In: Economics
An incumbent is a person who is currently in office, lets say a politician like governber or president. What the inefficient advantage of incumbency, in simple terms, is the fact that if the incumbent (lets say the governer) doesnt work at his best, and the outcome of his policies (lets say the reduction in crime) isnt as best as possible, the voters still vote for him because they think he is needed in the office to reduce crime further. The cost for the voter of electing a new governer is not as high as the cost of keeping the incumbent in office.
This can be seen from a game theory perspective. What does the reduction in crime depend upon? The state of the world (lets say factors such as recession, cultural habits etc.) and the policy of the governer (stricter laws, more budget for police forces etc.). The governer her the full information about these things and can act accordingly. The voter has only partial information and has to decide only on the basis of outcome of the policies of the governer (did the crime reduce?). If the voter had full information, of course he/she wouldve voted on the basis of policies of the governer too. But the voter doesnt. Now, as a politician, lets say I am percieved to be very strong in the area of crime reduction. Given that, is it in my best interest to perform at my best all the times? Maybe not because if I actually reduce crime by what was expected, the voters might not have any incentive to vote for me again (the utility of me now is low). But if there is still some crime, and people think I am good at reducing crime, they would like to elect me. The cost for them to elect the opposition is higher! The utility of me still being in office is higher!
So, in summary, for a governmer percieve to be tough on crime-
probability of being selected in high crime- 1
probability of being selected in low crime- 0
It is clear that the politican will not reduce crime completely as then voters would not have any utility for him.