  We study the classical and quantum values of one- and two-party linear games,
an important class of unique games that generalizes the well-known XOR games to
the case of non-binary outcomes. We introduce a ``constraint graph" associated
to such a game, with the constraints defining the linear game represented by an
edge-coloring of the graph. We use the graph-theoretic characterization to
relate the task of finding equivalent games to the notion of signed graphs and
switching equivalence from graph theory. We relate the problem of computing the
classical value of single-party anti-correlation XOR games to finding the edge
bipartization number of a graph, which is known to be MaxSNP hard, and connect
the computation of the classical value of more general XOR-d games to the
identification of specific cycles in the graph. We construct an orthogonality
graph of the game from the constraint graph and study its Lov\'{a}sz theta
number as a general upper bound on the quantum value even in the case of
single-party contextual XOR-d games. Linear games possess appealing properties
for use in device-independent applications such as randomness of the local
correlated outcomes in the optimal quantum strategy. We study the possibility
of obtaining quantum algebraic violation of these games, and show that no
finite linear game possesses the property of pseudo-telepathy leaving the
frequently used chained Bell inequalities as the natural candidates for such
applications. We also show this lack of pseudo-telepathy for multi-party
XOR-type inequalities involving two-body correlation functions.
