Mechanism design with level-k types: Theory and an application to bilateral trade

This paper studies mechanism design under the level-k solution concept. The first result gives a general necessary condition for a social choice rule to be level-k implementable. In some environments, this necessary condition is equivalent to Bayesian incentive compatibility, making level-k implementation more restrictive than Bayesian implementation. The second result shows that this is not a general implication. In the bilateral trade environment ex post efficient trade is always possible under level-k implementation. Further, ex post efficient trade is possible in a mechanism that is robust to different specifications of beliefs about the levels of reasoning of others and to any specification of beliefs about payoffs.