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In the context of game theory, Nash Equilibrium refers to a situation where no player can benefit by unilaterally changing their strategy, provided the strategies of the other players remain unchanged. In Bertrand's duopoly, each firm aims to set a price for its product. If Firm A sets a price, Firm B will adjust its price to capture the market. If both firms set the same price, they split the market equally. The concept of Nash Equilibrium in this scenario means both firms choose a pricing strategy where neither benefits from changing their price if the other's price remains constant.
For our problem, if both firms set their prices equal to their unit cost, which is 'c', neither has an incentive to lower the price because they would make a loss. Similarly, setting a higher price would result in losing market share to the competitor. Therefore, setting prices at \(c, c\) provides a stable outcome where no firm gains by unilaterally changing their price.
This situation exemplifies the unique Nash Equilibrium for this Bertrand duopoly model.

A demand function tells us the quantity of goods that will be demanded at different price levels. In this problem, the demand function is denoted by \(D(p)\) and given as any function for which \(D(p) \geq 0\) for all \(p\). This means the demand cannot be negative and exists for any price. Furthermore, there is a price \(\bar{p}\) where demand remains positive as long as the price does not exceed \(\bar{p}\).
This specific formulation ensures that demand decreases as price increases, which is a normal expectation in economics — higher prices typically lead to lower demand. For instance, a linear demand function like \(D(p)=\alpha-p\) tells us that demand decreases linearly with price. Here, \(\alpha\) represents a base demand level when the price is zero, and the negative \(p\) coefficient shows the decrease in demand with each unit increase in price.
Understanding the demand function is crucial because it determines how much market share a firm can capture at different prices, which directly influences their pricing strategies in the Bertrand duopoly model.

Game Theory is a framework for understanding strategic interactions among rational decision-makers. It is used to model situations where outcomes depend on the interactions of multiple agents, each trying to maximize their own payoff.
In the Bertrand duopoly setting, game theory helps us analyze how two firms choosing prices simultaneously impacts their market shares and profits. Each firm’s strategy must consider the competitor’s potential actions. For example, if one firm lowers its price below the competitor, it captures the entire market. However, if both firms continuously undercut each other, they may end up selling at a price equal to their cost, leading to zero economic profit.
Game theory concepts like Nash Equilibrium provide a way to find stable outcomes in these competitive scenarios. By examining how firms respond to each other's pricing strategies, game theory helps us understand that the equilibrium in Bertrand duopoly occurs when both firms set their prices equal to their unit cost, resulting in no firm having an incentive to deviate from this pricing strategy.