Competitive effect of cross-shareholdings in all-pay auctions with complete information

This paper investigates the competitive effect of cross-shareholdings in winner-take-all all-pay auctions with two asymmetric bidders. We show that cross-shareholdings may paradoxically create a “pro-competitive” effect and elicit more effort than a standard contest without cross-ownership. This observation runs in contrast to the anti-competitive effect that cross-shareholdings usually create in standard oligopolistic settings (such as Cournot or Bertrand competitions). Both bidding costs and the sizes of cross-shares affect the resultant total effort non-monotonically. Neither a cross-share nor a higher bidding cost necessarily decreases effort supply. A complete account of equilibrium bidding behaviors is provided and the necessary and sufficient conditions under which cross-shareholdings lead to higher or lower levels of overall effort are identified. However, the pro-competitive effect comes at a loss of efficiency.