Question: Consider Bertrand's duopoly game in which the set of possible prices is discrete. Does the sequences of prices under best response dynamics converge to a Nash equilibrium when both prices initially exceed c+1? What happens when both prices are initially equal to c?
Attempt: In best response dynamics, each player is choosing based on what the other player has played previously. So, when prices initially exceed c+1, each player will choose something lower until it converges to c. However in the discrete case, the Nash equilibrium is c+1 so that both players will receive a positive payoff instead of just a zero payoff. So would prices actually converge to c+1 in this case? I'm having a hard time seeing this.
I also don't know where to start if both prices are initially equal to c.
here is some more working that I have done. I am confused in the case where they both start at c if they will remain at c or if they will end up at c+1.
Assuming that $c+1$ is the minimum bid greater than $c$, I agree with your analysis. If both prices exceed $c+1$, we'll get to $c+1$ through best response dynamics and then stay there in the NE.
If prices start at $c$ (at which point profits are zero), then a deviation to $c-1$ would yield a loss. A deviation to $c+1$ would lead to you getting no market share, and thus also profits of zero. Therefore, $(c,c)$ is also a NE -- there is no strict incentive to deviate. In terms of best response dynamics, your answer depends on how you define the dynamics. If players play randomly over all options which best respond to the opponent's last play, then we'll eventually end up at $(c+1,c+1)$, because $BR(c)=\{c,c+1,c+2,\ldots\}$. If instead players only change their prices when they have strict incentives to do so, then $(c,c)$ can be be stable under best response dynamics.