Question

Problem 2. A variant of Hotelling’s model of electoral competition can be used to analyze the choices of product characteristics by competing firms in situations in which price is not a significant variable. (Think of radio stations that offer different styles of music, for example.) The set of positions is the range of possible characteristics for the product, and the citizens are the consumers rather than voters. Consumers’ tastes differ; each consumer buys (at a fixed price, possibly zero) one unit of the product she likes best. The model differs from electoral competition in that each firm’s objective is to maximize its market share, rather than to obtain a market share larger than that of any other firm. You can show that when there are two firms, the unique Nash equilibrium is (m, m) (both firms offer the consumers’ median favorite product). That is we have the median voter theorem but in a different context. For this assignment, show that when there are three firms there is no Nash equilibrium. (Hint: Start by arguing that when there are two firms whose products differ, either firm is better off making its product more similar to that of its rival.)

Answer 1

To explain the concept of Nash Equilibrium in a variant of Hoteling’s model of electoral competition let us first start with defining clearly what Nash Equilibrium is, Nash Equilibrium is named after its inventor, John Nash. It is considered one of the most important concepts of Game Theory, which attempts to determine mathematically and logically the actions that participants of a game should take to secure the best outcomes for themselves. The Nash Equilibrium can be incorporated into a wide range of disciplines, from economics to the social sciences. Now keeping in mind the model of electoral competition it can be said the when there are only two firms competing to have a larger pie of the market share rather to get more share than the other firm Nash Equilibrium exists because the strategy taken by one firm has no effect on the strategy taken by the other firm, thus even though the two firms knows each other strategies they stick to it and tries to win over the market.

But when a third party enters the scene Nash Equilibrium cease to exist because now based on the strategy the third firm adopts the existing two firms have to change their old strategy so that their individual objectives could be fulfilled.