"Run the risk" is an expression that refers to exposing oneself to danger, harm or loss. Fortunately, there are a variety of synonyms that can be used in place of this phrase. For example, you might "court danger," "tempt fate," "gamble," "take a chance," "venture," "hazard," or "put oneself in harm's way." Each of these phrases implies that the individual is taking a calculated risk that could result in negative consequences. It's important to consider which synonym best suits the situation at hand, especially when communicating complex ideas in writing or speaking.