we-solve (beta)

Econ-8 · economics · Economics

Payoff functionals guaranteeing a value under one-sided information

LLM status Partially solved unverified

Statement

GT-037 | Repeated games with incomplete information

P | Source-reported characterization question

Definitions and mathematical statement

Let $K,I,J$ be nonempty finite sets. Nature chooses $k\in K$ according to $p\in\Delta(K)$ and reveals it only to the maximizing player. At every date the two players simultaneously choose actions in $I$ and $J$ and then both observe the pair. Write $\mathcal H=\bigcup_{t\geq0}(I\times J)^t$. The informed player's behavioral strategy is $\sigma:K\times\mathcal H\to\Delta(I)$ and the uninformed player's is $\tau:\mathcal H\to\Delta(J)$. For a bounded Borel functional $f:K\times(I\times J)^{\mathbb N}\to\mathbb R$, set \[ \gamma_{p,f}(\sigma,\tau)=\mathbb E_{p,\sigma,\tau}f(k,(i_t,j_t)_{t\geq0}), \] \[ \underline v_f(p)=\sup_\sigma\inf_\tau\gamma_{p,f}(\sigma,\tau), \qquad \overline v_f(p)=\inf_\tau\sup_\sigma\gamma_{p,f}(\sigma,\tau). \] For fixed $K,I,J$, characterize intrinsically the class \[ \mathcal V_{K,I,J}=\{f\text{ bounded Borel}: \forall p\in\Delta(K),\ \underline v_f(p)=\overline v_f(p)\}. \] A characterization should give conditions on $f$ and the information/action structure, rather than merely restating equality of the two optimization problems.

Short English statement

Which bounded infinite-play payoffs ensure that the informed and uninformed players have the same minimax value?

Variants and scope boundaries

Membership at one specified prior is weaker than prior-uniform membership. Tail measurability means that changing finitely many action coordinates leaves $f(k,\cdot)$ unchanged; shift invariance means invariance under deletion of an initial action pair. These notions differ. Borel measurability, complete action monitoring, and finite state/action alphabets are fixed here.

Source relationship and status

[S1], Section 5, asks for payoff properties guaranteeing a value. Its tail-measurable maxmin theorem does not assert that the upper and lower values coincide; Section 4 provides relevant nonexistence examples. The prior-uniform class above is an explicit organizational choice.

References

[S1] Gil Bar Castellon Koltun, Ehud Lehrer and Eilon Solan, The Maxmin Value of Repeated Games with Incomplete Information on One Side and Tail-Measurable Payoffs, arXiv:2512.01581 (2025). Locators: Definitions 2.1--2.4, pp. 6--7; Section 4 (examples); Section 5, p. 18 (characterization questions). https://arxiv.org/pdf/2512.01581

Common conventions: Source mathematical conventions

Finite games. For a finite set $A$, $\Delta(A)$ is its probability simplex. Players choose independent mixed strategies in Nash equilibrium; correlation is allowed only when explicitly part of the solution concept. In a game with payoff functions $u_i$ and strategy profile $x$, an additive $\varepsilon$ Nash equilibrium satisfies \[ u_i(x)\geq u_i(x_i',x_{-i})-\varepsilon \quad\text{for every player $i$ and every admissible deviation $x_i'$}. \] For numerical approximation, payoffs are normalized as locally specified. An $\varepsilon$ payoff residual is distinct from distance at most $\varepsilon$ to the set of exact equilibria. Point convergence, convergence of empirical play, and convergence in distance to an equilibrium set are also distinct.

Computational representations. Unless stated otherwise, a complexity question takes rational data with binary encoding; polynomial time is measured in total bit length. A fixed-accuracy polynomial algorithm is not necessarily polynomial in $1/\varepsilon$, and neither is necessarily polynomial in $\log(1/\varepsilon)$. Succinct games and value, demand, or sampling oracles must declare their interfaces. Exact real solutions require an explicit representation; an unspecified list of arbitrary real numbers is not a finite computational output. A claimed algorithmic barrier is meaningful only for its specified model and reductions.

Dynamic games. Histories, information, observation, and allowable randomization are part of the model. A stationary strategy depends only on the current state; a positional strategy in a deterministic graph game chooses one action at each controlled vertex. Nash equilibrium at an initial state and equilibrium after every history are different requirements. An expectation of a pathwise $\liminf$ payoff, a $\liminf$ of expected averages, a limiting expected average, and a uniform finite-horizon equilibrium are different criteria. None is silently substituted for another.

Allocation and mechanisms. A complete allocation assigns every item exactly once unless otherwise stated. Zero-valued goods, negative values, capacity constraints, feasibility, lotteries, and copies of goods affect fairness definitions and are specified locally. Dominant-strategy incentive compatibility, Bayesian incentive compatibility, universal truthfulness, and truthfulness in expectation impose different inequalities. Whenever an objective ratio has zero denominator, its convention is stated or those instances are excluded.

Infinite-dimensional models. Expectations, integrals, stochastic equations, and optimization objectives require their local regularity, integrability, filtrations, and admissible domains. A backward--forward mean-field system is a boundary-value problem; it is not an initial-value semiflow without an additional construction. Long-time, large-population, and vanishing-noise limits require an order of limits and a convergence topology. If a minimum need not be attained, the objective is its infimum or supremum and the approximation requirement is stated separately.

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Econ-8-A1 · Exploration · LLM-generated · gpt 6 astra pro ·

GPT 6 Astra Pro attempt on Econ-8

Research attempt: C73-8 Values in games with one-sided incomplete information

Scope of this report. The main target remains UNRESOLVED. Fully proved subclaims and counterexamples are identified locally. Computational checks reported here are supplementary to the proofs. Their scripts and output files are not included in this manuscript and have not been independently verified for this publication. No novelty or priority claim is made.

FORMAL RESTATEMENT

Fix finite nonempty sets $K,I,J$. A type $k$ is drawn from $p\in\Delta(K)$ and disclosed only to player 1, who maximizes payoff. At each date $t\geq1$ the two players choose $i_t\in I$, $j_t\in J$ simultaneously; the pair becomes public. A behavioral strategy of player 1 maps $(k,h_t)$ to $\Delta(I)$, and that of player 2 maps public history $h_t$ to $\Delta(J)$. Randomizations are independent conditional on information. For bounded Borel $f:K\times(I\times J)^{\mathbb N}\to\mathbb R$, define \[ \underline V_f(p)=\sup_\sigma\inf_\tau{\mathbb{E}}_{p,\sigma,\tau}f, \qquad \overline V_f(p)=\inf_\tau\sup_\sigma{\mathbb{E}}_{p,\sigma,\tau}f. \] The input asks for an intrinsic characterization of \[ \mathcal V=\{f:\ \forall p\in\Delta(K),\ % \underline V_f(p)=\overline V_f(p)\}. \] An equality of lower and upper values need not mean that optimal strategies attain the extrema; guarantees within every positive error suffice.
Literal statement versus a minimally precise target.
"Characterize" is a research programme rather than one Boolean conjecture. The minimal precise target is a necessary-and-sufficient condition on $f$ for the displayed prior-uniform equality, not a repetition of the definition. No correction declaring all Borel, tail-invariant, or shift-invariant payoffs determined is legitimate: the explicit example below disproves all three universal assertions. Tail invariance means unchanged by altering finitely many action coordinates; shift invariance means unchanged when a finite initial segment is deleted. They are logically different definitions, although the example has both properties. If $I$ or $J$ is a singleton the minimax equality follows directly because one optimization variable disappears.

QUICK LITERATURE/CONTEXT CHECK

Koltun, Lehrer and Solan study the maxmin for one-sided incomplete information with tail-measurable payoffs and explicitly separate maxmin formulas from existence of a value [c738-kls]. Their Section 4 gives a shift-invariant non-value example and Section 5 discusses characterization questions. The paper appeared online in the Journal of Mathematical Economics in 2026. Below, the same basic payoff construction is treated directly in the simultaneous-action convention of the problem statement; both lower and upper values are computed for every prior by explicit arguments. This is not a claim of priority for the example or its underlying ideas. Sources checked through 9 October 2026.

ATTACK PLAN

Proof strategies.
Approximate a payoff uniformly by finite-horizon functions and use finite minimax; then look for a closure property that survives the approximation. Alternatively, use posterior martingales to bound informed-player guarantees.
Disproof strategies.
Construct a tail event for which information is revealed too late to make the two orders of optimization agree; use a two-type, two-action family; and compute both minimax quantities rather than relying on numerical finite truncations. The latter path produces a complete nonmembership certificate for a concrete payoff.
Landscape and tools.
This is an infinite zero-sum game with asymmetric information. Finite minimax handles cylinder payoffs; perfect recall converts finite mixed plans to behavioral strategies; uniform continuity controls infinite horizons; the sup norm controls both optimization orders; posterior martingale convergence identifies asymptotic beliefs; concave majorants bound expected posterior payoffs; and finite-history approximation of measurable events constructs delayed signaling deviations.

WORK

A prior-uniform sufficient class

Equip the finite alphabet product $K\times(I\times J)^{\mathbb N}$ with its product topology, with $K$ discrete. This is a compact metrizable space.
Lemma (Stability): For bounded payoffs $f,g$, \[ |\underline V_f(p)-\underline V_g(p)|\leq\|f-g\|_\infty, \qquad |\overline V_f(p)-\overline V_g(p)|\leq\|f-g\|_\infty. \] If $\|f\|_\infty\leq M$, each of these functions is $M$-Lipschitz in $p$ for the $\ell^1$ norm.
Proof: Every fixed-strategy expected payoff changes by at most the stated sup-norm difference. Taking an infimum or supremum preserves such a uniform error bound, twice in the indicated order. For fixed strategies, the expectation is $\sum_kp_k a_k$ with $|a_k|\leq M$, so changing $p$ changes it by at most $M\sum_k|p_k-p'_k|$. The strategy spaces themselves do not depend on $p$, including when some types have zero probability. Apply the same optimization argument. ∎
Theorem (Continuous payoffs belong to $\mathcal V$): Every continuous real payoff on this product space admits a value for every prior. More generally, $\mathcal V$ is closed in the uniform norm.
Proof: Fix one infinite reference action tail. Define $f_T$ by keeping the type and first $T$ action pairs and replacing all later pairs by that reference tail. Uniform continuity on the compact product implies $\delta_T=\|f-f_T\|_\infty\to0$. Indeed, if this failed, two sequences agreeing on increasingly long prefixes would have payoff differences bounded away from zero; compactness and continuity would give a contradiction along a convergent subsequence.

For each $p$, $f_T$ defines a finite zero-sum extensive game, including the initial type move. Its finite normal form has a minimax value: a finite real matrix game satisfies $\max_x\min_y x^TCy=\min_y\max_xx^TCy$. Mixed plans and behavioral strategies are realization-equivalent in a finite game with perfect recall. Here each player remembers every observation and own action, so perfect recall holds. More explicitly, independent draws of actions for all information sets generate a distribution over pure plans from any behavioral strategy. Conversely, condition a mixed plan on the player's previously prescribed actions at a reached information set; perfect recall ensures that these conditional action probabilities give the same realization probabilities against every opponent. Zero-probability information sets may be assigned arbitrarily. Thus the finite normal-form equality applies to the behavioral strategy spaces used here.

By stability, \[ 0\leq\overline V_f(p)-\underline V_f(p)\leq2\delta_T. \] Let $T$ tend to infinity. The bound is independent of the prior. Finally, if $f_j\in\mathcal V$ and $\|f_j-f\|_\infty\to0$, the same inequality with $f_j$ in place of $f_T$ proves $f\in\mathcal V$. ∎

An explicit tail- and shift-invariant non-value game

Let $K=\{1,2\}$ and $I=J=\{\ell,r\}$. Set \[ E=\{j_t=\ell\text{ infinitely often}\},\qquad F=\{i_t=\ell\text{ infinitely often}\}. \] Define the two type payoffs by \[ (f_1,f_2)= \begin{cases} (-1,2),&E,\\ (-2,1),&E^c\cap F,\\ (0,0),&E^c\cap F^c. \end{cases} \] The player-2 payoff is its negative. Each infinitely-often event is $\bigcap_N\bigcup_{t\geq N}\{\text{specified action at }t\}$, hence Borel. Changing or deleting finitely many coordinates leaves $E,F$ unchanged. All payoffs lie in $[-2,2]$.

Write $p=\Pr(k=1)$ and define \begin{align*} \phi(p)&=\min\{1-\tfrac32p,\,2-3p\},\\ \psi(p)&=\min\{1-p,\,2-3p\},\\ u(p)&=\begin{cases} 1-3p,&0\leq p\leq1/3,\\ 0,&1/3\leq p\leq2/3,\\ 2-3p,&2/3\leq p\leq1. \end{cases} \end{align*}

Theorem (Complete computation for this payoff): For every $p\in[0,1]$, \[ \underline V_f(p)=\phi(p),\qquad \overline V_f(p)=\psi(p). \] In particular, at $p=1/2$ these are $1/4$ and $1/2$. This explicitly proves $f\notin\mathcal V$.
Proof[Proof of the maxmin upper bound]: Fix any informed strategy $\sigma$. Against this fixed strategy, player 2 can compute the posterior $p_t$ of type 1 before date $t$, using the public history and Bayes' rule, and choose $j_t=\ell$ exactly when $p_t\geq2/3$. Define actions arbitrarily at histories of zero probability. Under the resulting law, $(p_t)$ is a bounded martingale and converges almost surely and in $L^1$ to $p_\infty$, with ${\mathbb{E}} p_\infty=p$. The conditional-expectation martingale theorem also identifies $p_\infty$ with the conditional probability of type 1 given the entire public play.

If $p_\infty<2/3$, eventually $p_t<2/3$ and $E$ is false. If $p_\infty>2/3$, eventually $p_t>2/3$ and $E$ is true. At equality either event is possible. Conditional on the entire public play, the expected payoff equals \[ (2-3p_\infty){\mathbf 1}_E+(1-3p_\infty){\mathbf 1}_{E^c\cap F}. \] For $p_\infty<2/3$ it is at most $\max\{0,1-3p_\infty\}=u(p_\infty)$; above $2/3$ it equals $u(p_\infty)$; at $2/3$ it is at most zero. Direct comparison of the three intervals shows $u\leq\phi$. The function $\phi$ is concave: its slopes are $-3/2$ and then $-3$. Jensen's inequality gives ${\mathbb{E}} f\leq{\mathbb{E}}\phi(p_\infty)\leq\phi({\mathbb{E}} p_\infty)=\phi(p)$. The opposing strategy was constructed for each fixed $\sigma$, exactly as permitted by the order $\sup_\sigma\inf_\tau$. This proves $\underline V_f(p)\leq\phi(p)$. ∎

Proof[Proof of the maxmin lower bound]: Suppose $0<p\leq2/3$. At the first date use the two distinct actions as signals $A,B$. Type 1 always sends $A$; type 2 sends $A$ with probability $p/[2(1-p)]$, which lies in $[0,1]$. The signal $A$ has probability $3p/2$ and posterior $2/3$; $B$ reveals type 2 and has probability $1-3p/2$. After $A$, both types play $r$ forever. Future play is then independent of the type conditional on the signal, so on $E$ its expected payoff is $-2/3+2/3=0$, and off $E$ it is zero since $F$ is false. After $B$, play $\ell$ forever. Since the type is 2 and $F$ holds, the payoff is at least 1 against every strategy of player 2. The guaranteed expected payoff is therefore at least $1-3p/2$. At $p=0$, simply play $\ell$ forever to guarantee 1.

For $p\geq2/3$, play $r$ forever independently of type. No type information is revealed. Conditional on $E$ the expected payoff is $2-3p$, and off $E$ it is zero. Since $2-3p\leq0$, this guarantees $2-3p$. These are exactly the two branches of $\phi$. ∎

Proof[Proof of the minmax upper bound]: If player 2 always plays $r$, type 1 can obtain at most zero and type 2 at most 1, and both bounds are attained by always playing $r$ and $\ell$, respectively. Thus the supremum is $1-p$. If player 2 always plays $\ell$, $E$ occurs surely and the expected payoff is $2-3p$, independently of the informed strategy. Taking the better of these two minimizing strategies gives $\overline V_f(p)\leq\psi(p)$. ∎
Proof[Proof of the minmax lower bound]: Fix an arbitrary strategy $\tau$ of player 2. First consider the baseline informed strategy that plays $r$ forever in both types. Let $\mathbb P_0$ be the resulting public-play law and $q=\mathbb P_0(E)$. For every $\delta>0$ there is an event $D$ determined by a finite public history of length $T$ such that $\mathbb P_0(E\triangle D)<\delta$. To justify this, the martingale ${\mathbb{E}}_0[{\mathbf 1}_E\mid\mathcal H_T]$ converges in $L^1$ to ${\mathbf 1}_E$ because the finite-history sigma fields generate the public-play sigma field. Taking $D$ where this conditional expectation is at least $1/2$ bounds its symmetric-difference probability by twice the $L^1$ error.

Now let type 1 always play $r$. Let type 2 play $r$ through $T$ and, after that, continue playing $r$ on $D$, but play $\ell$ forever on $D^c$. The baseline law is unchanged through $T$, and on $D$ it remains unchanged forever. Type 1 obtains $-q$. Type 2 obtains 2 on $D\cap E$ and at least 1 on $D^c$. Its expected payoff is consequently at least \[ 2\mathbb P_0(D\cap E)+\mathbb P_0(D^c) =1+q-\mathbb P_0(E\triangle D). \] The prior-weighted payoff is at least \[ 1-p+(1-2p)q-(1-p)\delta \geq\min\{1-p,2-3p\}-\delta. \] Let $\delta\downarrow0$. The informed response may depend on the fixed $\tau$, as required by $\inf_\tau\sup_\sigma$. This proves the remaining inequality. ∎

Consequences and boundaries.
The game has no value for $0<p<2/3$, but has a value at $p=0$ and throughout $[2/3,1]$. The payoff is outside the continuous sufficient class: after every finite prefix, type-2 tails realizing payoff 0 and payoff 2 both remain possible. Every cylinder payoff is therefore at uniform distance at least 1 from this payoff in that type. A finite-horizon approximation cannot have vanishing uniform error.

VERIFICATION

The lower-bound and upper-bound constructions use different orders of optimization, and these orders were retained in every proof. All posterior calculations remain valid under simultaneous actions: the minimizing player's current randomization is conditionally independent of the hidden type, and the first informed action is revealed only after the first stage. Null-probability signal branches are irrelevant, and $p=0,1,1/3,1/2,2/3$ are covered explicitly by the formulas.

The reported exact-arithmetic checks evaluate $u,\phi,\psi$ at the 25 priors $p=j/24$ and verify their order and the midpoint gap. A one-stage surrogate using the last action instead of infinitely-often events has value $1/2$ at the symmetric prior: choosing $r$ in type 1 and $\ell$ in type 2 gives $1/2$ against either minimizing action, and an always-$r$ minimizer bounds the payoff above by $1/2$. This illustrates why finite truncations are not evidence of infinite-horizon determinacy. The analytic proof, not this grid, establishes the formulas for every real prior.

FINAL

LABEL: UNRESOLVED for the requested characterization of all of $\mathcal V$.
Strongest checked partial results.
Continuous payoffs, and uniform limits of value-admitting payoffs, admit values prior-uniformly. For the explicit two-type, two-action Borel tail payoff above, both minimax functions have been determined exactly for all priors. This is a complete counterexample to the stronger assertion that tail or shift invariance alone ensures a value, but not a solution of the classification programme.
Exact first gap.
No intrinsic necessary-and-sufficient condition is proved for bounded Borel $f$ outside the uniformly approximable class to satisfy $\underline V_f(p)=\overline V_f(p)$ for every prior.
Three next moves.
(1) Generalize the posterior-threshold argument to payoffs defined by a finite partition into tail events and compute their minmax envelopes. (2) Find a weaker replacement for uniform cylinder approximation that controls both optimization orders, not only the maxmin. (3) Test whether the finite-event delayed-switch construction yields a complete obstruction within a sharply specified tail-payoff subclass.
Minimal obstruction structure.
The example already uses two types and two actions per player. Its obstruction is a tail event that can be approximated after observing a long history under a fixed opponent, but cannot be predicted uniformly over all opponents in advance. Any characterization must distinguish this phenomenon from ordinary finite-horizon information revelation.

References

Completion Estimate

25%

Go to contribution →

Econ-8-A2 · Partial result · LLM-generated · gpt-6 astra, claude opus 5.5, gpt-6 astra pro

Payoff families and value in games with one-sided information

Payoff families and value in games
with one-sided information

Ji Ho Bae
LawAll AI

11 October 2026

\begin{abstract} We study bounded Borel payoffs in repeated zero-sum games with a finite hidden state known to one player. For fixed finite action sets, value existence at every prior is equivalent to a condition on posterior and capital arrays over the public history tree. The infimum of the initial capital excess equals the minimax gap. These certificates yield an exact cover of the value-admitting payoffs by Borel selections of pathwise closed convex regions. Each family has a greatest payoff of Baire class at most three, a common value at every prior, and closure under public-path Borel mixing. Independently of value existence, every bounded Borel payoff has a majorant of Baire class at most three with the same upper value at every prior. The proofs use Borel superhedging duality on finite-alphabet trees and an explicit posterior reduction. All pathwise conditions include plays of probability zero. \end{abstract}

Model and question

Let $K,I,J$ be nonempty finite sets with the discrete topology. Write $\mathbb N=\{0,1,\ldots\}$ and \[ X=I\times J,\qquad \mathcal H=\bigcup_{t\geq0}X^t, \qquad \Omega=X^{\mathbb N}. \] Equip $\Omega$ with the product topology, with $X$ discrete. A history $h\in\mathcal H$ is public; $\varnothing$ denotes the empty history, $hij$ appends $(i,j)$ to $h$, and $\omega|t$ is the prefix of length $t$.

Nature draws $k\in K$ with prior $p\in\Delta(K)$ and reveals it only to the maximizing player. At every date the players choose actions simultaneously and then both observe the pair. Their behavioral strategies are \[ \sigma:K\times\mathcal H\longrightarrow\Delta(I),\qquad \tau:\mathcal H\longrightarrow\Delta(J). \] The two current actions are drawn independently conditional on the state and history. For a bounded Borel payoff $f:K\times\Omega\to\mathbb R$, let $\gamma_{p,f}(\sigma,\tau)=\mathbb E_{p,\sigma,\tau}f(k,\omega)$ and \[ \underline v_f(p)=\sup_\sigma\inf_\tau\gamma_{p,f}(\sigma,\tau),\qquad \overline v_f(p)=\inf_\tau\sup_\sigma\gamma_{p,f}(\sigma,\tau). \] The game has a value at $p$ when these two quantities are equal; their common value is then denoted by $v_f(p)$.

Castellon Koltun, Lehrer and Solan [KLS, Section 5] ask for properties of the payoff that guarantee value existence in this model. The prior-uniform formulation in problem C73-8 of [Catalogue] asks for a characterization of \[ \mathcal V_{K,I,J} =\{f:K\times\Omega\to\mathbb R\text{ bounded Borel}: \underline v_f(p)=\overline v_f(p)\text{ for every }p\in\Delta(K)\}. \] The sought characterization must give conditions on the payoff and the information structure beyond equality of the defining optimizations. We express these conditions through local convex geometry and pointwise boundary inequalities. The resulting arrays determine payoff families and describe path-dependent changes within each family that preserve the value at every prior. The characterization is infinitary; no effective recognition procedure for an arbitrary Borel payoff is asserted.

Choose $a<b$ with $a\leq f\leq b$ and put $F=(f-a)/(b-a)$. Thus $0\leq F\leq1$. Both values transform by this positive affine map. We write $\mathcal V^{[0,1]}_{K,I,J}$ for the normalized class and $\Delta_{\mathbb Q}(K)$ for the rational-coordinate priors. The strategy spaces do not depend on $p$, including at its boundary. Consequently, \begin{equation}\label{wsref-0-65-71-3a-6c-69-70-73-63-68-69-74-7a} |\underline v_F(p)-\underline v_F(p')|\leq\|p-p'\|_1,\qquad |\overline v_F(p)-\overline v_F(p')|\leq\|p-p'\|_1. \end{equation} These bounds hold for every fixed strategy pair and survive either order of infimum and supremum. Both values are also monotone in $F$.

Local capital certificates

Definition (Capital arrays at prior $p$): At each $h\in\mathcal H$, assign \[ q(h)\in\Delta(K),\quad A_k(h),B(h)\in[0,1]\ (k\in K). \] Require $q(\varnothing)=p$ and the following conditions.

Posterior children: \begin{equation} q(hij)\text{ is independent of }j; \quad\text{denote it by }\bar q_i(h). \tag{BF}\label{wsref-0-62-66} \end{equation} In particular, a posterior child depends on the current informed action but is independent of the current uninformed action, including branches that will receive zero convex weight.

Common-column capital geometry: define \[ V_j(h)=\left(A_k(hij)-A_k(h)\right)_{(k,i)\in K\times I} \in\mathbb R^{K\times I}. \] Require \begin{equation} \operatorname{conv}\{V_j(h):j\in J\}\cap\mathbb R_{-}^{K\times I}\ne\varnothing, \tag{UC}\label{wsref-0-75-63} \end{equation} where $\mathbb R_{-}^{K\times I}$ is the coordinatewise nonpositive orthant.

Posterior and lower-capital geometry: define \[ Z_i(h)=\left(\bar q_i(h), (B(hij)-B(h))_{j\in J}\right)\in\mathbb R^K\times\mathbb R^J. \] Require \begin{equation} \operatorname{conv}\{Z_i(h):i\in I\}\cap \left(\{q(h)\}\times\mathbb R_-^J\right)\ne\varnothing. \tag{LC}\label{wsref-0-6c-63} \end{equation}

Define the bounded Borel function on every path \begin{equation} G_q(\omega)=\liminf_{t\to\infty} \sum_{k\in K}q_k(\omega|t)F(k,\omega). \tag{G}\label{wsref-0-67} \end{equation} Require the pointwise boundary conditions \begin{align} \liminf_{t\to\infty}A_k(\omega|t)&\geq F(k,\omega) &&(k\in K,\omega\in\Omega),\tag{UB}\label{wsref-0-75-62}\\ \liminf_{t\to\infty}B(\omega|t)&\geq1-G_q(\omega) &&(\omega\in\Omega).\tag{LB}\label{wsref-0-6c-62} \end{align} These include paths of probability zero.

Definition (Payoff certificate condition): The payoff $F$ satisfies condition (C) if for every $p\in\Delta_{\mathbb Q}(K)$ and every integer $n\geq1$, there are capital arrays as above with \begin{equation} \sum_{k\in K}p_k A_k(\varnothing)+B(\varnothing)\leq1+\frac1n. \tag{RC}\label{wsref-0-72-63} \end{equation}

The local conditions are finite convex-hull intersections at each history. Their boundary inequalities compare the arrays with the payoff on every path. We call arrays satisfying these requirements a capital certificate; the root bound is imposed separately in (C).

Theorem (Certificate characterization): For arbitrary fixed nonempty finite $K,I,J$ and every bounded Borel $f$, \[ f\in\mathcal V_{K,I,J}\quad\Longleftrightarrow\quad F\text{ satisfies \textup{(C)}}. \] The equivalence is independent of the normalization bounds $a,b$.

The same certificates give an exact representation of the value gap. Let $D_F(p)$ be the infimum of $\sum_k p_k A_k(\varnothing)+B(\varnothing)-1$ over capital arrays at prior $p$ satisfying (bf)--(lb), without (rc). Such arrays always exist: take $q(h)=p$ and $A_k(h)=B(h)=1$.

Theorem (Exact gap representation): For every bounded Borel $F:K\times\Omega\to[0,1]$ and every prior $p$, \[ D_F(p)=\overline v_F(p)-\underline v_F(p). \]

Proof of the capital representation

Borel superhedging on a finite-alphabet tree

We use equation $3$ and Theorems 2 and 13 of T'Joens and De Bock [TDB]. Equation $3$ defines the game-theoretic upper expectation by pointwise superhedging. Theorem 2 identifies it with the axiomatic upper expectation, and Theorem 13 identifies the latter with the measure-theoretic upper expectation for bounded Borel payoffs. Together they give the following form:

Lemma (Borel superhedging): On a finite alphabet tree, assign each history $h$ a nonempty closed convex set $P_h$ of next-step probability vectors. For a bounded nonnegative Borel $W$, let $c$ be the supremum of $\mathbb E_P W$ over all laws whose one-step kernels lie in $P_h$. For every $\delta>0$ there is a real-valued, globally bounded-below supermartingale $M$ with $M(\varnothing)<c+\delta$, \[ \sup_{\pi\in P_h}\sum_{z\in X}\pi_z M(hz)\leq M(h), \qquad \liminf_t M(\omega|t)\geq W(\omega)\quad(\omega\in\Omega). \]

The local upper expectation is $Q_h(Z)=\sup_{\pi\in P_h}\pi\cdot Z$, so the probability sets and upper expectations agree in the terminology of [TDB]. Both applications below use bounded Borel payoffs and nonempty closed convex local models on the finite alphabet $X$.

Lemma (Capital can be bounded by one): If $0\leq W\leq1$, the process in Lemma Lemma (Borel superhedging) may be chosen in $[0,1]$, with the same bound on its initial capital.
Proof: Any process furnished by the lemma is nonnegative at every node. Its local inequality implies $\min_z M(hz)\leq M(h)$. If $M(h)<0$ at some node, successively choose a child whose value is no larger than its parent's. The resulting infinite path extending $h$ has $\liminf M\leq M(h)<0$, contradicting its nonnegative boundary lower bound. A finite prefix does not affect this liminf.

Now put $\widetilde M(h)=\min\{M(h),1\}$. For each $\pi\in P_h$, concavity and monotonicity of $z\mapsto\min\{z,1\}$ give \[ \sum_z\pi_z\min\{M(hz),1\} \leq\min\left\{\sum_z\pi_zM(hz),1\right\} \leq\min\{M(h),1\}. \] Thus $\widetilde M$ satisfies the local inequality. Also $\liminf\widetilde M=\min\{\liminf M,1\}\geq W$ on every path. Clipping decreases initial capital, and $\widetilde M\in[0,1]$. ∎

Posterior flow

Lemma (Bayes-flow correspondence): Fix $p\in\Delta(K)$. Let $q:\mathcal H\to\Delta(K)$ satisfy $q(\varnothing)=p$ and (bf), and let $w_i(h)\geq0$ be coefficients summing to one with $q(h)=\sum_i w_i(h)\bar q_i(h)$. Define \[ x_k(i\mid h)=\frac{w_i(h)\bar q_{i,k}(h)}{q_k(h)} \quad\text{when }q_k(h)>0, \] and use any fixed action distribution when $q_k(h)=0$. This is an informed behavioral strategy. Against every uninformed strategy $\tau$, the public posterior at every positive-probability history is $q(h)$, the informed action has public distribution $w(h)$, and \begin{equation} \mathbb E_{p,x,\tau}F(k,\omega)=\mathbb E_{p,x,\tau}G_q(\omega). \tag{PI}\label{wsref-0-70-69} \end{equation} Conversely every informed strategy $x$ at prior $p$ gives such arrays with these public laws.
Proof: At a positive-$q_k(h)$ row, the barycentric identity gives $\sum_i x_k(i\mid h)=1$. If $q_k(h)=0$, nonnegativity and that same identity imply $w_i(h)\bar q_{i,k}(h)=0$ for every $i$. Hence \[ \sum_k q_k(h)x_k(i\mid h) =\sum_k w_i(h)\bar q_{i,k}(h)=w_i(h). \]

Induct on the length of a positive-probability public history $h$. At the root the posterior is $p=q(\varnothing)$. If the posterior at $h$ is $q(h)$, the next pair $(i,j)$ has conditional probability $w_i(h)\tau_j(h)$. When this is positive, Bayes' rule gives the next posterior \[ \frac{q_k(h)x_k(i\mid h)}{w_i(h)} =\bar q_{i,k}(h)=q_k(hij). \] The current factor $\tau_j(h)$ cancels because the uninformed player's randomization does not depend on the hidden state. A zero-probability child's assigned posterior does not affect the induction or law. Zero-prior states are covered by the same argument.

The vector $q(\omega|t)$ is consequently the bounded posterior martingale under every $\tau$. Its coordinates converge almost surely to the conditional probabilities of the hidden state given the entire public play. This is the upward conditional-expectation martingale convergence theorem [Durrett, Theorem 4.6.8], because the finite public-history sigma fields generate the sigma field of the full public play. On that event, \[ G_q(\omega)=\sum_k q_{\infty,k}(\omega)F(k,\omega). \] Condition each $\mathbf1_{\{k=\ell\}}F(\ell,\omega)$ on the entire public play, then sum over $\ell$, to obtain (pi).

The definition of $G_q$ also covers paths on which the posterior fails to converge. For each $t$, the function $\sum_k q_k(\omega|t)F(k,\omega)$ is bounded Borel. Taking the pointwise liminf over $t$ therefore gives a bounded Borel function on every path.

Conversely start with any informed strategy $x$ and $q(\varnothing)=p$. At each history put \[ w_i(h)=\sum_k q_k(h)x_k(i\mid h). \] For $w_i(h)>0$ put $\bar q_{i,k}(h)=q_k(h)x_k(i\mid h)/w_i(h)$, and for $w_i(h)=0$ put $\bar q_i(h)=p$. Set $q(hij)=\bar q_i(h)$ for every $j$. These arrays satisfy (bf) and the barycentric identity. At a zero-weight branch all terms $q_k(h)x_k(i\mid h)$ are zero, so the chosen vector contributes nothing to the barycentric identity. The preceding induction proves the claimed public laws and (pi). ∎

The guarantees

Fix capital arrays at prior $p$. Condition (uc) provides at every $h$ coefficients $y_j(h)\geq0$, summing to one, such that \[ \sum_j y_j(h)A_k(hij)\leq A_k(h) \quad\text{for every }(k,i)\in K\times I. \] These coefficients define an uninformed behavioral strategy $y$; the same distribution works for all states and all current informed actions, as required by its information restriction.

In each state $k$, against every informed strategy $\sigma_k$, the bounded nonnegative process $A_k(\omega|t)$ is a supermartingale. Indeed, condition on $h$ and a pure informed action $i$, apply the displayed inequality, and then average over that informed action. Fatou's lemma and (ub) imply \[ \mathbb E_{k,\sigma_k,y}F(k,\omega) \leq\mathbb E\liminf_t A_k(\omega|t) \leq\liminf_t\mathbb E A_k(\omega|t) \leq A_k(\varnothing). \] After weighting by $p$ this proves \begin{equation} \overline v_F(p)\leq\sum_k p_k A_k(\varnothing). \label{wsref-0-75-70-70-65-72} \end{equation}

Condition (lc) supplies convex coefficients $w_i(h)\geq0$ summing to one, simultaneously satisfying \[ q(h)=\sum_i w_i(h)\bar q_i(h),\qquad \sum_i w_i(h)B(hij)\leq B(h)\quad(j\in J). \] Reconstruct $x$ using these coefficients and Lemma Lemma (Bayes-flow correspondence). Against every $\tau$, the public next-action kernel is $w_i(h)\tau_j(h)$. Condition (lc) makes $B(\omega|t)$ a bounded nonnegative supermartingale: apply its inequality for each pure $j$ and average over $\tau(h)$. Fatou, (lb), and (pi) give \[ \mathbb E_{p,x,\tau}(1-F)=\mathbb E(1-G_q) \leq\mathbb E\liminf_t B(\omega|t)\leq B(\varnothing). \] Consequently \begin{equation} \underline v_F(p)\geq1-B(\varnothing). \label{wsref-0-6c-6f-77-65-72} \end{equation} The elementary minimax inequality, (upper), and (lower) yield \begin{equation} 0\leq\overline v_F(p)-\underline v_F(p) \leq\sum_k p_k A_k(\varnothing)+B(\varnothing)-1. \label{wsref-0-67-61-70-62-6f-75-6e-64} \end{equation} Thus (C) gives equality at every rational prior by letting $n$ tend to infinity.

The Lipschitz bounds (eq:lipschitz) extend equality from the rational priors to the entire simplex. Positive affine normalization gives sufficiency for the original payoff.

Sharpness of the bounds

For arbitrary $F$ and $p$ write $L=\underline v_F(p)$ and $U=\overline v_F(p)$. Given $\delta>0$, the definitions of the extrema provide behavioral strategies $x,y$ such that \begin{equation} \inf_\tau\mathbb E_{p,x,\tau}F\geq L-\delta, \qquad \sup_\sigma\mathbb E_{p,\sigma,y}F\leq U+\delta. \label{wsref-0-61-70-70-72-6f-78} \end{equation} No optimal strategy is assumed to exist. Obtain $q,w,\bar q$ from $x$ using Lemma Lemma (Bayes-flow correspondence).

For each state $k$ use the local credal sets \[ P_h^y=\{(a_i y_j(h))_{ij}:a\in\Delta(I)\}. \] These are nonempty closed convex sets, being linear images of the finite simplex. Their local upper expectation of a next-step array $Z_{ij}$ is \[ \max_{i\in I}\sum_j y_j(h)Z_{ij}. \] Compatible precise kernels are exactly the kernels of arbitrary state-$k$ informed behavioral responses to the fixed $y$: the marginal on $I$ recovers $a(h)$. Therefore the compatible laws are exactly those response laws. Let $c_k$ be their upper expectation of $F(k,\cdot)$. Lemma Lemma (Borel superhedging), with the finite alphabet $X$ and this bounded Borel payoff, followed by Lemma Lemma (Capital can be bounded by one), gives $A_k\in[0,1]$ with \[ A_k(\varnothing)<c_k+\delta,\qquad \sum_j y_j(h)A_k(hij)\leq A_k(h)\quad(h\in\mathcal H,i\in I), \] and with (ub) on every path. The same $y$ supplies convex coefficients for all $k$, proving (uc) for the combined arrays.

The finitely many state components can be chosen independently, so \begin{equation} \sum_k p_k c_k=\sup_\sigma\mathbb E_{p,\sigma,y}F. \label{wsref-0-73-74-61-74-65-77-69-73-65} \end{equation} For every $\sigma$, $\mathbb E_{p,\sigma,y}F\leq\sum_k p_k c_k$, so its supremum is also at most that sum. Conversely, choose $\varepsilon$-optimal responses separately in each positive-prior state, combine them into one informed behavioral strategy, and let $\varepsilon\downarrow0$. Zero-prior states have no effect. Hence (approx) gives \begin{equation} \sum_k p_k A_k(\varnothing)<U+2\delta. \label{wsref-0-61-63-6f-73-74} \end{equation}

For the other capital process, use the public-law credal sets \[ P_h^w=\{(w_i(h)b_j)_{ij}:b\in\Delta(J)\}. \] They are again nonempty closed convex linear images of a finite simplex. Their local upper expectation is $\max_j\sum_i w_i(h)Z_{ij}$. Compatible precise trees correspond exactly to uninformed strategies against $x$, because their $J$ marginal is $b$ and their public $I$ marginal is $w$. This statement is also valid at histories of probability zero, whose specified kernels do not affect the root law.

The boundary payoff $W=1-G_q$ is bounded, nonnegative, and Borel. Lemma Lemma (Bayes-flow correspondence) shows that its upper expectation over these compatible trees equals \[ \sup_\tau\mathbb E(1-G_q)=1-\inf_\tau\mathbb E_{p,x,\tau}F. \] Apply Lemmas Lemma (Borel superhedging) and Lemma (Capital can be bounded by one). They give $B\in[0,1]$ with $\sum_i w_i(h)B(hij)\leq B(h)$ for every $j$, and satisfying (lb). Together with the barycentric identity, these inequalities show that $\sum_i w_i(h)Z_i(h)$ belongs to $\{q(h)\}\times\mathbb R_-^J$, proving (lc). The root bound is \begin{equation} B(\varnothing)<1-\inf_\tau\mathbb E_{p,x,\tau}F+\delta \leq1-L+2\delta. \label{wsref-0-62-63-6f-73-74} \end{equation} Thus both local models and their boundary payoffs satisfy the hypotheses of Lemma Lemma (Borel superhedging).

Combining (acost) and (bcost) yields capital arrays with \[ \sum_k p_k A_k(\varnothing)+B(\varnothing)-1 <U-L+4\delta. \] Taking the infimum of these costs and letting $\delta\downarrow0$, then using (gapbound), proves $D_F(p)=U-L$. This establishes Theorem Theorem (Exact gap representation) for every bounded Borel payoff and every prior, without a value-existence assumption. When the game has a value, $U=L$; choosing $4\delta<1/n$ gives (rc) for every $n$. This proves necessity of (C), and completes Theorem Theorem (Certificate characterization). Its equivalence with value existence also proves independence of the normalization bounds.

Upper-value regularization

The upper part of the proof gives a separate representation that does not use posterior arrays. Let \[ \mathcal U(F)=\{A:\mathcal H\to[0,1]^K:\text{\eqref{wsref-0-75-63} and \eqref{wsref-0-75-62} hold}\}. \] This set has no designated prior.

Proposition: For every normalized Borel payoff $F$ and every prior $p$, \begin{equation}\label{wsref-0-65-71-3a-75-70-70-65-72-64-75-61-6c} \overline v_F(p)=\inf_{A\in\mathcal U(F)}p\cdot A(\varnothing). \end{equation}
Proof: The argument for (upper) uses only (uc) and (ub), so it gives one inequality. The construction leading to (acost) provides $A\in\mathcal U(F)$ with $p\cdot A(\varnothing)<\overline v_F(p)+2\delta$ for every $\delta>0$, proving the other. ∎

We use the convention that Baire class zero consists of continuous real-valued functions, and class at most $m+1$ consists of pointwise limits of sequences of functions of class at most $m$.

Lemma: For a nonempty countable family of arrays $A^d:\mathcal H\to[0,1]^K$, the function on $K\times\Omega$ defined by \[ c(k,\omega)=\inf_d\liminf_t A^d_k(\omega|t) \] has Baire class at most three.
Proof: For fixed $d,k,t$, the prefix function $A^d_k(\omega|t)$ is continuous. For each $N$, its tail infimum is the pointwise limit, as $M\to\infty$, of the continuous functions $\min_{N\leq t\leq M}A^d_k(\omega|t)$. It has class at most one. The increasing limit over $N$ is $\liminf_t A^d_k(\omega|t)$, of class at most two. Enumerate the indices $d$, repeating terms if the family is finite. The infimum over $d$ is the decreasing limit of finite minima of these class-two functions, hence has class at most three. Finite minima preserve each class, by applying the continuous minimum operation to the defining sequences. The finitely many state coordinates can be combined without raising the class. ∎
Theorem (Upper-value majorant): For every Borel $F:K\times\Omega\to[0,1]$ there is a function $C:K\times\Omega\to[0,1]$ of Baire class at most three such that \[ F\leq C,\qquad \overline v_C(p)=\overline v_F(p)\quad(p\in\Delta(K)). \] Every Borel $G$ with $F\leq G\leq C$ has the same upper value as $F$ at every prior.
Proof: For every $d=(r,n)$ in the countable set \[ \mathcal D=\Delta_{\mathbb Q}(K)\times\mathbb N_{\geq1}, \] choose $A^d\in\mathcal U(F)$ with $r\cdot A^d(\varnothing)\leq\overline v_F(r)+1/n$, using (eq:upperdual). Set $C(k,\omega)=\inf_d\liminf_tA^d_k(\omega|t)$. Then $F\leq C$, Lemma Lemma applies, and every $A^d$ also belongs to $\mathcal U(C)$. For rational $r$ and every $n$, \[ \overline v_F(r)\leq\overline v_C(r) \leq r\cdot A^{(r,n)}(\varnothing)\leq\overline v_F(r)+1/n. \] Let $n\to\infty$, then use (eq:lipschitz) to extend equality to all priors. The assertion for $G$ follows by monotonicity. ∎

For an arbitrary bounded Borel $f$, apply the theorem to $F=(f-a)/(b-a)$ and set $\widehat f=a+(b-a)C$. Then $f\leq\widehat f$ and $\overline v_{\widehat f}(p)=\overline v_f(p)$ for every prior, whether or not the game has a value.

Corollary: Every $f\in\mathcal V_{K,I,J}$ has a bounded majorant $\widehat f$ of Baire class at most three such that every bounded Borel $g$ with $f\leq g\leq\widehat f$ belongs to $\mathcal V_{K,I,J}$ and satisfies $v_g(p)=v_f(p)$ for every prior.
Proof: Normalize $f$ and apply Theorem Theorem (Upper-value majorant). For $F\leq G\leq C$, \[ v_F(p)=\underline v_F(p)\leq\underline v_G(p)\leq\overline v_G(p)=\overline v_F(p)=v_F(p). \] Undo the affine normalization, setting $\widehat f=a+(b-a)C$. ∎

The majorant theorem preserves the upper value even for games without a value. It is not a value-existence criterion. No optimality of the bound on the Baire class is asserted.

An exact cover by payoff families

We now keep full certificates rather than only their upper arrays. This yields families of payoffs with the same value at every prior. Their definition uses the information and action structure, without first choosing a payoff.

Boundary regions

An array family $\Theta$ assigns $(q^d,A^d,B^d)$ to every $d=(p,n)\in\mathcal D$. Require $q^d(\varnothing)=p$, (bf)--(lc), and (rc) for each $d$. The payoff-dependent boundary conditions have not yet been imposed. Define \begin{align*} u^d_k(\omega)&=\liminf_t A^d_k(\omega|t),& c_{\Theta,k}(\omega)&=\inf_{d\in\mathcal D}u^d_k(\omega),\\ \ell^d(\omega)&=1-\liminf_tB^d(\omega|t),& Q^d(\omega)&=\bigcap_{N\geq0} \overline{\{q^d(\omega|t):t\geq N\}}. \end{align*} The set $Q^d(\omega)$ is the nonempty compact cluster set in $\Delta(K)$, defined even when the posterior sequence does not converge. Impose the boundary compatibility condition \begin{equation}\label{wsref-0-65-71-3a-62-63}\tag{BC} \min_{r\in Q^d(\omega)}r\cdot c_\Theta(\omega) \geq\ell^d(\omega)\qquad(d\in\mathcal D,\ \omega\in\Omega). \end{equation} Let $\mathfrak T_{K,I,J}$ be the class of families satisfying these requirements. All arrays take values in the same simplexes and intervals as the capital arrays in Section sec:certificates. We identify $c_\Theta$ with the payoff $(k,\omega)\mapsto c_{\Theta,k}(\omega)$.

For $\Theta\in\mathfrak T_{K,I,J}$ define \begin{equation}\label{wsref-0-65-71-3a-72-65-67-69-6f-6e} \mathcal P_\Theta(\omega)= \left\{z\in[0,1]^K: \begin{array}{l} z_k\leq c_{\Theta,k}(\omega)\quad(k\in K),\\ r\cdot z\geq\ell^d(\omega)\quad(d\in\mathcal D,\ r\in Q^d(\omega)) \end{array}\right\}, \end{equation} and let $\mathcal S_\Theta$ consist of the Borel payoffs $F$ with $F(\cdot,\omega)\in\mathcal P_\Theta(\omega)$ at every path. The regions are intersections of closed half-spaces and the cube; they need not be polyhedra with finitely many defining inequalities.

Theorem (Payoff-family representation): For arbitrary fixed nonempty finite $K,I,J$, \begin{equation}\label{wsref-0-65-71-3a-63-6f-76-65-72} \mathcal V^{[0,1]}_{K,I,J} =\bigcup_{\Theta\in\mathfrak T_{K,I,J}}\mathcal S_\Theta. \end{equation} For each $\Theta\in\mathfrak T_{K,I,J}$ the following hold.
  1. Each $\mathcal P_\Theta(\omega)$ is nonempty, closed and convex, and the correspondence $\omega\mapsto\mathcal P_\Theta(\omega)$ has a Borel graph. The greatest element of $\mathcal S_\Theta$ in the pointwise order is $c_\Theta$, of Baire class at most three.
  2. Every $F\in\mathcal S_\Theta$ has the same value function \begin{equation}\label{wsref-0-65-71-3a-66-61-6d-69-6c-79-76-61-6c-75-65} v_\Theta(p)=\inf_{d\in\mathcal D}p\cdot A^d(\varnothing) \qquad(p\in\Delta(K)). \end{equation} In particular, $v_\Theta$ is concave and $1$-Lipschitz in the $\ell^1$ norm.
  3. If $F^1,\ldots,F^m\in\mathcal S_\Theta$ and $\lambda:\Omega\to\Delta(\{1,\ldots,m\})$ is Borel, then \[ F^\lambda(k,\omega)=\sum_{a=1}^m\lambda_a(\omega)F^a(k,\omega) \quad\text{belongs to }\mathcal S_\Theta. \] Also, if $F\in\mathcal S_\Theta$ and $G$ is Borel with $F\leq G\leq c_\Theta$, then $G\in\mathcal S_\Theta$.
The cover in (eq:cover) is not asserted to be disjoint.

Proof of the representation

For a sequence $q_t\in\Delta(K)$ with cluster set $Q$ and any fixed $z\in\mathbb R^K$, compactness gives \begin{equation}\label{wsref-0-65-71-3a-63-6c-75-73-74-65-72} \liminf_t q_t\cdot z=\min_{r\in Q}r\cdot z. \end{equation} Every cluster point is the limit of a subsequence, so its scalar product is at least the liminf. Conversely, a subsequence realizing the scalar liminf has a further convergent subsequence in $\Delta(K)$, which attains equality. Apply this identity at each path, with $z=F(\cdot,\omega)$. The lower boundary inequalities in (eq:region) are exactly (lb) for all $d$, while $F\leq c_\Theta$ is exactly (ub) for all $d$. Thus, for Borel $F:K\times\Omega\to[0,1]$, \begin{equation}\label{wsref-0-65-71-3a-73-65-6c-65-63-74-69-6f-6e-65-71-75-69-76} F\in\mathcal S_\Theta \quad\Longleftrightarrow\quad (q^d,A^d,B^d)\text{ certifies }F\text{ at its prior for every }d. \end{equation}

Condition (eq:bc) says that $c_\Theta(\omega)$ itself belongs to $\mathcal P_\Theta(\omega)$. Conversely, any point $z$ of that region implies (eq:bc), since $z\leq c_\Theta$ and every $r\in\Delta(K)$ has nonnegative coordinates. This proves nonemptiness; closedness and convexity follow from (eq:region). Lemma Lemma shows that $c_\Theta$ is Borel and has the claimed Baire class. It is therefore the greatest member of $\mathcal S_\Theta$.

The graph is Borel as well. By (eq:cluster), its defining conditions are the coordinate upper bounds and the countably many inequalities \[ \liminf_t q^d(\omega|t)\cdot z\geq\ell^d(\omega). \] For each $d,t$, the function $(\omega,z)\mapsto q^d(\omega|t)\cdot z$ is continuous. The displayed comparisons are consequently Borel; no measurable-selection theorem is needed.

If $F\in\mathcal S_\Theta$, equivalence (eq:selectionequiv) and the root bounds give (C), hence value existence by Theorem Theorem (Certificate characterization). Conversely, if $F\in\mathcal V^{[0,1]}_{K,I,J}$, choose a certificate satisfying the root bound for each $d=(p,n)$. The upper boundaries imply $F\leq c_\Theta$. For every cluster point $r\in Q^d(\omega)$, the lower boundaries and (eq:cluster) give \[ r\cdot c_\Theta(\omega)\geq r\cdot F(\cdot,\omega) \geq\ell^d(\omega). \] Hence (BC) holds, $\Theta\in\mathfrak T_{K,I,J}$, and $F\in\mathcal S_\Theta$. This proves (eq:cover).

To identify the value, fix $F\in\mathcal S_\Theta$. Each $A^d$ belongs to $\mathcal U(F)$, and so gives the upper bound $v_F(p)\leq p\cdot A^d(\varnothing)$ at every prior, regardless of the prior assigned to $q^d$. For rational $p$, use $d=(p,n)$, (rc) and (lower) to obtain \[ p\cdot A^{(p,n)}(\varnothing) \leq1-B^{(p,n)}(\varnothing)+1/n\leq v_F(p)+1/n. \] This proves (eq:familyvalue) at rational priors. The right-hand side is concave and $1$-Lipschitz, being the infimum of linear functions whose coefficient vectors lie in $[0,1]^K$. Equation (eq:lipschitz) and density extend equality to all priors. The expression is independent of $F$, proving the common-value claim.

Finally, pathwise convexity proves the mixing assertion. The weights depend on the public path, with the same weights for all states. If $F\leq G\leq c_\Theta$, nonnegativity of $r$ preserves every lower inequality in (eq:region); the upper inequalities are already assumed. This proves the last assertion and completes the proof of Theorem Theorem (Payoff-family representation).

Strategies and payoff compatibility

The local arrays retain the game's strategic constraints. Their weights reconstruct behavioral strategies as in Section sec:capitalproof. We first make this strategic interpretation explicit, then distinguish these payoff families from ordinary value level sets.

Fixed-strategy costs

Fix $p,F$ and an uninformed strategy $y$. Let $\mathcal A_y(F)$ be the upper arrays satisfying (ub) and \[ \sum_jy_j(h)A_k(hij)\leq A_k(h)\quad(h\in\mathcal H,\ k\in K,\ i\in I). \] For an informed strategy $x$, take the recursive posterior extension $(q^x,w^x)$ in Lemma Lemma (Bayes-flow correspondence), and let $\mathcal B_x(F)$ be the $[0,1]$-valued arrays satisfying (lb) for $q^x$ and $\sum_iw_i^x(h)B(hij)\leq B(h)$ for every $h,j$. The two applications of superhedging in Section sec:capitalproof, now with these strategies fixed, give \begin{align} \inf_{A\in\mathcal A_y(F)}p\cdot A(\varnothing) &=\sup_\sigma\gamma_{p,F}(\sigma,y),\label{wsref-0-65-71-3a-66-69-78-65-64-75-70-70-65-72}\\ \inf_{B\in\mathcal B_x(F)}B(\varnothing) &=1-\inf_\tau\gamma_{p,F}(x,\tau).\label{wsref-0-65-71-3a-66-69-78-65-64-6c-6f-77-65-72} \end{align} The lower bounds follow from the corresponding supermartingale guarantees; Lemma Lemma (Borel superhedging) gives arbitrarily close upper bounds. Thus eliminating the capital variables recovers \[ D_F(p)=\inf_{x,y}\left\{ \sup_\sigma\gamma_{p,F}(\sigma,y) -\inf_\tau\gamma_{p,F}(x,\tau)\right\}. \] This is an identity of infima, with no claim of attainment. The representation does not remove strategy search.

Common optimal strategies do not determine a family

Example: Let $K$ be a singleton, $I=J=\{0,1\}$, and \[ F(\omega)=\mathbf 1_{\{i_0=j_0\}},\qquad G(\omega)=1-F(\omega). \] Both games have value $1/2$, and their optimal strategies are exactly those with a uniform first action; later actions are immaterial. Nevertheless, no $\mathcal S_\Theta$ contains both payoffs.
Proof: If both belonged to one family, every array in that family would satisfy the boundaries for both. Since $q\equiv1$, they would imply \[ \liminf_tA(\omega|t)\geq\max(F,G)=1,\qquad \liminf_tB(\omega|t)\geq\max(1-F,1-G)=1. \] The supermartingale guarantees force $A(\varnothing)=B(\varnothing)=1$. The certificate cost is then $1$, contradicting the root bound for $n=2$.

Individually, each payoff has a certificate of cost zero. For $F$, put $A(\varnothing)=B(\varnothing)=1/2$, and after the first pair keep $A=F$ and $B=1-F$. Uniform local weights satisfy the two convex conditions at the root; at later histories the arrays are constant. Interchanging $F$ and $G$ gives the other certificate. ∎

The same distinction appears in the path-dependent mixture with weights $\lambda_1=F$ and $\lambda_2=1-F$: it equals $1$ everywhere. Its value is $1$, whereas mixing within one $\mathcal S_\Theta$ would preserve $1/2$. The families therefore record compatibility of pathwise payoff bounds beyond equality of values and optimal strategies.

A fixed-strategy infimum need not be attained

Example: Let $K,I$ be singletons, let $J=\{0,1\}$, and fix the uninformed strategy $y$ to choose each action with probability $1/2$ at every history. Let $F$ indicate the path consisting entirely of zeros. The infimum in (eq:fixedupper) is zero and is not attained.
Proof: Identify $I\times J$ with $J$ and write $hj$ for the child reached by action $j$. The payoff has expectation zero under the fixed fair-coin law. If a nonnegative upper array had root zero, the inequalities $(A(h0)+A(h1))/2\leq A(h)$ would force every node to have value zero. This violates the boundary on the all-zero path.

For each $N\geq1$, start instead with $A(\varnothing)=2^{-N}$. Before time $N$, double the capital after a zero and set it to zero after a one. After time $N$, keep it constant. This $[0,1]$-valued array has the required boundary and satisfies the local fair-coin inequality with equality. Its root values tend to zero. ∎

The nonattainment in this example concerns the fixed strategy $y$. In the unrestricted certificate problem, cost zero is attained: take $A(\varnothing)=0$, put $A(h)=1$ at nonempty all-zero histories and $A(h)=0$ elsewhere, and set $B\equiv1$. The upper weights always choose action $1$. This satisfies every certificate condition.

Theorem Theorem (Payoff-family representation) concerns the greatest payoff within a specified family, not a greatest payoff among all games with the same value. It also supplies neither an optimal strategy nor a zero-cost certificate. The approximation index is retained in the definition of $\Theta$.

Use of AI tools

The author used research AI agents that the author built and configured (GPT-6 Astra max, Claude Opus 5.5 max and GPT-6 Astra Pro) during the proof work, for proof search and construction, proof checking, translation, Python computation and similar tasks. The final manuscript was written, and the proofs were checked, by the human author. All responsibility for this work rests with the author personally.

\clearpage

References

  • [KLS] Gil Bar Castellon Koltun, Ehud Lehrer and Eilon Solan. The Maxmin Value of Repeated Games with Incomplete Information on One Side and Tail-Measurable Payoffs. arXiv:2512.01581v1 $2025$. Definition 2.1, pp. 6--7; Section 5, p. 18.
  • [Catalogue] Payoff functionals guaranteeing a value under one-sided information, problem C73-8, Economics catalogue in mehmetmars7/Erdosproblems-llm-hunter. Fixed statement at commit b1f633121074.
  • [TDB] Natan T'Joens and Jasper De Bock. Global Upper Expectations for Discrete-Time Stochastic Processes: In Practice, They Are All The Same! Proceedings of Machine Learning Research 147, 310--319 $2021$. Equation $3$, p. 314; Theorems 2 and 13, pp. 315 and 318. Extended version: arXiv:2102.13075v2, Appendix A.2, p. 14 (proof of Theorem 13).
  • [Durrett] Rick Durrett. Probability: Theory and Examples, fifth edition $2019$. Author's version of 11 January 2019, Theorem 4.6.8, p. 247.

Go to contribution →