Achievable Outcomes in Smooth Dynamic Contribution Games

This paper studies a class of dynamic voluntary contribution games in a setting with discounting and neoclassical payoffs (differentiable, strictly concave in the public good, and quasilinear in the private good). An achievable profile is the limit point of a subgame perfect equilibrium path -- the ultimate cumulative contribution vector of the players. A profile is shown to be achievable only if it is in the undercore of the underlying coalitional game, i.e., the profile cannot be blocked by a coalition using a component-wise smaller profile. Conversely, if free-riding incentives are strong enough that contributing zero is a dominant strategy in the stage games, then any undercore profile is the limit of achievable profiles as the period length shrinks. Thus, in this case when the period length is very short, (i) the set of achievable contributions does not depend on whether the players can move simultaneously or only in a round-robin fashion; (ii) an efficient profile can be approximately achieved if and only if it is in the core of the underlying coalitional game; and (iii) any achievable profile can be achieved almost instantly.