# Mistake in the "price war" game analysis

Five years ago when analyzing rational strategies in a game I have made a mistake. I was so fascinated that for some parameter sets mixed strategy can be used, that I have forgotten, that pure strategy equilibria might still exist and be more attractive than the mixed strategy equilibrium. In this post, I share the updated app.

## Interactive app

The app below shows \( p(N) \) plot, where dependent variable \( p \) is the probability that \( p_1 \) price will be charged in a market with two competing sellers and \( 100 \) customers. The independent variable \( N \) is the number of customers each seller could serve. Note that manufacturing cost is fixed at \( p_0 = 0.5 \). See the original post for a more detailed discussion of the underlying model.

One could see \( N \) as being an indicator of competition: if \( N=100 \) both sellers try to take over the market, and thus low price is being charged. On the other hand, if \( N = 50 \) both sellers are content on sharing the market and high price is charged by both. For intermediate \( N \) there are three equilibria: two pure strategy equilibria (high price and low price), and one mixed strategy equilibria. In the original post I have only taken into account the mixed strategy equilibrium, but the updated app shows all three curves.