A VotingSystem for chosing one option from a set (eg electing a president or MP). 

The voter is presented with a ballot form which lists the options, each with a tickable box. The voter then ticks the boxes corresponding to the options they would accept - this could be none, one, some or all of the boxes. The number of votes cast for every option is then totalled, and the one with the most (the highest approval) is selected.

The merit of this system is that it abolishes TacticalVoting.

''Except that it doesn't. The voter still has an incentive to i) vote for candidates they don't approve of to help ensure that candidates they strongly disapprove of don't win; ii) not vote for candidates they do approve of in case they beat the voter's favourite candidate.''

True. The GradedAcceptanceVote fixes this to some extent.

''Approval voting is a bit like a degenerate case of preferential voting with ties allowed, with only two different preferences possible. Preferential systems allow the voter to express far more information, and don't require this tactically troublesome distinction between approved and not approved.''

A preference ballot does contain more information that an acceptance ballot (N! options rather than 2^N options). However, the "tactically troublesome distinction between approved and not approved" is replaced by similarly vexing problems about the ordering.

ScientificAmerican asked some voting boffins what the best system is, and they (mostly) said AcceptanceVoting:

http://www.sciam.com/askexpert/math/math2.html

It is used by the unicode consortium (see 10.7):

http://www.unicode.org/unicode/consortium/utc-procedures.html

And the capital PC user's group (see XVIII):

http://www.cpcug.org/user/comm/election.txt

And (apparently) the Institute of Electrical and Electronics Engineers, the American Statistical Association, the Mathematical Association of America and the Institute for Management Science.

A more sophisticated version - call it GradedAcceptanceVote - might allow a gradation of acceptance - for example, giving each option a score out of 10.

A variation that was developed but never used allowed election of a parliament by approval voting in such a way that defined constituencies (in fact multiple sets of partitioning constituencies) were fairly represented:

http://bcn.boulder.co.us/government/approvalvote/cavote.html

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CategoryVoting