The Nash equilibrium of the classical Bertrand model is that the price of both firms equals to their marginal cost.
Now, if one of the firms has a cost advantage, i.e., the marginal cost of the firm $i$ is $c_i$ and $c_1<c_2$, and their profit functions are
$$
\pi_i(p_1,p_2)=\begin{cases}
0,&p_i>p_j,\\
q(p_i-c_i)/2,&p_i=p_j,\\
q(p_i-c_i),&p_i<p_j.
\end{cases}
$$
where $i=1,2$ and $j=3-i$.
I think the equilibrium should be $p_1$ is "a little bit" less than $c_2$ and the firm 2 doesn't sell anything. But, as they can set their price "continuously", is the word "a little bit" available or "right" to describe a Nash equilibrium?
Best Answer
No, if prices are continuous, no equilibrium exists. This is usually called the "open set problem". As you write in your explanation the player with the cost advantage wants to undercut the other player by a small amount, say $\epsilon$ to $c_2-\epsilon$. However, for all $\epsilon>0$, there is a better strategy. For example, setting $p_1=c_2-\epsilon/2$. With a similar argument you can rule out mixed strategies.
There are usually two approaches to fix this. First, assume that prices have to be discrete. In this case, the equilibrium is that the weak player chooses the lowest price on the price grid that is weakly above $c_2$ and the strong player chooses the next lower price on the price grid. Second, assume that ties are broken in favor of the stronger player. That is, if both players set the same price, the strong player gets all the demand. This can make sense if you interpret the Bertrand game as a first-price auction.