In certain jurisdictions (e.g. Honk Kong, Sweden), when a buyer acquires a certain level of ownership in the company (30% in both HK and Sweden), it must make a mandatory offer for all of the remaining shares of the company. The consideration also depends on the jurisdiction, but usually is either the last price or the highest price paid by the buyer in the last 6 months.
Mandatory offer situations might differ depending on the way the ownership threshold was breached (was it a small purchase in the open market or a large, still ongoing tender offer). In any case, the key here is to understand the incentives of the buyer (will it be willing to proceed with a further offer or will it try to evade it) and the possible risk of the mandatory offer getting waived. For example, in HK it is possible to escape the mandatory offer only if 75% of the total and 50% of independent shareholders approve the whitewash waiver. Therefore, sometimes, the buyers take precautions and include such a condition in the original tender offer beforehand.