Willpower and Game Theory
Are you ready for the Prisoners Dilemma Diet?
Posted Apr 11, 2014
A Theory of Temptation
Suppose you've started a new diet plan, and your co-worker brings donuts to the work meeting.
This gives you a choice. You can stick to your diet, or you can eat some donuts.
What will you do?
Before you answer, consider the fact that you will likely face many similar temptations before reaching your weight loss goal.
And pretend for a moment that, while you are responsible for making the present choice, you will not be making those future choices. Instead, future versions of yourself will. In other words, think of these future selves being entirely different people facing their own temptations.
Now suppose you assign the following values to your possible outcomes:
A. A thinner self in the future + donuts now = 10
B. A thinner self in the future + no donuts now = 7
C. A fatter self in the future + donuts now = 3
D. A fatter self in the future + no donuts now = 0
And suppose you reason along these lines:
If all the future versions of myself stick to the diet, then I should make an exception for myself and eat donuts, because I will then get my favorite outcome instead of my second favorite (10 units instead of 7).
If, on the other hand, all the future versions of myself will defect on the diet, then I'll be the lone sucker if I resist these donuts. I should eat donuts and get my third favorite outcome instead of my fourth favorite (3 units instead of 0)
Either way, I (the self making this decision right now) am better off eating donuts.
Now, if every version of your self thinks this way, the outcome is clear. You will repeatedly break your diet and will not see a thinner version of yourself for the foreseeable future.
Some Hope for Willpower?
But what if we could assure you of the following two things:
Condition 1: If you cooperate now, future versions of yourself facing similar choices will cooperate as well.
Condition 2: If you defect now, later versions of yourself facing similar choices will defect as well.
If you had confidence in these two conditions, then your set of available outcomes would be reduced. Outcome A would no longer be available, because, if you defect, your future selves will, too. And outcome D won't be available, because if you cooperate, your future selves will as well.
So your decision would be reduced to a choice between outcome B and outcome C. And that's a no-brainer. If you stick to the diet, you’ll get an outcome valued at 7, and, if you eat donuts, you’ll get an outcome valued at 3. The choice is simple. You stick to the diet.
This gives us a glimmer of hope. If our outcome preferences resemble those above, and we can find a reason to think that conditions 1 and 2 hold, or even that they are likely, then we might be able to get ourselves to resist the donut and stick to the diet.
Resolution to the Rescue
In his book “Breakdown of Will”, George Ainslie claims that resolutions (sets of personal rules) are the glue that makes willpower work (when it does).
Resolutions help to coordinate the decisions of successive versions of the self that are locked into a bargaining game with each other.1
When we start a diet, for instance, we adopt new rules for eating. And typically we start out with some optimism about our ability to follow these rules.
What produces this optimism? One possibility is that it’s produced by a (mostly subconscious) simulation of the bargaining game that will take place among our future selves as they face the series of temptations that will challenge the resolution.
If we feel like the rules will provide a basis for cooperation, then we are optimistic -- and feel a great sense of resolve. If we think widespread defection likely, then we are not optimistic, and probably don’t adopt the diet in the first place.
But is this optimism well-grounded? Do resolutions provide sufficient willpower to help us achieve our goals?
Our experience tells us the answer is: “Sometimes, and sometimes not”.
Sometimes we set new rules for ourselves, follow them without exception, and they take us all the way to our goal.
Sometimes we follow the rules well-enough (with occasional lapses) and eventually get to our goal a little behind schedule.
And sometimes our rules don’t get us all the way to our goal. Somewhere along the way we fall off the wagon and never get back on again.
If this is how things work, two questions seem apt: why do resolutions work when they do? and why do they stop working sometimes?
Why Do Resolutions Work?
Resolutions sometimes translate into willpower, because they sometimes give us confidence that conditions 1 and 2 are in effect.
To see why, we must consider how a current self’s decision will affect the decisions of future selves.
When a future self looks back at the decisions of previous selves, she will take their decisions as evidence for predicting how selves in her future will decide.
If this future self looks back and sees defections, she loses confidence in the power of the resolution to produce cooperation, and expects future selves to defect as well. This will drive her to defect.
Thus, condition 2 seems to hold.
On the other hand, if a future self looks back and sees a series of cooperative choices, she will use this as a basis for predicting future cooperation as well. That prediction, combined with condition 2 holding, will lead her to cooperate.
Thus, condition 1 seems to hold as well.
In general, an earlier self’s choice will serve as evidence that future selves will use to estimate how likely it is that selves in their future will cooperate with them. If you defect, you make it more likely that future selves will also defect. And if you cooperate, you make it more likely that future selves will cooperate.
Resolutions can coordinate expectations like this because they are explicitly placed in the forefront of attention of every self that will face a temptation in the series of relevant temptations. The resolution becomes a common touchpoint in the bargaining logic.
Notice also that this model predicts that there will be a tendency for our resolve to strengthen over time. The more times we cooperate, the greater our confidence that future selves will cooperate as well.
Unfortunately, this strengthening force is opposed by weakening forces (as we know all too well from experience).
Why Do Resolutions Fail?
Here are three reasons our resolutions can lose force over time:
1. Exception Creep
Sometimes we come up against circumstances that make it especially tempting to break the resolution. For instance, if we’re on a diet, and Thanksgiving rolls around, we might think “hey, it’s Thanksgiving. My family will think I’m completely OCD if I stick to my diet today.”
So we make an exception and go off our diet for a special holiday.
This might be alright. If the exception is rare enough, and important enough, the exception might not poison the bargaining logic much.
After all, if a future self looks back and sees only one defection -- the one that happened on the special holiday -- and then looks ahead and sees that there aren’t many special holidays in the future, it might still expect most of the future selves to cooperate, and so it can still feel confident about getting outcome B by cooperating without worrying too much that it will get stuck instead with outcome D.
But you must be careful. If you make an exception on Thanksgiving, and then Christmas, that might be alright. But then it’s your birthday, and your children’s birthdays, and then it’s because your friend came to town, and then it’s because you’re feeling a little under the weather, and then it’s because you’re in a bad mood, and, before you know it, no future self has any confidence that cooperation will pay off, because other selves are making exceptions all over the place.
The resolution is then broken, and will likely not be trusted again any time soon.
2. Mutual cooperation becomes less attractive.
But also note that the success of resolutions depends on outcome B being assigned a higher value than outcome C. And these values can change over time.
For instance, if your original goal was to lose 30 pounds, and you’ve lost 25 of them, losing that last 5 might not seem nearly as important at this point as losing 30 seemed at the outset.
Perhaps the 7 units of value you assigned to the 30-pound-lighter version of you at the outset slipped along the way to a 6, and then a 5 as the gap between your current weight and the goal weight continued to shrink.
That might be one reason “the last 5 pounds” are notoriously the toughest to lose.
3. Mutual defection becomes more attractive.
And the values can change from the other direction as well. At the outset the mutual defection case might have a value of 3, but, as you lose weight, you might find yourself getting hungrier as time goes on, as is often reported by dieters losing substantial amounts of weight. In that case the value of the mutual defection outcome starts to rise, perhaps becoming a 4, and then a 5 and then a 6.
At some point, through some combination of a falling value for the mutual cooperation outcome and a rising value for the mutual defection outcome, it might no longer seem worth it to stay on the diet (regardless of what we expect future selves to do).
So resolutions can help us resist temptation as long as our preferences for certain bargaining outcomes have a certain structure, and as long as we remain confident that future selves will choose as we do.
And these conditions are often met.
And resolutions can also lose their power when we start to allow too many exceptions, when mutual cooperation loses its luster, and when mutual defection grows more enticing.
Well, it might be that understanding these things will help us develop more willpower.
For instance, while making resolutions, we might consider how to prevent exceptions from creeping in by making the line between allowed behavior and prohibited behavior very clear and crisp (what Ainslie calls a “bright line”).
We might also spend more time anticipating exceptions (like special holidays) ahead of time, and setting up special rules for those occasions so that they lose their ability to undermine the general confidence in the resolution.
We might consider ways of keeping the mutual defection outcome from growing in value over time. For instance, if we are dieting, we might try a lower reward diet that allows our set point to fall with our weight, so we don’t get as hungry in later stages of the diet.
And we might consider ways to keep the value of the mutual cooperation outcome high as well.
And, at the very least, this model of temptation and willpower gives us another way to understand what's going on behind the curtain as we face temptations along the way as we pursue our goals.
1 The view presented here is very similar to Ainslie’s view, but isn’t exactly Ainslie’s view. Ainslie’s model has competing interests bargaining with each other. The present model has successive selves bargaining with each other. It seems to me that the successive self model yields a more intuitive analysis — as given above. However, Ainslie’s worry seems to be that successive selves disappear after each moment, and don’t persist long enough to have true interests in future outcomes. Ainslie proposes that interests are what persist through time, and so they must be the entities bargaining with each other. However it’s difficult, in my view, to get an “intuitive” analysis using a bargaining-interest model.
I suspect that a more complete account will involve both interests and successive selves. A good analogy would be a legislative body. The legislature is composed of individuals with competing interests, but it also must think about how it’s ability to follow its own procedures will set precedent for future versions of the legislative body. Interests lobby and bargain within the legislative body (one level of bargaining), and the legislature sets precedent for itself as a body (a second level of bargaining — and the one presented here).