Extension Theorems for General Lotto Games With Applications to Network Security

Abstract

The General Lotto game is a well-studied model where two opposing players strategically allocate a limited amount resources to multiple contests. In the classic setup, each contest represents an individual battlefield with an associated value, and the objective is to maximize the accumulated value by winning individual battlefields. In this paper, we consider scenarios beyond the classic setup, where (i) the notion of a battlefield is generalized to individual contests, (ii) success on a contest can depend on securing subsets of battlefields, and (iii) the winner of a battlefield can be based on alternate winning rules other than the classic winner-take-all rule. Our main results demonstrate that having an equilibrium solution to a single contest scenario can provide immediate equilibrium characterizations for two important generalizations: 1) the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">linear-count extension</i> , where performance is measured by the cumulative weight of the contests that are won. 2) The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">weakest-link extension</i> (best-shot), where one player must win <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">all</i> (at least one) of the constituent contests in order to earn any value. We demonstrate the applicability of these extensions to a general class of network path defense problems. In particular, our results can generate exact solutions to series-parallel networks, as well as provide bounds on equilibrium payoffs bounds for arbitrary networks. To conclude, we provide equilibrium solutions to a novel class of contests with non-linear favoritism winning rules.