We define two impartial games, the Relator Achievement Game $\texttt{REL}$ and the Relator Avoidance Game $\texttt{RAV}$. Given a finite group $G$ and generating set $S$, both games begin with the empty word. Two players form a word in $S$ by alternately appending an element from $S\cup S^{-1}$ at each turn. The first player to form a word equivalent in $G$ to a previous word wins the game $\texttt{REL}$ but loses the game $\texttt{RAV}$. Alternatively, one can think of $\texttt{REL}$ and $\texttt{RAV}$ as make a cycle and avoid a cycle games on the Cayley graph $Γ(G,S)$. We determine winning strategies for several families of finite groups including dihedral, dicyclic, and products of cyclic groups.