# Interpretation of Nash equilibrium as a potential stable point of a dynamic process

I'm reading an article called "The Nash equilibrium: A perspective" by Holt and Roth, and the below paragraph caught my attention.

When the goal is prediction rather than prescription, a Nash equilibrium can also be interpreted as a potential stable point of a dynamic adjustment process in which individuals adjust their behavior to that of the other players in the game, searching for strategy choices that will give them better results.

Is this paragraph referring to repeated games, or can this view of the Nash equilibrium be thought to apply to single-shot games as well? It mentions "game" (i.e. singular). However, I don't see how you can adjust strategies in a single game. On the other hand, if it is saying that a single-shot game is equivalent to a hypothetical series of steps that are taken, then that would be a very interesting take on what constitutes a "game", and one which I would love to know more about.

• I recommend looking up "evolutionary game theory". Commented Jun 19, 2020 at 18:45

As an example, consider Cournot competition. Supposse, two firms compete in quantities. The figure below (also from the wikipedia page) depicts the reaction functions (or best-response functions). If firm 2 sets quantity $$q_2$$, the best response by firm 1 is to set $$q_1= R1(q_2)$$. We have a Nash equilibrium where these two functions intersect, i.e., when all firms play a best response against each other. The NE is at $$(q1,q2)$$ in the figure.
Suppose firm 1 thinks: "I believe firm 2 sets quantity $$q'_2. Hence, I will respond with $$q'_1=R1(q'_2)$$." Next firm 1 would think: "But wait, if I set $$q'_1$$, then firm 2 would not set $$q'_2$$, but $$q''_2=R2(q'_1)$$. This means I should set $$q''_1=R1(q''_2)$$." Next: "But wait, then firm 2 would set $$q'''_2=R2(q''_1)$$..." and so on. This though process converges to the NE $$(q1,q2)$$. Similarly, we arrive at the same NE when starting with $$q'_2>q2$$. If you want to, you can also think about the firms actually setting these quantities in repeated interactions, making only one of the steps of the thought experiment, and thereby they would also arrive at the NE and stay there.