Let me address the criticism that the Nash equilibrium is of questionable real-life significance.

I'll begin by openly admitting something that theorists often are reluctant to admit: One big reason for the obsession with Nash equilibria is that the proof of their existence is so beautiful. It is a fixed-point theorem, and fixed-point theorems send shivers down the spine of mathematicians. So they would study Nash equilibria even if Nash equilibria had no real-world significance. That much I am happy to concede.

But the notion that Nash equilibria have no real-word significance because your opponents might not play an equilibrium strategy is a common misconception. The concept of a Nash equilibrium arises naturally if you set out to find not just a strategy that will beat your current opponent, but the *best* or the *optimal* strategy.

Surely you can see the practical value of finding the optimal strategy. But there is a catch: what does *optimal* mean? Does there even *exist* an optimal strategy? Maybe, maybe not, but if there *is* an optimal strategy then surely you would want to find it. So let's think about what *optimal* might mean.

As a general principle, if you're not sure how exactly to define X, it can be a fruitful tactic to begin by listing some things that X is *not*. For example, wouldn't you agree that if Alice and Bob are playing each other, and one of them—say Alice—realizes that by deviating from her current strategy, she can do better (as long as Bob continues to do what he's doing), then Alice's current strategy cannot possibly be optimal? If *optimal* means anything at all, then it must mean that you can't improve on it.

If you accept that argument, then you are led directly to the notion of a Nash equilibrium. If Alice and Bob are both playing optimally, then it means that neither one can do better than what they're currently doing by deviating (unless the other player also deviates). And that is precisely the definition of a Nash equilibrium.

Just to be clear, I'm **not** claiming that "optimal = Nash equilibrium." I'm **only** claiming that **if** there is such a thing as an "optimal strategy" then a **necessary** condition is that when both players play optimally, the result will be a Nash equilibrium. Thus a starting point for finding an optimal strategy (again, if such a thing even exists) is to find all the Nash equilibria.

Moreover, I would claim that Nash equilibria are of real-world significance *for expert players*. Expert players are always looking for an edge. Imagine a pool of expert players who are adopting various strategies, but they are not at a Nash equilibrium. What that means is that at least one of them has the opportunity, at least in principle, of switching to a new strategy and gaining an edge over the others. That's what expert players live for! So in such a situation, if the game is not hopelessly complicated, there will be a tendency for the experts to converge to a Nash equilibrium, as they keep trying to outsmart each other.

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