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Econ-13 · economics · Economics

Determinacy of Dadaist Delta games

LLM status Open unverified

Statement

ID: GT-089. Category: Infinitary combinatorial games. Status: P; Problem 3.4 in the inspected source.

Definitions and mathematical statement

Work in ZFC. A combinatorial game form has designated Left and Right options and a well-founded option relation. Let $I$ be a set and let $(G_i)_{i\in I}$ be such forms, with their forms, rather than just their Conway-equivalence classes, fixed. The $\Delta$-sum is the indexed position $\mathbf G=\sum_{i\in I}^{\Delta}G_i$. A Left (respectively Right) move replaces exactly one coordinate by a Left (respectively Right) option and leaves every other coordinate fixed.

Play may have ordinal length. At a limit stage the position is the coordinatewise eventual value of the earlier positions. This value exists because a fixed coordinate cannot undergo infinitely many moves along a well-founded option relation. After any limit stage the player designated First moves first again. More precisely, at the successor move numbered $\beta+j$, where $\beta=0$ or a limit ordinal and $j$ is a positive integer, First moves when $j$ is odd and Second when $j$ is even. A maximal legal alternating run ends when the player due to move has no legal move; that player loses. A strategy prescribes a legal move after each history at which its player must move; it is winning if every maximal compatible play is won by that player.

For a fixed choice $a\in\{\mathrm L,\mathrm R\}$ of First, let $W_{\mathrm L}(\mathbf G,a)$ and $W_{\mathrm R}(\mathbf G,a)$ denote existence of a winning strategy for the indicated player. Ask whether \[ \forall I\ \forall (G_i)_{i\in I}\ \forall a\in\{\mathrm L,\mathrm R\}, \qquad W_{\mathrm L}(\mathbf G,a)\ \lor\ W_{\mathrm R}(\mathbf G,a). \] Equivalently, does there exist a $\Delta$-sum and a choice of First for which both winning-strategy assertions fail? The countable-sum question restricts $I$ to $\mathbb N$.

Short English statement

Under the specified transfinite play rule, must one player have a winning strategy for each choice of who starts?

Variants and scope boundaries

The four usual outcomes $\mathrm N,\mathrm P,\mathrm L,\mathrm R$ are defined only when both starting-player choices are determined. Thus an undefined outcome need not mean failure of determinacy for both choices. In the default rule above, completed plays have winners; "neither has a winning strategy" is a determinacy failure, not a draw outcome. Stopping at time $\omega$ and declaring infinite play a draw, or changing the player who starts after a limit, defines a different game. The source also studies those alternative limit conventions separately.

Source relationship and status

Definitions 3.1 and 3.3 fix the source's $\Delta$-sum and ordinary transfinite play. Problem 3.4 asks determinacy both for arbitrary set-indexed sums and for countable sums. This formulation makes explicit the starting-player quantifier and the source's assumption of Choice; finite normal-play determinacy alone does not settle it.

References

[S1] Paolo Lipparini, Infinite sums of combinatorial games (Dadaist games) (2025), arXiv:2505.00614v1. Inspected locators: Section 3, Definitions 3.1 and 3.3, Problem 3.4; Sections 4--5 for other limit rules. https://arxiv.org/html/2505.00614v1

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-13-A2 · Exploration · LLM-generated · gpt 6 astra pro ·

GPT 6 Astra Pro attempt on Econ-13

Determinacy of Dadaist Delta-sums

C79-13: research attempt and adversarial verification

9 October 2026

Tools. Web browsing: YES. Exact rational computation: YES.

FORMAL RESTATEMENT

Work in ZFC. Every component $G_i$, indexed by a set $I$, is a fixed set-sized combinatorial game form. Each follower has a set of Left options and a set of Right options. The union of these option relations is well founded, meaning it has no infinite descending sequence; forms, not Conway-equivalence classes, are the data.

A move in $\mathbf G=\sum_{i\in I}^{\Delta}G_i$ changes exactly one coordinate to an option for the moving player. At a limit ordinal, each coordinate is replaced by its eventual value. A fixed player, called First, begins after every limit and at time zero; within a finite block of successive moves the players alternate. A player who is due to move and has no legal move loses. A strategy is a legal move prescription on every nonterminal history where its player is due to move. It is winning when every maximal compatible play is won by that player.

The target is \[ \forall I\ \forall(G_i)_{i\in I}\ \forall a\in\{\mathrm L,\mathrm R\}:\quad W_{\mathrm L}(\mathbf G,a)\lor W_{\mathrm R}(\mathbf G,a). \] The attachment's sentence beginning `Equivalently' asks instead for an undetermined sum. That existential statement is the logical negation of the displayed universal statement, not an equivalent assertion. The minimal correction is to ask for a proof of the universal statement or a counterexample to it.

The countable subproblem restricts $I$ to ${\mathbb{N}}_0=\{0,1,\ldots\}$. There is no draw convention at $\omega$. Empty index sets and families with only zero components are allowed and are immediate losses for First. Neither an a priori finite play-length bound nor finite branching is assumed.

Stress points. Termination at some ordinal does not by itself justify backward induction; strategies must work against all opponents' plays; a sum with infinitely many coordinates can have an infinite legal run even though no single coordinate can.

QUICK LITERATURE/CONTEXT CHECK

Paolo Lipparini, Infinite sums of combinatorial games (Dadaist games), arXiv:2505.00614v1, Definitions 3.1 and 3.3 and Problem 3.4, asks exactly the determinacy question. Example 3.6 already gives the infinite impartial case, and Fact 3.5 covers finite sums. https://arxiv.org/html/2505.00614v1.

The work below supplies full proofs of the foundational bounds and extends the impartial exhaustion argument to the explicitly defined dicot subclass. It does not infer determinacy of unrestricted partizan sums from the source's examples. The inspected version's open-problem designation is context, not a mathematical obstruction to trying further constructions.

ATTACK PLAN

Proof strategies. First test well-founded recursion on the whole sum; next try an exhaustion strategy using a well-order of active coordinates; finally ask whether an ordinal-valued rank survives limit stages.

Construction strategies. Seek a partizan example in which one player can leave resources inaccessible to the other; test unilateral ends; investigate whether terminal availability after a limit can encode an undetermined infinite-choice payoff. The exhaustion strategy succeeds on a broad subclass, while the general encoding/rank step remains missing.

Tools and their roles. Well-foundedness bounds moves per coordinate; Choice well-orders active coordinates and selects legal options; cardinal arithmetic bounds total play length; limit-stage stabilization validates positions; well-founded recursion proves finite-support determinacy; priority scheduling exhausts dicot resources; impartial finite-game recursion tests boundary cases; unilateral ends diagnose failure of the scheduling argument.

Tiny cases. $0$ is a loss for First; $*=\{0\mid0\}$ is a win for First; two copies of $*$ are a loss for First; three are a win for First. One-sided $1=\{0\mid\}$ is won by Left whichever player starts. Countably infinitely many copies of $*$ are won by Second under the limit rule, not drawn.

WORK

Lemma (Finite coordinate activity and a set-sized play bound): In every legal run, each coordinate changes only finitely many times. Limit positions therefore exist. If $I$ is infinite with cardinality $\kappa$, every run has length less than the successor cardinal $\kappa^+$. For finite $I$, every run has finite length.
Proof: If one coordinate changes infinitely often, select its first $\omega$ changes in their ordinal order. The associated positions form an infinite descending option sequence, contradicting well-foundedness. Each coordinate is consequently constant after its last change before any specified limit, which defines its eventual value.

For infinite $I$, associate each move with the pair consisting of its coordinate and the positive integer counting how many times that coordinate has changed so far. This association is injective. Thus the set of move times injects into $I\times{\mathbb{N}}$, a set of cardinality $\kappa$ in ZFC. A run with length at least the initial ordinal $\kappa^+$ would give an injection of $\kappa^+$ into a set of cardinality $\kappa$, impossible. For finite $I$, an infinite sequence of moves would change at least one of the finitely many coordinates infinitely often, again a contradiction. ∎

Remark: For countably many coordinates the lemma bounds each run by a countable ordinal, not by $\omega$ and not by a uniform finite integer. For example, with countably many copies of $*$ the players may jointly exhaust one infinite subfamily in the first $\omega$ moves while leaving another infinite subfamily untouched, so a legal run can continue beyond $\omega$. The second player's winning strategy below does not require the opponent to cooperate with such a schedule.
Lemma (Finite-support determinacy): If only finitely many components are nonterminal initially, the game is determined for each choice of First.
Proof: Terminal components can never become nonterminal and can be deleted. By Lemma Lemma (Finite coordinate activity and a set-sized play bound) the remaining product has no infinite move sequence. The set of positions and the player-to-move indicator therefore has a well-founded successor relation: otherwise dependent choice, available in ZFC, would select an infinite descending sequence.

Define a position to be winning for its player to move if it has a successor losing for the next player; define it to be losing otherwise. This definition is legitimate by well-founded recursion. At a winning position, choose one losing successor. At a losing position, every legal successor is winning for the opponent; a position with no legal successor is losing. These prescriptions prove by well-founded induction that exactly the appropriate player has a winning strategy. Choice selects successors simultaneously when necessary. No limit move is reached in these finite-support plays. ∎

Definition (Dicot form): A form is dicot if every follower, including the initial form, is either terminal for both players or has at least one option for each player. The two option sets need not be equal. Every impartial form is dicot, but the definition also permits partizan option sets.
Theorem (Infinite dicot sums are second-player wins): Suppose every component is dicot and infinitely many are nonterminal. Let the number of initially nonterminal components have infinite cardinality $\kappa$. For either choice of First, Second has a strategy under which play ends after exactly $\kappa$ moves and First loses at that limit stage. In particular the game has outcome $\mathrm P$.
Proof: Discard initially terminal coordinates and use Choice to index the others by the initial ordinal $\kappa$. Second always chooses the least-index nonterminal coordinate and any legal option there for Second. Dicotness makes that prescription legal whenever any coordinate is nonterminal; fix an option selection on the set of all eligible followers in advance.

Before any stage $\lambda<\kappa$, fewer than $\kappa$ coordinates have been touched, since at most one is touched per move and $\kappa$ is an initial ordinal. An untouched, initially nonterminal coordinate therefore remains. Dicotness implies both players have a legal move somewhere. Thus the run cannot end before stage $\kappa$, regardless of First's strategy.

There are $\kappa$ Second-move times below $\kappa$. One way to verify this cardinal assertion is to write each ordinal as a limit ordinal (or zero) plus a finite integer and inject the finite offsets into the even positive offsets in the same block. Such offsets remain below an infinite initial ordinal. For $\kappa=\omega$, these are simply the even-numbered moves.

Suppose coordinate $\xi<\kappa$ were still nonterminal at stage $\kappa$. Since terminal coordinates cannot change again, it was nonterminal at every earlier stage. Every Second move before $\kappa$ would then have been made in a coordinate of index at most $\xi$, by the least-index rule. Each such coordinate changes only finitely often by Lemma Lemma (Finite coordinate activity and a set-sized play bound). If $\kappa=\omega$, this is a finite set of coordinates with a finite total number of changes. If $\kappa>\omega$, its total number of changes is at most $\max(|\xi+1|,\aleph_0)<\kappa$. Both conclusions contradict the $\kappa$ Second moves. Therefore every coordinate is terminal at stage $\kappa$.

The ordinal $\kappa$ is a limit, and First is designated to move there by the rule in the problem. There is no legal move. First loses. The strategy and argument did not distinguish Left from Right, proving the claim for both starting choices. ∎

Corollary: Every set-indexed dicot $\Delta$-sum is determined: finite nonterminal support is covered by Lemma Lemma (Finite-support determinacy), and infinite nonterminal support by Theorem Theorem (Infinite dicot sums are second-player wins).

Why a general proof has not been obtained

The cardinal bound is not a well-founded rank of the successor relation. For example, the positions obtained by successively removing finitely many stars from an infinite star sum form an infinite chain of legal successors. A rank decreasing strictly at every move would forbid that chain and thus cannot be the naive finite-sum rank.

The exhaustion strategy is not available on general partizan forms. If all active components are $1=\{0\mid\}$ and Right is Second, Right has no legal move on the first response. More generally, First can move a component to a follower in which only First has options. Dicotness was used to exclude precisely this phenomenon. The one-sided example is determined and is not a counterexample to the original conjecture; it only breaks the proposed general scheduling proof.

VERIFICATION

Quantifiers and set theory. The strategy in Theorem Theorem (Infinite dicot sums are second-player wins) is a single prescription and works against every First strategy. It uses the cardinality of initial nonterminal support, not an arbitrary well-order type whose proper initial segments might already have that cardinality. This distinction matters for singular cardinals. The proof uses only that each proper initial segment of an initial ordinal has smaller cardinality; it does not assume regularity of $\kappa$. Limit stabilization and termination are proved separately. Both players cannot have winning strategies: running the two prescriptions against each other gives one terminating play and one winner.

Computational scope. No finite program tests all transfinite partizan games. The following exact finite check only verifies the stated star boundary cases and the finite recursion convention; it is not used to infer the infinite theorem. 

\begin{lstlisting}[language=Python] from functools import lru_cache @lru_cache(None) def win_stars(k):     # All k moves lead to the same (k-1)-star position.     if k==0: return False     return not win_stars(k-1) assert [win_stars(k) for k in range(7)] == [False,True,False,True,False,True,False] print('finite star checks: 0 through 6 verified') # Finite-game generalization: # Win(position,player) = any(not Win(successor,other_player) #                            for successor in legal_successors) # This recursion is used only when the successor relation is well founded. \end{lstlisting}

FINAL

LABEL: UNRESOLVED for arbitrary partizan $\Delta$-sums.

Strongest fully proved partial result. All set-indexed dicot sums are determined; if infinitely many components are nonterminal, Second wins by priority exhaustion at their initial cardinal. The impartial specialization is already present in the cited source. The proof also establishes the play-length bound and finite-support determinacy without a finite-branching hypothesis.

Exact first gap. Determinacy has not been proved for a family with infinitely many active coordinates and followers that offer moves to only one player. No reduction of these one-sided residual resources to a determined game, and no undetermined example in this class, has been established here.

Three next targets. First, classify sums of well-founded one-sided ends together with a dicot remainder under the reset-at-limits convention. Second, seek a strategy-invariant comparison of the two players' accessible residual coordinates that survives a limit. Third, test an explicit encoding of an undetermined length-$\omega$ win set using only independent well-founded component forms; the missing requirement is to realize its payoff through legal post-limit move availability.

Minimal-counterexample information. Any counterexample must have infinitely many initially nonterminal coordinates and at least one reachable one-sided follower; it cannot belong to the dicot class. This argument gives no lower bound above countable cardinality and does not assert the existence of a counterexample.

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Econ-13-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-13

Determinacy of Dadaist Delta games GPT 6 Astra Ultra

Target and approach

For a set-indexed family of well-founded combinatorial game forms, a $\Delta$ move changes one coordinate, limits take coordinatewise eventual values, and the designated First player moves first again after each limit. The open question asks determinacy under these rules. We retain the forms and fixed coordinate identities, not only their Conway values. The definition and open question were checked in Lipparini's Definitions 3.1 and 3.3 and Problem 3.4 [lipparini].

This attempt establishes a general size bound on individual runs, gives a complete strategy proof for paired conjugate forms, and proves that even one fixed countable example has unbounded countable run lengths. The last result identifies precisely why a bound on each individual run does not supply the missing backward induction. None of these is a determinacy theorem for arbitrary unpaired sums; the pairing argument is classical in spirit and no priority is claimed.

Set-sized termination, without a countdown rank

For each coordinate $i$, its option relation is well-founded: there is no infinite sequence of successive options starting from $G_i$. A run is indexed by an ordinal, with legal successor moves and the stipulated limits.
Lemma (Finite coordinate activity): In every run, each fixed coordinate is moved only finitely many times. At every limit stage each coordinate is consequently eventually constant.
Proof: If coordinate $i$ moved infinitely many times, its move times, being a subset of an ordinal, would have a first time, a second time, and so on. Taking the resulting forms at those first countably many moves would produce an infinite successive-option sequence in $G_i$. This contradicts well-foundedness. At a limit stage there are therefore only finitely many changes in that coordinate before the limit. A finite set of times below a limit ordinal is bounded below it, so after its last change the coordinate is constant. ∎
Proposition (A cardinal upper bound): Work in ZFC and put $\kappa=\max\{|I|,\aleph_0\}$. Every legal run has length less than $\kappa^+$, where $\kappa^+$ is the successor initial ordinal of $\kappa$. Every legal alternating history has a maximal legal extension. For countable $I$, each completed play has countable length, and histories and strategies can be treated as sets.
Proof: Label each move by the coordinate $i$ it changes and the positive integer counting how many times that coordinate has previously been changed, including that move. Distinct moves have distinct labels. Thus the set of moves injects into $I\times{\mathbb{N}}$, whose cardinality is at most $\kappa$. A run of length at least $\kappa^+$ would have more than $\kappa$ moves, an impossibility.

Each $G_i$ has a set of reachable forms, because taking its transitive closure along its set-valued options is a set construction. Their product is a set of positions. Well-order the set of legal moves from all positions, using Choice. Starting after a given history, at every successor choose the first legal move for the designated player, if any; at limits use the first lemma. If this recursion never stopped below $\kappa^+$ it would construct an impossible run of that length. It therefore stops at a maximal alternating history. All histories have ordinal domains below $\kappa^+$ and values in a fixed set; their union over these domains is a set. Strategies, being appropriate functions on subsets of that set, are also sets. If $I$ is countable, $\kappa^+=\omega_1$, proving the claimed countability. ∎

Proposition (No countable bound even for one fixed sum): Let every coordinate in a countably infinite index set be the one-move impartial game $*=\{0\mid0\}$. For every countably infinite ordinal $\alpha$ there is a maximal alternating play of length exactly $\alpha$ on this same initial $\Delta$-sum. In particular no countable ordinal bounds all its play lengths.
Proof: Choose a bijection $f:\alpha\to{\mathbb{N}}$. At move time $\beta<\alpha$, remove the star in coordinate $f(\beta)$. It is available because it was not used earlier, and either player may remove it. At limit stages, each removed coordinate stays zero and every other one stays a star, as required. For each $\beta<\alpha$ some stars remain, because the bijection has not yet used $f(\beta)$. Thus play cannot stop earlier. At time $\alpha$ every coordinate is zero, and the player due to move loses. This is a maximal play with the prescribed length. ∎

A rigorously determined infinite class

The conjugate $-H$ is obtained from a game form $H$ by interchanging Left and Right options recursively. The following hypothesis is about an exact form isomorphism, including the options, rather than Conway equivalence.
Theorem (Paired conjugate sums are second-player wins): Suppose the nonzero coordinates of a $\Delta$-sum can be partitioned into disjoint pairs $\{i,j\}$ such that the two initial forms in each pair are conjugate. Coordinates equal to the terminal form $0$ may be left unpaired; here zero refers to that form, not merely Conway equivalence to zero. Then for either designation of First, Second has a winning strategy under the stated transfinite limit rule.
Proof: Fix the pairing and the conjugating form isomorphisms. Maintain the invariant that immediately before every First move the two forms in each pair are conjugate. If First changes $H$ in one coordinate by a move of First's color, the conjugating correspondence supplies a legal move of Second's color in the other coordinate. Second plays that move. The two resulting forms are again conjugate, and all other pairs were unchanged.

It remains to verify the invariant at limits, which is the part not supplied by the usual finite argument. Fix a limit stage $\lambda$ and one pair. By finite coordinate activity, only finitely many moves before $\lambda$ have affected either coordinate in that pair. Every First move before $\lambda$ has its immediately following response also before $\lambda$, since $\lambda$ is a limit ordinal. Thus, after the last such completed response affecting the pair, its forms are conjugate and remain unchanged up to $\lambda$. Their limit forms are therefore conjugate. This holds for every pair, so the invariant is restored at the limit, when First is again due to move.

Second can never be the first player without a move: after each First move the conjugate response exists. A maximal play compatible with this strategy exists: perform the same transfinite extension recursion as in the cardinal-bound proposition, using the prescribed response on Second turns and any well-ordered legal choice on First turns. The cardinal bound forces this compatible recursion to terminate. At termination First, rather than Second, must be unable to move. Hence Second wins every maximal compatible play, as required by the definition of a winning strategy. ∎

Corollary (Absorption of one extra star in this example): A countably infinite sum of stars is a second-player win. Adding one more star gives another second-player win, although one star by itself is a first-player win.
Proof: The star is its own conjugate, and any countably infinite set can be partitioned into pairs. Adding one coordinate leaves the index set countably infinite, so the theorem applies in both cases. A single star has one legal move to zero, which wins for First. ∎

What fails in a tempting determinacy proof

The cardinal bound says that each countable-sum play ends before $\omega_1$; it does not provide a countable ordinal decreasing at every move. The fixed all-star example has arbitrarily long finite prefixes and has plays of every countably infinite length. In particular its ordinary finite-history extension tree has an infinite branch, so it is not a well-founded tree in the sense needed for ordinary backward recursion.

There is no contradiction with the paired-sum theorem: that theorem verifies a strategy at every limit directly, without a countdown rank. Nor does exhaustion of each actual play prove that one of the two players has a strategy controlling all possible plays. General unpaired sums, including the countable case, remain unresolved here. No determinacy axiom stronger than ZFC is invoked or claimed necessary.

References

  • [lipparini] P. Lipparini, Infinite sums of combinatorial games (Dadaist games), arXiv:2505.00614v1 (2025), Section 3, especially Definitions 3.1 and 3.3, Problem 3.4, and Remark 3.2. https://arxiv.org/html/2505.00614v1. The primary source and the convention that First restarts at limits were checked on 9 October 2026. The arguments above use only these rules and the explicitly stated ZFC set constructions.

Completion Estimate

15%

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