Basic game theory fact that many people get wrong
Consider an imperfect-information extensive-form game, such as poker. We know that every such game has at least one Nash equilibrium, by Nash’s existence theorem. (Technically Nash’s theorem applies just to perfect-information normal-form games, but since every extensive-form game can be converted to a (significantly larger) normal-form game, the theorem applies straightforwardly also to extensive-form games). Nash equilibrium is the central solution concept in game theory. Informally, a set of strategies for all players is a Nash equilibrium if no player can improve their expected payoff by deviating to another strategy.
Back to the example of poker. Is it possible under a Nash equilibrium strategy that a player can play one hand in a way that is losing a small amount but enables us to gain a larger amount with other hands? Intuitively it seems very plausible that this could be the case, and many people I have spoken with (including successful high-stakes poker players who often apply game theory to their play) believe it. In fact, I was a bit confused about this initially as well when I was starting grad school. It seems very plausible that it could be optimal to give up a small amount of value with one hand enabling us to make it back and more with other hands. For example, if we only bet aggressively with our very strong hands, then our opponents would always fold when we bet (unless they have an extremely good hand) and we would never make money from weaker hands. So clearly we will need to “bluff” with some weaker hands in order to ensure that the opponents will sometimes pay us off when we have our strong hands.
There is a common misconception that these “bluffs” lose a small amount of money, which is compensated by how much we gain with our strong hands. However, in a Nash equilibrium strategy this is not the case. Every action we take, whether it is an aggressive “bluff” or a passive action, either obtains a greater or equal value to the value of every other possible action. Purely based on intuition this is far from obvious, though it is basically trivial when you look at the definition of Nash equilibrium: suppose a player makes a “bluff” with hand x in a situation, where the value of not bluffing (i.e., “checking”) is slightly higher. Then the player could switch their strategy to check with x instead and improve their payoff, contradicting the fact that the initial strategy is a Nash equilibrium.
It is pretty remarkable that there exists a set of strategies for all players such that all players are simultaneously playing optimally with every individual hand, whether it is a strong/weak hand and aggressive/passive action. It is far from trivial that this is the case, and this is part of why Nash’s existence theorem and the concept of Nash equilibrium are so powerful.
Of course, if you are playing against an opponent who is “irrational” and not following a Nash equilibrium strategy, you may be able to trick them by making some apparent mistakes early on in order to exploit them in the future. However, when following a Nash equilibrium strategy, the phenomenon of playing one hand suboptimally in order to gain a future benefit with other hands does not occur.
