Until secret votes were imposed throughout democracies, outright vote buying was not uncommon. Even today, there are suspicions that some elections or referendums are influenced by vote buying. In fact, delivering campaign promises to particular segments of the polity is a particular form of vote buying that is openly practiced today. So what if vote buying were generalized and accepted?
Eddie Dekel, Matthew Jackson and Asher Wolinsky at this question under two scenarios: outright vote auctioning and campaign promises, both compared to a status quo scenario where the electorate honestly votes according to its preferences. Campaign promises can only be delivered upon successful election, whereas vote buying implies payment is contingent only on (observable) votes. Two parties compete in a sequential and alternating bidding process.
The two scenarios lead to dramatically different distributions of favors: with campaign promises, only the electors close to the median voter matter, and those get substantial promises. Vote buying is substantially cheaper for the winning party and is not influenced by voter preferences. This implies that vote buying is inefficient: the richer party wins no matter what.