Cournot Duopoly
The Third Great Sesame Street War is nearing an end. Cookie Monster’s Fungeon is brimming to its capacity, and the war is straining his business. It’s become evident that keeping everyone in the Fungeon is no longer sustainable. As the first prisoners in the Fungeon, Cookie Monster selects you and Bartholomew to be the first to taste freedom, but only after his final challenge: revive his dying global cookie monopoly. You and Bartholomew will compete to sell cookies. By the end of the challenge, you will earn 100% of the revenue as thanks for revitalizing the cookieconomy. You will be freed whatever happens, so as the capitalist you are, you only care about the money.
You and Bartholomew only have one responsibility: decide how many cookies to sell. However, if there are too many cookies circulating, the value of each cookie decreases.
Specifically, if you sell $a$ cookies and Bartholomew sells $b$ cookies, then the price of each cookie is $1200 - a - b$ dollars. (These are high-end cookies, after all.) Thus, you will generate $$a(1200 - a - b)$$ dollars of revenue.
As it turns out, customers are willing to buy fractional amounts of cookies, so Cookie Monster allows you to pick any real number of cookies to sell. You know Bartholomew, who has the same experience playing Cookie Monster’s ridiculous games, will act shrewdly in his best interest.
How many cookies will you sell?
cookies, you think aloud. A stroke of genius. To make sure it wasn’t a hallucination, you decide to write your calculations down:
cookies, you think aloud. Then, you quickly discard that idea, realizing its foolishness. I can't believe I ever thought that would be a good idea. It's like someone's putting these ideas into my head, trying to control me, Archibald thinks. For clarity, you write your calculations down:
We are trying to maximize $a(1200 - a - b) = -a^2 + (1200 - b) a.$ This is a quadratic with respect to $a.$ Recall that the maximum value of the function is located at the vertex of this parabola, or $$a = \frac{1200-b}{2}.$$ Alternatively, use the first partial derivative with respect to $a$ to identify the maximum.
Thus, Archibald’s best response to the action b is the action $a = (1200 - b) / 2.$ Symmetrically, Bartholomew’s best response to the action a is the action $b = (1200 - a) / 2.$ Let’s plot these on a graph, with Archibald’s curve in red and Bartholomew’s curve in blue:
The optimal strategy may not be obvious, but there are certainly strategies that are never optimal. Note that from Archibald’s best response curve, he should never reasonably sell over 600 cookies, as such a decision is only the best response if Bartholomew sells a negative number of cookies. Similarly, Bartholomew should never reasonably sell over 600 cookies. So, we can ignore all strategies that involve Archibald or Bartholomew selling over 600 cookies. Graphically, we can ignore everything outside of the green box:
With our new possible set of action profiles, note that Archibald should never sell fewer than 300 cookies, as such actions are only best responses to when Bartholomew sells over 600 cookies. Similarly, Bartholomew should never sell fewer than 300 cookies. We can restrict our green box once again:
But our new green box is merely a scaled-down version of the original. We can indefinitely shrink our reasonable strategy set by iteratively deleting these dominated strategies, ultimately converging at the intersection point of these two curves: both parties should sell 400 cookies.
Of course! When both best response curves intersect, neither party will regret their decision. It simply boils down to the Nash equilibrium.
You name your mathematical discovery the Cournot Duopoly after your favorite theoretical heat engine (but you forget exactly how it’s spelled). For other examples of analyzing competitive duopoly through game theory, look up Bertrand’s model (where firms can set prices) or Stackelberg’s model (where firms don’t set quantities simultaneously).
But is there a way to make even more money?
You rush down the Fungeon labyrinth to tell Cookie Monster your choice, thinking about the money you’re about to make. If you both sell $400$ cookies, you’ll make $400(1200-800) = 160,000$ entire doubloons! Bartholomew is right on your tail. He’s certainly reached the same conclusion. But as you navigate the brutalist floor plan together, an idea strikes you. After all, if you’ve learned anything from the Butter Cookie or Snickerdoodle game, it’s that you may have something to gain from coordination. You turn to your archnemesis.
Bartholomew! What if, instead of maximizing our individual profits, we maximize our combined profit and split the bag?
You do the calculations quickly in your head. If you two agree to sell $x$ cookies in total, you’ll make $x(1200 - x)$ dollars combined. This is a quadratic which is maximized at $x = 600,$ making a total of $360,000$ dollars. That’s $180,000$ dollars for each of you, an entire $20,000$ more.
As the corrupt business owner that he is, Bartholomew agrees to establish a trust with you. Both of you tell Cookie Monster to sell $300$ cookies and escape the Fungeon together. As you and Bartholomew step outside to feel the sun on your skin for the first time in years, you celebrate your new friendship, profits, and knowledge of game theory.
Unfortunately, you are immediately beamed up and abducted before everything fades to black.