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Trying to calculate the prices charged by each firm, I don't understand why the solution argues that all firms set the same price. If I use the assumption that they are the same, I can figure out the value. I just don't understand why they should be the same.

For those unfamiliar with a Salop circle, Levy and Reitzes describe one, depicted below with evenly spaced firms (A through F). Consumers are uniformly distributed along a circle of unit circumference (say, on every point of the circle is a consumer, and the circle has a circumference of $1$) and served by one firm only. Firms only compete directly with their neighbors. Using profit maximization strategy and finding the market segements for each firm (through finding the indifferent consumers), the price set by firms is given by

$$ p_i=\frac{1}{4} (p_{i+1}+p_{i-1} ) + \frac{1}{2} (t/N^2+c), $$ where $p_{i+1}$ ($p_{i-1}$) is the price set by the clockwise (counterclockwise) rival, $t$ represents the cost consumers experience per unit travel, $N$ is the number of firms in the market, and $c$ is the marginal production cost (this is the same for all firms).

Salop circle with evenly spaced firms

But here is the part that I don't understand. From this, it is then argued that with a Bertrand-Nash equilibrium, firms cannot directly influence the prices of their rivals. And therefore the prices should be the same.

This I already find strange: in the formula above, you would naturally argue that the prices set by firms $i+1$ and $i-1$ have an influence on that of $i$. And even if you were to assume they cannot influence each other's prices directly, how would you interpret the equation above? And how could you derive that the prices set by all firms are the same? Would it be derived mathematically from the formula above, or would it be argued by economic intuition in words?

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tl;dr: Prices are identical because firms are identical.

Profits of firm $i$ depend on own price $p_i$ and on neighbors' prices $p_{i-1}$ and $p_{i+1}$. Prices are set simultaneously, therefore competition in this market is described by a (normal form) game. A strategy is a price, and your formula for $p_i$ is the best response function, showing firm $i$'s profit maximizing price, given its neighbors' prices. Since consumers are distributed uniformly along the circle and the firms are equidistant and have the same marginal costs, the game is symmetric. It is therefore natural to look for a symmetric Nash equilibrium as a solution of the game. To do this, you assume that your own price and your neighbors' prices are symmetric, i.e., identical, while at the same time your price is the best response to their prices. This gives you the common price of all $N$ firms in symmetric Nash equilibrium.

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