knows that his cost is high and will find it optimal to wait until T.20
If T > c∕2, the mixed strategy equilibrium of Lemma 3b(ii) does not exist. The
intuition is as follows. If there were such an equilibrium, the uninformed individual
i would update his beliefs about his cost following the action of this rival, and
if the game reaches a point in time — ∣ + T — δ, δ positive but small, i would
know almost with certainty that his cost is high. But then, i would not concede in
(—∣ + T — δ, — ∣ + Tj, but instead wait until T.
For T ≥ c∕2, there is an equilibrium where the uninformed individual concedes
immediately (qi (0) = 1). Moreover, contrary to the case of private values, there is
an equilibrium where q<,∙b (0) = 1 and qjH (T) = qi (T) = 1. Here, i knows that his
cost is high if there is no immediate concession of j and thus finds it optimal to wait
until T. In turn, j cannot profitably deviate given that qi (T) = 1.
As in the private values case, we consider the 2 × 2 game of information acquisition
defined by the payoffs in the war of attrition.
Proposition 3 Consider the game of information acquisition with common values
and suppose that Assumption 1 holds.
(i) If T < c∕2, only one individual acquires information in equilibrium;
(ii) if T ≥ c∕2, dependent on the equilibrium selection in the war of attrition (case
(N, I)), only one individual or both individuals acquire information in equilibrium.
In case of T ≥ c∕2, the war of attrition in case (N, I) has two diametrically
opposed equilibria, and decisions on information crucially depend on which of the
equilibria is played. For a small T, however, as in the case of private values, one
individual strategically chooses to remain uninformed of the cost of provision, and
in turn the informed individual concedes immediately if the (common) cost is low.21
20There is no further pure strategy equilibrium because, even if qjL (T) = qjH (T) = 1, i’s
best response is q⅛ (T) = 1. Moreover, there is no mixed strategy equilibrium: intuitively, if j∣.
randomized and i provided the good at some t > 0, i would know that his expected cost would be
higher than c (it becomes more likely that the cost is high); thus, i prefers to wait until T.
21As in Proposition 1, there are two asymmetric equilibria where exactly one individual acquires
information and one symmetric equilibrium where both individuals randomize their information
acquisition decision.
23