  Large language models (LLMs) have been increasingly employed for
(interactive) decision-making, via the development of LLM-based autonomous
agents. Despite their emerging successes, the performance of LLM agents in
decision-making has not been fully investigated through quantitative metrics,
especially in the multi-agent setting when they interact with each other, a
typical scenario in real-world LLM-agent applications. To better understand the
limits of LLM agents in these interactive environments, we propose to study
their interactions in benchmark decision-making settings in online learning and
game theory, through the performance metric of \emph{regret}. We first
empirically study the {no-regret} behaviors of LLMs in canonical
(non-stationary) online learning problems, as well as the emergence of
equilibria when LLM agents interact through playing repeated games. We then
provide some theoretical insights into the no-regret behaviors of LLM agents,
under certain assumptions on the supervised pre-training and the rationality
model of human decision-makers who generate the data. Notably, we also identify
(simple) cases where advanced LLMs such as GPT-4 fail to be no-regret. To
promote the no-regret behaviors, we propose a novel \emph{unsupervised}
training loss of \emph{regret-loss}, which, in contrast to the supervised
pre-training loss, does not require the labels of (optimal) actions. We then
establish the statistical guarantee of generalization bound for regret-loss
minimization, followed by the optimization guarantee that minimizing such a
loss may automatically lead to known no-regret learning algorithms. Our further
experiments demonstrate the effectiveness of our regret-loss, especially in
addressing the above ``regrettable'' cases.
