Bluematter.

Two murder trials take place within weeks of each other. The first one is almost purely a procedural matter: there is overwhelming evidence that the accused is guilty and no reasonable man could argue otherwise. Not so in the second case, where while the evidence point towards a reasonably high probability of guilt, some jurors and members of the public still have their doubts (reasonable or otherwise).

Both men are convicted - and sentenced to the same number of years in prison.

Repeat this same scenario a few thousand times, and the defendants that fall in the second category will have served a lot more 'undeserved' years in prison than those in the first. In fact, given that 'society' determines its optimal sentencing policy by taking into account the probability a miscarriage of justice takes place, those that are indeed guilty will enjoy a more lenient sentence than they would have been awarded in a world with perfect information.

This is a classic example of the inefficiency arising by awarding an average (rather than probability weighed) performance payout to all agents under conditions of uncertainty. I take it as given that society prefers someone who is undeservedly in jail to at least be freed sooner rather than later. In so far as the balance of opinion within a jury is correlated with probability of guilt (and it better be, because otherwise we should discard the notion of a 'fair trial' all together) our approach to sentencing is grossly inefficient and unfair.

Postscript: I am ignoring potential issues arising from strategic interactions amongst jury members here, however these should not affect the basic thrust of the argument.