From Not Exactly Rocket Science, here is a writeup on an experiment comparing two systems for enforcing cooperation:  punishments and rewards.  Subjects were organized into groups of four who repeatedly played a simple public goods game.  Each subject in the group could contribute some money to a common pool which would then be multiplied and divided equally among the group members.  It is efficient for the group if all members contribute, but each individual group member would do better by free-riding:  keeping his money and enjoying the benefits of the others’ contributions.

Playing this game repeatedly encourages cooperation:  if one subject is seen to free-ride, the others can respond by contributing less the next time.  Forseeing this, the subjects are induced to keep contributing in order to avoid such a breakdown.  This is all standard.

Now what happens when at the end of each round you give the players the additional option to punish the others by spending $x in order to reduce the other’s income by $3x?  As you would expect this adds to the threat and further enhances cooperation.  But as it turns out, punishments are not as effective as rewards.  An identical setup with the exception that spending $x increases the other’s income by $3x leads to even higher payoffs.

Here’s how to think about the game theory behind this.  We start by considering all options at the players’ disposal and ask what’s the most money they can make as a group if they are nice.  Then we ask an even more important question:  what’s the least they can make if they are nasty.  What matters for cooperation is not so much these amounts separately, but the size of the difference between these.  This difference measures how strong is the threat of a breakdown in cooperation.  If the difference is big enough, it will provide enough incentive to cooperate.

That is, there really is no such thing as a carrot.  Its all about the stick.  What the experimenters are calling a carrot is really just additional scope for cooperation.  When we ask much they can earn by cooperating, taking all options into account, we add this “carrot” into the calculation:  cooperation means contributing to the public good and giving rewards (remember that you pay $x to reward $3x, so the “reward” is just an extra public good added on to the original one.)

Since withdrawing the reward and literally imposing a punishment both reduce the opponent’s payoff by the same amount, the incentive to cooperate is exactly the same in the two treatments.  We should therefore expect that the level of contribution to the public good (net of the reward/punishment addendum) should be the same and the extra payoffs from the reward treatment comes simply from the fact that the rewards are added on.

Aha.  The left panel shows contributions to the public good net of rewards/punishments.  Blue is the reward treatment, red is the punishment treatment.  The right panel shows total payoffs.