Why does Game Theory seem to ignore the impact that collectively altering a player's possible strategies can have on their utility?

Question

A recurring theme I've noticed in Game Theory is the tendency to dish out ideas for collective action, with the stated goal of tipping or forcing society toward a strategy vector with better (perhaps even Pareto dominating) payoffs, but which actually replace the initial game with a new game, usually reducing options for the players.

For example, discussion of the Prisoner's Dilemma is often qualified with the observation that players attaining better-than-Nash-Equilibrium payoffs (i.e., firms colluding to keep prices high) might be good or bad depending on perspective. However, I have never heard it acknowledged that exogenous removing one of a player's strategies (either cooperate or defect) can affect the utility of their remaining strategy.

Similarly, in problems where the Nash Equilibrium of each player choosing whether to drive across a bridge or take a train leads to a suboptimal payoff in terms of aggregate travel time, issuing bridge licences to limit the number of drivers is often proposed. While it would be trivial to modify the payoffs to reflect, i.e., logarithmic utility of travel time rather than travel time itself, I have never heard it acknowledged that utility might also depend on whether or not the decision to drive or take the train is coerced. I see no reason to assume a player's utility obtained by choosing drive should be equal, or even approximate equal, to their utility obtained by being assigned drive, just as I see no reason to assume choosing a life partner yields similar utility to being assigned that same life partner in an arranged marriage. The entire game at least appears to have been replaced by a new and somewhat dissimilar game, but the difference between choosing drive and being assigned drive tends to be glossed over, treated as some philosophical consideration exogenous to Game Theory.

Does everything in Game Theory still apply if we let the utility function implicit in the payoffs of a game depend on what specific interventions are proposed to collectively move between outcomes, or do we have to treat the resulting game as genuinely new? In the latter case, are there good reasons to expect the new game to resemble the initial game closely enough to warrant how frequently this point gets glossed over in Game Theory?

Answer 1

A basic assumption of game theory is that strategy choices lead to outcomes and players have preferences over outcomes only. If players in a given strategic choice situation are known to also have preferences over their choice sets (e.g., preferring more options to choose from, even if the additional options do not have preferred consequences), then these choice sets become part of the outcomes of the game that is really being played, which is different from the initial choice situation. In this sense, yes, you "have to treat the resulting game as genuinely new".

So the problem you refer to is not a problem of game theory, but of the art of adequately modeling a given choice situation as a game. As to whether there are "good reasons to expect the new game to resemble the initial game closely enough", there is no general answer to this question, it rather depends on the specific situation you want to model.