Volunteering and the Strategic Value of Ignorance

In the analysis of the war of attrition, if individual i knows his cost of provision,
we will denote by i_L (⅛) player i with low (high) cost of provision. Moreover, we will
have to allow individuals to randomize their concession time. Consequently, a mixed
strategy of an uninformed individual i ∈ {1,2} will be a cumulative distribution
function F_i. Moreover, q_i (t) will be the probability that i concedes exactly at t,
and it will be employed to describe pure strategies. If i acquires information, we
denote by F_iL (F_iH ) the distribution function that corresponds to the mixed strategy
i chooses when his cost is low (high). Again, we will use q_iL (t) and q_iH (t) to describe
type-contingent pure strategies in case i acquires information.

Mixed strategies that individuals choose in the different continuation games will
exhibit a common structure. For this purpose, we define a function Φ as

( 1 — (1 — q⁰) e (,  0 ≤ t < t

Φ (t; c,f,q⁰) = <  1 — (1 — q⁰) (ɪ,  t ≤ t<T

[ 1,                  t > T

Φ (t; c, t^, q⁰) describes a cumulative distribution function of concession times t with
positive mass in the interval (0,t), no mass in (t, T), and possibly a mass point at
zero (of size q⁰) and/or a mass point at T.

No individual knows his cost of provision. If neither of the individuals knows
his true provision cost, both choose their waiting time based on their expected cost
c, and the volunteering game is strategically equivalent to the war of attrition with
complete information.⁹

Consider individual i and suppose that j waits until T with probability one. If
i concedes in t_i < T, his expected payoff is V — c — t_i. For t_i = T, he gets a payoff
of V — c/2 — T. Thus, if T < c/2, t_i = T is strictly preferred to any t_i < T,
and there is an equilibrium where both wait until T with probability one, which
is the unique equilibrium. If, however, T > c/2, i’s best response to t₇∙ = T is to

⁹This holds because individuals are assumed to be risk-neutral and the payoffs are linear in the
provision cost. Thus maximizing expected payoffs is equivalent to the maximization based on the
expected cost.

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