I could be wrong but those cognitive processes you refer to might be "elaboration likelihood model" basically:
comment by Viliam
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I just quoted the article, but it makes a lot of sense to me that people probably use "willpower" to refer to a few fundamentally different things, which is why we still have neither a solid theory of "willpower" nor ways to train it reliably.
There is probably the component of "does this person react immediately, or do they slow down and consider things first?" Which may be partially a habit, and partially biological.
Then there is a question of the mental model the person has. If one person believes that "X means Y", while other believes that "X means Z", of course they are going to react differently when X happens. This model may refer to external or internal world.
For example, if you make a decision to stop smoking, but then you forger and smoke one cigarette, does your model of the world suggest that "it's okay, mistakes happen at the beginning, the important thing is to throw the rest of the cigarettes away and persevere at your decision", or does it say "well, this means you failed completely, you might as well smoke the remaining 19 in the box, because you are a loser anyway, it doesn't make a difference anymore"?
But other example, perhaps more relevant for the marshmallow test is "when other people tell me that if I do X now, I will get a reward in the future, are they usually telling the truth, or are they usually lying?" Because if the child e.g. has a parent who is a habitual liar, with such experience it is rational to grab the one marshmallow while it is there, instead of trading it for a promise of two marshmallows in the future.
Which means that the ways to "increase willpower" would include, perhaps depending on the situation:
- general techniques to "cool down", physiologically, such as breathe deeply, take a walk, always sleep before making an important decision;
- techniques to overcome cognitive biases, especially the ones you happen to be more prone to, for example write down the alternatives, or call a trusted friend and talk with them;
- improve your knowledge of human psychology and reality in general; for example realize that "being 90% successful in your decision to quit smoking" is still a huge improvement over the status quo; and when some people keep telling you otherwise, it is strategical to avoid talking with these people about important things (it may help to notice that they are probably losers in the similar areas of life, so you don't want to take advice from them);
- calibrate yourself to reality, for example by keeping a diary or using some kind of logging system that tells you "if I did X, the usual outcome was Y"; i.e. learn which mechanisms are trustworthy, and by regularly reminding yourself of the fact start actually trusting them on the emotional level;
- probably some other things.
comment by Applesauce
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Thank you for eleaborating. And I agree, a person's schema does influence their thinking as you described in your examples.