Three Is Company, Two’s a Crowd: A Third Rival’s Deescalating Effect in the Dollar Auction∗

This paper demonstrates a pivotal effect of adding a third rival to bilateral rivalry in the dollar auction, a dynamic model of wars of attrition. Any recursive subgame perfect equilibrium (SPE) involving three rivals on path dynamically Pareto dominates (for the bidders) any recursive SPE involving only two rivals on path, due to a free-rider effect. In any such trilateral-rivalry equilibrium, the gap by which the third-place rival lags behind the frontrunner may collapse or expand, depending on whether the former manages to leapfrog; when the gap reaches the maximum sustainable by the equilibrium, the second-place rival pauses his own costly escalation bid, without conceding to the frontrunner, in the hope to free-ride the third-place bidder’s leapfrogging effort to top the frontrunner thereby repeating the trilateral escalation cycle. ∗We thank seminar participants at the University of British Columbia, National University of Singapore, Singapore Management University and Lingnan University College at Sun Yatsen University for their comments. The financial support from the Social Science and Humanities Research Council of Canada, Insight Grant R4809A04, is gratefully acknowledged. †Ivey Business School, The University of Western Ontario, London, ON, Canada, N6A 3K7, [email protected]. ‡Department of Economics, The University of Western Ontario, London, ON, Canada, N6A 5C2, [email protected].