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In this chapter we investigate how rationality criteria for extensive form games react to certain transformations of games, that is, functions which map an extensive form game to some new game. In particular, it is analyzed whether the set of conjecture profiles selected by the criterion in the new game “coincides” with the set of conjecture profiles selected in the original game. If this is the case, we say that the criterion is invariant to the transformation under consideration. Invariance is a desirable property for a rationality criterion if one believes that the transformation does not affect the “strategic features” of the game.

In this article three different types of loss aversion equilibria in bimatrix games are studied. Loss aversion equilibria are Nash equilibria of games where players are loss averse and where the reference points-points below which they consider payoffs to be losses-are endogenous to the equilibrium calculation. The first type is the fixed point loss aversion equilibrium, introduced in Shalev (2000; Int. J. Game Theory 29(2):269) under the name of 'myopic loss aversion equilibrium.' There, the players' reference points depend on the beliefs about their opponents' strategies. The second type, the maximin loss aversion equilibrium, differs from the fixed point loss aversion equilibrium in that the reference points are only based on the carriers of the strategies, not on the exact probabilities. In the third type, the safety level loss aversion equilibrium, the reference points depend on the values of the own payoff matrices. Finally, a comparative statics analysis is carried out of all three equilibrium concepts in 2 x 2 bimatrix games. It is established when a player benefits from his opponent falsely believing that he is loss averse.

In this paper we look for a solution to a land division problem that could be applied to different types of disputes when the arbitrator has a very limited information about the agents’ preferences. The solution must be fair and efficient under the constraint of the limited information available to the arbitrator. To this scope, we propose to use the concept of equal-opportunity equivalence defined by Thomson (Soc Choice Welfare 11:137–156 1994). We prove the existence of an efficient and equal opportunity equivalent allocation for a land division problem and we present a simple procedure to implement a rule that selects such allocation at each preference profile.