The University of Sydney Business School, Discipline of Business Analytics
Abstract
When only the moments (mean, variance or t-th moment) of the underline distribution are known, numerous
max-min optimization models can be interpreted as a zero-sum game, in which the decision maker (DM)
chooses actions to maximize her expected profit while Adverse Nature chooses a distribution subject to
the moments conditions to minimize DM’s expected profit. We propose a new method to efficiently solve
this class of zero-sum games under moment conditions. By applying the min-max inequality, our method
reformulates the zero-sum game as a robust moral hazard model, in which Adverse Nature chooses both the
distribution and actions to minimize DM’s expected profit subject to incentive compatibility (IC) constraints.
Under quasi-concavity, these IC constraints are replaced by the first-order conditions, which give rise to
extra moment constraints. Interestingly, these extra moment constraints drastically reduce the number of corner points to be considered in the corresponding semi-infinite programming models. We show that in the equilibrium, these moment constraints are binding but have
zero Lagrangian multipliers and thus facilitate closed-form solutions in several application examples with
different levels of complexity. The high efficiency of the method enables us to solve a large class of zero-sum
games and the corresponding max-min robust optimization models