C. Karp has shown that if $\alpha$ is an ordinal with $\omega^\alpha = \alpha$ and $A$ is a linear ordering with a smallest element, then $\alpha$ and $\alpha \bigotimes A$ are equivalent in $L_{\infty\omega}$ up to quantifer rank $\alpha$. This result can be expressed in terms of Ehrenfeucht-Fraisse games where player $\forall$ has to make additional moves by choosing elements of a descending sequence in $\alpha$. Our aim in this paper is to prove a similar result for Ehrenfeucht-Fraisse games of length $\omega_1$. One implication of such a result will be that a certain infinite quantifier language cannot say that a linear ordering has no descending $\omega_1$-sequences (when the alphabet contains only one binary relation symbol). Connected work is done by Hyttinen and Oikkonen in [H] and [O].