TO Hybrid Defensive Resource Allocations in the Face of Partially Strategic Attackers in a Sequential

convex in c, since the linear combination of the continuous and strictly convex functions pi(ci)xi’s is continuous and strictly convex. Finally, the summation of the two continuous and strictly convex functions is continuous and strictly convex. Remarks: Note that we assume that the defender and the attacker have the same target valuations xi’s. If that does not hold, the objective function as defined in Equation (11) is not continuous. One can construct an example where the objective function for the defender is discontinuous. For example, targets 1 and 2 value 200 and 100 to the defender and value 199.99 and 200 to the attacker. When the defender increases the defense to target 2 by such a small amount that the attacker will attack target 1 instead. The expected payoff for the attacker changes continuously while the expected payoff for the defender changes discontinuously. By assuming the same target valuations, the game is zero-sum and both the defender’s and the attacker’s payoffs change continuously in c. If the target valuations differ for the defender and attacker, then the expected loss of the defender becomes pk(ck)xk where k ∈ argmax i pi(ci)yi with yi’s being the attacker’s values, which needs not be continuous.