The Representation and Framing of Structure

[This is post 5 in the "Structure and Cognition" series; links to all the posts can be found here]

I. 

A major motivation for starting this blog was to clarify my thoughts about an idea that wouldn’t leave me alone. I see it everywhere, which is ironic, because the idea is about seeing (or not seeing) common structure lurking beneath different-seeming surfaces. 

Framing is a big topic in the psychology of decision making. The way choices are presented can influence the decisions we make. This is why food sold in the supermarket is more likely to say '99% fat free' than '1% fat.' Sanitizing wipes 'kill 99.9% of germs,' they don’t 'only leave 0.1% of germs alive.' Obviously, these values are identical, so it shouldn’t matter one way or the other, but it does. 

From the perspective of traditional economics, this is insane. People should prefer whatever they actually like – every product should have (or be translatable to) a utility value. You should gain some amount of utility, whatever that amount might be for you, from yogurt or Lysol, etc., and use that to make your decisions. If people prefer 99% fat free yogurt to 1% fat yogurt, that implies that there are 2 different utility values for the same product (or that we don’t make decisions based on a coherent sense of utility at all – I don’t know which of these traditional economists would hate least). 

The classic example of framing effects in decision making comes from Kahneman and Tversky (1981). They presented one group of participants with the following scenario (Brackets show the breakdown of participants’ choices): 

Imagine that the U.S. is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two alternative programs to combat the disease have been proposed. Assume that the exact scientific estimate of the consequences of the programs are as follows:

Problem 1:

If Program A is adopted, 200 people will be saved. [72 percent]

If Program B is adopted, there is 1/3 probability that 600 people will be saved, and 2/3 probability that no people will be saved. [28 percent]

Which of the two programs would you favor?

A second group was presented with another set of choices: 

Problem 2:

If Program C is adopted 400 people will die. [22 percent]

If Program D is adopted there is 1/3 probability that nobody will die, and 2/3 probability that 600 people will die. [78 percent]

Which of the two programs would you favor? 

The issue, which is easy to see when looking at both problems simultaneously, is that A is the same as C and B is the same as D. You can choose the gamble (B/D) or the sure thing (A/C), whichever you prefer. What you can’t do is choose A and D, but of course that’s what most people did. 

The standard story here is that people are inconsistent and let information that should be irrelevant sway their decision making. The more specific explanation is that losses are more painful than gains are good. Losing $100 hurts more than finding $100 would feel good. Gambling to save all 600 isn’t worth the chance of losing everyone, because 200 lives saved for sure is pretty good. On the other hand, losing 400 people for sure seems incredibly bad and even gambling on losing everyone is worth the risk. 

Sometimes there’s some pushback about linguistic implicature where the argument is that Problem 1 seems more optimistic or maybe the lack of explicit comment in Program A about the fate of the other 400 people leads participants to assume those people may have a chance to live.

My interest here is not whether people are irrational or what precise psychological mechanism causes the preference swap. The issue I’ve been building to from my first post is the profound influence that framing has on cognition and problem solving. 

This might seem like a trivial point, but it might start to feel more important when you connect it to the effects of structure discussed in the previous posts. We’ve seen that environmental structure can have a dramatic effect on behavior and decisions. Well, structure doesn’t just present itself to the mind – it needs to be represented. Depending on their framing, problems may be represented very differently, and the right representation can mean the difference between easily getting the right answer and a futile struggle that goes nowhere. 

Take the following math problem, supposedly presented to John von Neumann by another mathematician: 

“Two trains 200 miles apart are moving toward each other; each one is going at a speed of 50 miles per hour. A fly starting on the front of one of the trains flies back and forth between them at a rate of 75 miles per hour. It does this until the trains collide and crush the fly to death. What is the total distance the fly has traveled?”

To solve this problem, it seems like you have to sum up the infinite series of distances the fly travels as it flies back and forth between each train. This is an arduous computational process, to say the least. In fact, though, much of the work can be avoided if you realize that the two trains will collide in 2 hours, which means the fly will spend that same amount of time flying at its speed of 75 miles per hour, yielding a total distance traveled of 150 miles. 

Von Neumann immediately replied with the correct answer. When the mathematician suggested that he must have known the “trick” solution, as most people try to sum the infinite series, von Neumann reportedly responded “what do you mean? That’s how I did it.”

The point, for those of us less mathematically gifted than von Neumann (aka, everyone), is that the problem is extremely difficult to solve when it is framed in terms of distance, but trivial if framed in terms of time. 

As is the case with many of the problems cognition needs to solve, framing is intractable. Computationally speaking, there isn’t enough time before the universe burns out to cycle through every possible framing of every (or probably any) problem. It would be nice if we could just pick the best framing for every problem we face, but that’s not possible.

Here’s a quick illustration of the power of framing on thought: Which of the following words doesn't belong: 

    “skyscraper”, “cathedral”, “temple”, “prayer” 

Most people say “prayer” is the odd one out. But how about here: 

    “prayer”, “cathedral”, “temple”, “skyscraper” 

These are the same 4 words, but arranged differently, it now seems like “skyscraper” is the outlier. 

Maybe the best way to answer this question would be to rearrange the words into the 24 different possible orders and more objectively assess possible outliers, but that’s so impractical it’s almost inconceivable. 

In fact, we don’t just avoid switching frames, sometimes we almost can’t. Take the famous candle problem: on a table in front of you there is a candle, a book of matches, and a box of thumbtacks. How can you fix and light a candle on a wall so that the candle doesn’t drip wax onto the table below? 

This task is really hard for most people because they can’t escape the intuitive representation of the presented objects. To solve it, you need to stop thinking of the box of thumbtacks as a container and start thinking about it as a separate object. If you pin the box to the wall, it can serve as a shelf for the candle. Here too, framing can help achieve a better representation: when the nouns, including “box” are underlined, the percentage of people solving the problem almost doubles. Phrasing the box of tacks as a “box and tacks” also helps performance. 

That we basically take framings as givens has a bunch of interesting implications. For one, it means we might struggle with a problem in one framing when the exact same problem is easy when framed differently. 

The Wason selection task (1966) is a famous task in the psychology of deductive reasoning. There are 4 cards below: E, K, 2, and 3. The task is to test the rule: “if a card has a vowel on one side, then it has an even number on the other side.” Which cards do you have to turn over to test this rule? 

 In Wason’s study, less than 10% of participants got the right answer. Most people think you have to turn over the E and 2. Some people just turn over the E. These responses are, you will probably be unsurprised to learn, not right. 

The rule doesn’t require anything about even numbers, like the 2. It (implicitly) allows a consonant to have an even number on the other side. The only constraint is that vowels must have even numbers on their other side. The answer, then, is to turn over the E and the 3. If the E has an odd number or if the 3 has a vowel on the opposing side, the rule is wrong. The other cards can’t prove anything one way or the other. (Logically, the rule is 'if p, then q.’ Proving it wrong requires ‘p and not q.’ Turning over the 2 implies ‘not p and q,’ which isn’t a violation of the rule.)   

The twist is that people easily solve this problem when it’s about a social situation rather than an abstract logic puzzle – that is, when it’s framed differently. 

Below is another set of 4 cards: Beer, Soda, 25, and 17, where the numbers are ages of drinkers at a bar. Here the rule to be tested is “if someone is drinking alcohol, they are over 21.” This problem has the same logical structure to the first formulation, but everyone gets that one wrong and this one right. It should be intuitive that we don’t care what the 25-year-old is drinking – we only need to test the Beer and 17 cards. 

 

This result is often interpreted as evidence for a special cheating detection system in the brain that is great at noticing when people break social rules. I don’t have evidence to support this claim, so I won’t belabor this point, but I suspect this an effect of the framing giving us a representation that allows us to solve the problem using less working memory.*

Herbert Simon demonstrated how sensitive problem solving is to framing with his studies of the Tower of Hanoi problem. The Tower of Hanoi is a puzzle comprising 3 pegs and (in the example below) 4 rings. The task is to move all the rings from A to C with the following restrictions: you can only move one ring at a time, and you cannot place a larger ring on top of a smaller one. 

 

Simon and Hayes (1977) formulated various problems that were isomorphic to this problem (i.e, the underlying structure was the same), but that had different cover stories framing the problem. Instead of pegs and rings, there were 3 monsters and 3 globes of different sizes that could shift in various ways according to the rules of “monster etiquette” which prohibited, e.g., moving more than one globe/monster at a time. 

In one version of the problem, the monsters are standing on the globes in the wrong order and had to move themselves to the right globes. In another, they’re holding the wrong globes and need to pass them back and forth in the right way. In others, the globes/the monsters can change sizes in accordance with monster etiquette. 

Though all versions of the problem had the same structure, the framing had a significant impact on participants’ ability to solve them. In one experiment, the average times to arrive at the solution ranged from 13.7 minutes to 25.3 minutes depending on the framing. Failures to solve the problem at all within the allotted hour time limit were also dependent on condition: 0/19 failed one problem formulation while 8/24 failed in another framing. 

Zhang (1991) studied a variant of the Tower of Hanoi problem with different framings. The first was called the Waitress and Oranges problem: 

“A strange, exotic restaurant requires everything to be done in a special manner. Here is an example. Three customers sitting at the counter each ordered an orange. The customer on the left ordered a large orange. The custom in the middle ordered a medium sized orange. And the customer on the right ordered a small orange. The waitress brought all three oranges in one plate and placed them all in front of the middle customer (as shown in Picture 1).

Because of the exotic style of this restaurant, the waitress had to move the oranges to the proper customers following a strange ritual. No orange was allowed to touch the surface of the table. The waitress had to use only one hand to rearrange these three oranges so that each orange would be placed in the correct plate (as shown in Picture 2), following these rules:

• Only one orange can be transferred at a time. (Rule 1)

• An orange can only be transferred to a plate in which it will be the largest. (Rule 2)

• Only the largest orange in a plate can be transferred to another plate. (Rule 3)”

 

A second version of this problem was identical, but substituted cups of coffee for oranges, with the cups beginning in a stack with the smallest on the bottom. 

Despite the two problems possessing identical structure, the oranges version took more than twice as long to solve, required about twice as many moves to solve, and caused more errors. The difference is that in the coffee cups problem, rules 2 and 3 are required by the physical constraints of the external world. You can’t put a smaller cup in a larger cup because it will spill (Rule 3). That’s why they have to start stacked from smallest to largest, unlike the oranges, which are just sitting in a bowl. For the same reason, you can’t transfer a cup to a location where it won’t be largest (Rule 2).** 

When all 3 rules need to be juggled in working memory at the same time, the problem is much harder to solve. When 2 of the rules can be offloaded into the structure of the world, it’s much easier to solve the problem. This is what I mean about environmental structure helping cognition and framing partially determining that structure.

The coffee cup problem is easier because you only have to focus on one rule, whereas keeping track of 3 rules at once is a lot for working memory to handle. A similar benefit can be achieved by organizing information in ways that aid representation and relieve the burden on working memory. 

Newell and Simon (1972) describe a game of “number scrabble” with the following rules: 

“A set of nine cardboard squares (pieces), like those used in the game of scrabble, in placed, face-up, between the two players. Each piece bears a different integer, from 1 to 9, so that all nine digits are represented. The players draw pieces alternately from the set. The first player who holds any subset of exactly three pieces, from among those he has drawn, with digits summing to 15, wins. If all the pieces are drawn from the set without either player obtaining three whose digits sum to 15, the game is a draw” (p. 59). 

This seems like a difficult game to play. You have to hold various combinations of digits that sum to 15 in order to recognize which you need and which your opponent needs, if you are going to block them when they have 2 out of 3 numbers needed to win. 

With some clever strategizing, you can trap your opponent by picking numbers that give you more than one way to get 15. For example, if you have 8 and 2, and 1, 6 and 7 are still available, you can guarantee a win by picking the 6 (assuming your opponent doesn’t win on the next turn). 6 can be used in conjunction with either of your other numbers and those remaining on the board to create 15: 8 + 6 + 1 = 15 and 6 + 2 + 7 = 15. Your opponent can’t block you by picking either the 7 or the 1, because you can just choose the other. If this was a lot to follow, hang on a second, it should get a lot easier. 

Let’s shift to a simpler game: tic-tac-toe. Though it isn’t immediately apparent, this is exactly the same game. If the numbers are laid out to form a “magic square,” where each row and column sum to 15, the two games’ isomorphic structure (and how I managed to write the previous paragraph) is revealed. 

 

Again, the point here is that despite these problems being logically identical, it's a lot easier to play tic-tac-toe than number scrabble. If this is the case, we might hope to just translate one framing into another, but because trying out infinite framings (what other formats could the numbers map on to? Letters? Shapes?) is impossible, we don't do this. We don't even recognize that we could, because the surface framing obscures the underlying structure. 

Note that it isn’t that writing things down helps, per se. We often do use writing to change a problem’s representation and reduce the cognitive effort required to solve it. For example when we write down a math problem there is less information we need to keep track of as we solve the problem. But what kind of visual representation matters. Arabic numerals are quite useful for things like multiplying, unlike Roman numerals, where looking at a written formulation of the problem is not likely to help.

(Incidentally, adding Roman numerals is easier than you might expect. Given two numbers, combine all the letters from each one, regroup them so all of the same letters are next to each other, and then simplify by reducing, e.g., IIIII to V. For example, 

306 + 238 = 

CCCVI + CCXXXVIII =

CCCVICCXXXVIIII = 

CCCCCXXXVVIIII = 

DXXXXIIII (Norman, 1993)).

As another example of how a visual representation may or may not help reduce working memory demands, take the following graphs. (The graphs plot the percentage of houses per state that exceed the requirements for radon exposure.) 

The first graph is hardly easier to use than a list of states with their corresponding percentages written down next to them. The shading is arbitrary, so any attempt to compare multiple states requires a constant shifting of attention back and forth between the key and the map to update the contents of working memory. Try to compare Arizona, North Dakota, and Kansas and note how you have to maintain the association between each state and its percentage as you check what the next state’s shading corresponds to on the key. 

 

Norman (1993) calls this an “unnatural mapping.” The graph below has a natural mapping. Here, the shading is meaningful and it’s easy to compare states without referring back to the key over and over. (Or, at least, it would be, if I could find scans of these graphs of greater than potato-quality). 

The difference is that the ordered density of shading conveys the same increasing pattern as the percentages they represent. The random patterns on the first graph don’t map neatly onto the percentages, so they need to be manually checked each time they are referenced. 

 

III. 

One of the other interesting consequences of framing obscuring common underlying structure between problems is that many problems we face may turn out to have the same structure and we may never notice. For example, take the deep similarity between squirrels foraging for acorns in a group of trees and what you are likely to do if I ask you to list as many animals as you can from memory. 

When animals forage, they want to gain the most food for the lowest cost. If a squirrel finds a tree with a lot of acorns, it is worth staying and harvesting acorns from that tree. Leaving the tree to search for another one takes time and effort and may not pay out if the next tree doesn’t have acorns. But the longer the squirrel stays at a single tree, the more it depletes the acorns in that tree, making leaving in search of better options more enticing. 

The squirrel doesn’t want to stay too long at a given tree if it can do better by exploring others, but it also doesn’t want to leave a good tree too early. The marginal value theorem (Charnov, 1976) predicts that animals foraging in “patchy” environments should switch to a new patch when the rate of reward in their current patch falls below the average rate of return in the environment as a whole. In a rich environment, switching is likely to have a high payoff, whereas in a poor environment, one should stay longer if you manage to find a good tree. Another important factor is travel time: if there’s a long distance between trees, that’s more effort and opportunity cost from not harvesting while you’re traveling. On the other hand, shorter distances mean transitions should occur sooner, because the cost of switching patches is reduced. 

This is a theory of optimal foraging behavior (as opposed to a description of what animals actually do), but it happens to be very accurate in predicting the behavior of many animals. 

And it turns out that optimal foraging is related to other, very different seeming tasks, like “foraging” in memory. If I did ask you to list as many kinds of animals as you could from memory, what happens when you try to do this? 

Probably, without meaning to, you list a bunch of animals of a certain type, like pets, before shifting to another category like farm animals, maybe followed by predators or birds. This behavior is exactly the same as the squirrel with the trees: if a category keeps giving you ideas, stay and exploit it. When it starts to run dry, “move” to a different category and get what you can there. 

Ok, so two random things are similar, big deal. Except models of students studying have also applied optimal foraging theory to characterize how people avoid studying things they already know (returning to a previously visited, depleted patch) and choose to study the easiest facts first, and switch to studying a different “patch” when the items to be studied are too difficult (Metcalfe & Jacobs, 2010). 

Optimal foraging has also been used to explain mood and mood disorders like depression, where motivation is viewed as tracking rewards in the environment. When your efforts repeatedly meet with frustration, it’s akin to traveling to a whole bunch of trees and finding nothing at each one. Under these circumstances, the rational thing to do may be to do nothing and conserve energy (Nesse, 2019). 

Also, “information foraging,” where “patches” are reddit, twitter, youtube, etc. Because the travel times between these patches are basically 0, it’s a lot easier to switch as soon as the task you’re currently engaged with becomes the tiniest bit boring. This may partially explain our short attention spans in the modern world (Rosen & Gazzaley, 2016) 

When I started this blog, I wanted to talk about structure and especially its framing and how that can cause us to fail to notice isomorphic structure across problems. In my first post, I wrote about how I couldn’t think of an idea for the introductory post for weeks even though it turned out that the title I had in mind was basically describing the reason I was struggling (and the solution). Even though I had two ideas in mind, and they were basically the same idea, I didn’t realize because I couldn’t frame them the right way. Now, I hope I have. 

* This will hopefully be more clear by the end of the post, but the gist is you already know what violations of the social rule look like, they're "compiled" in the mind. Possible violations of the logical rule, on the other hand, need to be simulated and effortfully compared to the text of the rule. If I tell you "a 17 year old is drinking," you already know to check more closely. If I tell you "a 3 has a vowel on the other side," it takes some work to figure out whether that's ok - do you even still remember the rule? 

[Edit: When I first wrote this post, I speculated about cheating detection being due to the framing of the problem rather than an evolutionary module specialized to detect violations of social rules. It turns out Kahneman and Tversky (1996) made the same suggestion.]

** Zhang (1991) also had an intermediate case with donuts placed on pegs, just like the standard Tower of Hanoi. In this version, only rule 3 was external (i.e., constrained by the external world) and performance was intermediate between the oranges and coffee cups problems.  

References:

Charnov, E. L. (1976). Optimal foraging, the marginal value theorem. Theoretical population biology, 9(2), 129-136.

Gazzaley, A., & Rosen, L. D. (2016). The Distracted Mind: Ancient brains in a high-tech world. Mit Press.

Tversky, A., & Kahneman, D. (1996). On the reality of cognitive illusions. Psychological Review, 103(3), 582-591.

Metcalfe, J., & Jacobs, W. J. (2010). People's study time allocation and its relation to animal foraging. Behavioural processes, 83(2), 213-221.

Nesse, R. M. (2019). Good Reasons for Bad Feelings: insights from the frontier of evolutionary psychiatry. Penguin.

Newell, A., & Simon, H. A. (1972). Human Problem Solving. Englewood Cliffs, NJ: Prentice-Hall.

Norman, D. (1993). Things That Make Us Smart: Defending human attributes in the age of the machine. Diversion Books.

Simon, H. A., & Hayes, J. R. (1977). The understanding process: Problem isomorphs. Cognitive Psychology, 8(2), 165-190.

Tversky, A., & Kahneman, D. (1981). The framing of decisions and the psychology of choice. Science, 211(4481), 453-458.

Wason, P. C. (1966). Reasoning. In B. M. Foss (ed.), New Horizons in Psychology, 1. Harmondsworth: Penguin Books. 

Zhang, J. (1991). The interaction of internal and external representations in a problem solving task. In Proceedings of the thirteenth annual conference of cognitive science society (Vol. 88, p. 91). Erlbaum Hillsdale, NJ.