<p>Let <span class="math-container">$D(p)$</span> for the good when the selling price is <span class="math-container">$p$</span> and let <span class="math-container">$c$</span> be the unit cost for both firms. Let <span class="math-container">$p^m$</span> 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 <span class="math-container">$p_1$</span> and <span class="math-container">$p_2$</span> are posted by the firms, the selling price is <span class="math-container">$\min\{p_1,p_2\}$</span> at both firms.</p>
<p>Then for posted prices <span class="math-container">$p_1$</span> and <span class="math-container">$p_2$</span>, the demand for firm <span class="math-container">$1$</span> is</p>
<p><span class="math-container">$$D_1(p_1,p_2)=\begin{cases} \frac{1}{2}D(p_1) &amp; p_1\leq p_2\\\frac{1}{2}D(p_2) &amp;p_1&gt;p_2 \end{cases}$$</span></p>
<p>The profit of firm <span class="math-container">$1$</span> is
<span class="math-container">$$\pi_1(p_1,p_2)=\begin{cases} \frac{1}{2}D(p_1)(p_1-c) &amp; p_1\leq p_2\\\frac{1}{2}D(p_2)(p_2-c) &amp;p_1&gt;p_2 \end{cases}$$</span></p>
<hr />
<p><strong>Why is there no strictly dominant strategy?</strong></p>
<p>Suppose firm <span class="math-container">$1$</span> posts a price of <span class="math-container">$p_1'$</span>. If firm <span class="math-container">$2$</span> posts a price of <span class="math-container">$p_2\leq p_1'$</span> then all prices <span class="math-container">$p_1\geq p_2$</span> yield the same profit for firm <span class="math-container">$1$</span>, i.e. <span class="math-container">$\frac{1}{2}D(p_2)(p_2-c)$</span>. So posting the price <span class="math-container">$p_1'$</span> is not a strictly dominant strategy.</p>
<hr />
<p><strong>Why is posting the price <span class="math-container">$p^m$</span> a weakly dominant strategy?</strong></p>
<p>As explained above, for any price <span class="math-container">$p_2\leq p^m$</span>, all prices <span class="math-container">$p_1\geq p^m$</span> yield the same profit, and so <span class="math-container">$p^m$</span> is not strictly dominant.</p>
<p>On other hand, if <span class="math-container">$p_2&gt;p^m$</span>, then firm <span class="math-container">$1$</span>'s profit from posting a price <span class="math-container">$p^m$</span> is <span class="math-container">$\frac{1}{2}(D(p^m)-c)$</span> (i.e. half the monopoly profit).</p>
<ul>
<li>If instead firm <span class="math-container">$1$</span> posted a price <span class="math-container">$p_1\in(p^m,p_2]$</span> or <span class="math-container">$p_1&lt;p^m$</span>, firm <span class="math-container">$1$</span>'s profit would be</li>
</ul>
<p><span class="math-container">$$\frac{1}{2}(D(p_1)-c)&lt;\frac{1}{2}(D(p^m)-c)$$</span></p>
<ul>
<li>If instead firm <span class="math-container">$1$</span> posted a price <span class="math-container">$p_1&gt;p_2$</span>, firm <span class="math-container">$1$</span>'s profit would be</li>
</ul>
<p><span class="math-container">$$\frac{1}{2}(D(p_2)-c)&lt;\frac{1}{2}(D(p^m)-c)$$</span></p>
<p>Thus posting a price of <span class="math-container">$p^m$</span> always gives firm <span class="math-container">$1$</span> at least as high a profit as any other posted price, regardless of firm <span class="math-container">$2's$</span> posted price (and for prices <span class="math-container">$p_2&gt;p^m$</span> a strictly higher profit than any other price). This makes <span class="math-container">$p^m$</span> a weakly dominant strategy.</p>
<hr />
<p>It seems that each firm pricing at <span class="math-container">$p^m$</span> 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 <span class="math-container">$p^m$</span> than any other price if the other firm's posted price is greater than <span class="math-container">$p^m$</span>.</p>

CC BY-SA 4.0
2023-11-11T11:31:49.687Z
2023-11-11T11:31:49.687Z
smcc
<p>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.</p>
<p>In that case, can there be any dominant strategy, weak or strong?</p>
<p>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.</p>

CC BY-SA 4.0
2023-11-05T21:39:58.057Z
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?
Giskard
CC BY-SA 4.0
2023-11-06T00:01:00.637Z
What is the game?
Michael Greinecker
<game-theory><industrial-organisation><pricing>
Can There Be a Dominant Strategy If Both Firms Choose to Match the Lowest Price in the Market in a Bertrand Model of Competition?
DR109
# Can There Be a Dominant Strategy If Both Firms Choose to Match the Lowest Price in the Market in a Bertrand Model of Competition?

(Asked by: username_0 on 2023-11-05)

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.

**Tags:** game-theory, industrial-organisation, pricing
> **Comments:**
> **username_1**: 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?  
> **username_2**: What is the game?  

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## 1 answers

### Answer by username_3 on 2023-11-11. Score: 0

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
$$\pi\_1(p\_1,p\_2)=\begin{cases} \frac{1}{2}D(p\_1)(p\_1-c) & p\_1\leq p\_2\\\frac{1}{2}D(p\_2)(p\_2-c) &p\_1>p\_2 \end{cases}$$

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**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.

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**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.

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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 $p^m$.


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