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We study security games in which a defender commits to a mixed strategy forprotecting a finite set of targets of different values. An attacker, knowingthe defender's strategy, chooses which target to attack and for how long. Ifthe attacker spends time $t$ at a target $i$ of value $\alpha_i$, and if heleaves before the defender visits the target, his utility is $t \cdot \alpha_i$; if the defender visits before he leaves, his utility is 0. The defender'sgoal is to minimize the attacker's utility. The defender's strategy consists ofa schedule for visiting the targets; it takes her unit time to switch betweentargets. Such games are a simplified model of a number of real-world scenariossuch as protecting computer networks from intruders, crops from thieves, etc.  We show that optimal defender play for this continuous time security gamesreduces to the solution of a combinatorial question regarding the existence ofinfinite sequences over a finite alphabet, with the following properties foreach symbol $i$: (1) $i$ constitutes a prescribed fraction $p_i$ of thesequence. (2) The occurrences of $i$ are spread apart close to evenly, in thatthe ratio of the longest to shortest interval between consecutive occurrencesis bounded by a parameter $K$. We call such sequences $K$-quasi-regular.  We show that, surprisingly, $2$-quasi-regular sequences suffice for optimaldefender play. What is more, even randomized $2$-quasi-regular sequencessuffice for optimality. We show that such sequences always exist, and can becalculated efficiently.  The question of the least $K$ for which deterministic $K$-quasi-regularsequences exist is fascinating. Using an ergodic theoretical approach, we showthat deterministic $3$-quasi-regular sequences always exist. For $2 \leq K < 3$we do not know whether deterministic $K$-quasi-regular sequences always exist.

Quasi-regular sequences and optimal schedules for security games