IFlowNets: Extending Generative Samplers to Learn Strategies in Incomplete Information Games

Conor M. Artman; Nicholas Di; Scott Perkins

REVIEW 4 major objections 6 minor 13 references

IFlowNets: Extending Generative Samplers to Learn Strategies in Incomplete Information Games

T0 review · 4 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper proves that generative flow networks can learn valid strategies in incomplete-information games — such as poker — by repairing the flow constraints: summing over hidden histories in an information set instead of averaging over…

desk verdict The paper has a clean negative result about expectation-based aggregation, but its proposed fix drops the belief weighting over hidden histories, so the central claim of expected reward-proportional sampling does not hold. read the letter →

arxiv 2608.05422 v1 pith:Y322RV5O submitted 2026-08-05 cs.LG cs.MA

Conor M. Artman , Nicholas Di , Scott Perkins This is my paper · ORCID

classification cs.LGcs.MA

keywords generativeflownetworksincompleteinformationgamesexpecteddetailedbalanceinfostateaggregationmatchingtrajectoryquantalresponseequilibriumcounterfactualregretminimization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Generative flow networks learn to sample objects with probability proportional to a reward, but the constraints that make them work break down when a player cannot see the full history. This paper shows that in incomplete-information games, the expectation-based flow constraints of AFlowNets cannot produce a valid density: satisfying flow matching over an information set would force the environment distribution to put probability 1 on every hidden history at once. The fix is to replace the expectation over hidden histories with a sum, and to add an infostate aggregation condition $F(I)=\sum_{h\in I}F(h)$ that treats flow at an infoset as the total flow of its constituent histories. With these corrected constraints, the trajectory balance objective is valid and each player's strategy samples actions proportional to expected reward, recovering the complete-information AFlowNet guarantee as a special case. The authors report preliminary results on RPS+, Kuhn poker, and Leduc poker where IFlowNets match or beat CFR and deep-RL baselines in exploitability and speed.

What carries the argument

The engine of the argument is the infostate aggregation property combined with the corrected agent-transition constraint. In an extensive-form game of incomplete information, an information set $I$ bundles histories $h$ that a player cannot distinguish; the property $F(I)=\sum_{h\in I}F(h)$ declares that the flow through the infoset is the total flow of its hidden histories, not an average. Coupled with $F(I)P_{\text{agent}}(a|I)=\sum_{h'\ni a}F(h')$, this makes valid the expected flow matching identity $F(I)=\sum_{I'\in\operatorname{Ch}(I)}F(I')$ over infosets, which is the exact condition under which the trajectory balance objective trains a sampler to produce reward-proportional terminal distributions.

What would settle it

Construct a small perfect-recall game with an infoset containing two histories that lead to different rewards, train an IFlowNet with the aggregation property replaced by averaging over hidden histories, and check whether the learned policy still samples terminal nodes proportional to expected reward; the paper's theory predicts it will not, while a version with the sum will.

Watch

Extended reading notes

Core claim

At the core of the paper is a negative result and a repair. The negative result (Theorem 4) proves that directly importing the expected-detail-balance (EDB) constraints from AFlowNets into incomplete-information games — by taking expectations over the uncertain history — is impossible: the resulting flow-matching condition over an infoset would require $P_{\text{env}}(h)=1$ for every $h$ in the infoset, which no probability distribution can satisfy. The repair is the generalized expected detailed balance (GEDB) constraints, which add the infostate aggregation property $F(I)=\sum_{h\in I}F(h)$ and replace the expected agent transition with a sum over hidden histories, $F(I)P_{\text{agent}}(a|I)=\sum_{h'\ni a}F(h')$. The paper proves these constraints induce flow matching over infosets (Lemma 5), which legitimizes the trajectory balance objective for training generative samplers in these games, and that in complete-information settings the GEDB constraints reduce exactly to the EDB constraints, so IFlowNets strictly generalize AFlowNets.

Load-bearing premise

The load-bearing premise is that the flow at an information set must equal the sum of flows of the hidden histories inside it ($F(I)=\sum_{h\in I}F(h)$); the paper introduces this as a self-consistency condition rather than deriving it from payoffs, and the validity of the training objective depends on it and on perfect recall.

Editorial extensions

If this is right

The trajectory balance objective becomes a valid loss for training generative samplers in any perfect-recall extensive-form game of incomplete information, without needing recursive search or explicit regret matching.
IFlowNets strictly extend AFlowNets: in complete-information settings the GEDB constraints collapse to the EDB constraints, so every AFlowNet solution is recovered.
Each player's learned policy samples actions proportional to the expected branch-adjusted reward, which places the resulting joint play in the family of agent quantal response equilibria for extensive-form games.
Preliminary experiments indicate that IFlowNets reach exploitability comparable to OS-MCCFR on Kuhn poker and better than OS-MCCFR, DeepCFR, and NFSP on Leduc poker, while running faster than the deep baselines.
The GEDB fixed point is unique for each player given the others' policies, a property the naive expectation-based constraints lack.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct test of the load-bearing assumption would be to drop the aggregation property $F(I)=\sum_{h\in I}F(h)$ from the training objective and measure whether the resulting policy stops matching expected-reward-proportional play; the theory predicts a clear failure.
Because the core issue is expectation-versus-sum at hidden states, the same repair may extend to partially observable and stochastic environments beyond games, where generative samplers currently assume state-level flow matching.
The paper sketches but does not test the QRE interpretation; a natural follow-up is to compare IFlowNets' equilibria to classical agent-QRE estimates to see whether the generative sampler recovers known equilibrium benchmarks.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Thanks for the report. I want to flag one thing before anything else: the stress-test concern is correct and it's load-bearing. The paper's GEDB constraints (C3-C4) replace belief-weighted expectations with unweighted sums over hidden histories. In a simple example where Nature reaches h1 with 0.9 and h2 with 0.1 and the agent only sees the infoset, the induced policy at the infoset is 1/2 for both actions, independent of the Nature probabilities. The expected-reward-proportional policy would put roughly 0.9 on the action that wins in h1. So Lemma 5 does prove algebraic flow matching over the infoset graph, but that is not the game-theoretic object the paper needs. The terminal reward-proportional claim in Appendix D is unsupported.

The paper does something genuinely useful before that: Theorem 4 (in Appendix E.1) cleanly shows that a direct expectation-based extension of AFlowNets is inconsistent, because it forces the smoothing distribution to be a point mass. That negative result is worth publishing on its own. The GEDB constraints also recover the EDB constraints when infosets are singletons, which is a nice sanity check. The writing is clear for a workshop paper and the authors are explicit about what is preliminary.

The soft spots beyond the core flaw: The existence and uniqueness proofs (Theorems 7-10) are sketches; the recurrence in Theorem 7 does not fully specify flows on individual hidden histories, and C4 is introduced as an ansatz rather than derived. The empirical section is thin: single runs, no code or seeds, and hyperparameters were selected on the exploitability metric itself. In Kuhn Poker the reported exploitability (0.087) is worse than OS-MCCFR (0.068) and slower in iterations/sec, so the abstract's "comparable or better" overstates the results. Leduc looks better, but the selection bias makes it hard to trust.

Who should read this: anyone working on extending GFlowNets to imperfect-information games. The negative result and the belief-weighting problem are the useful parts; the proposed solution as written is not correct. I'd send it to peer review—the failure mode is subtle and worth a serious referee, and a major revision that fixes the aggregation (for example by weighting with the reach probabilities) could produce a valuable paper. But as it stands, the central claim is not established.

Referee Report

4 major / 6 minor

Summary. The paper proposes Information Flow Networks (IFlowNets), a generative-flow-network approach for extensive-form games of incomplete information. The authors introduce an "infostate aggregation property" (C4) and a sum-based action-flow constraint (C3), forming their Generalized Expected Detailed Balance (GEDB) conditions. They prove that the expectation-based DEDB constraints cannot satisfy flow matching over infostates, that the GEDB constraints do satisfy such flow matching, and that the GEDB constraints recover the EDB constraints in complete-information settings. They then present preliminary experiments on RPS+, Kuhn poker, and Leduc poker comparing against OS-MCCFR, NFSP, and DeepCFR.

Significance. The problem of extending GFlowNets to imperfect-information games is timely, and the algebraic observation in Lemma 5 is clean: the GEDB constraints do imply an additive flow-matching identity over infostates. If the constraints also induced expected-reward-proportional sampling in the game-theoretic sense, the paper would provide a useful new method and a practical route to approximate quantal response equilibria. The paper is also commendable for using standard OpenSpiel benchmarks and for making its training objective explicit. However, the load-bearing game-theoretic interpretation is not established: the additive aggregation in C3/C4 erases the player's belief over hidden histories, so the resulting policies are not expected-reward-proportional in general. This undermines the abstract's central claim that IFlowNets produce valid densities corresponding to player strategies.

major comments (4)

[§3.2, Eqs. (C3)-(C4) and Lemma 5] The GEDB aggregation is unweighted over hidden histories. Consider a single-agent incomplete-information DAG in which Nature reaches h1 with probability 0.9 and h2 with probability 0.1, the agent observes only the infoset I={h1,h2}, and the two actions Up/Down yield terminal rewards exp(u(h1,Up))=e, exp(u(h2,Up))=1, exp(u(h1,Down))=1, exp(u(h2,Down))=e. Equation (C3) gives F(I)P(Up)=e+1=F(I)P(Down), hence P(Up)=1/2, independent of Nature's probabilities. A belief-weighted expected-reward-proportional policy would put substantially more mass on Up. Thus the constraint system does not implement expected reward proportional sampling in the game-theoretic sense, and the claims in the abstract and in Appendix D that IFlowNets produce valid player strategies or QREs are unsupported.
[Appendix E.1, Theorem 4] The proof of Theorem 4 assumes that the target of the DEDB constraints is unweighted flow matching over infosets, i.e., Σ_a Σ_h F(h,a). That is precisely the contested aggregation rule C4. If the correct game-theoretic object is belief-weighted flow matching, Σ_a Σ_h P(h|I)F(h,a), then the contradiction P_env(h)=1 disappears and the DEDB constraints may be valid. Consequently, the paper's claim that "previously established constraints" are inadmissible for incomplete-information games is not established; the argument is circular with respect to the very definition at issue.
[Appendix E.4, Theorem 7, Eq. (20)] The recurrence in Eq. (20) defines F(I) for agent infostates, environment infostates, and terminal states, but it does not define the history-level flows F(h) that appear in constraint C4. In the Nature example above, C4 only fixes the sum F(h1)+F(h2), and C2 constrains only a weighted combination; infinitely many assignments of F(h1) and F(h2) satisfy the stated constraints. Hence the claimed existence and uniqueness of a (F,P_agent) pair is not demonstrated, and Theorem 9 inherits this gap. The proof sketch in Theorem 7 that the recurrence induces a unique F is insufficient.
[Tables 1-2 and Appendix F] The empirical results are reported from single runs with no error bars, no seed variance, and no standardized definition of an iteration across methods. The abstract's claim that IFlowNets "perform comparably to or better than" OS-MCCFR, NFSP, and DeepCFR is therefore not yet supported by the evidence. This is less severe than the theoretical issues above, but it should be addressed with multiple seeds and a controlled compute or iteration budget in any revision.

minor comments (6)

[Title page] The affiliation contains the typo "Deptartment".
[Appendix F] The text says "32 hiddent units" and Figure 4's caption says "utilites"; both should be corrected.
[§3.2 and Appendix E] The notation "∀a, h', I∈ I_agent" together with "h'∋a" is ambiguous; the authors should define h'=(h,a) and state explicitly over which h∈I the sum ranges.
[Appendix F] The displayed trajectory balance objective is missing a closing parenthesis: the expression opens with log( but never closes before the final "!".
[Appendix D] The claim that Lemma 5 implies P_i(I_i) ∝ E_{P_-i}F_i(I_i) needs a derivation; Lemma 5 as stated is purely algebraic and does not by itself justify the expectation over P_-i.
[References] The reference "Madan et al." lacks a year and venue in the bibliography.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is an algebraic consequence of explicitly introduced self-consistency constraints, and the empirical comparisons use external baselines.

full rationale

The paper's theoretical chain is self-contained in the sense that its central constraints are introduced explicitly rather than being derived from the conclusion they are later said to imply. C4 is openly labeled a 'self-consistency condition' (Section 3.1), and Lemma 5 derives infoset flow matching by summing C3 over actions and applying C4; this is a logical implication of proposed axioms, not a hidden restatement of a fitted prediction. No parameters are fitted inside the derivation itself, and the empirical section compares against external baselines (OS-MCCFR, NFSP, DeepCFR) and external exploitability utilities from OpenSpiel, so the reported results are not circularly generated by the model's own training objective. The Appendix D discussion of quantal response equilibria is explicitly deferred ('we do not empirically or theoretically study AQRE properties of IFlowNets in this work'), so it is not load-bearing. The skeptical concern that C3/C4 aggregate hidden histories without belief weights P(h|I) is a substantive correctness risk for the 'expected reward proportional sampling' claim, but it is not a circularity: that claim is not defined as C4, nor is it imported from the present authors' prior work. The only external results invoked, such as Jiralerspong et al.'s Proposition 5, come from a different authorship, and the paper does not use self-citations as its load-bearing justification. Overall, no step reduces by construction to its own inputs.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The theoretical derivation adds no new physical entities. It relies on standard game-theoretic assumptions (perfect recall, DAG history structure) and one new modeling axiom, the infostate aggregation property C4. The only fitted quantity is the learned normalizing constant Z_theta.

free parameters (1)

Normalizing constant Z_theta = learned during training
A learned scalar in the trajectory balance objective (Appendix F) that scales the flow to match the empirical flow marginal. It is standard GFlowNet practice but is a number fitted to data.

assumptions (3)

domain assumption Perfect recall: each player remembers their own past actions
Assumed in Section 2 to make infostates and child infostates well-defined; Lemma 5 and the GEDB constraints C3 and C4 rely on the fact that taking an action a from any history in an infostate leads to histories in a single child infostate.
domain assumption Positive terminal rewards via an exp transform
Rewards are set to exp(utility) (Appendix F) because GFlowNets require positive rewards; this transformation is chosen by the authors, not forced by the game.
ad hoc to paper Infostate aggregation property F(I) = sum_{h in I} F(h)
Introduced as Generalization #1 (Eq. C4) as a self-consistency condition. It is not derived and is load-bearing for Lemma 5 and the flow-matching guarantee.

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Cite this review

Pith. "Pith review of IFlowNets: Extending Generative Samplers to Learn Strategies in Incomplete Information Games." pith.science (2026). https://pith.science/paper/Y322RV5O

@misc{pith2026260805422,
  author       = {Pith},
  title        = {Pith review of: IFlowNets: Extending Generative Samplers to Learn Strategies in Incomplete Information Games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y322RV5O}},
  note         = {Machine review of arXiv:2608.05422}
}

read the original abstract

While many algorithms blend reinforcement learning (RL) with counterfactual regret (CFR) methods to leverage tradeoffs in computational speed and performance, there are fewer investigations into generative sampling frameworks in game theoretic applications in incomplete information games. We extend a generative flow network framework, Adversarial Flow Networks (AFlowNets), to incomplete information games, called Information Flow Networks (IFNs). We prove that previously established constraints for generative flow networks in complete information games are inadmissible for obtaining valid densities (corresponding to player strategies) and a valid training objective. We show that our proposed generalization, IFlowNets, alleviates this issue and strictly generalizes AFlowNets. In preliminary results for three standard game environments, IFlowNets perform comparably to or better than Outcome Sampling Monte Carlo Counterfactual Regret (OSMCCFR) and standard RL-based methods in performance and speed.

Figures

Figures reproduced from arXiv: 2608.05422 by the authors.

**Figure 2.** Left: Illustration of why infostate aggregation property necessary. Right: Illustration of higher intra-graph uncertainty in incomplete information games. (See below for details.) Left Scenario of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** (a) KLD of IFN policy vs. NE strategy (b) Exploitability in RPS+ [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: IFN exploitability and KLD from CFR solution. (Exploitability and CFR computed with OpenSpiel utilites in Kuhn OpenSpiel environment.) 18 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 9 canonical work pages

[1]

Then the two distributions characterizingP F , the forward policy, areP agent, the agent’s policy, andP env, the environment dynamics

ProofIn analogy to the EFlowNets case of AFlowNets, consider a single agent navigating an incomplete information game treeGby alternating moves with a Nature player, encoded byP env. Then the two distributions characterizingP F , the forward policy, areP agent, the agent’s policy, andP env, the environment dynamics. In parallel to Jiralerspong et al. (202...

work page 2024
[2]

9 Artman Di Perkins Appendix C

Therefore,P 1 must take this into account and pool the total flow mass associated withh 1 andh 2 in a way that still induces expected reward proportional sampling. 9 Artman Di Perkins Appendix C. Review of Expected & Adversarial Flow Networks Adversarial Flow Networks (AFlowNets) are collections of alternating EFlowNets, so we start with EFlowNets. EFlowN...

work page 2024
[4]

doi: 10.1038/s41598-025-86899-8

ISSN 2045-2322. doi: 10.1038/s41598-025-86899-8. URLhttps://www.nature.com/ articles/s41598-025-86899-8. Publisher: Nature Publishing Group. Harold W. Kuhn and Albert William Tucker.Contributions to the Theory of Games, Volume I. Princeton University Press, March

work page doi:10.1038/s41598-025-86899-8
[5]

(2024) as long as it satisfies the key requirement of flow matching, and by Lemma 5, the GEDB induce flow matching on infosets

The proof proceeds similarly to Proposition1in Jiralerspong et al. (2024) as long as it satisfies the key requirement of flow matching, and by Lemma 5, the GEDB induce flow matching on infosets. Consequently, as in Proposition5, we may once again consider the longest trajectory over the DAG, and it satisfies a recurrence induced by the GEDB. Remark 8Note ...

work page 2024
[6]

Population-based Evaluation in Repeated Rock-Paper-Scissors as a Benchmark for Multiagent Reinforcement Learning

URL http://arxiv.org/abs/2303.03196. arXiv:2303.03196 [cs]. Kanika Madan, Jarrid Rector-Brooks, Maksym Korablyov, Emmanuel Bengio, Moksh Jain, Andrei Nica, Tom Bosc, Yoshua Bengio, and Nikolay Malkin. Learning GFlowNets From Partial Episodes For Improved Convergence And Stability. Richard D. McKelvey and Thomas R. Palfrey. Quantal Response Equilib- ria fo...

work page Pith review arXiv
[7]

Similar to Theorem 7, we start with the recurrence on the flow functionF i induced by the GEDB

Here, we will refer toP −i asP j,j̸=i. Similar to Theorem 7, we start with the recurrence on the flow functionF i induced by the GEDB. Fi(I) =    P I ′∈Ch(I) Fi(I ′), I∈ I iP I′∈Ch(I) Fi(I ′)Fj (I ′)P I′∈Ch(I) Fj (I ′) I∈ Ij, j̸=i P h∈I Fi(h), I∈ I i Ri(I)I∈ Z (24) By Theorem 7, we know that this recurrence induces an unique (Fi, Pi) pair, bu...

work page 2024
[8]

arXiv:2206.05825 [cs]

URLhttp://arxiv.org/abs/2206.05825. arXiv:2206.05825 [cs]. 8 Generative Sampling in Incomplete Information Games Appendix A. Extended Note on Necessity of Generalization #1: Information Set Aggregation In incomplete information games, information set aggregation (Eqn. D4) is not a trivial condition to omit. This reasoning behind this is the following.Allg...

arXiv 2023
[13]

(2024), by constructionF 1(z) =R 1(z) andF 2(z) =R 2(z) for allz∈Z, soF(z) def.=F 1(z)F2(z) C1=R 1(z)R2(z) for allz∈Z

15 Artman Di Perkins Theorem 10(Inn= 2player zero-sum IFNs, if agent policies(P 1, P2)and flows(F 1, F2) jointly satisfy existence and uniqueness with respect to the GEDB, thenF(I) ∆=F 1(I)F 2(I) satisfies flow matching with respect to the joint rewardR(x) ∆=R 1(x)R2(x).) Proof Following Jiralerspong et al. (2024), by constructionF 1(z) =R 1(z) andF 2(z) ...

work page 2024

Show all 13 references

[1998]

doi: 10.1023/A:1009905800005

ISSN 1386-4157, 1573-6938. doi: 10.1023/A:1009905800005. URL https://www.cambridge.org/core/journals/experimental-economics/ article/abs/quantal-response-equilibria-for-extensive-form-games/ 7 Artman Di Perkins CECA1F0028DCE25FA77566303C5EA1EA. Publisher: Cambridge University ...

doi
[2020]

arXiv:1908.09453 [cs]

URLhttp://arxiv.org/abs/1908.09453. arXiv:1908.09453 [cs]. Marc Lanctot, John Schultz, Neil Burch, Max Olan Smith, Daniel Hennes, Thomas Anthony, and Julien Perolat. Population-based Evaluation in Repeated Rock-Paper- Scissors as a Benchmark for Multiagent Reinforcement Learni...

1908 arXiv
[2023]

arXiv:2307.01422 [cs]

URLhttp://arxiv.org/abs/2307.01422. arXiv:2307.01422 [cs]. Marco Jiralerspong, Bilun Sun, Danilo Vucetic, Tianyu Zhang, Yoshua Bengio, Gauthier Gidel, and Nikolay Malkin. Expected flow networks in stochastic environments and two-player zero-sum games, March

arXiv
[2024]

arXiv:2310.02779 [cs]

URLhttp://arxiv.org/abs/2310.02779. arXiv:2310.02779 [cs]. Behbod Keshavarzi and Hamidreza Navidi. Comparative analysis of extensive form zero sum game algorithms for Poker like games.Scientific Reports, 15(1):2917, January

arXiv
[2025]

doi: 10.1016/j.geb.2025.02.008

ISSN 0899-8256. doi: 10.1016/j.geb.2025.02.008. URLhttps://www. sciencedirect.com/science/article/pii/S0899825625000211. Tristan Deleu and Yoshua Bengio. Generative Flow Networks: a Markov Chain Perspective, July

2025 doi

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Then the two distributions characterizingP F , the forward policy, areP agent, the agent’s policy, andP env, the environment dynamics

ProofIn analogy to the EFlowNets case of AFlowNets, consider a single agent navigating an incomplete information game treeGby alternating moves with a Nature player, encoded byP env. Then the two distributions characterizingP F , the forward policy, areP agent, the agent’s policy, andP env, the environment dynamics. In parallel to Jiralerspong et al. (202...

work page 2024

[2] [2]

9 Artman Di Perkins Appendix C

Therefore,P 1 must take this into account and pool the total flow mass associated withh 1 andh 2 in a way that still induces expected reward proportional sampling. 9 Artman Di Perkins Appendix C. Review of Expected & Adversarial Flow Networks Adversarial Flow Networks (AFlowNets) are collections of alternating EFlowNets, so we start with EFlowNets. EFlowN...

work page 2024

[3] [4]

doi: 10.1038/s41598-025-86899-8

ISSN 2045-2322. doi: 10.1038/s41598-025-86899-8. URLhttps://www.nature.com/ articles/s41598-025-86899-8. Publisher: Nature Publishing Group. Harold W. Kuhn and Albert William Tucker.Contributions to the Theory of Games, Volume I. Princeton University Press, March

work page doi:10.1038/s41598-025-86899-8

[4] [5]

(2024) as long as it satisfies the key requirement of flow matching, and by Lemma 5, the GEDB induce flow matching on infosets

The proof proceeds similarly to Proposition1in Jiralerspong et al. (2024) as long as it satisfies the key requirement of flow matching, and by Lemma 5, the GEDB induce flow matching on infosets. Consequently, as in Proposition5, we may once again consider the longest trajectory over the DAG, and it satisfies a recurrence induced by the GEDB. Remark 8Note ...

work page 2024

[5] [6]

Population-based Evaluation in Repeated Rock-Paper-Scissors as a Benchmark for Multiagent Reinforcement Learning

URL http://arxiv.org/abs/2303.03196. arXiv:2303.03196 [cs]. Kanika Madan, Jarrid Rector-Brooks, Maksym Korablyov, Emmanuel Bengio, Moksh Jain, Andrei Nica, Tom Bosc, Yoshua Bengio, and Nikolay Malkin. Learning GFlowNets From Partial Episodes For Improved Convergence And Stability. Richard D. McKelvey and Thomas R. Palfrey. Quantal Response Equilib- ria fo...

work page Pith review arXiv

[6] [7]

Similar to Theorem 7, we start with the recurrence on the flow functionF i induced by the GEDB

Here, we will refer toP −i asP j,j̸=i. Similar to Theorem 7, we start with the recurrence on the flow functionF i induced by the GEDB. Fi(I) =    P I ′∈Ch(I) Fi(I ′), I∈ I iP I′∈Ch(I) Fi(I ′)Fj (I ′)P I′∈Ch(I) Fj (I ′) I∈ Ij, j̸=i P h∈I Fi(h), I∈ I i Ri(I)I∈ Z (24) By Theorem 7, we know that this recurrence induces an unique (Fi, Pi) pair, bu...

work page 2024

[7] [8]

arXiv:2206.05825 [cs]

URLhttp://arxiv.org/abs/2206.05825. arXiv:2206.05825 [cs]. 8 Generative Sampling in Incomplete Information Games Appendix A. Extended Note on Necessity of Generalization #1: Information Set Aggregation In incomplete information games, information set aggregation (Eqn. D4) is not a trivial condition to omit. This reasoning behind this is the following.Allg...

arXiv 2023

[8] [13]

(2024), by constructionF 1(z) =R 1(z) andF 2(z) =R 2(z) for allz∈Z, soF(z) def.=F 1(z)F2(z) C1=R 1(z)R2(z) for allz∈Z

15 Artman Di Perkins Theorem 10(Inn= 2player zero-sum IFNs, if agent policies(P 1, P2)and flows(F 1, F2) jointly satisfy existence and uniqueness with respect to the GEDB, thenF(I) ∆=F 1(I)F 2(I) satisfies flow matching with respect to the joint rewardR(x) ∆=R 1(x)R2(x).) Proof Following Jiralerspong et al. (2024), by constructionF 1(z) =R 1(z) andF 2(z) ...

work page 2024