A bat and a ball together cost one dollar and ten cents. The bat costs one dollar more than the ball. How much does the ball cost?
If you said ten cents, you are in good company. Most people say ten cents. It feels right. It arrives instantly, without effort, and it seems so obviously correct that checking it feels unnecessary.
But ten cents is wrong. If the ball costs ten cents and the bat costs one dollar more, the bat costs one dollar and ten cents, and the total is one dollar and twenty cents. The correct answer is five cents. The bat costs one dollar and five cents, the ball costs five cents, the total is one dollar and ten cents.
This is the most famous example of attribute substitution — a cognitive phenomenon identified by Daniel Kahneman and his colleague Shane Frederick that sits at the heart of how our brains handle complex decisions. When faced with a difficult question, the brain automatically and unconsciously substitutes an easier question, answers the easier question, and presents that answer as the response to the original question. The substitution happens so smoothly that we never realize a different question was answered.
When I first encountered attribute substitution in the psychological literature — in the context of research by Gustav Kuhn and Alice Pailhes on why certain procedures in magic are so effective at maintaining the illusion of fairness — I realized I had found the missing piece in a puzzle I had been turning over for months.
The Substitution Mechanism
The way attribute substitution works is elegantly simple in principle, even though its implications are vast.
The brain encounters a complex or unfamiliar question — what Kahneman calls the target question. This question is difficult to answer because it requires information the brain does not have, or because processing the available information would require sustained analytical effort.
Rather than engaging System 2 to work through the difficulty, the brain defaults to System 1 and looks for a related but simpler question — what Kahneman calls the heuristic question. This simpler question addresses something adjacent to the original question but is much easier to answer. System 1 answers the heuristic question and serves up that answer as if it were the answer to the target question.
The person experiencing this substitution has no awareness that a switch has occurred. They believe they have answered the original question. They feel confident in their response. The substitution is completely invisible from the inside.
The bat-and-ball problem demonstrates this perfectly. The target question is “how much does the ball cost given these mathematical constraints?” This requires algebra — System 2 work. The heuristic question that System 1 substitutes is “how can I split $1.10 into two parts that feel right?” And the answer to that simpler question is $1.00 and $0.10. The constraint about the bat costing “one dollar more” is too complex for System 1 to process, so it gets quietly dropped.
The Performance Application
Now translate this to the context of a spectator making a choice during a performance.
The target question the spectator faces is often something like: “Is this choice genuinely free? Am I being influenced? Is the procedure fair? Could the outcome be predetermined regardless of what I choose?”
These are complex, analytical questions. Answering them properly would require the spectator to pause, analyze the procedure, consider alternative explanations, and evaluate the logical structure of what is happening. This is heavy System 2 work, and it requires both time and motivation.
The heuristic question that System 1 substitutes is much simpler: “Did I intend this? Does the outcome feel like it matches my intention? Am I comfortable?”
If the answer to the heuristic question is yes — and it almost always is, because performers design situations where intentions feel satisfied — then the spectator concludes that the answer to the target question is also yes. The choice was free. The procedure was fair. Everything is as it appears.
The substitution is what makes unfamiliar procedures feel fair even when a careful analysis would reveal inconsistencies. The spectator never performs the careful analysis because System 1 intercepts the question and provides a satisfying answer before System 2 can engage.
The Cross-Cut Example
The research on this is remarkable. Kuhn and Pailhes studied a specific card selection procedure and found results that initially seemed impossible.
In the study, participants went through a procedure where the outcome was predetermined — the participant’s action had no effect on which card was ultimately selected. The researchers tested the procedure under different conditions: with misdirection, without misdirection, with a time delay, without a time delay.
Their expectation was that misdirection and time delays would be essential for the procedure to work. Surely, without these elements, participants would realize that their action did not actually determine the outcome.
To their astonishment, the procedure worked just as well without misdirection or time delay. Out of ninety participants, only four understood what had happened, regardless of condition. Even when the researchers added deliberate cues that should have revealed the procedure — numbering the backs of cards with a permanent marker, for instance — only seventeen percent of participants figured it out.
The explanation, the researchers concluded, is attribute substitution. The participants did not analyze the procedure. They substituted the complex question (“did my action actually cause this outcome?”) with a simpler one (“did I perform an action? did an outcome follow?”). Since the answer to the simpler question was yes, they concluded that the answer to the complex question was also yes.
The critical insight is that the substitution works because the procedure feels familiar enough to pattern-match to a simpler mental model. The participant performs an action. An outcome appears. The brain maps this onto its simplest available template: “I did something, and this is the result.” The causal gap between what the participant actually did and what actually determined the outcome is invisible because the brain never bothers to examine it.
Why Familiarity Amplifies the Effect
This connects to something I observed in my own performances long before I had the vocabulary to explain it.
Choices embedded in familiar-feeling procedures produce higher confidence in the spectator than choices embedded in unusual procedures. When the overall structure of the interaction maps onto something the spectator has encountered before — playing a card game, making a selection from a menu, answering a question — the familiarity provides a ready-made mental model that System 1 can use for its substitution.
“I cut the deck and got a card” maps onto a familiar game template. The spectator’s brain already has a model for what cutting a deck means: it means you are choosing where to divide the cards, and the card at that position is yours. This model is so ingrained that the brain applies it automatically, without checking whether the current situation actually works that way.
“I performed an unfamiliar series of actions and eventually a card appeared” does not map onto any familiar template. Without a ready-made mental model, System 1 has nothing to substitute. System 2 is more likely to engage, and once System 2 starts analyzing the procedure, the illusion of fairness becomes much harder to maintain.
I noticed this in hotel room practice sessions in Vienna, where I would work through different presentations of the same underlying effect. Versions that used familiar-feeling procedures produced a sense of ease and naturalness in my test audiences. Versions that used novel procedures — even if they were more clever from a technical standpoint — produced hesitation and analytical scrutiny.
The consulting parallel is exact. In strategy work, I always tell clients that the best process is the one that feels natural to the people going through it. If a decision-making process feels like something participants have done before, they engage smoothly. If it feels alien, they resist. The same principle applies to spectators evaluating whether a choice was fair.
The Hard-Question Dodge
There is another dimension of attribute substitution that I find particularly relevant to performance: the tendency to substitute questions about objective properties with questions about subjective feelings.
When someone asks you “are you happy with your life?” the question is complex, multidimensional, and would require extensive reflection to answer properly. Studies show that people typically substitute a simpler question: “how do I feel right now?” The current mood serves as a proxy for overall life satisfaction, even though the two are only loosely related.
In the performance context, the target question “was this procedure fair?” gets substituted with “do I feel comfortable with what just happened?” And “do I feel comfortable?” is influenced by factors that have nothing to do with procedural fairness: the performer’s warmth, the social atmosphere, the pace of the interaction, the spectator’s mood.
This means that everything surrounding the moment of choice — the rapport, the humor, the conversational ease — contributes to the spectator’s conclusion that the choice was fair. Not because these factors make the choice actually fairer, but because they make the spectator feel more comfortable, and that comfort is what System 1 evaluates when answering the substituted question.
I think about this constantly when I am designing the conversational context around a choice moment. The rapport building, the jokes, the shared laughter — all of these serve an obvious social function. But they also serve a cognitive function: they create a state of comfort that System 1 will interpret as evidence of fairness when it processes the substituted question.
The Flushtration Count Paradox
There is a demonstration from the research that illustrates the power of attribute substitution in an almost absurd way.
Researchers tested a specific visual counting procedure. The procedure was designed so that the audience should be able to detect an inconsistency — in fact, the researchers went out of their way to make the inconsistency detectable, even numbering the items and explaining what was happening.
Despite all of this, participants consistently failed to detect the inconsistency. Their brains substituted the complex question (“do the numbers add up correctly across this sequence?”) with a simpler one (“does this look like a normal counting procedure?”). Since the visual appearance of the procedure matched the template for “normal counting,” the brain answered yes and moved on.
This result haunted me. Even when the evidence of the substitution was literally written on the objects in permanent marker, the brain could not help itself. The substitution is not a choice. It is not laziness. It is the fundamental way System 1 processes information. It finds the closest matching template, evaluates against that template, and delivers a verdict — all before System 2 has a chance to intervene.
The Implications Are Humbling
I want to be honest about how this research made me feel when I first internalized it.
On one level, it was exciting. Understanding attribute substitution gave me a deeper appreciation for why certain approaches to performance work and others do not. It explained patterns I had observed but could not articulate.
But on another level, it was humbling. Because attribute substitution does not just affect spectators in performance situations. It affects all of us, all the time, in every decision we make.
When I evaluate a business proposal, I am subject to attribute substitution. When I decide whether to trust someone, I substitute the hard question (“is this person trustworthy?”) with an easier one (“do they seem trustworthy?”). When I assess whether a strategy will work, I substitute the complex analysis with a simpler feeling: “does this feel right?”
The bat-and-ball problem is not a trick played on stupid people. It is a demonstration of a universal cognitive mechanism that operates in all of us, including the most educated and analytically sophisticated. Kahneman himself has said that knowing about cognitive biases does not make you immune to them. At best, it allows you to catch them after the fact — but the initial substitution happens automatically, before awareness has a chance to intervene.
For me, this insight bridges my two worlds — consulting and magic performance — more completely than any other single concept. Both domains involve situations where people make judgments about complex situations. Both domains involve the risk that simple heuristics will be substituted for careful analysis. And both domains reward the practitioner who understands the substitution mechanism and designs accordingly.
In consulting, I design processes that account for attribute substitution — adding explicit checkpoints where System 2 is prompted to verify System 1’s conclusions. In performance, I design experiences that work with attribute substitution — creating conditions where System 1’s substituted answer serves the experience rather than undermining it.
The mechanism is the same. The direction of the design is different. And understanding both directions makes me better at both.
When a spectator makes a choice and concludes it was free, they are answering a substituted question. They are answering “did I intend this?” instead of “was the procedure fair?” And because the intention was real and the feeling of comfort was genuine, the answer comes back: yes. Freely chosen. No doubt.
A bat and a ball together cost one dollar and ten cents. The ball costs five cents. And the spectator’s choice was free.
Both of these facts are surprising. Only one of them matters to the spectator. And that is exactly how it should be.