Abstract
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.
Highlights
As an analytic tool, it can be useful to view the “rules of the game” governing many strategic scenarios as parameters that can be changed
decision maker (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, 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
Summary
It can be useful to view the “rules of the game” governing many strategic scenarios as parameters that can be changed. While there have been efforts into combining envelope theorems and comparative static analysis in games [5], our work is the first to consider the relation between DM’s willingness to pay to change the parameter values in the case where DM’s decision to change the parameter values is observed, with the case where that decision is unobserved. Our analysis is not restricted to considering a player’s willingness to change parameters that specify endogenous information structures and partitions of information sets; we provide a fully general analysis that can be applied to any parameter specification that can arise endogenously Another important difference is that we allow arbitrary solution concepts by the players of the game
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