The true cost of acting rational is the difference between acting truly rational versus acting purely rational in a situation.
First we have to make the distinction between what is factual truth or strategically optimal and what an agent believes is true or strategically optimal. For non-rational agents these are different, there is at least one instance where what they believe is not what is true or how they act is not optimal. For rationalists, what they believe has to be proven true and they must act optimally.
A situation with only rational agents, they would find the optimal cumulative payoff and distribute it optimally in-line with potentially different goals. This assumes rational agents can agree on an optimal distribution strategy, possibly based on incurred costs (time, resources spent) and goal priority, and have non-conflicting goals.
Non-rational agents may have conflicting goals, be greedy and not achieve optimal distribution, and may not find or implement the strategy to achieve the best cumulative payoff. Each of these areas identifies costs of non-rationality.
In a situation in which rationalist agents must work with non-rationalist agents there will be a cost of non-rationality no matter how the rationalist acts. In these situations a pure rationalist will act differently than a true rationalist. Truly rational agents take into account both factual truth and non-rational agents beliefs and strategies when deciding strategy. The true rationalists will pursue strategies that achieve a sub-optimal but minimum cost. Purely rational agents only take into account factual truth when deciding strategy and does not account for non-rational agents beliefs or strategies. Pure rationalists will incur costs above this sub-optimal minimum. Thus, the true cost of acting rational is the difference between the cost of acting purely rational and acting truly rational.
The word "rational" is overloaded with associations, so let me be clear: to me [here], more "rational" means better believing what is true, given one's limited info and analysis resources.
Rationality certainly can have instrumental advantages. There are plenty of situations where being more rational helps one achieve a wide range of goals. In those situtations, "winnners", i.e., those who better achieve their goals, should tend to be more rational. In such cases, we might even estimate someone's rationality by looking at his or her "residual" belief-mediated success, i.e., after explaining that success via other observable factors.
But note: we humans were designed in many ways not to be rational, because believing the truth often got in the way of achieving goals evolution had for us. So it is important for everyone who intends to seek truth to clearly understand: rationality has costs, not only in time and effort to achieve it, but also in conflicts with other common goals.
Yes, rationality might help you win that game or argument, get promoted, or win her heart. Or more rationality for you might hinder those outcomes. If what you really want is love, respect, beauty, inspiration, meaning, satisfaction, or success, as commonly understood, we just cannot assure you that rationality is your best approach toward those ends. In fact we often know it is not.
The truth may well be messy, ugly, or dispriting; knowing it make you less popular, loved, or successful. These are actually pretty likely outcomes in many identifiable situations. You may think you want to know the truth no matter what, but how sure can you really be of that? Maybe you just like the heroic image of someone who wants the truth no matter what; or maybe you only really want to know the truth if it is the bright shining glory you hope for.
Be warned; the truth just is what it is. If just knowing the truth is not reward enough, perhaps you'd be better off not knowing. Before you join us in this quixotic quest, ask yourself: do you really want to be generally rational, on all topics? Or might you be better off limiting your rationality to the usual practical topics where rationality is respected and welcomed?