2x2 delegation games with implementability in weakly undominated SPNE

In this paper we study delegation environments based on Fershtman, Judd, and Kalai ([1987], hereafter FJK). By imposing a certain assumption on the notion of implementability, called implementability with mutually rational agents, they show that every efficient outcome can be fully implemented in subgame perfect Nash equilibrium (SPNE). For their analysis, we first argue that FJK's model can and should be interpreted as a problem in mechanism design. With this in mind, we first modify their model so that agents' participation decision is explicitly built in. Then, we argue that FJK uses a non-standard solution concepts in the mechanism design literature - in order to attain full implementability, they strengthen notion of implementability, instead of using a refined equilibrium concept. In response, we follow the standard mechanism design approach - employ a refined SPNE (weakly undominated SPNE, U-SPNE) as the equilibrium concept, while keeping the notion of the implementability unchanged. By applying U-SPNE, we show that in certain classes of 2x2 games - including prisoners' dilemma, coordination games, and battle of sexes - every efficient outcome is fully implementable in U-SPNE.