Computing mixed strategies equilibria in presence of switching costs by the solution of nonconvex QP problems

Abstract

In this paper we address a game theory problem arising in the context of network security. In traditional game theory problems, given a defender and an attacker, one searches for mixed strategies which minimize a linear payoff functional. In the problem addressed in this paper an additional quadratic term is added to the minimization problem. Such term represents switching costs, i.e., the costs for the defender of switching from a given strategy to another one at successive rounds of the game. The resulting problem is a nonconvex QP problem with linear constraints. We prove that, though simple when one of the two terms in the objective function is missing, the problem is, in general, a NP-hard one. The proof is based on a reduction from the decision version of the clique problem. Next, we discuss different reformulations and bounds for such problem. Finally, we show, through an extensive set of computational experiments, that the most promising approach to tackle this problem appears to be a branch-and-bound approach where a predominant role is played by an optimality-based domain reduction, based on the multiple solutions of LP problems at each node of the branch-and-bound tree. The computational cost per node is, thus, high but this is largely compensated by the rather small size of the corresponding tree.

Repeated Games with Switching Costs: Stationary vs History Independent Strategies

“…As mentioned before, a direct result of Proposition 6 is that the optimal strategy comes from a finite set and is robust to the exact value of c. Unlike Corollary 1 however, this time it is also true for the best responses of Player 1 (since she best-responds purely). This can provide the theoretical explanation to the reported numerical results from Rass and Rainer [2014] and Liuzzi et al [2020], that slightly changing c (or α, in their model) has a small effect if any on the optimal strategies.…”

“…Second, although static strategies are considered by many (Schoenmakers et al [2008] is one example, but also works such as Bomze et al [2020] present a similar mathematical model to describe a different problem), the main focus is on the computational aspects of the optimal static strategies or of an approximation to it [Liuzzi et al, 2020]. We provide a theoretical background to the usage of static strategies and are able to explain several phenomena that these empirical works report, such as the fact that the optimal strategies only change slightly (if at all) in response to a small change in the ratio between the switching costs and the stage payoffs.…”

“…Another natural method to include switching costs in the payoff is by considering some weighted average between the stage payoff and the switching costs. This was done by Rass and Rainer [2014] and others [Rass et al, 2017, Wachter et al, 2018, Liuzzi et al, 2020, who studied a model where players use static strategies and the payoff is a weighted average of the switching costs and the costs of the game itself: αx T Ay p1 ´αqy T Sy, with α P r0, 1s. This is slightly different from our model.…”

“…Hence, when solving for ṽpcq, only the sum of symmetric off-diagonal elements of S matter and not their exact values. This, for example, is the reason why Liuzzi et al [2020] assumed S is a symmetric matrix.…”

“…We call strategies that utilize the same mixed action at each time step "static strategies" (see also Schoenmakers et al [2008]). Although the problem of finding an optimal static strategy 2 is NP-hard, there are algorithms that find efficiently an approximate solution [Liuzzi et al, 2020]. In this paper we compare the optimal payoff in static strategies to the value of the game (achievable through stationary strategies) in different scenarios, and provide bounds on the loss incurred due to applying this constraint on strategies.…”