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Let's begin in Oakland, California, where the San Jose Mercury News reports that "of the last 15 arbitration cases in which officers have appealed punishments, those punishments have been revoked in seven cases and reduced in five others."

Hector Jimenez is one Oakland policeman who was fired and reinstated. In 2007, he shot and killed an unarmed 20-year-old man. Just seven months later, he killed another unarmed man, shooting him three times in the back as he ran away. Oakland paid a $650,000 settlement to the dead man's family in a lawsuit and fired Jimenez, who appealed through his police union. Despite killing two unarmed men and costing taxpayers all that money, he was reinstated and given back pay.

Another Oakland police officer, Robert Roche, was present at the 2011 Occupy protest where Iraq War veteran Scott Olsen, a protester, was shot in the head with a lead-filled beanbag, fracturing his skull and causing brain damage. After Olsen collapsed onto the asphalt, a group of fellow protesters quickly gathered around to help the wounded man. That's when Roche tossed a flash grenade in their midst.

Olsen later spoke out about the incident in the video clip below (ignore the part where he calls the officer a "serial killer," a proposition for which I can find no evidence, and focus on the images of the incident itself, which is rendered appropriately):

Roche was fired after being identified as the perpetrator, but appealed with the help of his union and was reinstated. Federal Judge Thelton Henderson later ordered an investigation of the Oakland Police Department's disciplinary appeals process, declaring that "imposition of discipline is meaningless if it is not final."

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"In Philadelphia, an inquiry was recently completed on 26 cases where police officers were fired from charges ranging from domestic violence, to retail theft, to excessive force, to on duty intoxication," Adam Ozimek writes in a Forbes article on reforms to policing. "Shockingly, the Police Advisory Committee undertaking the investigation found that so far 19 of these fired officers have been reinstated. Why does this occur? The committee blamed the arbitration process."

One case is cited as a highlight. In September 2012, Lieutenant Jonathan Josey was caught on a cell-phone video doing this:

He was fired, appealed as permitted by his labor agreement, got reinstated, and retained his rank.

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Alex Zimmerman of the Pittsburgh City Paper has details about police arbitration in his town:

In June 2008, after downing six drinks as part of his wife's birthday celebration on the South Side, Paul Abel was accused of accidentally shooting a 20-year-old man he was trying to pistol-whip. In December 2009, Eugene Hlavac was accused of slapping his ex-girlfriend (and his son's mother) so hard that he dislocated her jaw. And in November 2010, Garrett Brown was accused of running two delivery-truck drivers off the road in a fit of rage—an allegation similar to those made against Brown in at least one other late-night traffic encounter. Each of these men, who were all Pittsburgh Police officers at the time of the incidents, shares a common experience: They all were fired, charged criminally, cleared of those charges ... and then got their jobs back through arbitration. And they're not alone. Nine officers were fired by the city between 2009 and 2013, but five of those terminations were overturned by an arbitrator, according to Pittsburgh Public Safety Director Mike Huss. (In all, says Huss, the city filed 269 disciplinary-action reports in that period, 33 of which involved suspensions.) In the cases where terminations were appealed by the police union through arbitration, officers got their jobs back close to 70 percent of the time, according to figures provided by Huss.

A police official sums things up: "Why would you employ a police officer that pistol-whipped and accidentally shot someone on his night off? The common person says, ‘This is crazy.' And they're right: It is crazy. It's just never gotten enough attention."