Version 1.0 — 21 October 2026. Prepared for editorial review.
1. Question, provenance, and scope
Thompson [7], Dalkey [2], and Elmes–Reny [3] give transformations under which finite extensive-form games with the same reduced normal form are strategically equivalent. Mailath, Samuelson, and Swinkels [6] recover extensive-form structure inside that normal form. For rational solution concepts this is a coherent position. For evolutionary dynamics it is not the whole story. Cressman [1] shows that replicator and best-response flows on a normal-form simplex can ignore subgames, unreached information sets, and linkage among roles. The Wright manifold is his geometric name for statistical independence across roles.
This note places that comparison in a sheaf. An asymmetric game determines a small site of role sets. Behavior strategies are the descent object for an interaction-separated cover. The Wright manifold is a section of the projection from joint mixtures back to that object, not the descent object itself. The new statements proved here are the two-role independence criterion, invariance of linkage leaves under role-additive replicator dynamics, exponential decay of the two-role linkage defect under an explicit revision dynamic, the chain-store linearization, and two exact Jacobian obstructions. A nine-strategy sunspot supplies a numerical sign change. Classical learning theorems in [1] are cited rather than re-proved. In particular, we do not re-prove subgame monotonicity, the general ESSet classification, or the characterization of playwise bandit rules.
The ambient $(\infty,1)$-topos is the ordinary topos of sheaves of spaces on this site [5, Ch. 6]. Every theorem below uses set-valued presheaves, ordinary limits, and finite-dimensional linear algebra. No higher homotopy group of the topos is computed. The accompanying verify.py expands the polynomial identities and the sunspot Jacobian in exact or forty-decimal arithmetic.
2. The subgame site
Let the role set be $U=\{1,\ldots,N\}$, with symmetric meeting weights $\rho(k,\ell)=\rho(\ell,k)\ge 0$ and finite action sets $S_k$, each of cardinality at least $2$. The category $\mathcal C_\Gamma$ is the poset of nonempty subsets of $U$, ordered by inclusion.
A family of inclusions $\{I_\alpha\hookrightarrow I\}$ is a $J_{\mathrm{sub}}$-cover when the sets $I_\alpha$ cover $I$ and $\rho$ vanishes between $I_\alpha\setminus I_\beta$ and $I_\beta\setminus I_\alpha$ whenever $\alpha\ne\beta$. Thus roles that meet with positive weight are not separated into different members of a cover. If $\rho(k,\ell)=0$ for all $k\ne\ell$, the singleton family covers $U$.
Two presheaves of sets record strategies. The normal-form presheaf sends $I$ to the simplex $\Delta\!\left(\prod_{k\in I}S_k\right)$. The behavior presheaf sends $I$ to $\prod_{k\in I}\Delta(S_k)$, with restriction by dropping coordinates. On a cover by disjoint singletons, a matching family for the behavior presheaf is exactly a tuple of local mixtures, so that presheaf satisfies descent and its value is the ordinary product. The same cover does not give unique gluing for the normal-form presheaf.
Example 2.1. Take $N=2$, $S_1=S_2=\{0,1\}$, and $\rho(1,2)=0$. Marginalization $\Delta^3\to\Delta^1\times\Delta^1$ is surjective, and the fiber over $(1/2,1/2)$ contains both the product state $(1/4,1/4,1/4,1/4)$ and the correlated state $(1/2,0,0,1/2)$. These agree locally and disagree globally, so the normal-form presheaf is not a sheaf. Its sheafification on this cover is the behavior product.
With empty overlaps, the Čech diagram contributes no pairwise matching data. The limit is the ordinary product. That is the only descent statement used below.
3. The Wright manifold
Write a joint mixture as $p\in\Delta(S)$, $S=\prod_k S_k$, and write $p^k\in\Delta(S_k)$ for its marginal on role $k$. Kuhn's theorem [4] is the reason behavior strategies have this product shape under perfect recall; the map below is only the marginal projection associated with that description. Define
$$ \operatorname{proj}:\Delta(S)\longrightarrow\prod_{k=1}^N\Delta(S_k),\qquad p\longmapsto(p^1,\ldots,p^N). $$
The Segre section $\sigma$ sends $(p^1,\ldots,p^N)$ to the independent joint $p_i=\prod_k p^k_{i_k}$. Its image $W=\sigma\!\left(\prod_k\Delta(S_k)\right)$ is the Wright manifold [1]. For a subset $I$ and multi-indices $i,j$, write $i\setminus j$ for the multi-index that takes its $I$-coordinates from $j$ and the others from $i$.
Proposition 3.1. For $N=2$ and $p$ in the interior of $\Delta(S_1\times S_2)$, the following are equivalent: $p\in W$; $p_{ij}p_{k\ell}=p_{i\ell}p_{kj}$ for all indices; and $p_{ij}=p^1_i p^2_j$.
Proof. A product joint satisfies the cross identity because both sides equal $p^1_i p^1_k p^2_j p^2_\ell$. Conversely, fix $(i,j)$ and sum $p_{ij}p_{k\ell}=p_{i\ell}p_{kj}$ over $k$ and $\ell$. The left side is $p_{ij}$. The right side is $p^1_i p^2_j$. $\square$
The dynamics below use only this two-role case. verify.py checks the binary summation identity. The same summation suggests an induction for $N>2$, which is not proved here.
For positive $p$ define the cross ratios $K_{ij,k\ell}(p)=(p_{ij}p_{k\ell})/(p_{i\ell}p_{kj})$. Algebra gives $K_{ij,k\ell}=K_{k\ell,ij}=K_{i\ell,kj}^{-1}$. Proposition 3.1 says that $p\in W$ exactly when every ratio equals $1$. These ratios are not, on their own, a cocycle in the cohomology of an arbitrary site: the three-factor multiplication rule sometimes claimed for them is not an identity.
4. Linkage leaves of the replicator
Specialize to two roles and a role-additive payoff. For matrices $A,B,C,D$ of the appropriate sizes,
$$ \pi((a,b),(c,d))=A_{ac}+B_{ad}+C_{bc}+D_{bd}, $$
and $\pi((a,b),p)$ is the $p$-expectation of this function. The separated form forces $\pi(a,b)+\pi(a',b')=\pi(a,b')+\pi(a',b)$. The single-population replicator is $\dot p_i=p_i(\pi(e_i,p)-\bar\pi(p))$.
Proposition 4.1. Let $K$ be constant and set $\Delta=p_{ab}p_{a'b'}-K\,p_{ab'}p_{a'b}$. Along any interior replicator trajectory, $\dot\Delta=(\pi(a,b)+\pi(a',b')-2\bar\pi)\,\Delta$. Every positive level of the cross ratios is invariant. In particular $W$ is invariant.
Proof. Additivity makes the payoff sum of $(a,b)$ and $(a',b')$ equal the payoff sum of the two splices. Differentiating a product $p_up_v$ along $\dot p_i=p_i(\pi_i-\bar\pi)$ multiplies that product by $\pi_u+\pi_v-2\bar\pi$. Both products in $\Delta$ receive the same factor, so $\dot\Delta$ equals that factor times $\Delta$. $\square$
The identity is an equality of polynomials in the payoff entries and is checked symbolically in verify.py. On $W$, marginalization conjugates this flow to the corresponding two-role replicator. Off $W$, invariance of a leaf says that linkage is a constant of motion. It does not by itself make the leaf stable.
5. Exponential attraction to $W$
Best replies are unnecessary for the following decay. Let $b(t)$ and $c(t)$ be any measurable selections of probability vectors, and define
$$ \dot p_{ij}=\frac12\bigl(p^2_j b_i-p_{ij}\bigr)+\frac12\bigl(p^1_i c_j-p_{ij}\bigr). $$
The simplex is forward-invariant. Summing gives the marginal equations $\dot p^1_i=\frac12(b_i-p^1_i)$ and $\dot p^2_j=\frac12(c_j-p^2_j)$.
Proposition 5.1. For $L_{ij}=p^1_i p^2_j-p_{ij}$, one has $\dot L_{ij}=-L_{ij}$. Thus $L_{ij}(t)=L_{ij}(0)e^{-t}$.
Proof. Substitute the marginal equations and the formula for $\dot p_{ij}$:
$$ \begin{aligned} \dot L_{ij} &=\frac12(b_i-p^1_i)p^2_j+\frac12 p^1_i(c_j-p^2_j)\\ &\quad-\frac12(p^2_j b_i-p_{ij})-\frac12(p^1_i c_j-p_{ij})\\ &=-L_{ij}. \end{aligned} $$
The cross terms cancel. $\square$
verify.py repeats this cancellation in the polynomial ring. The same decay holds when $b$ and $c$ are measurable best-reply selections. This equation is not the classical best-response differential inclusion. The distance from $p(t)$ to $W$ decays exponentially for any such selection.
Existence of $\lim_{t\to\infty}p(t)$ is a separate question. The defect vanishes, but the marginal equations can cycle, in which case $p(t)$ approaches a moving point of $W$ and the limit of states need not exist. There is still a topological retraction that does not use the vector field: each $\operatorname{proj}$-fiber is a transportation polytope, hence convex, and contains the product state $\sigma(\operatorname{proj}(p))$. The straight-line homotopy from $p$ to that product state is a strong deformation retraction of $\Delta(S)$ onto $W$.
6. A non-hyperbolic component
Consider the chain-store bimatrix with rows $N,E$ and columns $R,A$,
$$ \begin{pmatrix} 1,4 & 1,4 \\ 0,0 & 2,2 \end{pmatrix}. $$
Let $x=\mathrm{Prob}(E)$ and $y=\mathrm{Prob}(A)$. The two-population replicator is
$$ \dot x=x(1-x)(2y-1),\qquad \dot y=y(1-y)(2x). $$
Proposition 6.1. The Nash equilibria are the segment $\{(0,y):0\le y\le 1/2\}$ and the strict equilibrium $(1,1)$. On $x=0$ the Jacobian has eigenvalues $0$ and $2y-1$.
Proof. Player 2 strictly prefers $A$ whenever $x>0$, and is indifferent when $x=0$. Player 1 strictly prefers $E$ when $y>1/2$, strictly prefers $N$ when $y<1/2$, and is indifferent when $y=1/2$. The stated set is exactly the set of mutual best replies. Differentiating the vector field and setting $x=0$ gives the Jacobian
$$ \begin{pmatrix} 2y-1 & 0 \\ 2y(1-y) & 0 \end{pmatrix}, $$
whose characteristic polynomial is $\lambda(\lambda-2y+1)$. $\square$
For $y<1/2$ the transverse eigenvalue is negative. At $y=1/2$ both eigenvalues vanish. The zero eigenvalue along the segment is the unreached-choice phenomenon: if the first player never enters, the second player's action is payoff-irrelevant. The normal form still displays the segment. What it does not record is which information set is unreached. That is the sense in which passage to the normal form discards the explanation of the degeneracy, not the degeneracy itself.
7. Two exact obstructions to stability descent
Throughout this section, failure of descent means a concrete comparison of rest points. A behavior state can be a local evolutionary equilibrium while a joint state lying over it is a linearly unstable rest point of the normal-form replicator.
7.1. Linkage off an attracting behavior equilibrium
Let $\varepsilon>0$ and let $H$ denote $\begin{pmatrix}-1&2\\0&1\end{pmatrix}$. Payoffs on strategies $\{0,1\}^2$, ordered $(00,01,10,11)$, are
$$ \pi((a,b),(c,d))=\varepsilon H_{ac}+B_{ad}+C_{bc}+\varepsilon H_{bd}, $$
with $B=\begin{pmatrix}5&1\\4&2\end{pmatrix}$ and $C=\begin{pmatrix}4&0\\6&-2\end{pmatrix}$. For $|\lambda|\le 1/4$ define
$$ p(\lambda)=\Bigl(\tfrac14+\lambda,\ \tfrac14-\lambda,\ \tfrac14-\lambda,\ \tfrac14+\lambda\Bigr). $$
Both marginals of $p(\lambda)$ equal $(1/2,1/2)$. The linkage defect $p_{00}p_{11}-p_{01}p_{10}$ equals $\lambda$, so $\lambda=0$ is the Wright point in this fiber.
Theorem 7.1. Every $p(\lambda)$ is a replicator rest point. In coordinates $(z_{00},z_{01},z_{10})$, with the last coordinate restoring mass $1$, the characteristic polynomial of the replicator Jacobian along this line is
$$ \chi(t)=t\Bigl(t^2+(\varepsilon+2\lambda)t+Q(\varepsilon,\lambda)\Bigr), $$
where $Q=(\varepsilon^2(1-16\lambda^2)-32\lambda^2+2)/4$. The nonzero eigenvalues are
$$ -\frac{\varepsilon}{2}-\lambda\pm\frac12\sqrt{16\varepsilon^2\lambda^2+36\lambda^2+4\varepsilon\lambda-2}. $$
Proof. Role-additive payoff depends on a joint opponent state only through its marginals, and every $p(\lambda)$ has the same marginals. Direct evaluation gives every pure strategy the payoff $\varepsilon+5$, so the replicator vanishes on the line. The displayed polynomial is the exact characteristic polynomial of the reduced Jacobian, expanded symbolically by verify.py. The quadratic formula gives the eigenvalues, and the discriminant simplifies to the radicand shown. $\square$
At $\lambda=0$ the conjugate pair has real part $-\varepsilon/2<0$. At the rational point $\varepsilon=1/10$, $\lambda=-1/10$, still inside the simplex, the real part equals $1/20>0$ and the radicand equals $-1049/625<0$. The behavior equilibrium is linearly stable inside $W$ and the normal-form rest point over it at this linkage is linearly unstable.
7.2. A local ESS section with a repelling joint fiber
Now let every role pair meet under one matrix. With $H$ as above and $\varepsilon\in[0,1/2]$, set
$$ \begin{aligned} \pi(f_O,f_I;o_O,o_I) &=\varepsilon H_{f_O o_O}+\Bigl(\frac12-\varepsilon\Bigr)H_{f_O o_I}\\ &\quad+\Bigl(\frac12-\varepsilon\Bigr)H_{f_I o_O}+\varepsilon H_{f_I o_I}. \end{aligned} $$
The four situation weights sum to $1$. Conditional on the focal role being owner, which has probability $1/2$, the opponent is owner with probability $2\varepsilon$ and intruder with probability $1-2\varepsilon$. The induced owner-role matrix is therefore $2\varepsilon H+(1-2\varepsilon)H=H$. The same holds for the intruder role.
Lemma 7.2. The unique ESS of $H$ is $(1/2,1/2)$.
Proof. Against $q$, the payoff gap between the first and second rows is $1-2q_1$, so the unique interior equilibrium is $q_1=1/2$. For $v=(1,-1)$, one has $v^\top H v=-2<0$, hence the equilibrium is an ESS. $\square$
Let strategies be ordered $HH,HD,DH,DD$, and set
$$ p(t)=\Bigl(\tfrac14-t,\ \tfrac14+t,\ \tfrac14+t,\ \tfrac14-t\Bigr),\qquad |t|\le\frac14. $$
Both role marginals equal $(1/2,1/2)$ along this segment, so the segment is the joint fiber over the local ESS section of Lemma 7.2.
Theorem 7.3. The replicator vanishes at every $p(t)$. At $t=0$, in coordinates $(z_{HH},z_{HD},z_{DH})$, its Jacobian is
$$ \begin{pmatrix} -1/4 & -1/8 & -1/8\\ 0 & 1/8-\varepsilon/2 & \varepsilon/2-1/8\\ 0 & \varepsilon/2-1/8 & 1/8-\varepsilon/2 \end{pmatrix}. $$
The eigenvalues are $-1/4$, $0$, and $1/4-\varepsilon$. For $\varepsilon<1/4$ the pure states $HD$ and $DH$ are strict Nash equilibria.
Proof. verify.py differentiates the replicator and checks both the Jacobian and that the vector field vanishes on $p(t)$. The matrix is block triangular. Its lower-right block equals $\begin{pmatrix}a&-a\\-a&a\end{pmatrix}$ with $a=1/8-\varepsilon/2$, which has eigenvalues $0$ and $2a=1/4-\varepsilon$. The remaining eigenvalue is $-1/4$. The zero eigenvector is tangent to $p(t)$.
The column of payoffs against pure $HD$ is $(1/2,\ 1-2\varepsilon,\ 2\varepsilon,\ 1/2)$. Its gaps over the other three pure payoffs are $1/2-2\varepsilon$, $1-4\varepsilon$, and $1/2-2\varepsilon$, all positive precisely when $\varepsilon<1/4$. The case $DH$ is symmetric. $\square$
For $\varepsilon<1/4$ the transverse eigenvalue $1/4-\varepsilon$ is positive, so the joint fiber over the local ESS is linearly repelling, while $HD$ and $DH$ are strict equilibria elsewhere in the simplex. For $\varepsilon>1/4$ that transverse eigenvalue is negative, so the same fiber is normally attracting. The sign change at $\varepsilon=1/4$ is the exact threshold in this model.
8. A numerical sunspot witness
Let $A$ be the Rock–Scissors–Paper matrix
$$ A=\begin{pmatrix}0&6&-4\\-4&0&4\\2&-2&0\end{pmatrix}. $$
Lemma 8.1. The vector $p_0=(10,8,11)/29$ is the unique interior Nash equilibrium of $A$. The reduced replicator Jacobian there has eigenvalues $-2/29\pm 2\sqrt{879}\,i/29$.
Proof. All three coordinates of $Ap_0$ equal $4/29$, and the coordinates of $p_0$ are positive and sum to $1$. The two payoff-gap equations together with $p_1+p_2+p_3=1$ have coefficient determinant $-29$, so the solution is unique. The eigenvalues are the exact spectrum of the $2\times 2$ reduced Jacobian, computed symbolically in verify.py. Their common real part is negative, so the equilibrium is linearly asymptotically stable. $\square$
Form the nine-strategy sunspot game whose pure strategies are pairs $(a,b)\in\{0,1,2\}^2$ and whose payoff is $\pi((a,b),(c,d))=\frac12 A_{ac}+\frac12 A_{bd}$. Let $x=(-4,1,3)$ and $y=(-2,-1,3)$, and set
$$ p(r)_{ab}=p_{0,a}p_{0,b}+r\,x_a y_b. $$
Both $x$ and $y$ sum to $0$, so $p(r)$ has mass $1$ and both marginals equal $p_0$ for every $r$. At $r=0$ the state lies on $W$. For small positive $r$ it lies in the same equilibrium fiber and off $W$. Because every joint with these marginals equalizes payoffs, the whole fiber consists of replicator rest points.
verify.py builds the reduced $8\times 8$ Jacobian by a centered difference of step $10^{-18}$ at forty decimal places. At $r=0$ the dominant complex pair agrees with $-1/29\pm i\sqrt{879}/29$ to absolute error below $10^{-9}$, which is the Lemma 8.1 pair scaled by the explicit factor $1/2$ in the payoff. The real part of that pair is negative at $r=9/1000$ and positive at $r=1/100$, with the state still interior. This is a numerical witness of a transverse stability change on the equilibrium fiber, not a closed form for the crossing and not an identification with any particular decimal in [1, §4.6.1].
9. Reproducibility, authorship, and limits
verify.py uses Python, SymPy, and mpmath only. It checks the chain-store polynomial, the two-role independence identity, the replicator and revision cancellations, the cross-ratio symmetries of §3, the characteristic polynomials and eigenvalues of §7, Lemma 8.1, and the sunspot sign change of §8. A failure exits nonzero. The script does not import a computer-algebra certificate into a proof assistant. The sunspot crossing remains a floating-point sign witness; the algebraic statements are exact.
The responsible party is the owner who submits this note. The public byline is Anonymous until that owner replaces it. Grok assisted with the organization of the argument, symbolic expansion, and the numerical Jacobian. No referee report is claimed, and submission does not assert editorial acceptance.
The site $J_{\mathrm{sub}}$ sees only role sets and meeting weights. It does not encode an arbitrary game tree, perfect recall, or continuous-time subgame consistency. Playwise imitation, in the sense of [1], is the classical statement that a good bandit rule may replace an action only at the reached locus; that theorem is not re-proved here. Extending the sheaf from parallel role sets to the exit-path category of a tree is left open.
References
1. Ross Cressman. *Evolutionary Dynamics and Extensive Form Games*. MIT Press, 2003. doi:10.7551/mitpress/2884.001.0001. 2. Norman Dalkey. Equivalence of information patterns and essentially determinate games. In *Contributions to the Theory of Games, II*, Annals of Mathematics Studies 28, Princeton University Press, 1953, pp. 217–243. Earlier RAND paper P-265, 1952. 3. Susan Elmes and Philip J. Reny. On the strategic equivalence of extensive form games. *Journal of Economic Theory* 62(1), 1–23, 1994. 4. H. W. Kuhn. Extensive games and the problem of information. In *Contributions to the Theory of Games, II*, Annals of Mathematics Studies 28, Princeton University Press, 1953, pp. 193–216. 5. Jacob Lurie. *Higher Topos Theory*. Annals of Mathematics Studies 170, Princeton University Press, 2009. 6. George J. Mailath, Larry Samuelson, and Jeroen M. Swinkels. Extensive form reasoning in normal form games. *Econometrica* 61(2), 273–302, 1993. doi:10.2307/2951552. 7. F. B. Thompson. *Equivalence of Games in Extensive Form*. RAND Research Memorandum RM-759, 1952. doi:10.7249/RM759.