  An unconventional approach for optimal stopping under model ambiguity is
introduced. Besides ambiguity itself, we take into account how ambiguity-averse
an agent is. This inclusion of ambiguity attitude, via an $\alpha$-maxmin
nonlinear expectation, renders the stopping problem time-inconsistent. We look
for subgame perfect equilibrium stopping policies, formulated as fixed points
of an operator. For a one-dimensional diffusion with drift and volatility
uncertainty, we show that every equilibrium can be obtained through a
fixed-point iteration. This allows us to capture much more diverse behavior,
depending on an agent's ambiguity attitude, beyond the standard worst-case (or
best-case) analysis. In a concrete example of real options valuation under
volatility uncertainty, all equilibrium stopping policies, as well as the best
one among them, are fully characterized. It demonstrates explicitly the effect
of ambiguity attitude on decision making: the more ambiguity-averse, the more
eager to stop -- so as to withdraw from the uncertain environment. The main
result hinges on a delicate analysis of continuous sample paths in the
canonical space and the capacity theory. To resolve measurability issues, a
generalized measurable projection theorem, new to the literature, is also
established.
