  We evaluate the best-response algorithm in the context of pure-integer convex
quadratic games. We provide a sufficient condition that if certain interaction
matrices (the product of the inverse of the positive definite matrix defining
the convex quadratic terms and the matrix that connects one player's problem to
another's) have all their singular values less than 1, then finite termination
of the best-response algorithm is guaranteed regardless of the initial point.
Termination is triggered through cycling among a finite number of strategies
for each player. Our findings indicate that if cycling happens, a relaxed
version of the Nash equilibrium can be calculated by identifying a Nash
equilibrium of a smaller finite game. Conversely, we prove that if every
singular value of the interaction matrices is greater than 1, the algorithm
will diverge from a large family of initial points. In addition, we provide an
infinite family of examples in which some of the singular values of the
interaction matrices are greater than 1, cycling occurs, but any mixed-strategy
with support in the strategies where cycling occurs has arbitrarily better
deviations. Then, we perform computational tests of our algorithm and compare
it with standard algorithms to solve such problems. We notice that our
algorithm finds a Nash equilibrium correctly in every instance. Moreover,
compared to a state-of-the art algorithm, our method shows similar performance
in two-player games and significantly higher speed when involving three or more
players.
