Why is signaling commitment a strategic entry barrier?

This is a game theoretic argument for strategic entry barrier. Whether there is entry or not depends very much on what entrants expect what incumbent will do (and vice-versa) and then base their optimal decisions on it. The idea is that if the incumbent has incurred a large sunk investment such as production facilities, the incumbent signals there will be a price war if someone enters the market. One key assumption (more details below): when there is a price war, incumbent can realize economies of scale due to the sunk investment (for example, has invested in overcapacity, so can adjust output easily with decreasing costs).