Volunteering and the Strategic Value of Ignorance

Lemma 1 Consider the war of attrition for a given information structure. In any
equilibrium of the war of attrition, there is zero probability that individual i with cost
c_i provides the public good in ( — ^f + T, T).

For a large T, it will always be an equilibrium of the volunteering game that an
individual j volunteers immediately. In this case, the equilibrium strategy of i is not
uniquely determined, and he may choose a concession time t_i ∈ (—c_i∕2 + T, T), given
that in equilibrium he will not provide the public good. Any t_i ∈ (—c_i∕2 + T,T),
however, is weakly dominated, and whenever there is positive probability that j waits
until T, individual i (with cost c_i) strictly prefers t_i = T to any t_i ∈ (—c_i∕2 + T, T).
If T < c_i∕2, we have —Ci/2 + T < 0, and i prefers the random selection in T to a
contribution in any t_i < T. Lemma 1 holds independently of c_i being i’s true or
expected cost of provision; therefore, it can also be employed if individual i decides
not to become informed.

In what follows, we will focus on the case of an intermediate time limit T:

Assumption 1 ɪ < T < ɪ.

As will become clear in the remainder of this section, Assumption 1 implies
that an individual with high cost will find it optimal to wait until T, accepting the
consequence that he might be randomly chosen to fulfill the task. An individual with
low cost will prefer an early concession if the rival waits sufficiently long.⁸

Building on this assumption, we first determine the equilibria of the volunteering
game conditional on the decisions in stage 1, and we then analyze the incentives to
become informed in a 2 × 2 game defined by the ex ante expected payoffs in the war of
attrition. Ex ante expected payoffs are defined as the individuals’ expected payoffs
given the decisions on information, but before they find out about their provision
cost.

⁸This assumption ensures the strategic role of the information acquisition because the equilibrium
of the volunteering game will crucially depend on the individuals’ decisions whether or not to find
out about their cost of provision. If T > c∏/2, there is always an equilibrium of the war of
attrition where one individual concedes immediately, independently of the decisions in stage 1 and
the individuals’ true provision cost. If T < ¾∕2, in the unique equilibrium of the war of attrition,
both individuals wait until T independently of the stage 1 decisions and their true cost.

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