shown that in a generic perfect-information game the set of outcomes that are consistent with common certainty of rationality *(CCR)* at the beginning of the game coincides with the set of outcomes that survive one deletion of weakly dominated strategies and then iterative deletion of strongly dominated strategies. Thus, the backward induction outcome is not the only outcome that is consistent with *CCR*. In particular, cooperation in Rosenthal's (1981) centipede game, and fighting in Selten's (1978) chainstore game are consistent with *CCR* at the beginning of the game. Next, it is shown that, if in addition to *CCR*, there is *CC* that each player assigns a positive probability to the true strategies and beliefs of the other players, and if there is *CC* of the support of the beliefs of each player, then the outcome of the game is a Nash equilibrium outcome.