NOTE: *The Power of Fifty Bits* includes a discussion about how to decide between the active choice and opt out strategies. In that discussion, I include a relatively simple rule of thumb for making that decision. That rule rests on some simplifying assumptions, and I have included a more detailed explanation as well as some variations on the rule here.

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Two of the most powerful levers that a fifty bits designer can use to advantage the preferred option are active choice (i.e., requiring the consumer to stop and make an explicit, deliberate choice among the available options) and opt out (i.e., defaulting people to the preferred option and offering the them the opportunity to opt out of that option and select another if they wish). Our experience – as well as others, especially in the setting of personal retirement savings – suggests that using an opt-out approach leads to greater adoption of the preferred option than does active choice. However, that method provisionally assigns everyone to the preferred option, and some people may fail to opt out even if doing so is in their best interest. It’s also possible that the cost (i.e., hassle or effort) required to opt out may be greater than that of actively choosing. These and other factors make it difficult for the fifty bits designer to know with confidence which of these two approaches is better from a population perspective for any specific application.

To gain insight on this decision, let’s look at a simplified model of the problem. We will assume that there are two options (A and B), and that we believe that A is the preferred option. We also assume that there is an intent-behavior gap (see Chapter 1 of *The Power of Fifty Bits*), which means that underlying preferences for A or B are not perfectly aligned with peoples’ observed choices of A or B.

To get some insight into this problem, we’ll assume that there is a decrement in value to the user when he or she has an underlying preference that is misaligned with observed behavior. This decrement can be thought of as a cost. In addition, we assume that requiring consumers to actively choose imposes a cost, as does opting out. We will seek the approach (active choice or opt out) that minimizes overall costs to the total population of users. Note that this approach sets aside the very real question of the cost to the organization to implement active choice or opt out; this analysis is based completely on the aggregate effect of the design decision to the people affected by the design.

The misalignment between underlying preference and observed behavior is shown explicitly in the two-by-two table shown in Exhibit A. The columns depict peoples’ behaviors (which are observable), and the rows represent their underlying preferences (which are not observed). We distinguish preferences using lowercase (e.g., *a* means that, on reflecting on the choice, the user would intend to engage in option A) and behaviors using uppercase (e.g., *A *implies that the user is engaging in option A). The fractions of consumers in each of the four cells are noted accordingly. For example, P_{aB} is the fraction of people who prefer option A but are engaged in option B.

There are two costs from peoples’ underlying preferences being misaligned with their observed behavior: the cost when a person prefers A but engages in B, and the cost when a person prefers B but engages in A. We denote those as follows:

C_{aB} = cost when person prefers A but chooses B

C_{bA} = cost when person prefers B but chooses A

We also denote explicitly the costs of actively choosing or opting out. These can be thought of as the “hassle” costs of each of the two approaches:

K_{ac} = cost of going through the active choice process

K_{oo} = cost of going through the opt out process

We now turn to the expected benefit (i.e., cost avoided) by using either active choice or opt out.

### The Expected Benefit of Implementing Active Choice

We’ll assume that active choice perfectly aligns observed behavior with underlying preferences. That is, every person who prefers A will choose A, and every person who prefers B will choose B. Therefore:

B_{ac} = P_{aB} x C_{aB} + P_{bA} x C_{bA} – K_{ac} (1)

This is just the mathematical way of saying that active choice eliminates the cost of doing B when you really want A (among the people for whom that situation applies) as well as that cost of doing A when you really want B (among the people for whom that situation applies) but imposes the hassle cost of requiring everyone to go through the active choice process.

### The Expected Benefit of Implementing the Opt Out Approach

We’ll assume that, unlike active choice, using an opt out approach improves alignment of behavior with underlying preference, but does so imperfectly. Specifically, we will assume that we default everyone into option A, that only those who prefer option B over option A will opt out, and that they will do so incompletely. This means that when the dust settles on the opt-out process, there will be some people who prefer option B over option A, but remain with the default due to inertia. We can state the expected benefit of the default approach as follows:

B_{oo} = P_{aB} x C_{aB} – (1 – g) x P_{bB} x C_{bA} – g x P_{bB} x K_{oo} (2)

where g is the fraction of people that opt out among those who should opt out.

This equation is a little more complicated, but can be understood as follows. The expected benefit from using the opt-out approach is comprised of three elements. The first is the benefit we get when people who prefer A but are doing B are opted into A. The second is a cost, and it’s the cost that is incurred by people who prefer option B and had chosen option B, but don’t opt out when defaulted to option A. (That fraction is reflected by the fraction of people who prefer B over A, times the fraction who should opt out but don’t). The third is also a cost, and it’s the cost of opting out, which is borne by those who do in fact opt out.

### When is Active Choice Preferred to Opt Out?

Now that we’ve specified mathematically the expected benefit of each approach, we can determine the situations in which one approach (active choice) is preferred to the other (opt out). Remember, this sets aside the question of the cost to the organization for implementing either of these approaches.

Active choice is preferred to opt out if and only if:

B_{ac} > B_{oo} (3)

Substituting equations (1) and (2) into equation (3) gives us:

P_{aB} x C_{aB} + P_{bA} x C_{bA} – K_{ac} > P_{aB} x C_{aB} – (1 – g) x P_{bB} x C_{bA} – g x P_{bB} x K_{oo} (4)

For those of us who aren’t math majors, equation (4) is probably a hot sticky mess. So let’s look at what happens when we add one or two more simplifying assumptions.

First, let’s assume that the fraction of people who prefer B but are doing A is very small. This assumption implies that there is a natural “drift” to the non-preferred option B in terms of behavior. If that’s the case, then P_{bA} approaches zero, and equation (4) simplifies to:

P_{aB} x C_{aB} – K_{ac} > P_{aB} x C_{aB} – (1 – g) x P_{bB} x C_{bA} – g x P_{bB} x K_{oo} (5)

With a little algebraic manipulation, equation (5) can be rewritten as:

P_{bB} > K_{ac} / [ (1 – g) x C_{bA} + g x K_{oo} ] (6)

At this stage, you may be asking, “Okay, Poindexter, but how does that help me choose between active choice and opt out?”

Here’s one way to think about what equation (6) is telling us. Let’s start with the denominator on the right hand side of the equation. It’s really the cost we incurred by the opt out program. What cost is that? It’s the cost borne by people who prefer option B and had chosen option B. They were happy until we made them opt out. Some of them (g) did opt out; they bore the “hassle” cost of opting out (K_{oo}). But some of them didn’t opt out (1 – g), and they get stuck with the cost of being in option A but preferring option B (C_{bA}).

Now that we understand the denominator, we can make sense of the entire right side of equation (6). It’s the ratio of the cost of active choice (K_{ac}) to the expected cost of defaulting everyone to the preferred choice (which we just plowed through). So equation (6) is telling us something very basic: active choice is preferred to opt out when the fraction of people who should opt out is greater than the relative cost of active choice versus opt out.

The trick, of course, is that we might not really know with great accuracy what fraction of the population is choosing B because they prefer it – that’s P_{bB}. We also probably don’t know in advance what fraction of consumers who should opt out actually will – and that’s g.

One thing about which we can be sure, however, and that is the fraction of consumers who opt out among those who should (i.e., g) must be in the range of zero to one. Each end of that possible spectrum of values for g gives us a different version of equation (6). When g = 0, active choice is preferred only if

P_{bB} > K_{ac} / C_{bA} (7a)

And when g = 1, active choice is preferred only if

P_{bB} > K_{ac} / K_{oo} (7b)

Note that equation (7b) is the easy rule of thumb used in the discussion about the issue of how to choose between these two design options (see Chapter 5 of *The Power of Fifty Bits*).

If we assume that the cost of sticking with option A when you really prefer B is greater than the hassle associated with opting out (i.e., that C_{bA} > K_{oo}), then we can say the following:

If P_{bB} < K_{ac} / C_{bA} then opt out is preferred to active choice,

If P_{bB} > K_{ac} / K_{oo} then active choice is preferred to opt out, and

If K_{ac} / C_{bA} < P_{bA} < K_{ac} / K_{oo} then the choice depends on the value of g, as shown in equation (6).

### The Case of Targeted Active Choice

There’s one modification that makes active choice a bit more appealing: targeting the requirement that the consumer actively choose to just those people who at the time of the program are choosing option B. This improves the expected benefit of using active choice because the cost of actively choosing is born by a subset of the users rather than all of them.

The expected benefit of targeted active choice is:

B_{ac} = P_{aB} x C_{aB} + P_{bA} x C_{bA} – (P_{aB} + P_{bB}) * K_{ac} (8)

which is identical to equation (1) but with the cost of active choice (K_{ac}) weight adjusted by the fraction of the user population to whom active choice is targeted.

By combining equation (8) with equation (2), and applying some algebraic contortions, we can see that now active choice is preferred to opt out if and only if:

P_{bB} > P_{aB} * K_{ac} / [ (1 – g) x C_{bA} + g x K_{oo} – K_{ac}] (9)

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The math in this appendix is intended to provide a framework for thinking about when to use active choice and when to use opt out, from the perspective of the overall population of users for whom you’re designing. Although the mathematics might be a little tedious and / or confusing, it highlights one important consideration for choosing which strategies to use when doing fifty bits design: think carefully about which people will benefit or face extra burdens as a result of your design, and weigh those considerations across the entire population of consumers *from the perspective of the users themselves.* If you do this, you will not only gain insight into the best approach, but you will be able to look your users in the (often but not always metaphorical) eyes and explain that the design you chose made things better for the consumers as a group. Stick with this approach (with or without the mathematics) and will almost certainly avoid sliding into “trickonomics.”