Volunteering and the Strategic Value of Ignorance

points of Fi are restricted to t_i = 0 and t_i = T.

We proceed in two steps: first we show that the mass points at zero are uniquely
determined, and second we prove that (10) and (11) constitute an equilibrium.

Step 1 : From (10), it follows that b_i < 1 implies F_i (t_i) < 1 for all t_i < T :
whenever i chooses a mixed strategy, there is strictly positive probability that t_i = T.
In particular, we have F_i (—c/2 + T) < 1, which implies that Fj_l (—c/2 + T) = 1.
This is due to the fact that there is a strictly positive probability that i waits until
T, and, as in the case of F_i above, Fj_l must be continuous on (0,T). However, as
Fi is constant in (—c/2 + T, T) and c^ < c, j^ strictly prefers tj_L = —c/2 + T to all
tj_L > —c/2 + T, and therefore Fj_l (—c/2 + T) < 1 contradicts the nonexistence of
interior mass points.

With (11), min (b_i, b_jL) = 0, and Fj_l (—c/2 + T) = 1, we get

bj_L is strictly decreasing in T with lim_τ∣_δ∕₂ bj_L = 1 and lim_τ↑c∕2-ci∏p_H bj_L = 0. Hence,
c/2 < T < c/2 — clnp_H is a necessary condition for the existence of a mixed strategy
equilibrium.²⁴

Step 2 : It remains to show that (10), (11) and (12) indeed constitute an equilib-
rium. Consider first individual i and suppose that j follows Fj_l and Fj_h , respectively.
For any t_i ∈ (0, —c/2 + T], i’s expected payoff is
which is equal to V — (1 — p_L) exp (—1/2 + T/c) c. If i concedes in T, he gets

r~F^+ττ 1 — p_L 1 i ₊ τ-      _t c /c ∖

— Pl         X------' ^{2 +} ð dx — (1 — Pl)(t+ T)

Jo          Pl c                        22 J

²⁴To be precise, if T = c/2 — clnpχ, we get bj_L = 0 and b_i > 0 is not uniquely determined.
Hence, there exists a continuum of mixed strategy equilibria where i’s payoff is υ — c, as in the pure
strategy equilibrium. We omit this case in order to simplify the exposition.

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