Volunteering and the Strategic Value of Ignorance



points of Fi are restricted to ti = 0 and ti = 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 bi1 implies Fi (ti) 1 for all ti < T :
whenever i chooses a mixed strategy, there is strictly positive probability that ti = T.
In particular, we have
Fi (c/2 + T) 1, which implies that Fjl (—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 Fi above, Fjl must be continuous on (0,T). However, as
Fi is constant in (c/2 + T, T) and c^ < c, j^ strictly prefers tjL = c/2 + T to all
tjL c/2 + T, and therefore Fjl (c/2 + T) 1 contradicts the nonexistence of
interior mass points.

With (11), min (bi, bjL) = 0, and Fjl (c/2 + T) = 1, we get

bi = 0 and bjL = — 1 (1 Pl) exp
Pl L


(-1∙∙ T)∣


(12)


bjL is strictly decreasing in T with limτδ2 bjL = 1 and limτc∕2-ci∏pH bjL = 0. Hence,
c/2 < T < c/2 clnpH 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 Fjl and Fjh , respectively.
For any
ti (0, c/2 + T], i’s expected payoff is
which is equal to
V (1 pL) exp (1/2 + T/c) c. If i concedes in T, he gets

- Pl


pti

Jo


1

x-


- pL 1 -i + τ-x

-C 2 '

Pl c


— I, τ-ti

dx (1 pL) c- 2+ c


(c + ti)


r~F+ττ 1 pL 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 bjL = 0 and bi > 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.

29



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