Takeover contests, toeholds and deterrence

We consider a setting in which two potential buyers, one with a prior toehold and one without, compete in a takeover modeled as an ascending auction with participating costs. The toeholder is more aggressive during the takeover process because she is also a seller of her own shares. The non-toeholder anticipates this extra-aggressiveness of the toeholder. Thus, he is deterred from participating unless he has a high valuation for the target company. This leads to large inefficiency losses. For many configurations, expected target returns are first increasing then decreasing in the size of the toehold.