Casting Infinite Tsukuyomi over the Shinobi World

(Academic version: Robust Narrative Persuasion.)

The Rinne Sharingan reflected on the moon sends one public pattern across the shinobi world. Every shinobi sees the same omen. Priors determine which dream-template becomes most plausible, so one field of moonlight produces individualized dreams.

Anime still: NARUTO Official Site.

Madara casts a menu of dream-templates in the moonlight before the waking world reveals its next omen. Devastation and cooperation emerge from the waking world itself. After observing one of them, each shinobi selects the template that makes the omen most plausible under his prior, updates within that template, and chooses between waking agency and the dream sustained by the God Tree.

The same moonlight reaches every shinobi. The same menu interprets every omen. Heterogeneous priors turn that common menu into different selected templates and different posteriors. Infinite Tsukuyomi is therefore a public narrative system that generates individualized dreams through self-selection.

Anime still: NARUTO Official Site.

Settings

The waking world

Let the world state be

$$\Omega=\{H,L\},$$

where:

• $H$ means the waking world has entered terminal conflict, making permanent immersion in a peaceful dream attractive to a shinobi;
• $L$ means the waking world still contains a viable path to peace, making waking agency attractive.

Each shinobi chooses

$$a\in\{0,1\},$$

where $a=1$ means entering and remaining in Infinite Tsukuyomi and $a=0$ means preserving waking agency.

Each shinobi has a prior

$$\theta\in[0,1],$$

where $\theta$ is his prior on state $H$.

The shinobi world observes one public omen from

$$S=\{d,c\},$$

where $d$ is devastation and $c$ is cooperation. Devastation includes renewed war, mass defection, and battlefield collapse. Cooperation includes a credible ceasefire, coordinated village action, and successful reconciliation.

A fixed waking-world process $\pi$ generates these omens. Madara controls their interpretation through the menu carried by the moonlight.

The dream-entry threshold

A shinobi enters the dream when his posterior on terminal conflict reaches the action threshold

$$\tau:=\frac{|u_L|}{u_H+|u_L|}.$$

Here $u_H>0$ is the gain from dream entry in the terminal-conflict state, while $u_L<0$ is the loss from dream entry when waking peace remains viable.

The main example takes

$$\tau=\frac12.$$

Dream entry then follows whenever the selected template makes terminal conflict at least as likely as a recoverable world.

Dream-templates

A dream-template $m$ assigns probabilities to each omen in each world state. After observing omen $s$, a shinobi with prior $\theta$ evaluates template $m$ by

$$\operatorname{Fit}(s,m,\theta) = (1-\theta)m(s\mid L)+\theta m(s\mid H).$$

He selects the template with the highest fit. If the selected template is $m^{*}$, his posterior becomes

$$\widehat\mu_s(\theta) = \frac{\theta m^{*}(s\mid H)} {(1-\theta)m^{*}(s\mid L)+\theta m^{*}(s\mid H)}.$$

The omen is public. The template is selected individually. This is how one moonlight pattern generates many private dreams.

Anime still: NARUTO Official Site, Episode 647.

An ambiguous shinobi world

Let $F$ denote the distribution of priors across the shinobi world. Madara observes its mean $\widehat\theta$ and a lower bound $\sigma^{2}$ on dispersion:

$$\Pi(\widehat\theta,\sigma^{2}) = \left\{ F: \mathbb{E}_{F}[\theta]=\widehat\theta, \quad \operatorname{Var}_{F}(\theta)\geq\sigma^{2} \right\}.$$

For every moonlight menu, the least favorable distribution in this set determines the guaranteed rate of dream entry. Madara chooses the menu with the highest guarantee.

One omen, one persuasion threshold

For each omen $s$, the selected posterior rises with the prior. Shinobi with sufficiently high priors enter the dream, while those with lower priors preserve waking agency. The menu therefore creates one persuasion threshold after each omen:

$$\theta_s = \inf\left\{\theta:\widehat\mu_s(\theta)\geq\tau\right\}.$$

A lower threshold means that the omen draws a larger part of the shinobi world into Infinite Tsukuyomi.

Under binary world states, one threshold-setting template per omen is sufficient. With devastation and cooperation, the moonlight menu needs at most two templates. Extra templates can win a fit comparison while producing a weaker posterior, delaying dream entry for some priors.

The moonlight frontier

The two thresholds move together because every template assigns probabilities to both devastation and cooperation. A template built to dominate the interpretation of devastation also enters the fit contest after cooperation.

At $\tau=1/2$, three threshold pairs summarize the central choices:

$$\left(\frac14,\frac14\right), \qquad \left(0,\frac12\right), \qquad \left(\frac12,0\right).$$

The symmetric pair $(1/4,1/4)$ spreads persuasive force across both omens. The pair $(0,1/2)$ gives devastation complete coverage and lets cooperation persuade shinobi whose priors already exceed one half. The pair $(1/2,0)$ reverses those roles.

This frontier is the discipline imposed by one public moon. Greater coverage after one omen shifts the remaining threshold burden toward the other omen.

A devastation-first Infinite Tsukuyomi

Represent a binary-omen template by

$$m=\bigl(m(d\mid H),m(d\mid L)\bigr).$$

Consider the menu

$$M_d^{\mathrm{bd}}=\{m^{d},m^{u}\},$$

with

$$m^{d}=(1,0)$$

and

$$m^{u}=(0,0).$$

The first template treats devastation as decisive evidence of terminal conflict. The second assigns cooperation probability one in both states, preserving the shinobi’s prior after cooperation.

After devastation, every interior prior selects $m^{d}$, and the selected posterior equals one. The devastation threshold is therefore

$$\theta_d=0.$$

After cooperation, $m^{u}$ wins the fit comparison and preserves the prior. Dream entry then occurs at

$$\theta_c=\frac12.$$

The menu implements

$$(\theta_d,\theta_c) = \left(0,\frac12\right).$$

The cooperation-first menu reverses the construction and implements $(1/2,0)$.

The higher-value omen receives complete persuasive coverage. The other omen relies on the share of shinobi whose priors already lie above the dream-entry threshold.

Robust Infinite Tsukuyomi

For a fixed threshold pair, the least favorable distribution places as much prior mass as possible near the dream-entry threshold. Mass at the threshold raises the mean and helps satisfy the dispersion requirement while contributing little to the guaranteed share strictly above the threshold.

Anime still: NARUTO Official Site.

At $\tau=1/2$, robust casting selects a boundary pair. If devastation carries the larger payoff weight, the optimal thresholds are $(0,1/2)$. If cooperation carries the larger payoff weight, the optimal thresholds are $(1/2,0)$.

Write $w_d$ and $w_c$ for the values of dream entry after devastation and cooperation. Let

$$T\left(\frac12\right)$$

denote the smallest share of priors strictly above one half among all distributions satisfying the mean and dispersion restrictions. The guaranteed value of the optimal boundary menu is

$$V^{*} = \max\{w_d,w_c\} + \min\{w_d,w_c\} T\left(\frac12\right).$$

The first term comes from the omen with complete coverage. The second comes from the prior mass that the moment restrictions force above the remaining threshold.

Why greater dispersion helps Madara

A higher lower bound on dispersion removes concentrated prior distributions from the ambiguity set. The least favorable shinobi world then needs more mass away from the middle. Some of that mass reaches the high-prior side and enters the dream after the partially covered omen.

Greater required dispersion therefore raises the guaranteed value of Infinite Tsukuyomi. The result comes from the geometry of the ambiguity set: stronger dispersion information leaves fewer hostile population compositions available.

The Infinite Tsukuyomi paradox

A higher average prior can reduce guaranteed dream entry.

Take

$$\tau=\frac12$$

and

$$\sigma^{2}=0.10.$$

When the average prior is $0.15$, the moment restrictions force at least

$$2\left(0.10-0.15(0.50-0.15)\right)=0.095$$

of the population strictly above one half.

When the average prior rises to $0.25$, the forced upper-tail share becomes

$$2\left(0.10-0.25(0.50-0.25)\right)=0.075.$$

The average belief in terminal conflict rises, while the guaranteed high-prior share falls from $9.5\%$ to $7.5\%$.

Nature creates this reversal by moving prior mass from the low end toward $1/2$. That movement raises the average prior. It also concentrates more shinobi at the boundary between waking agency and dream entry, reducing the share strictly above the threshold.

The result distinguishes average receptiveness from guaranteed receptiveness. Infinite Tsukuyomi gains from priors that cross the dream-entry threshold, while a higher mean can be produced by mass that approaches the threshold and remains there.

Why the boundary spell wins

A frontier-interior menu leaves both thresholds positive. The least favorable distribution can then place a large share of shinobi at very low priors and a smaller share at very high priors, satisfying the moment restrictions while limiting dream entry after both omens.

A boundary menu fully covers the more valuable omen across the entire prior spectrum. This guaranteed base dominates the gains from spreading persuasive force more evenly across devastation and cooperation.

Robustness therefore changes the casting strategy. A known shinobi world may reward balanced persuasion. An ambiguous shinobi world rewards complete insurance after the omen with the larger value.

Infinite Tsukuyomi and Bayesian persuasion

Bayesian persuasion lets a sender design the information structure that generates observations. Narrative persuasion keeps the waking-world process fixed and lets the sender design the menu through which observations are interpreted.

Infinite Tsukuyomi belongs to the second environment. The waking world generates devastation or cooperation. The moonlight determines which interpretation becomes most compelling for each prior. A shinobi may select one template after devastation and another after cooperation, so the two posteriors arise from different likelihood models.

The spell’s commitment power comes from the public menu. Madara fixes the available dream-templates before the omen appears, and every template participates in the fit contest after every omen. The moon reshapes meaning through a common ex ante architecture.

Selected reading

• Song Li, “Robust Narrative Persuasion”.
• Joshua Schwartzstein and Adi Sunderam, “Using Models to Persuade,” American Economic Review 111(1), 2021, pp. 276–323.
• Chiara Aina, “Tailored Stories,” 2025.
• Emir Kamenica and Matthew Gentzkow, “Bayesian Persuasion,” American Economic Review 101(6), 2011, pp. 2590–2615.
• Ian Ball and Deniz Kattwinkel, “Robust Robustness”.