The value of randomized strategies in distributionally robust risk averse network interdiction games

Conditional Value at Risk (CVaR) is widely used to account for the preferences of a risk-averse agent in the extreme loss scenarios. To study the effectiveness of randomization in interdiction games with an interdictor that is both risk and ambiguity averse, we introduce a distributionally robust network interdiction game where the interdictor randomizes over the feasible interdiction plans in order to minimize the worst-case CVaR of the flow with respect to both the unknown distribution of the capacity of the arcs and his mixed strategy over interdicted arcs. The flow player, on the contrary, maximizes the total flow in the network. By using the budgeted uncertainty set, we control the degree of conservatism in the model and reformulate the interdictor's non-linear problem as a bi-convex optimization problem. For solving this problem to any given optimality level, we devise a spatial branch and bound algorithm that uses the McCormick inequalities and reduced reformulation linearization technique (RRLT) to obtain convex relaxation of the problem. We also develop a column generation algorithm to identify the optimal support of the convex relaxation which is then used in the coordinate descent algorithm to determine the upper bounds. The efficiency and convergence of the spatial branch and bound algorithm is established in the numerical experiments. Further, our numerical experiments show that randomized strategies can have significantly better in-sample and out-of-sample performance than optimal deterministic ones.