How does the kinked-demand curve explain price rigidity in an oligopoly?

Price rigidity is a situation which is characterized by less movement of prices. It is experienced by oligopolists because they face a demand curve that has different elasticity levels at different prices. The reason behind such a type of demand curve is that the rival businesses are unlikely to equal another firm's price rise, though they may equal a price cut. So, the moves of one of the manufacturers affect the decisions of other manufacturers also. If one of the manufacturers reduces the price to gain market share, others may do the same. Though this move might result in profit in the short-run, it would lead to a loss in the long run. So, a manufacturer would not find it profitable to alter his/her prices. This is the same for the situation in case of increasing the price. Due to alternatives present in the market, no manufacturer would opt to increase his/her prices. So, in oligopolies, a manufacturer cannot change prices immediately when something happens, and this is price rigidity.