Final answer:
The alter ego doctrine applies to option 2, where creditors seek to hold shareholders directly liable for corporate debts, indicating a piercing of the corporate veil.
Step-by-step explanation:
The alter ego doctrine may be invoked in circumstances where the separation between the corporation and its shareholders is so blurred that the corporation ceases to be a distinct entity. This can occur when the corporate form is abused or used for unjust purposes, such as fraud or evasion of existing obligations. Out of the options provided, the scenario most fitting the invocation of this doctrine would be option 2: when unpaid creditors are trying to collect from shareholders a debt owed by the corporation.
This scenario implies that the corporate veil is being pierced to hold shareholders directly responsible for the corporation's obligations, which is core to the alter ego doctrine. In contrast, options 1, 3, and 4 deal with actions on behalf of the corporation, mismanagement, and fraud by a third party, respectively, which do not typically invoke the alter ego doctrine unless those actions are also specifically related to the misuse of the corporate form.