I've taken this example from some lecture slides. The slides state there is no Nash equilibrium. I suspect there is also no dominant strategy for either player. Is this true?
Two players $i$ and $j$ simultaneously call either $heads$ or $tails$. If they call the same, player $i$ wins. Otherwise player $j$ wins.
The pay-off matrix is:
$\begin{matrix} i/j & H & T\\ H & 1,-1 & -1,1 \\ T & -1,1 & 1-1 \end{matrix}$
I think there are no dominant strategies because for $i$ the utility of (H,H) $\geq$ (T,T) $\gt$ (H,T) $\geq$ (T,T) where (A,B) denotes $i$ played A and $j$ played B. (1 $\geq$ 1 $\gt$ -1 $\geq$ -1). A similar situation occurs for $j$.
Does that reasoning seem correct?
Thanks!