Bertrand Duopoly Equilibrium for Discrete Prices There are two identical firms, $1$ and $2$, with zero marginal costs. They produce homogenous product, which is demanded by a unit mass of identical consumers, each of which has inelastic unit demand with a reservation price of $2$. Prices are constrained to take only integer values. Using standard game theoretic reasoning, determine whether or not each possible pair $(p_1,p_2)$ can be regarded as Bertrand Nash equilibrium.


Would the Nash equilibria be $(p_{1}^{*},p_{2}^{*})=$ $\{(0,0),(1,1)\}$. 
For symmetric prices, $\{(2,2)\}$ since the demand would be $0$, I can see how unilateral deviation to $1$ could increase profits for a firm. 
However for prices $\{(0,0),(1,1)\}$, no unilateral profits are possible. For other non symmetric price pairings, unilateral deviations are possible. Therefore, the Nash equilirium prices are $\{(0,0),(1,1)\}$. 
Is my reasoning correct?
The problem formulation admits the following Normal Form representation. We can reject any strategy involving price greater than 2, as demand falls to zero and such strategies are strictly dominated by those for which prices are either 1 or 2.

    0         1         2

0 [0,0]     [0,0]     [0,0]    

1 [0,0]   [0.5,0.5]   [1,0]

2 [0,0]     [0,1]     [1,1]


With this setup you can see that there are three Nash equilibria: (0,0), (1,1) and (2,2). At (0,0), firms are indifferent to the outcome of a higher price, and will not deviate. At (1,1), Firm 1 will not deviate to 2, nor will Firm 2 deviate to 2, as this cedes the market to the opponent. At (2,2) firms are indifferent to reducing price to 1.  

(0,1), (0,2), (1,0), and (2,0) are strictly dominated by deviating to a price-matching strategy, splitting the market between firms.

Similarly, (1,2) and (2,1) fail to be Nash equilibria because whichever firm opens with a price of 2 can capture half the market by deviating to a price of 1.