It just takes one business trying to maximize profits by gaining market share to begin the process. If it were possible to have 100% of business collude in price fixing, it would be happening. It's not happening. The desire to maximize profits leads to the attempt to gain market share.
You're not quite there yet. Should be "The desire to maximize profits leads to attempts to improve margins". Sometimes that leads to attempts to gain market share in order to either (i) improve margins by amortizing fixed costs over more units, or (ii) get a dominant, semi-monopolistic position so as to set the market price (Jack Welch's approach). Sometimes it just leads to raising the price.