Rick Blaine (Humphrey Bogart), in the first 23 minutes of Casablanca, is shown to be an incredible enigma, a mercurial and wounded figure who deliberately eschews attachment to any cause or person, choosing for one reason or another to be strictly business with regards to his business and the varying forces at play in Casablanca. This is mostly conveyed in the first part of the film with his interactions with both Ugarte (Peter Lorre) and Captain Renault (Claude Rains), when an issue of neutrality is made apparent in his club.
When we first see Rick, he is shown to be very protective of his bar, kicking out someone he finds to be impetuous and risky. When he first sees Ugarte, Ugarte treats him as a friend, which Rick reacts to begrudgingly, reacting with clipped negatives and quips to earnest attempts to engage with him. This shows just how guarded Rick is as a character, as he is very selective with who and what he gets himself involved in. His placement at a chess table during his conversation with Ugarte symbolically likens him to a strategist, someone who always plays both sides and never makes any risky moves.
After his conversation with Renault, which is cordial yet distant, his final expression of his attempt to emotionally distance himself from the conflict comes when Ugarte runs toward him, begging for his help. Doing nothing and saying nothing to him, he allows Ugarte to be hauled away. When someone says to him, "When they come for me, I hope you'll be more of a help," he replies coolly, "I stick my neck out for nobody." This solidifies his desire to keep his head down as much as possible in these trying circumstances, and to not risk anything of himself.