# Can There Be a Dominant Strategy If Both Firms Choose to Match the Lowest Price in the Market in a Bertrand Model of Competition?

In a Bertrand duopoly model where firms choose to price match ie match the lowest price in the market, any price p∈[c,Pm] (where c is cost and Pm is the monopoly price) can be a Nash. Hence, there are multiple Nash equilibria.

In that case, can there be any dominant strategy, weak or strong?

My first instinct is to say no because if both firms did have a dominant strategy, then there wouldn't be multiple Nashes. But my confusion stems from why both firms don't simply choose to charge Pm, because there is no incentive to undercut the other firm in a price matching scenario.

• Hi! As currently posed, the question seems ill-defined: what is a 'strategy' here, what is the set of possible strategies? Also, I like your intuition, but have you tried applying the rigorous definition of weakly/strongly dominant strategy? Commented Nov 5, 2023 at 21:39
• What is the game? Commented Nov 6, 2023 at 0:01

Let $$D(p)$$ for the good when the selling price is $$p$$ and let $$c$$ be the unit cost for both firms. Let $$p^m$$ denote the unique profit-maximizing monopoly price (assuming one exists). Assume that all buyers go to the firm with the lower selling price and that if the selling price are equal each firm gets half of the buyers. Assume firms price match so that the whatever prices $$p_1$$ and $$p_2$$ are posted by the firms, the selling price is $$\min\{p_1,p_2\}$$ at both firms.

Then for posted prices $$p_1$$ and $$p_2$$, the demand for firm $$1$$ is

$$D_1(p_1,p_2)=\begin{cases} \frac{1}{2}D(p_1) & p_1\leq p_2\\\frac{1}{2}D(p_2) &p_1>p_2 \end{cases}$$

The profit of firm $$1$$ is $$\pi_1(p_1,p_2)=\begin{cases} \frac{1}{2}D(p_1)(p_1-c) & p_1\leq p_2\\\frac{1}{2}D(p_2)(p_2-c) &p_1>p_2 \end{cases}$$

Why is there no strictly dominant strategy?

Suppose firm $$1$$ posts a price of $$p_1'$$. If firm $$2$$ posts a price of $$p_2\leq p_1'$$ then all prices $$p_1\geq p_2$$ yield the same profit for firm $$1$$, i.e. $$\frac{1}{2}D(p_2)(p_2-c)$$. So posting the price $$p_1'$$ is not a strictly dominant strategy.

Why is posting the price $$p^m$$ a weakly dominant strategy?

As explained above, for any price $$p_2\leq p^m$$, all prices $$p_1\geq p^m$$ yield the same profit, and so $$p^m$$ is not strictly dominant.

On other hand, if $$p_2>p^m$$, then firm $$1$$'s profit from posting a price $$p^m$$ is $$\frac{1}{2}(D(p^m)-c)$$ (i.e. half the monopoly profit).

• If instead firm $$1$$ posted a price $$p_1\in(p^m,p_2]$$ or $$p_1, firm $$1$$'s profit would be

$$\frac{1}{2}(D(p_1)-c)<\frac{1}{2}(D(p^m)-c)$$

• If instead firm $$1$$ posted a price $$p_1>p_2$$, firm $$1$$'s profit would be

$$\frac{1}{2}(D(p_2)-c)<\frac{1}{2}(D(p^m)-c)$$

Thus posting a price of $$p^m$$ always gives firm $$1$$ at least as high a profit as any other posted price, regardless of firm $$2's$$ posted price (and for prices $$p_2>p^m$$ a strictly higher profit than any other price). This makes $$p^m$$ a weakly dominant strategy.

It seems that each firm pricing at $$p^m$$ is the most likely outcome of this game as neither firm can do better with any other posted price regardless of the other firm's posted price, and can do strictly better with $$p^m$$ than any other price if the other firm's posted price is greater than $$p^m$$.