Constitutionally, the answer to the ‘how many’ question is five: Bulgaria, Peru, Senegal, Taiwan and Tanzania. In practice, though, the answer is four. In 2009 Senegal passed a constitutional amendment creating the position of vice president. However, President Abdoulaye Wade, who was behind the reform, never appointed anyone. So, the answer to the how many question is fairly straightforward.
The answer to the ‘what do they do?’ question is more difficult. Peru and Tanzania have a prime minister and collective cabinet responsibility, but, in practice, they operate as presidential systems, even if Tanzania has a one-party dominant system as opposed to Peru’s typical Latin American fragmented party system. So, the role of the vice-president in these countries resembles those in equivalent presidential systems.
In Taiwan, as far as I can tell, the role of the vice-president is merely to replace the president if the president dies, resigns, or is incapacitated. My understanding is that President Ma’s first vice-president, Vincent Siew, also met with representatives of mainland China at certain meetings, which was politically sensitive. However, the post seems to be mainly honorific. Any more information would be welcome.
In Bulgaria, the vice-president assists the president (Art. 92 -2). In addition, Art. 104 states that the president “shall be free to devolve to the Vice President the prerogatives established by Art. 98 items 7, 9, 10 and 11.”. They are the power to “7. appoint and remove from office other state officials, established by law”, “9. grant, restore, relieve from and withdraw Bulgarian citizenship;”, “10. grant asylum;”, “11. exercise the right to pardon”. Does the vice president exercise any of these powers? The answer is yes.
In Bulgaria, the vice-president’s use of the right to pardon has become politically controversial. For example, the Sofia News Agency, Novinite, reports that in the ten years from 2002-2012 Vice President Angel Marin signed 49 pardon decrees even when the Pardons Committee had recommended that the pardon should not be used. In total during this period the President and Vice-President together issued 529 pardons, including more than 200 people convicted for murder. Interestingly, since taking office in January 2012 the current Vice President, Margarita Popova, has issued only one pardon. When she did so, she identified very clear grounds for issuing the pardon, trying to differentiate her use of the pardon from her predecessor’s.
So, Vice Presidents are very rare under semi-presidentialism. They are even rarer still in semi-presidential countries where the president is not a strong figure. Bulgaria is the only country where there is a weak president as well as a vice president. However, even in this latter case, the role of the Vice President can be controversial.