  We study the extent to which it is possible to approximate the optimal value
of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally,
we prove lower bounds against the accuracy of FPC-interpretations that map
Unique Games instances (encoded as relational structures) to rational numbers
giving the approximate fraction of constraints that can be satisfied. We prove
two new FPC-inexpressibility results for Unique Games: the existence of a
$(1/2, 1/3 + \delta)$-inapproximability gap, and inapproximability to within
any constant factor. Previous recent work has established similar
FPC-inapproximability results for a small handful of other problems. Our
construction builds upon some of these ideas, but contains a novel technique.
While most FPC-inexpressibility results are based on variants of the
CFI-construction, ours is significantly different. We start with a graph of
very large girth and label the edges with random affine vector spaces over
$\mathbb{F}_2$ that determine the constraints in the two structures.
Duplicator's strategy involves maintaining a partial isomorphism over a minimal
tree that spans the pebbled vertices of the graph.
