How Much Would You Pay to Change a Game before Playing It?

Envelope theorems provide a differential framework for determining how much a rational decision maker (DM) is willing to pay to alter the parameters of a strategic scenario. We generalize this framework to the case of a boundedly rational DM and arbitrary solution concepts. We focus on comparing and contrasting the case where DM’s decision to pay to change the parameters is observed by all other players against the case where DM’s decision is private information. We decompose DM’s willingness to pay a given amount into a sum of three factors: (1) the direct effect a parameter change would have on DM’s payoffs in the future strategic scenario, holding strategies of all players constant; (2) the effect due to DM changing its strategy as they react to a change in the game parameters, with the strategies of the other players in that scenario held constant; and (3) the effect there would be due to other players reacting to a the change in the game parameters (could they observe them), with the strategy of DM held constant. We illustrate these results with the quantal response equilibrium and the matching pennies game and discuss how the willingness to pay captures DM’s anticipation of their future irrationality.