A few VERY CONCRETE examples would help a lot. I can't tell if you're just talking about holistic-optimization, where there are carryovers or correlation between subgames, so deviation from optimal in early games gives you better future subgames. Or whether you're talking about more decision-theory variance, based on opponent's predictive power and "who goes first, logically" questions.
In the situations commonly discussed around here, "proposer" and "responder" are misleading terms, because the whole problem is that the sequence of events doesn't match the sequence of decisions - once you introduce precommitment and prediction, you've messed with causality in a way that mixes up the terms. It's not clear that "optimal" even belongs in the term for this, so perhaps "strategically sub-optimal" or the like might work.
It's still the case that the long run is a strict sum of short runs, and "holistic" is the term I'd use for decision theories that include all effects of decisions, not just the ones in an identified subgame.
I’d like an easy way to distinguish the behavior of payoff-maximizing players who would or would not play a strategy in an extensive form game that deviates from the subgame-perfect (Bayesian) equilibrium strategy profile of the game (when their strategy is known to their opponent, and their opponent is also payoff-maximizing).
An example of what I’m interested in can be seen in an ultimatum game where a proposer presents a responder with an offer of the form , where and for are the respective payoffs of the proposer and the responder.
For now, let’s call a strategy subgame-optimal if for every subgame of the game, the strategy’s restriction to that subgame is still optimal within the subgame. In other words, at each decision node of the game, a payoff-maximizing player with a subgame-optimal strategy chooses the action which maximizes their expected payoff (as calculated at that decision node, rather than as calculated according to their prior). A payoff-maximizing player who can commit to a subgame-suboptimal strategy will play the action which maximizes the payoff they expected at their initial decision node, without the need to play a strategy that holds up to backward induction.
Say that the responder’s strategy is known to the proposer ahead of time, and the proposer is restricted to only subgame-optimal strategies. What strategy should the responder use? A (subgame-suboptimal) strategy to reject all proposals with a > ε for arbitrary ε would force the proposer to make an offer arbitrarily in the responder’s favor. But this is impossible for a responder who can only play subgame-optimal strategies. Updateless agents don’t have to worry about this problem, but updateful agents without commitment devices or values other than payoff maximization do.