Definition of 'Oversubscription Privilege'
A privilege provided to existing shareholders in a company when the company issues a rights or warrants offering. This enables shareholders to "subscribe" to purchase extra shares that are not picked up by the remaining shareholders.
When a company issues a rights or warrants offering, existing shareholders are given the option to maintain their current percentage of ownership in the company by acquiring the rights to further shares. If any current shareholders decide to not accept all of the rights for which they are eligible, these extra rights/warrants become available to the rest of the shareholders at a predetermined ratio (ie. the right to purchase 0.25 shares for every existing share held).
Investopedia explains 'Oversubscription Privilege'
There will always be a maximum number of rights or warrants available on the whole, and should there be more oversubscription demand than supply available, whatever is left over will be allocated on a pro-rata basis to those who pick up their oversubscription privilege.
Companies that decide to issue a rights or warrants offering do so in lieu of a secondary stock offering, which would be available to other investors as well. While the goal of all three offerings is to raise capital for the issuer, if a company bypasses conducting a secondary stock offering, then it shows investor demand is not high enough for them to do so.