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.