Competition

Webster's Dictionary of the English Language

·noun The act of seeking, or endeavoring to gain, what another is endeavoring to gain at the same time; common strife for the same objects; strife for superiority; emulous contest; rivalry, as for approbation, for a prize, or as where two or more persons are engaged in the same business and each seeking patronage;

— followed by for before the object sought, and with before the person or thing competed with.