Weak Monotonicity and Bayes-Nash Incentive Compatibility

An allocation rule is called Bayes-Nash incentive compatible, if there exists a payment rule, such that truthful reports of agents ’ types form a Bayes-Nash equilibrium in the direct revelation mechanism consisting of the allocation rule and the payment rule. This paper provides characterizations of Bayes-Nash incentive compatible allocation rules in social choice settings where agents have one-dimensional or multi-dimensional types, quasi-linear utility functions and interdependent valuations. The characterizations are derived by constructing complete directed graphs on agents ’ type spaces with cost of manipulation as lengths of edges. Weak monotonicity of the allocation rule corresponds to the condition that all 2-cycles in these graphs have non-negative length. For one-dimensional types and agents ’ valuation functions satisfying non-decreasing expected differences, we show that weak monotonicity of the allocation rule is a necessary and sufficient condition for the rule to be Bayes-Nash incentive compatibile. In the case where types are multi-dimensional and the valuation for each outcome is a linear function in the agent’s type, we show that weak monotonicity of the allocation rule together with an integrability condition is a necessary and sufficient condition for Bayes-Nash incentive compatibility