Nash Equilibrium: Seeing the Forest Through the Trees

Around the 1940s and 1950s, a defining moment in the development of economic thought was underway, led by academics such as Paul Samuelson and John Nash. They believed that human interactions could and should be described using the same tools that physical scientists use to describe the natural world. These would include the interactions between particles or the orbits of planets.

The aim of this article is to give a conceptual overview of one of the most influential ideas to come out of this movement, namely the Nash equilibrium.

Motivation, Please

Their motivation is easy to understand: Human beings, who themselves are nothing more than a complex composition of particles and molecules, should exhibit regularities in behavior. These should be describable in much the same way that physicists such as Albert Einstein and Werner Heisenberg described general relativity and quantum mechanics a few decades earlier. This approach would quickly come to replace the “old way” of explaining empirical phenomena using less precise verbal reasoning or rules of thumb. And these methods are still prevalent in related fields such as sociology and political science even today.

What is Nash Equilibrium?

Stated very simply, a Nash equilibrium is a state of affairs in which all parties find themselves pursuing a strategy from which they have no incentive to deviate.

Let’s unpack this a little bit. Suppose you are a tree (yes, a tree). Right next to you is a neighboring tree. You are both competing for sunlight. If you are both the same height, then you both share the same amount of sunlight.

Now, if your neighbor grows taller than you, they capture all of the sunlight, but they do so at the cost of requiring more resources to support larger branches. Nevertheless, if the benefit of more sunlight outweighs that cost, we should expect the tree (through evolution) to eventually adopt this strategy. However, if this occurs, you yourself will want to grow taller, until you both find yourselves sharing the sunlight once again. But this time there will be the extra added cost of supporting larger branches.

Wouldn’t it have been easier if you had both remained smaller in the first place? But then how could you be sure your neighbor wouldn’t decide to grow taller overnight in order to capture all of the sunlight in the morning? When does this back and forth “negotiation” finally conclude, where neither of you have an incentive to change your strategy the next day?

Such a situation is exactly what we call a Nash equilibrium, which you can learn more about in my ECON 4301 course. The example just described is an application of the field of evolutionary game theory. Before we explore the solution to this problem, let us consider one more example, this time from economics.

Firms’ Output Decisions

Suppose you are one of two companies competing with one another. Let’s say that you are “Firm 1”, and denote the amount of output that you produce by Q₁. Let’s let Q₂ denote the amount of output that the competing firm (Firm 2) produces.

According to the law of demand, the more output that either of you produce, the price should be lower in the market. To simplify matters, suppose that this price, which we denote by P, is determined by

P=10-Q₁-Q₂.

To further simplify matters, let’s assume that your cost of producing Q₁ units is simply Q₁. Your profits, which are given by total revenue minus total costs, can then be described by the equation

Profit = P x Q₁-Q₁

Because we have a nice equation for P from above, we can substitute that in and get

Profit = ( 10-Q₁-Q₂) x Q₁-Q₁ .

Finally, let’s further simplify life and assume that you and your competitor can either choose to produce 1 unit of the good, or 2 (so that both Q₁ and Q₂ can be equal to 1, or 2).

Questions, Questions

The question then is, how much should you choose to produce in order to maximize, given what your competitor is producing? And is there a level of production at which neither of you will want to change to a different level of production, or, in other words, a Nash equilibrium?

To investigate this, what we can do is formulate a “game plan” based on whether or not we observe our opponent producing Q₂=1 or Q₂=2. In the first case, if Q₂=1 , then if you choose to produce Q₁=1 , you will obtain a profit of

Profit = (10-1-1) x 1 – 1 = 7.

If instead you choose Q₁=1 in this case, you will obtain a profit of

Profit = (10-2-1) x 2 – 2 = 12.

Profit Maximization

We therefore conclude that if we observe our competitor producing one unit, then producing two units is optimal, as it gives us the highest profit. We can summarize our findings in the following table:

Now, we can do the same exercise under the assumption that Q₂=2, and figure out what our optimal response would be. After making these calculations and adding them to our table, we get the following:

Once again, we see that if Q₂=2, it is optimal to produce Q₁=2 in order to obtain the profit of 10 as opposed to the profit of 6 you would get if you produce one unit.

Into The Matrix

Lastly, we can do the same exact calculations from the perspective of our competitor, Firm 2. Given that they have the same cost structure as us, we can figure out what their optimal level of production would be based on what WE choose to produce. By writing in their profits next to ours in the table, we get what we call a “payoff matrix”, which looks like the following:

To read this table, we choose a level of output for each firm, and in the corresponding box, the number on the left-hand side gives Firm 1’s payoff, and the number on the right-hand side gives Firm 2’s payoff. For example, if Q₁=1 and Q₂=2, then Firm 1 would get a payoff of 6, and Firm 2 would get a payoff of 12, as this corresponds to the box in the upper right-hand corner.

Also notice a key feature of this situation: My profit may change simply by the competitor doing something different. Like the example involving trees above, it is not enough to choose a strategy that I am content with, but I must also take into consideration what my competitors or rivals are doing.

What Would Nash Do?

Let’s go ahead and figure out what the Nash equilibrium would be in this situation. Remember, we are looking for a combination of strategies, one for you and your opponent, so that neither party wants to switch strategies the next day.

Would Q₁=1, Q₂=1 be a Nash equilibrium? No. Notice that if your opponent is producing Q₂=1, then you would want to switch strategies from Q₁=1 to Q₁=2 in order to capture the higher profit (12 compared to 7).

What about  Q₁=2, Q₂=1? Again, no. Why? Because if you are producing Q₁=2, your competitor will have an incentive to switch from  Q₂=1 to Q₂=2 in order to increase profits from 6 to 10.

But Wait, There’s More

For the same reason, Q₁=1, Q₂=2 is ALSO not a Nash equilibrium, because you would have an incentive to increase production in order to increase your profits from 6 to 10.

Lastly, we see that Q₁=2, Q₂=2 does in fact satisfy the definition of being a Nash equilibrium. Holding fixed what the “other guy” is doing, switching strategies would result in a lower profit for both firms (going from 10 to 6). Therefore, the Nash equilibrium prediction of where the dust will settle at the end of the day will have both firms producing two units of the good.

How Can We Apply This Idea to Other Fields?

Let’s wrap up our discussion by revisiting the initial example with our trees. Let’s suppose that a tree can be either small (S) or large (L). Let’s assume that if both trees are the same size, they share the sunlight, receiving a benefit of ½ each. However, if one tree is large and the other is small, the large tree receives all of the sunlight, or a benefit of 1, while the small tree receives a benefit of 0. Lastly, suppose the cost of being large is given by some number, let’s call it c. Then we can construct a payoff matrix like above, with the following payoffs:

	S	L
S	½ , ½	0, 1-c
L	1-c, 0	½-c, ½ -c

Here we have a variety of possible outcomes, which all depend on c, or the cost of supporting larger branches. Suppose that c were relatively small, say c= ¼ . Then the payoff matrix would be:

	S	L
S	½ , ½	0, ¾
L	¾ , 0	¼ , ¼

Hip To Be Small?

Notice that the only Nash equilibrium in this situation is that trees eventually grow large, as the bottom right-hand corner is the only Nash equilibrium.

Paradoxically, both trees receive a higher payoff if they both stay small, given by the payoffs of ½ in the upper left-hand corner. But as long as one tree is small, the other tree has an incentive to grow large and capture the payoff of ¾. From there, it follows that the other tree has an incentive to follow suit, leading to them both being large. And once they are both large, neither tree has an incentive to be small again.

It’s All About Cost

What if the cost of supporting larger branches were higher, say c= ¾ ? Our payoff matrix would then look like the following:

	S	L
S	½ , ½	0, ¼
L	¼ , 0	-¼ , -¼

Here, the only Nash equilibrium is that trees remain small, as the top left-hand corner is the only Nash equilibrium. Thus, we see that the cost of growing large will be the key factor in what equilibrium is reached. This may provide researchers with a clue as to why some species of trees grow larger than others.

We see from the above two examples that a little bit of algebra goes a long way in simplifying our analysis of human interactions. Using the basic principle of a Nash equilibrium, economists, evolutionary biologists, political scientists, and researchers in a wide variety of fields have been able to arrive at more precise predictions.

In fact, see if you can write down a payoff matrix which describes a situation you are facing from your own experience. I’d be more than happy to take a look over it if you send it my way!

Eric Hoffmann
Associate and Pickens Professor of Economics