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 pro le is the limit point of a subgame perfect equilibrium path – the ultimate cumulative contribution vector of the players. A pro le is shown to be achievable only if it is in the undercore of the underlying coalitional game, i.e., the pro le cannot be blocked by a coalition using a component-wise smaller pro le. Conversely, if free-riding incentives are strong enough that contributing zero is a dominant strategy in the stage games, then any undercore pro le is the limit of achievable pro les 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 ef cient pro le can be approximately achieved if and only if it is in the core of the underlying coalitional game; and (iii) any achievable pro le can be achieved almost instantly.