  Introduced by Harris, Insko, Prieto Langarica, Stoisavljevic, and Sullivan,
the \emph{tipsy cop and drunken robber} is a variant of the cop and robber game
on graphs in which the robber simply moves randomly along the graph, while the
cop moves directed towards the robber some fixed proportion of the time and
randomly the remainder. In this article, we adopt a slightly different
interpretation of tipsiness of the cop and robber where we assume that in any
round of the game there are four possible outcomes: a sober cop move, a sober
robber move, a tipsy (uniformly random) move by the cop, and a tipsy (uniformly
random) move by the robber. We study this tipsy cop and tipsy robber game on
the infinite grid graph and on certain families of infinite trees including
$\delta$-regular trees %infinite binary trees with an infinite path rooted at
every vertex, and $\delta$-regular trees rooted to a $\Delta$-regular tree,
where $\Delta \geq \delta$. Our main results analyze strategies for the cop and
robber on these graphs. We conclude with some directions for further study.
