Lost opportunity of provision at T. The provision of many public goods is
allocated on a voluntary basis, but it is compulsory in the sense that one individual
has to contribute. In companies, for instance, a team leader may select one individual
if no one volunteers. Other public goods can only be provided within a certain period,
after which the opportunity of provision disappears.
Instead of assuming that at T one individual is randomly selected, suppose that
the investment opportunity disappears if no individual has conceded before T. In this
case, the incentive to wait until T in the war of attrition is weakened; the analysis,
however, qualitatively carries over from the previous section if we modify Assumption
1 on the time limit such that individuals with a high cost do not want to provide
the public good and individuals with a low cost prefer to concede. This requires the
time limit to be such that:
Assumption 1’ 0 < — (n — ⅛) < T < — (n — c∏).
Hence, high types have a (weakly) dominant strategy to wait until T. Moreover,
if T is sufficiently small (T < — (n — c)) and only individual j knows his contribution
cost, the uninformed individual i waits until T (as in Lemma 3a). For a larger T,
there is an equilibrium where i and jɪ, randomize on some interval [0, t∖ U{T} (similar
to Lemma 3b(ii)) and jL has a mass point at zero.
Proposition 4 Consider the game of information acquisition and suppose that at T
the opportunity of provision disappears. If Assumption 1’ holds and T is sufficiently
small, only one individual acquires information in equilibrium.
We do not provide a complete analysis of equilibria of the war of attrition22 and
incentives to acquire information, but we show that, whenever T is small, there is a
strategic value of ignorance: as in the previous section, remaining uninformed can be
used as a commitment not to concede if an individual’s expected cost of provision is
sufficiently high in relation to the payoff from waiting until T.
22A detailed analysis of equilibria of the war of attrition would build on Theorems 1-3 in Hendricks
et al. (1988).
24