A note on budget constraints and outside options in common agency

Bernheim and Whinston (Q J Econ 101:1–31, 1986) show that, in a common agency problem without budget constraints, the set of Nash equilibria with truthful strategies (TNE), the set of coalition‐proof Nash equilibria (CPNE), and the principal‐optimal core of the underlying coalitional game are non‐empty and all equivalent in payoff space. We show that, with budget constraints, none of Bernheim and Whinston’s (Q J Econ 101:1–31, 1986) results hold: (i) a CPNE may not exist, (ii) a TNE may not exist even when a CPNE exists, (iii) a TNE may not be a CPNE, and (iv) both TNE and CPNE payoffs are core allocations but are not necessarily principal‐optimal. However, when principals have outside options but no budget constraints, (i), and (iii) continue to hold but not for (ii) and (iv). In particular, a TNE always exists but the core may be empty.