The Chairperson and other members of the Board shall be appointed by the Secretary of State, from nominees approved in writing by the agencies to which this subchapter applies and the exclusive representative (if any) for each such agency. Each member of the Board shall be appointed for a term of 2 years, subject to renewal with the same written approvals required for initial appointment. In the event of a vacancy on the Board, an appointment for the unexpired term may be made by the Secretary of State in accordance with the procedures specified in this section. In the event of inability to obtain agreement on a nominee, each such agency and exclusive representative shall select 2 nominees and shall, in an order determined by lot, in turn strike a name from a list of such nominees until only one name remains. For purposes of this section, the nominee whose name remains shall be deemed to be approved in writing by each such agency head and exclusive representative.