How exactly is a Bertrand equilibrium defined? By 'given the price set by other firm' does this mean the firm knows its competitor's exact output? I had read 3 textbooks, but one describes that they have a precise expectation of what their competitor's output might be but this sentence here sounds as though the firm is informed about their competitor's exact output. Could someone explain? Thanks
What do we mean by "makes the best decision based on what the other player has decided to do"

Your question touches a little bit on the philosophical foundations for Nash Equilibrium. As you know, a Nash Equilibrium occurs when each player chooses the best response to the strategies chosen by others. In other, words, they act as if they know what the strategies of the other players will be, and play the best response accordingly—even if all strategies are chosen simultaneously.

Why does it make sense to model people behaving as if they knew everyone else's strategy? The whole point of looking for an equilibrium of a game is to try to find a way to 'solve' the game such that the solution gives a plausible prediction about how people might play the game. Suppose we have identified a way to compute the solution. If we can compute the solution then the players can compute the solution too (using the same method), and figure out what everyone's equilibrium strategy will be. If the players compute the solution and discover that their equilibrium behavior in the solution does not maximise their own payoff then they will refuse to play the solution! So any solution concept that (a) players can use to solve the game and (b) has one or more players behaving sub-optimally will be self-defeating. Flipped on its head, any solution concept that players are actually willing to go along with must necessarily involve them playing a best response—i.e must be a Nash Equilibrium.

One concise way to phrase this is that a player's strategy can be accurately predicted by the other players (using the same solution concept we are using the solve the game)! Note that this is slightly different to what you wrote in a comment ("the competitor's action can be accurately predicted"). We can use the equilibrium to predict strategies, but might not be able to predict actions (because the strategy might be a mixed strategy, meaning the action is random).



What is a Bertrand Equilibrium

Recall that to set up a static game of complete information we need to know (1) the players, (2) the actions, and (3) payoffs. To turn to your question, a Bertrand Equilibrium is just another name for the Nash Equilibrium of the following particular static game of complete information:


There are N firms (players)
Each firm's action is to choose a price, $p_i$
The payoffs are that the lowest price firm serves all consumers and earns $p_i-c$ per-consumer (where $c$ is the marginal cost). If $m$ firms tie at the lowest price then each of them serves $1/m$ of the consumers. firms that serve no consumers earn zero profit.


So, in a Bertrand equilibrium, using the principle of Nash, each firm acts as if it knows the others' price and chooses the best price in response. 



Characterising the Bertrand Equilibrium?

To find the Nash ("Bertrand") equilibrium of this game, we have to find a set of strategies (prices) such that each firm is playing a best response (i.e., such that no firm could profit by changing its price).

Claim There is no equilibrium in which the lowest firm's price is bigger than $c$.

To prove this claim, suppose on the contrary that $\min p_i&gt;c$. We will show that at least one firm can profitably change its strategy (and is therefore not playing a best response). Order the firms so that $p_1\leq p_2\leq p_3\leq\ldots\leq p_n$. There are two possibilities: 


All firms charge the same price (so $n$ ties at the lowest price). Then $n$'s profit is $(p_n-c)/m$. But suppose $n$ reduced its price by some amount $\Delta$. Then it would have a price strictly below anyone else and earn profit $p_n-\Delta-c)$. If $\Delta$ is small enough, this is a strict increase in $n's$ profit.
$p_n&gt;p_1$ so $n$ does not attract any consumers. $n$'s profit is zero. But $n$ could deviate to choosing a price of $p_1-\Delta$, yielding a profit of $(p_1-\Delta-c)$. Provided $\Delta$ is small enough, this is positive so $n$'s profit has increased. 


Claim At least two firms must set $p=c$ in equilibrium. All firms earn zero profit.

To prove this claim, start by noticing that if some firm sets $p_i&lt;c$ then at least one firm will make negative profits and would prefer to deviate to $p=c$ (thus guaranteeing zero profit).

Suppose that fewer than two firms set $p=c$. By the previous claim, we know at least one firm sets $p=c$. This means precisely one firm sets $p=c$, while all others set $p&gt;c$. But then the firm that sets $p=c$ earns profit $(p-c)=0$, but could get positive profit by increasing $p$ (just a little so it is still the cheapest firm). Because there is a profitable deviation, we can't have an equilibrium with fewer than two firms setting $p=c$.

Lastly, we know the firms that set $p=c$ earn zero profits. We also know that no firm sets $p&lt;c$ so any firm not setting $p=c$ must set $p&gt;c$, meaning it attracts no consumers and also makes zero profit.