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?
Best Answer
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.