Non-deterministic group contest with private information

I study a non-deterministic group contest à la Tullock in which contestants are privately informed about their valuation for winning the contest. A group of two and a single contestant compete against each other to win the contest. I analyze contestants' incentive to free-ride in the large group and I offer a sufficient condition for all contestants to exert strictly positive expected effort in equilibrium. Lastly, I apply this model to verify the validity of Olson's paradox, which asserts that large groups are less effective than small groups at winning a contest.