Why is game theory formulated in terms of equilibrium instead of winning strategies?

Question

Game theory, on the outset, seems to invite the questions,

"what can I do to win" or "how do I beat my opponent?"

So many people who are not familiar with game theory look to game theory as some sort of instruction manual to beat their opponents.

In practice, however, game theory says,

"if everybody plays according to a Nash equilibrium, then there is no incentive for me to change my strategy, no matter how horrible my current position is."

The second answer seems to be pessimistic and also places a very strong assumption which is that all the other players are playing at the Nash equilibrium, hence it has questionable real life significance.

So my question is, why the emphasis on equilibrium finding as opposed to winning strategies? Are these equivalent in some sense?

Answer 1

There's a few issues that need to be distinguished here. First, one can distinguish the question of how you find the winning strategy from the question of how you define what the winning strategy even is.

In game theory, the winning strategy is defined as a Nash equilibrium, basically making the assumption that each player should play assuming the other players are as skilled as possible and so will never make a move when another move is better.

After making this assumption, the question of finding a winning strategy is then often a combinatorial or computational problem and is studied in fields like combinatorial game theory, as Iosif Pinelis and David Speyer point out. I will mention that this computational problem is an active field of study as a practical programming problem for many games. Game theory itself is more devoted to finding the equilibrium in games that are less than purely competitive, often inspired by real-life circumstances.

But if I read your question right, you have a fundamental objection with the definition of a Nash equilibrium as the "best" strategy at all. One should understand how opponents really play, rather than how they would theoretically optimally play, and move to counter that.

Of course the problem is that this is a psychological or empirical question rather than one of mathematics. In games like poker you see different styles of play, with interaction between them, a "game theory optimal" approach following the mathematical theory and an "exploitative" approach that aims to do better by taking advantage of how others do worse (while taking the risk of doing worse itself).

Given that you don't know that opponents will play the optimal strategy and you don't know they will play according to any psychological theory, you may instead wish to pick a strategy that has some guarantee for how well it will do regardless of what the opponent plays. Presumably of all strategies, with all possible guarantees, you will pick the best one.

This is, for two player perfectly competitive games, equivalent to the Nash equilibrium. In fact it was invented first by von Neumann in his game theory, the generalization being the work of Nash. One can see this by noting that - and here is where the "pessimism" comes in - it asks for the strategy that does best assuming the opponent picks a counter-strategy that is the absolute worst for you, which is, in a perfectly competitive situation, also the counter-strategy that is best for them.

So the Nash equilibrium indeed provides a winning strategy for two-player zero-sum games, which are typically the kind of games people are most devoted to winning.

Answer 2

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.

Answer 3

The problem of finding a winning strategy (if one exists) is purely computational (with rare exceptions), and its solution very much depends on the game. So, this problem cannot be part of a general theory.

However, there are some general theorems (such as Nash's equilibrium existence theorem), which are applicable to large varieties of games.

Answer 4

I recommend you read "Harrington on Cash Games, Volume I" by Dan Harrington and Bill Robertie. It is not a formal math book, but rather a book on winning strategies for poker.

What I find fascinating is that, without any formal mathematical terminology, he basically re-invents and motivates the idea of a Nash equilibrium. For example he gives advice like "Fold this hand 40% of the time, call 40% of the time, and raise 20% of the time." The way he explains it is that you need to be unpredictable and keep your opponents guessing; otherwise your opponents will figure out your tendencies, learn what your bets mean, and exploit you.

The entire book is worth reading, at least if you enjoy poker. In his book Harrington cares only about winning strategies, and has no interest in developing formal mathematics for its own sake. So you might find it illuminating to see how he arrives at the same conclusions.

Answer 5

Consider this: there is a class of problems which are not usually considered pertinent to the game theory in which several players take turns and one of them can always win depending on the starting conditions (example). People view those as fun problems but do not really play them.

For all games, Nash equilibrium does not conflict with finding the winning strategy at all. For real-world applicability, all the steps are pretty much the same as finding one except for making the final decision. Imagine a game with incomplete information like poker: finding the Nash equilibrium involves calculating probabilities of what you could get vs what your opponents could get, and at some point, a decision is made based on how things would average out. If one replaces this last step with some psychology shenanigans mixed with personal preferences (opponent A is risk-aversive, opponent B is not and I feel playing games being more rewarding if I take extra risks as well as opposed to the mathematically optimal strategy), it gets exactly the kind of applicability someone might be seeking.

Answer 6

Equilibria and strategies go hand in hand. To figure out the equilibria, you have to propose strategies for each player and play them out to their logical end. The optimal strategies (which are not necessarily winning ones, e.g., if everyone loses), are the ones that lead to equilibrium, precisely because no one has the incentive to change strategies.

Answer 7

My guess is that the real reason is that game theory is not really about games.

Game theory as we know it today started with the book Theory of Games and Economic Behavior by von Neumann and Morgenstern, a mathematician and an economist. And economics is much more interested in finding an equilibrium than about beating your opponent.

(This is only a guess because I have not read the book and do not know very much about economics.)