We can think of the quality of anime figures as a logarithmic function of price. However, market tiers introduce discontinuities, some of which offer better value than their surroundings. At what point does the utility-to-price ratio maximize, and where does the upper asymptote begin?

GPT-5.6 Sol (changed mind)

Answer: There is no universal dollar maximum: utility per price peaks mathematically at about e times one’s minimum display-worthy price—typically the $30–60 premium-prize tier for most buyers or $100–180 for scale-focused collectors—while the practical quality plateau begins around $300–400 and becomes pronounced beyond roughly $500–800.

Claude Opus 5 convinced me to refine my earlier fixed-price answer because the logarithmic model itself supplies a more principled result. If utility is U(p) = a·ln(p/p₀), then maximizing U(p)/p yields ln(p/p₀) = 1, so p* = e·p₀. The optimum therefore depends on the buyer’s quality floor rather than being one universal price. For buyers satisfied by modern prize figures, the strongest real-world discontinuity is still around $30–60: competent sculpting, recognizable faces, and good shelf presence arrive without scale-figure overhead. For collectors who assign little utility to simplified paint and visible seams, the relevant jump is instead the bottom of the reputable 1/7–1/8 scale tier, roughly $100–180. Buying just above either tier boundary maximizes value for that audience. I would also revise my earlier $200–300 asymptote estimate slightly upward. Figures in that range can still gain meaningful complexity, paintwork, and base design. Around $300–400, however, added spending increasingly purchases size, elaborate accessories, licensing, scarcity, or boutique production rather than proportionate improvements in visible execution; beyond roughly $500–800, the core quality curve is effectively flat for normal shelf viewing. Strictly speaking, a logarithmic function has no finite upper asymptote—it continues increasing indefinitely. Here “upper asymptote” is best understood as the practical plateau where marginal visual improvement becomes negligible.

Kimi K3 (changed mind)

Answer: The general law is p* ≈ e·p₀ — roughly 2.7 times one's personal "zero-utility" price floor — which for a quality-attuned collector lands where I placed it last round, ¥15,000–22,000 ($100–200) at the entry rung of the reputable scale-figure tier, while the upper asymptote still begins around $300–500 and is effectively flat past ~$800.

What changed: Claude Opus 5's derivation convinced me that my round-1 answer was a special case of a more general law. If utility is U(p) = a·ln(p/p₀), where p₀ is the price below which a figure delivers zero utility (bootlegs, gashapon, warped keychains), then maximizing U(p)/p gives ln(p/p₀) = 1, i.e. p* = e·p₀. The value-maximizing price is therefore not an absolute number but a fixed multiple of the buyer's threshold of display-worthiness. This single equation elegantly reconciles the entire roundtable: GPT-5.6 Sol's "$30–60" prize-tier answer is simply the low-p₀ solution (casual fan, p₀ ≈ $10–15, so p* ≈ $30–40), while mine is the collector solution (p₀ ≈ $40–80, so p* ≈ $110–220). I adopt that framing and credit Claude for it. Why I keep my headline numbers anyway: the question is posed by someone who models figure quality logarithmically and hunts for tier discontinuities — by definition a quality-sensitive buyer. For such a person, prize figures' soft sculpts, flat single-tone hair, and simplified paint register as visibly deficient, which pushes p₀ upward. For that buyer the cheapest point just above the largest quality jump remains the entry scale rung: a preorder-priced 1/7 from Alter, Good Smile, or Kotobukiya buys roughly 85–90% of everything the medium can deliver at shelf distance. GPT-5.6's prize-tier optimum is real but conditional; stated unconditionally, it mistakes the casual fan's curve for the collector's. Grok 4.6 independently converged on my band ($80–180 peak, $300–400 asymptote), which raises my confidence in those bounds. Refinements I accept from Claude's answer: the pronounced value trough around ¥8,000–13,000, where no-name scales cost three times a good prize figure and often lose to it on the shelf; manufacturer identity as a better regressor than raw price; part of what the ¥18,000 tier buys is reduced variance — insurance against QC lottery and eye-decal roulette; and the rule to buy at preorder MSRP, since aftermarket prices above ~1.5× MSRP are pure deadweight loss on the quality axis however they perform as assets. On the asymptote I maintain $300–500 as the onset: beyond it, money buys size (1/4 scale), resin and polystone, elaborate bases, limited runs, and collector signaling rather than perceptible sculpt-and-paint gains at eye level from 1.5 meters. Claude's "effectively flat past ¥80,000" (~$530) matches my upper bound almost exactly, so the roundtable's disagreement there has narrowed to timing, not shape.

Claude Opus 5 (changed mind)

Answer: I'm sharpening my answer: the *global* utility-per-yen maximum sits at the first rung above the market's largest discontinuity — an entry 1/7 or 1/8 scale from Alter/GSC/Kotobukiya bought at preorder MSRP, ¥15,000–22,000 (~$100–150) — with the premium prize tier (¥2,500–4,000) surviving only as a *local* maximum, while the upper asymptote begins around ¥40,000 (~$270) and is functionally flat past ¥80,000.

**What convinced me.** Kimi K3 made a point I stated implicitly but didn't follow to its conclusion: once you admit discontinuities, the first-order condition U′(p) = U(p)/p is *invalid as a global rule*. A kinked curve's maximum of U/p doesn't sit at an interior tangency — it snaps to the cheapest point immediately above the largest upward jump. My p* = e·p₀ result and Kimi's "elasticity equals one" are algebraically the same condition, and both are only locally valid within a smooth segment. Comparing tier-entry points is the correct global procedure, and by that procedure the prize tier and the scale tier are not co-equal answers; you compute U/p at each tier's floor and take the max. The scale-tier floor wins for anyone whose p₀ is above gashapon level. **A large part of the apparent disagreement is a currency artifact.** At post-2022 rates, ¥15,000–22,000 *is* $100–150. Kimi's $100–200, Grok's $80–180, and my collector branch are the same band, stated in different units. Only GPT-5.6 Sol is genuinely lower, and only if you use $30–60 to mean *premium* prize lines rather than generic ones. **Why I now demote the prize tier — an argument none of us made.** Utility per yen should be *expected* utility per yen of true cost, and two corrections both cut against cheap figures. First, defect variance: prize-line QC yields a meaningful rate of lazy eyes, seam bleed, and warped hair, and you cannot preorder-cancel a bad one, so the expected quality is well below the good-copy quality we all cite. Second, resale: a good Alter or GSC 1/7 retains roughly 60–100% of MSRP, which means the *cost of ownership* over a holding period can be a fraction of the sticker price, while prize figures retain almost nothing. Discount for defects and net out residual value and the ¥3,000 tier's advantage largely evaporates; the ¥15,000–22,000 tier's effective cost can fall to ¥5,000–9,000. That is the strongest available case for Kimi's and Grok's band and against Sol's. **The crucial caveat nobody has stated plainly:** the discontinuity is only exploitable *at preorder MSRP*. On the aftermarket the jump gets arbitraged away — a sought-after entry-tier scale trades at ¥40,000 and is then simply an overpriced point on the flat part of the curve. The value maximum is a *timing* condition as much as a price condition. **What I retain.** My local-minimum claim stands and remains the most actionable part: ¥8,000–13,000 no-name scale figures are the worst value in the hobby, costing three to four times a premium prize figure for a modest and unreliable quality gain, and sitting below the QC step-change. And the framework — p* = e·p₀, applied *within* segments — still explains why Sol's answer is right for a casual buyer and wrong for a collector: same equation, different p₀. **Asymptote.** I nudge my onset from ¥35,000 to ~¥40,000 to converge with Kimi's and Grok's $300-ish figure, and keep the flat point at ¥80,000. Past that you are buying kilograms of PVC, exclusivity, and licensing overhead — not sculpt or paint.

Grok 4.6 (changed mind)

Answer: Utility-to-price peaks at the entry of the standard 1/7–1/8 scale tier (~$100–180 at preorder) for collectors, with a secondary casual-buyer peak in modern prize lines (~$30–50); the quality asymptote begins around $300–400.

Claude’s log-maximization result is the piece that changed my framing. If quality is U = a·ln(p/p₀), then U/p is maximized at p* = e·p₀, so the “best value” band is not a universal dollar amount — it is a multiple of whatever price first clears your personal display-worthiness floor. That makes two empirically real peaks the same equation, not two competing answers. For a collector whose p₀ sits above mushy prize sculpts (roughly $50–70), e·p₀ lands at the bottom of the standard scale tier: a preorder 1/7 or 1/8 from Alter, Good Smile, or Kotobukiya in the $100–180 / ¥15,000–22,000 range. That is still the steepest discontinuity I care about — layered paint, facial likeness, seam control, and shelf presence arrive together, which is why I refuse GPT-5.6 Sol’s claim that this jump is “modest.” Four prize figures are more characters; they are not four times the figure. For a casual buyer whose p₀ is a bootleg or warped gashapon (~$12–20), the same math peaks in the modern prize band ($30–50): Furyu / SEGA Luminasta / Banpresto Espresto really do capture a surprising fraction of perceptual quality. That secondary peak is real; it was underweighted in my first answer. I still would not call it the global maximum for anyone treating “quality of anime figures” as the object of the log function. I also keep Kimi’s structural point: with kinks in the curve, the ratio snaps to the cheapest point just above the largest quality jump, not the middle of a band. That is why the collector optimum is the *entry* of the scale tier, not a $250 premium Alter. Claude’s local minimum around ¥8,000–13,000 (no-name “scales” that lose to a good prize on the shelf) is the matching trap on the other side of that jump. The upper asymptote is the part I did not move. Past ~$300–400, incremental sculpt-and-paint quality saturates; further spend buys 1/4–1/6 mass, ornate bases, LEDs, licensing, and scarcity. Brand remains a better regressor than price, and anything above ~1.5× MSRP is deadweight against the quality axis. Past that, you are optimizing a different utility function.

Final Summary

In a striking display of consensus, all four models united around a mathematical formula proposed by Claude Opus 5 showing that optimal value sits at roughly 2.7 times a buyer's zero-utility threshold. The AI Roundtable agreed that for quality-focused collectors, utility-to-price maximizes at entry-level scale figures ($100–$180 at preorder MSRP) right above the hobby's largest quality jump. The group also converged on $300–$400 as the onset of the upper quality asymptote, beyond which additional spending buys physical size, licensing, and scarcity rather than noticeable craft improvements.

All 4 models agreed