Honda et al.: Measuring Which Options Are Worth Trying by How Much Uncertainty They Remove — Fukai Reads

TL;DR

"Is this option worth trying?" It is a question designers and puzzle makers face every day, and a new arXiv paper writes it down as a computational model. Honda and colleagues at the University of Tokyo propose the Bayesian Expected Uncertainty Reduction (B-EUR) model, which measures the value of trying a candidate action by a single quantity: how much the trial is expected to reduce your ignorance about the relation between actions and outcomes. Ignorance is expressed as entropy (an information-theoretic measure of how spread out a belief is), and the value of an action is the entropy before observing minus the average entropy expected to remain after observing.

The test bed is an abstract task called the Graph-Shape Guessing Game. A single curve is hidden; you pick a horizontal position and see only the height at that point. Repeat ten times and guess the overall shape. Two properties of the environment are varied: generalizability (how far knowledge from one observation extends to neighbouring candidates) and outcome discriminability (how clearly outcomes of different candidates can be told apart). The authors examine both through simulation and through two human experiments analysed on 44 participants.

One finding sits at the centre. The value of trying, the enjoyment, and the likelihood of being chosen all peak at intermediate generalizability. Too low, and seeing one point tells you nothing about its neighbours. Too high, and a few points settle everything. Either way the point of trying is killed. Note up front that this is an arXiv preprint posted on 6 August 2026, not yet peer reviewed and not yet widely discussed.

Introduction

The authors are Shimon Honda, Takuma Miyaguchi, Koji Koizumi and Takanori Sano — all of the Department of Mechanical Engineering, Graduate School of Engineering, the University of Tokyo — together with Tristan Briard of the Laboratoire Conception de Produits et Innovation (LCPI) at Arts et Métiers ParisTech in France, and corresponding author Hideyoshi Yanagisawa of the University of Tokyo. The paper is arXiv:2608.05642, classified under cs.AI, submitted on 6 August 2026. It is a preprint rather than a peer-reviewed publication, and its citation count can be taken as effectively zero at this point. The only DOI is the one arXiv assigns (10.48550/arXiv.2608.05642).

I picked it today because, although the stated domain is industrial product design, I spent the whole reading thinking about puzzle boards. Making puzzles is, at bottom, the job of arranging moves that are worth trying. A board full of cells that change nothing when you press them is boring; so is a board where pressing anywhere settles everything at once. This paper tries to explain both flavours of boredom as two ends of a single quantity.

The paper also comes with human experiments. It checks whether the peak the model predicts matches the shape of what people actually report as worth trying and enjoyable. Had it been a modelling paper alone I would not have written this much about it. Lining up model prediction, subjective rating and choice behaviour is what makes it worth the time.

Background

Design research has long argued that uncertainty is what drives exploration. Engineering design splits uncertainty in two: epistemic uncertainty, which comes from lack of knowledge and can be reduced by acquiring information, and aleatory uncertainty, which comes from inherent randomness and cannot be reduced in principle. This paper deals only with the former; the latter sits outside the model as observation noise.

The field also has the UDA model (Uncertainty Driven Action model — Cash and Kreye's framework, which treats design activity as a progression through information, knowledge-sharing and representation actions, with uncertainty perception governing selection and progression). Building on it, the authors write that it remains unclear exactly how changes in uncertainty perception determine action selection. B-EUR is explicitly scoped to cover only one piece of that gap: the mechanism for evaluating candidate actions before one is selected.

The other lineage is active inference (Friston and colleagues' extension of the free-energy principle — which treats the brain as a machine minimising prediction error — to action selection). There, the value of an action splits into pragmatic value (does it get you the outcome you want) and epistemic value (how much uncertainty do you expect the observation to remove). B-EUR carries that definition of epistemic value straight into design exploration. The authors also cite prior work (Abhyankar and Ganapathy, 2014) showing that more trial-and-error iterations do not necessarily improve learning or performance, and argue that we therefore need to identify which environmental properties make trials pay off.

Approach

The task works like this. A curve is hidden on screen; the participant picks a horizontal position with a slider, presses Confirm, and the height at that position appears as a point. Ten times. At the end they predict the height at a specified position and receive a rank from S+ to D according to accuracy — but the rank does not change their payment. The authors frame this as a variant of spatially correlated multi-armed bandit tasks (an experimental paradigm where you try options one at a time and learn), while stressing that unlike the standard version, obtaining high output values carries no value here. The point is to strip pragmatic value out of the task and leave only epistemic value.

The curves come from a Gaussian process (a mathematical tool that fixes how points correlate with one another and then samples smooth functions from that). There are two knobs. One is correlation length: turn it up and the curve becomes smoother, so one observation reaches further — generalizability rises. The other is amplitude: turn it up and the vertical swing widens, so differences between candidates become clearer — outcome discriminability rises. I am leaving the equations out here; what the paper does is turn these two knobs.

What the B-EUR model itself is learning is a table mapping "if I take this action, which outcome state comes next". It measures how vague that table is using entropy, predicts the average vagueness expected to remain after trying a given action, and takes the difference. That difference is the epistemic value. The agent computes it for every candidate and picks one under a softmax policy (a way of choosing that spreads probability smoothly in proportion to value). One design detail matters: instead of updating only the observed location, each observation updates beliefs about neighbouring locations too, with a width proportional to the correlation length. That follows prior work (Wu et al., Schulz et al.) showing humans generalise observations to nearby options under limited exploration.

Findings

The simulations yield four predictions. First, uncertainty before observing (prior entropy) rises with amplitude and falls with correlation length. Second, epistemic value follows an inverted U against correlation length, peaking at an intermediate value. Third, epistemic value is higher at larger amplitude. Fourth, epistemic value decays faster over time when correlation length is large and more slowly when amplitude is large. The authors fit an exponential to the time course to obtain a decay coefficient, and in the figure with amplitude fixed at 2.0 they report that at correlation length 0.01 the value stays low from the start, at 1.0 it drops sharply early, and at 0.25 it stays comparatively high.

The human experiments were run at the Design Engineering Laboratory, the University of Tokyo. Forty-six adults aged 20 to 39 were recruited; two were excluded for inconsistencies in the instruction procedure or problems in task performance, leaving 44 (23 men, 21 women) analysed. Experiments 1 and 2 together took about 70 minutes, and participants received a 2,000-yen gift card. The ethics approval number is KE26-13. Stimuli covered eight conditions: two amplitude levels (1.0, 2.0) and four correlation-length levels (0.020, 0.054, 0.147, 0.400). Analysis used linear and generalized linear mixed models in R (statistical methods that estimate effects while accounting for differences between participants), with ratings standardised within each participant.

In Experiment 1, perceived uncertainty rose with amplitude (b=0.106, SE=0.023, t=4.598, p<.001) and fell with correlation length (b=-0.441, SE=0.010, t=-42.743, p<.001), as predicted. But for the value of trying, the main effect of amplitude was not significant (b=-0.030, SE=0.054, t=-0.565, p=.572). The quadratic term of correlation length was negative and significant (b=-0.161, SE=0.014, t=-11.196, p<.001), confirming the inverted U. Enjoyment took the same shape: no effect of amplitude (b=-0.026, SE=0.098, t=-0.261, p=.794) and a negative significant quadratic term (b=-0.188, SE=0.049, t=-3.839, p<.001). The value of trying also fell as observations accumulated, and fell faster at larger correlation length (b=-0.056, SE=0.004, t=-12.558, p<.001).

Experiment 2 looked at choice. All eight conditions were shown side by side and participants picked the one that looked most enjoyable. Here amplitude mattered clearly (b=1.035, SE=0.080, z=12.906, p<.001). The quadratic term of correlation length was again negative and significant (b=-0.816, SE=0.046, z=-17.834, p<.001), so intermediate correlation length was again preferred. In short, outcome discriminability did not show up in ratings when conditions were judged one at a time, but did show up when they were compared side by side. The authors suggest this is because, in this task, the value of trying depended more on how far an observation generalises than on the output range itself, and offer the example that with a hidden sine wave, changing the amplitude would not stop you predicting once you had the wavelength and phase.

Where you can use this

First, the design of individual moves in deduction puzzles. In something like Sudoku or Nonograms, "how much of the board gets settled when I fill this cell" is exactly correlation length. A board where filling a cell moves nothing around it becomes brute force; a board where one cell cascades into the whole grid ends the moment you find the first move. In this paper's terms, both have low epistemic value. If I were building a daily logic puzzle, I would track the distribution of cells settled per move as a generation-time metric and reject both extremes.

Second, how you order a level pack. When you build a set of stages with PCG (Procedural Content Generation, the automatic generation of content), the similarity between adjacent stages plays the role of correlation length. Too similar and it becomes chores; too different and nothing learned in the previous stage carries over. In the design-implications section the authors write that you should vary elements that create differences in response while letting those differences generalise to similar options — that reads directly as level-design advice. For a puzzle game's tutorial run, the principle becomes: change one rule at a time, but make it a rule that still matters in later levels.

Third, the granularity of feedback. Amplitude was outcome discriminability. Whether you return only right-or-wrong or also how close the attempt was is a question on this axis. Here the mismatch between Experiments 1 and 2 becomes interesting. Amplitude did not affect the subjective value of trying when conditions were played one at a time, but it did affect choice when conditions were shown together. I will not overstate it, but it reads as a possibility that in level-select or mode-select screens, where candidates sit side by side, visible differences in outcome can raise how often something is picked.

Fourth, deciding how long a single play should be. The fourth simulation prediction — that epistemic value decays faster when generalizability is high — turns into the rule that the more widely information spreads across a board, the sooner the puzzle should end. The experiment showed the same: the value of trying fell faster at larger correlation length. When trimming move limits or timers later, I think this gives a defensible direction.

Fifth, a restatement of difficulty. This paper measures not difficulty but room to learn. If you re-describe a game's difficulty setting in terms of uncertainty removed per move rather than success rate, levels that are too easy and levels that are unfair line up at the two ends of the same axis. That is what the inverted U is organising.

Limitations

The authors name three limitations. First, the Graph-Shape Guessing Game does not reproduce the full design process; it carves out one aspect — learning action-outcome relations through trials — and B-EUR does not distinguish among, or model transitions between, the information, knowledge-sharing and representation actions of the UDA model. Second, the input-output relation is abstracted to a one-dimensional function. Real product evaluation spans vision, touch, sound and the feel of operation, and multiple dimensions such as functionality, usability, aesthetics, quality and satisfaction; the model does not explicitly capture correlations or interactions among them. Third, real exploration is not driven by epistemic value alone. Pragmatic value, resource constraints, expertise and organisational culture all intervene, and this study deliberately excludes pragmatic value, which puts it that much further from practice.

What follows is what I noticed on reading. First, correlation length has only four levels, spaced logarithmically between 0.02 and 0.4, while the simulation figures span 0.01 to 1.0. Whether the peak of the inverted U sits at the same place for model and humans cannot be read off this paper. The inverted U itself is tested by the sign of the quadratic term, but a significant quadratic term and a peak located inside the tested range are, strictly, different claims.

Second, the two environmental properties are not cleanly separated as experimental manipulations. Raising amplitude raises outcome discriminability and, at the same time, raises prior uncertainty itself — that is precisely the authors' own S1. So amplitude acts through at least two routes. Reading the null effect of amplitude on the value of trying in Experiment 1 as "outcome discriminability does not affect subjective value" is, I think, premature.

Two smaller points. Participants were instructed that their goal was to enjoy the game. For a design that uses enjoyment ratings as one of its main measures, that instruction could push ratings upward. And while 44 participants is not unusually small for this kind of experiment, the data come from one laboratory, one task and one cultural setting, and I found no mention of preregistration or replication in what I read. The inverted U is a shape that recurs constantly in behavioural science, which is exactly why it deserves caution. No more can be said than that it was observed under the conditions of this study.

How Fukai reads it

What follows is my own reading. I would place this study at the most implementation-facing end of a long lineage of attempts to measure interestingness — from Berlyne's arousal potential theory (the old hypothesis that pleasure peaks at intermediate novelty and complexity) through to curiosity research and its claim that we explore where learning looks possible. The authors touch on both in their discussion. But what struck me is that this paper measures interestingness not as a property of the object, but as a property of the relation between object and observer. Correlation length is a property of the curve; epistemic value depends on what you currently know and falls moment by moment. In the vocabulary of design criticism, this is less the automation of a difficulty curve than an attempt to define boredom as a function of time. Stop pinning a static number to a puzzle's difficulty and hold it instead as a function of the player's belief state — if you are willing to go that far, I read this formulation as having real use.

Closing

Read on its own, this paper may feel abstract. If you want a map, start on the active-inference side with expository work on Friston's free-energy principle, then read alongside the series of information-theoretic studies of affect coming out of the corresponding author's laboratory — the line that treats novelty and interest as quantities of information. From there you can see what the definition of epistemic value as expected uncertainty reduction is standing on.

If you would rather come at it from games, the entry point is human experiments on multi-armed bandit tasks and the exploration-exploitation trade-off, in particular the spatially correlated bandit work by Wu and by Schulz that this paper cites. The question of how far people generalise knowledge to neighbouring options under a limited budget has almost the same shape as the level-design question of what carries over from one stage to the next. I am, quietly, looking forward to the day those two can be discussed in the same words.

References

Papers and materials referenced in this article:

・Bayesian Expected Uncertainty Reduction (B-EUR) Model: A Computational Account of What Makes Design Options Worth Trying (Shimon Honda, Takuma Miyaguchi, Koji Koizumi, Takanori Sano, Tristan Briard, Hideyoshi Yanagisawa, 2026, arXiv preprint arXiv:2608.05642)

・DOI: 10.48550/arXiv.2608.05642 (assigned by arXiv; not a journal DOI)

・HTML version of the paper (full text, figures, appendix)

・Publication list of the Yanagisawa Laboratory (Design Engineering Laboratory), the University of Tokyo (related work by the corresponding author)

・Related work: Information-Theoretic Free Energy as Emotion Potential: Emotional Valence as a Function of Complexity and Novelty (Yanagisawa et al., 2020, arXiv:2003.10073)