  Strategic agents in incomplete-information environments have a conflicted
relationship with uncertainty: it can keep them unpredictable to their
opponents, but it must also be overcome to predict the actions of those
opponents. We use a multivariate generalization of information theory to
characterize the information processing behavior of strategic reasoning
experts. We compare expert and novice poker players --- "sharks" and "fish" ---
over 1.75 million hands of online two-player No-Limit Texas Hold'em (NLHE).
Comparing the effects of privately known and publicly signaled information on
wagering behavior, we find that the behavior of sharks coheres with information
that emerges only from the interaction of public and private sources ---
"synergistic" information that does not exist in either source alone. This
implies that the effect of public information on shark behavior is better
encrypted: it cannot be reconstructed without access to the hidden state of
private cards. Integrative information processing affects not only one's own
strategic behavior, but the ability of others to predict it. By characterizing
the informational structure of complex strategic interactions, we offer a
detailed account of how experts extract, process, and conceal valuable
information in high-uncertainty, high-stakes competitive environments.
