In a finite game, suppose player 1 has strategies $\{\alpha_1,\alpha_2\}$ and player 2 $\{\beta_1,\beta_2\}$ with payoffs as below.
\begin{array}{c|c|c} &\beta_1&\beta_2\\ \hline\\ \alpha_1&1,1&0,0\\ \hline \alpha_2&0,0&0,0 \end{array}
A friend of mine told me that the only Nash equilibria are $(\alpha_1,\beta_1)$ and $(\alpha_2,\beta_2)$. But I doubt that.
Considering mixed strategies, if $\sigma_1=(x,1-x),~x\in[0,1],$ is the mixed strategy of player 1 and $\sigma_2=(y,1-y),~y\in[0,1],$ is the mixed strategy of player 2, then player 1's payoff function is $U_1=xy$ and player 2's is $U_2=xy$. Therefore one of Nash's equilibria is $(\sigma_1,\sigma_2)=(1,0,1,0)$, corresponding to pure strategy equilibrium $(\alpha_1,\beta_1)$. But I found that other equilibria are $(\sigma_1,\sigma_2)=(x,1-x,0,1),~x\in[0,1]$.
Then there are equilibria other than $(\alpha_1,\beta_1),~(\alpha_2,\beta_2)$. Is there something wrong with my thought?
Your friend is correct. Consider this intuitively.
If the first player has any chance of playing $\alpha_1$, then the second player's dominant strategy is to play $\beta_1$ all of the time, since he gains a positive expected utility compared to zero utility from $\beta_2$. Similarly, if the second player plays $\beta_1$ with positive probability, the first player will always want to play $\alpha_1$. Only when the other plays his second strategy with $100\%$ probability does a player not gain incentive from deviating to his first strategy.
For the actual calculation: Let $p$ be player 1's probability of choosing $\alpha_1$, and let $q$ be player 2's probability of choosing $\beta_1$. For player 1 to mix, he must be indifferent between the two options. So $$\begin{cases} \pi_1(\alpha_1, q) = (1)(q) + (0)(1 - q) = q \\ \pi_1(\alpha_2, q) = (0)(q) + (0)(1 - q) = 0. \end{cases}$$ Setting $\pi_1(\alpha_1, q) = \pi_1(\alpha_2, q)$ gives us $q = 0$. A similar derivation gets us $p = 0$. So $(0, 1, 0, 1)$ is the unique MNE. Basically, in this case a player will not be willing to mix unless he has no chance of getting positive utility.