Assuming everything about the vehicles is the same - mass especially, but also shape, center of gravity, etc., such that the entire problem boils down to tire grip, you will lose.
See this answer for more information, but essentially, assuming your vehicle weight is low enough that you're not going to deform the competition surface, static friction is given by:
$$
F_{\mbox{friction}} = \mu_{\mbox{static}} P_{\mbox{tire}} A
$$
where $\mu_{\mbox{static}}$ is the static coefficient of friction, $P_{\mbox{tire}}$ is the pressure the tires exert on the ground, and $A$ is the contact surface area. However, because:
$$
P_{\mbox{tire}} = \frac{F_{\mbox{normal}}}{A}
$$
where $F_{\mbox{normal}}$ is the normal force of the vehicle, you wind up with:
$$
F_{\mbox{friction}} = \mu_{\mbox{static}} \frac{F_{\mbox{normal}}}{A} A
$$
or
$$
F_{\mbox{friction}} = \mu_{\mbox{static}} F_{\mbox{normal}}
$$
So, again, assuming everything else about the two robots is the same, and that you're not operating on a surface like mud or something else that will appreciatively deform under the tires, then the traction/friction force will be higher in the vehicle with a higher coefficient of friction.
That is, your tires should start slipping before theirs do, and since static friction is greater than dynamic friction, once your tires start to slip they get a significant "pushing" advantage over you.