A note on the definition of Bayesian Nash equilibrium of a mechanism when strategies of agents are costly actions

TL;DR

This paper revises the concept of Bayesian Nash equilibrium for mechanisms where agents' strategies are costly actions, showing that traditional direct implementation may not be feasible under these conditions.

Contribution

It introduces a profit-based equilibrium definition for costly actions and demonstrates limitations of direct mechanisms in such settings.

Findings

Bayesian Nash equilibrium should be based on profit functions for costly actions.

Direct mechanisms may fail to truthfully implement social choice functions when strategies are costly actions.

Traditional utility-based equilibrium concepts are insufficient for mechanisms with costly strategies.

Abstract

In mechanism design theory, a designer would like to implement a desired social choice function which specifies her favorite outcome for each possible profile of agents' types. To do so, the designer constructs a mechanism which describes each agent's feasible strategy set and the outcome function. Generally speaking, each agent's strategy in a mechanism has two possible formats: an action, or a message. In this paper, we focus on the former format and claim that the notion of Bayesian Nash equilibrium of a mechanism should be based on a profit function instead of the conventional utility function when strategies of agents are costly actions. Next, we derive the main result: Given a social choice function which can be implemented by an indirect mechanism in Bayesian Nash equilibrium, if all strategies of agents are costly actions, then it cannot be inferred that there exists a direct mechanism that can truthfully implement the social choice function in Bayesian Nash equilibrium.