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Stretching the Ratio
The general idea of stretching the ratio is this: If reinforcers are withdrawn slowly and gradually enough, they may not even be missed, and as a result, the behavior supported by those reinforcers may continue on, even in their absence.
For example, if you make a new acquaintance who is a laugh a minute, you are going to love being with that person, because after all, you are being reinforced at a rate of one reinforcer a minute, just for physically being where that person is located. That's a very good payoff.
Nonetheless, not surprisingly, research tells us that if the laughs were to stop abruptly, your desire to be with that person and your response of seeking them out would also come to a sudden halt.
However, the research tells us that if that person was to go from a laugh a minute to boring, very gradually, in tiny increments, over a time period of many months or years, you would be much more likely to continue to seek them out than if your interactions with them had suddenly become less rewarding all at once.
We see that same phenomena with a barking dog who gets in the habit of sounding off frequently because in the beginning he finds that by doing so, he can force people to pay attention to him. However, as his humans grow accustomed to the his vocalizations over time, they gradually respond less and less often. Which means that without meaning to, they stretch the ratio, thereby causing the dog to continue to bark, even though people no longer respond to his outbursts by offering the rewards that fueled the barking response at the outset.
We can see that stretching the ratio and fading are the inverse of one another, since in the course of fading a reinforcer, or whatever, you automatically, simultaneously, stretch the ratio.
Stretching the ratio is a behavioral procedure that falls in the category of operant conditioning.