Delay-tolerant networks (DTNs) are typically sparse ad hoc networks where node density is low and contacts between nodes in the network do not occur very frequently. The existing location privacy protection methods, which require mobile nodes to collectively change their pseudonyms in special regions called mix zones, may not work well in DTNs due to their unique characteristics, including low network density and limited contact duration. In this paper, we propose a novel cooperative location privacy protection scheme, which is called AVATAR, for sparse DTNs. The main idea of AVATAR is to generate a certain number of virtual nodes in the proximity of a node and allow both virtual and real nodes to make a coordinated pseudonym change in an enlarged region, which are named virtual mix zones. Each AVATAR participant benefits from increased location privacy protection at the cost of generating a series of signed position messages, which are named footprint signatures. To stimulate each node to contribute more footprint signatures to the virtual mix zones, AVATAR proposes a reward mechanism, which is modeled as a multiunit discriminatory auction game. Extensive simulations and analysis have been provided to demonstrate the effectiveness and efficiency of the proposed scheme.