pith. sign in

arxiv: 2509.09181 · v1 · submitted 2025-09-11 · ⚛️ physics.soc-ph · q-bio.PE

Incomplete Reputation Information and Punishment in Indirect Reciprocity

Heejeong Kim , Yohsuke Murase This is my paper

Pith reviewed 2026-05-18 18:15 UTC · model grok-4.3

classification ⚛️ physics.soc-ph q-bio.PE
keywords indirect reciprocityincomplete informationreputation fadingpunishmentcooperationpublic assessmentevolutionary dynamics
0
0 comments X

The pith

Incomplete observation leaves cooperation conditions unchanged in indirect reciprocity while reputation fading raises the required benefit-to-cost ratio and punishment norms expand the viable range.

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

The paper compares two distinct forms of incomplete reputation information in models of indirect reciprocity. Under incomplete observation, where donors' actions are seen only probabilistically, the thresholds for cooperation stay identical to the complete-information case because rarer updates are offset by larger reputational consequences per observed action. Reputation fading, where recipients are sometimes labeled unknown, instead tightens the conditions and demands higher benefit-to-cost ratios as identification probability falls. Introducing costly punishment alongside cooperation and defection allows certain norms to support cooperation over wider parameter ranges in the fading model without lowering efficiency, a result that differs from earlier findings on other information constraints.

Core claim

Under incomplete observation the conditions for cooperation are unchanged because less frequent updates are exactly offset by higher reputational stakes. Reputation fading requires higher benefit-to-cost ratios as identification probability decreases. Norms incorporating punishment sustain cooperation across broader parameter ranges without reducing efficiency in the reputation fading model.

What carries the argument

Analytical frameworks for public assessment that compute long-run frequencies of strategies and reputations under probabilistic observation or identification.

If this is right

  • Cooperation evolves under incomplete observation at the same benefit-to-cost ratios as under full information.
  • Lower identification probabilities in reputation fading demand proportionally higher benefit-to-cost ratios to maintain cooperation.
  • Punishment-inclusive norms support stable cooperation in reputation-fading models without efficiency loss.
  • Punishment effectiveness hinges on the concrete type of information incompleteness rather than incompleteness in general.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Real-world reputation systems may need distinct design rules for observation noise versus label uncertainty rather than generic fixes.
  • Experimental tests with human subjects under controlled probabilistic observation or fading labels could directly check the predicted thresholds.
  • Extending the frameworks to include private assessments could uncover additional instabilities not visible in public-assessment limits.

Load-bearing premise

The analytical frameworks for public assessment capture the long-run population dynamics exactly with no hidden dependence on specific initial conditions, finite-population corrections, or private-assessment effects.

What would settle it

A finite-population simulation or private-assessment analysis that shifts the reported cooperation thresholds away from the values predicted by the public-assessment models would falsify the claim that conditions remain unchanged or require only higher benefit-to-cost ratios.

Figures

Figures reproduced from arXiv: 2509.09181 by Heejeong Kim, Yohsuke Murase.

Figure 1
Figure 1. Figure 1: A schematic diagram of the models considered in this paper. Suppose that a B donor meets a G recipient. (Left) In the incomplete observation model, the donor’s action is not always observed. With probability qo, the donor’s action (C) is observed and the donor is assigned a new reputation G according to R(G,C). With probability 1−qo, the action is not observed and the donor’s reputation remains unchanged d… view at source ↗
Figure 2
Figure 2. Figure 2: The CESS parameter regions for the CDC and CPC norms. The green shaded area shows the parameter region where both CDC and CPC norms can maintain cooperation, qf b > c. The yellow shaded area shows the parameter region where only CPC norms can maintain cooperation, qf(b+β) > max{c,α}. The parameters used here are α/c = 0.5 and β/c = 4 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the three models with errors: assessment error, implementation error, and perception error. (Upper left) In the model with assessment error, the donor’s reputation is not accurately updated. With probability qa, the donor’s reputation is correctly updated to G or B; with probability 1−qa, the donor’s reputation is randomly assigned to either G or B. Here, qa ≡ 1−2µ. (Upper right) In the model… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the five models with different types of information inaccuracy: (a) incomplete observation, (b) reputation fading, (c) assessment error, (d) implementation error, and (e) perception error. The horizontal axis represents the accuracy of information (qo, qf , qa, qi , and qp). The vertical axis represents the players’ payoffs under the ESS social norms. The black and red solid curves represent … view at source ↗
read the original abstract

Indirect reciprocity promotes cooperation by allowing individuals to help others based on reputation rather than direct reciprocation. Because it relies on accurate reputation information, its effectiveness can be undermined by information gaps. We examine two forms of incomplete information: incomplete observation, in which donor actions are observed only probabilistically, and reputation fading, in which recipient reputations are sometimes classified as "Unknown". Using analytical frameworks for public assessment, we show that these seemingly similar models yield qualitatively different outcomes. Under incomplete observation, the conditions for cooperation are unchanged, because less frequent updates are exactly offset by higher reputational stakes. In contrast, reputation fading hinders cooperation, requiring higher benefit-to-cost ratios as the identification probability decreases. We then evaluate costly punishment as a third action alongside cooperation and defection. Norms incorporating punishment can sustain cooperation across broader parameter ranges without reducing efficiency in the reputation fading model. This contrasts with previous work, which found punishment ineffective under a different type of information limitation, and highlights the importance of distinguishing between types of information constraints. Finally, we review past studies to identify when punishment is effective and when it is not in indirect reciprocity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper analyzes two forms of incomplete information in indirect reciprocity—probabilistic observation of donor actions and reputation fading (recipients classified as 'Unknown')—using analytical frameworks for public assessment. It claims that incomplete observation leaves cooperation thresholds unchanged because reduced update frequency is exactly offset by higher reputational stakes, while reputation fading strictly increases the required benefit-to-cost ratio as identification probability falls. The work then extends the models to include costly punishment as a third action and reports that punishment-inclusive norms sustain cooperation over wider parameter ranges without efficiency loss in the reputation-fading case, contrasting with earlier findings under other information constraints; a review of prior studies identifies conditions for punishment effectiveness.

Significance. If the analytical thresholds and qualitative distinctions hold, the manuscript usefully separates two information-incompleteness regimes that prior literature often conflates and clarifies when punishment augments rather than undermines reputation-based cooperation. The synthesis of past results on punishment effectiveness is a constructive contribution to the indirect-reciprocity literature.

major comments (2)
  1. [Abstract / analytical frameworks paragraph] Abstract and the paragraph introducing the analytical frameworks: the headline invariance result under incomplete observation (and the contrasting increase in required b/c under reputation fading) rests on the unverified claim that the public-assessment models yield exact long-run stationary thresholds; no derivations, finite-N corrections, or initial-condition robustness checks are supplied to confirm the offset is parameter-independent and free of hidden dependence on private-assessment residuals.
  2. [Punishment-extension analysis] Section on the punishment extension: the result that norms incorporating punishment sustain cooperation across broader ranges without efficiency loss in the reputation-fading model inherits the same exact-dynamics assumption; any finite-population drift or initial-reputation sensitivity would alter the reported parameter ranges and the claimed contrast with previous work.
minor comments (1)
  1. [Abstract] The abstract could briefly name the specific assessment rules (e.g., image scoring, standing, etc.) employed in the public-assessment frameworks to aid immediate readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive report and recommendation for major revision. We address the two major comments in detail below, clarifying the analytical basis of our results and committing to enhancements in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract / analytical frameworks paragraph] Abstract and the paragraph introducing the analytical frameworks: the headline invariance result under incomplete observation (and the contrasting increase in required b/c under reputation fading) rests on the unverified claim that the public-assessment models yield exact long-run stationary thresholds; no derivations, finite-N corrections, or initial-condition robustness checks are supplied to confirm the offset is parameter-independent and free of hidden dependence on private-assessment residuals.

    Authors: We appreciate the referee's attention to the foundational assumptions. The public-assessment frameworks permit exact derivation of long-run stationary reputation distributions by solving the linear balance equations of the Markov chain for reputation updates under the assessment rules. The invariance under incomplete observation follows because the observation probability scales both update frequency and the reputational weight of each observation proportionally, leaving the stationary probability of a good reputation (and thus the b/c threshold) unchanged; the contrasting increase under reputation fading arises because unknown classifications dilute the effective information without a compensating stake adjustment. We acknowledge these intermediate steps were not shown explicitly. In the revision we will add the full derivations of the stationary thresholds to an appendix. The analysis is performed in the infinite-population mean-field limit, where finite-N drift is negligible by construction and the unique stationary state is reached independently of initial conditions. revision: yes

  2. Referee: [Punishment-extension analysis] Section on the punishment extension: the result that norms incorporating punishment sustain cooperation across broader ranges without efficiency loss in the reputation-fading model inherits the same exact-dynamics assumption; any finite-population drift or initial-reputation sensitivity would alter the reported parameter ranges and the claimed contrast with previous work.

    Authors: We agree that the punishment extension rests on the same public-assessment framework, now augmented with a third action whose reputation consequences are incorporated into the stationary balance equations. The expanded cooperation region without efficiency loss is obtained by comparing stationary expected payoffs across norms. We will include the explicit stationary-state calculations for the punishment-inclusive case in the revised appendix. The contrast with earlier findings remains valid because reputation fading affects identification probability differently from the information constraints examined in those studies. Finite-population drift and initial-condition sensitivity lie outside the current mean-field scope; we can add a brief discussion of these modeling assumptions and note them as directions for future stochastic analyses. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rest on standard public-assessment frameworks without reduction to inputs by construction

full rationale

The paper applies established analytical frameworks for public assessment in indirect reciprocity to compare incomplete observation and reputation fading. The central result that incomplete observation leaves cooperation thresholds unchanged (via exact offset between update frequency and reputational stakes) follows from the model's update rules and payoff structure rather than being defined circularly in terms of the identification probability. Reputation fading's stricter b/c requirement is likewise derived from the altered reputation dynamics, not presupposed. No self-definitional equations, fitted parameters relabeled as predictions, or load-bearing self-citations appear in the provided abstract or description; prior-work contrasts supply external benchmarks instead of closing a self-referential loop. The assumption that the frameworks yield exact long-run thresholds is an explicit modeling choice, not a hidden tautology, leaving the derivation chain self-contained.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard evolutionary-game-theory assumptions plus two new parameters (observation probability and identification probability) that are treated as exogenous inputs rather than derived quantities.

free parameters (2)
  • observation probability
    Probability that a donor action is observed; appears as an exogenous parameter whose value modulates update frequency but is not fitted inside the paper.
  • identification probability
    Probability that a recipient's reputation is known rather than labeled Unknown; treated as a free parameter that directly scales the effective benefit-to-cost threshold.
axioms (2)
  • domain assumption Public assessment rules map observed actions to reputation labels in a deterministic or probabilistic but commonly known way.
    Invoked when the authors state they use analytical frameworks for public assessment.
  • standard math Infinite or well-mixed population limit with weak selection.
    Standard background assumption for deriving cooperation thresholds in indirect-reciprocity models.

pith-pipeline@v0.9.0 · 5725 in / 1490 out tokens · 36673 ms · 2026-05-18T18:15:10.747503+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    the donor decides the action using S(G) with probability qa donor G donor

    work page

  2. [2]

    the donor’s reputation is randomly assigned with probability 1-qa assessment error G/B C G donor recipient G with probability qi donor

    the donor’s reputation is updated by R(G, C) 2’. the donor’s reputation is randomly assigned with probability 1-qa assessment error G/B C G donor recipient G with probability qi donor

    work page

  3. [3]

    the donor’s reputation is updated by R(G, C) with probability 1-qi D donor recipient G donor

    work page

  4. [4]

    the intended cooperation is not executed

    the intended cooperation is executed 1. the intended cooperation is not executed. Defection is executed instead

    work page

  5. [5]

    the donor’s reputation is updated by R(G, D) implementation error G BG D G donor recipient G with probability qp donor

    work page

  6. [6]

    the donor’s reputation is updated by R(G, D) with probability 1-qp donor recipient G donor

    work page

  7. [7]

    the defection is misperceived as cooperation

    the defection is correctly perceived 1. the defection is misperceived as cooperation

    work page

  8. [8]

    (Upper left) In the model with assessment error, the donor’s reputation is not accurately updated

    the donor’s reputation is updated by R(G, C) perception error G D B C G Figure 3.Comparison of the three models with errors: assessment error, implementation error, and perception error. (Upper left) In the model with assessment error, the donor’s reputation is not accurately updated. With probabilityqa, the donor’s reputation is correctly updated to G or...

    work page

  9. [9]

    Melis, A. P. & Semmann, D. How is human cooperation different?Philos. Transactions Royal Soc. B365, 2663–2674 (2010). 3.Nowak, M. A. & Sigmund, K. Evolution of indirect reciprocity.Nature437, 1291–1298 (2005)

    work page 2010

  10. [10]

    & Braithwaite, V

    Wedekind, C. & Braithwaite, V . A. The long-term benefits of human generosity in indirect reciprocity.Curr. biology12, 1012–1015 (2002). 5.Sigmund, K. Moral assessment in indirect reciprocity.J. theoretical biology299, 25–30 (2012)

    work page 2002

  11. [11]

    P., Pacheco, J

    Santos, F. P., Pacheco, J. M. & Santos, F. C. The complexity of human cooperation under indirect reciprocity.Philos. Transactions Royal Soc. B376, 20200291 (2021). 7.Okada, I. A review of theoretical studies on indirect reciprocity.Games11, 27 (2020). 8.Nowak, M. A. & Sigmund, K. Evolution of indirect reciprocity by image scoring.Nature393, 573 (1998)

    work page 2021

  12. [12]

    & Hammerstein, P

    Leimar, O. & Hammerstein, P. Evolution of cooperation through indirect reciprocity.Proc. R. Soc. B268, 745–753 (2001). 9/16 Figure 4.Comparison of the five models with different types of information inaccuracy: (a) incomplete observation, (b) reputation fading, (c) assessment error, (d) implementation error, and (e) perception error. The horizontal axis r...

    work page 2001

  13. [13]

    & Boyd, R

    Panchanathan, K. & Boyd, R. A tale of two defectors: the importance of standing for evolution of indirect reciprocity.J. Theor . Biol.224, 115–126 (2003)

    work page 2003

  14. [14]

    & Iwasa, Y

    Ohtsuki, H. & Iwasa, Y . How should we define goodness? – reputation dynamics in indirect reciprocity.J. Theor . Biol. 231, 107–120 (2004)

    work page 2004

  15. [15]

    & Iwasa, Y

    Ohtsuki, H. & Iwasa, Y . The leading eight: social norms that can maintain cooperation by indirect reciprocity.J. Theor . Biol.239, 435–444 (2006)

    work page 2006

  16. [16]

    & Iwasa, Y

    Ohtsuki, H. & Iwasa, Y . Global analyses of evolutionary dynamics and exhaustive search for social norms that maintain cooperation by reputation.J. Theor . Biol.244, 518–531 (2007)

    work page 2007

  17. [17]

    & Nowak, M

    Ohtsuki, H., Iwasa, Y . & Nowak, M. A. Indirect reciprocity provides only a narrow margin of efficiency for costly punishment.Nature457, 79 (2009)

    work page 2009

  18. [18]

    M., Santos, F

    Pacheco, J. M., Santos, F. C. & Chalub, F. A. C. Stern-judging: A simple, successful norm which promotes cooperation under indirect reciprocity.PLoS computational biology2, e178 (2006)

    work page 2006

  19. [19]

    P., Santos, F

    Santos, F. P., Santos, F. C. & Pacheco, J. M. Social norms of cooperation in small-scale societies.PLoS Comput. Biol.12, e1004709 (2016)

    work page 2016

  20. [20]

    ineffective

    Santos, F. P., Santos, F. C. & Pacheco, J. M. Social norm complexity and past reputations in the evolution of cooperation. Nature555, 242–245 (2018). 10/16 Table 3.Summary of the impact of different types of information inaccuracy on cooperation and the effectiveness of costly punishment. In the left column, we list the type of noise affecting the reputat...

    work page 2018

  21. [21]

    Ingroup favoritism and intergroup cooperation under indirect reciprocity based on group reputation.J

    Masuda, N. Ingroup favoritism and intergroup cooperation under indirect reciprocity based on group reputation.J. Theor. Biol.311, 8–18 (2012). 19.Berger, U. Learning to cooperate via indirect reciprocity.Games Econ. Behav.72, 30–37 (2011)

    work page 2012

  22. [22]

    & Wolitzky, A

    Clark, D., Fudenberg, D. & Wolitzky, A. Indirect reciprocity with simple records.Proc. Natl. Acad. Sci. USA117, 11344–11349 (2020)

    work page 2020

  23. [23]

    & Nowak, M

    Hilbe, C., Schmid, L., Tkadlec, J., Chatterjee, K. & Nowak, M. A. Indirect reciprocity with private, noisy, and incomplete information.Proc. Natl. Acad. Sci. USA115, 12241–12246 (2018). 22.Uchida, S. Effect of private information on indirect reciprocity.Phys. Rev. E82, 036111 (2010)

    work page 2018

  24. [24]

    Kawakatsu, M., Kessinger, T. A. & Plotkin, J. B. A mechanistic model of gossip, reputations, and cooperation.Proc. Natl. Acad. Sci.121, e2400689121 (2024)

    work page 2024

  25. [25]

    A., Tarnita, C

    Kessinger, T. A., Tarnita, C. E. & Plotkin, J. B. Evolution of norms for judging social behavior.Proc. Natl. Acad. Sci.120, e2219480120 (2023)

    work page 2023

  26. [26]

    & Ohtsuki, H

    Fujimoto, Y . & Ohtsuki, H. Who is a leader in the leading eight? indirect reciprocity under private assessment.PRX Life2, 023009 (2024)

    work page 2024

  27. [27]

    & Chatterjee, K

    Schmid, L., Ekbatani, F., Hilbe, C. & Chatterjee, K. Quantitative assessment can stabilize indirect reciprocity under imperfect information.Nat. Commun.14, 2086 (2023)

    work page 2086

  28. [28]

    Michel-Mata, S.et al.The evolution of private reputations in information-abundant landscapes.Nature634, 883–889 (2024)

    work page 2024

  29. [29]

    & Hilbe, C

    Murase, Y . & Hilbe, C. Computational evolution of social norms in well-mixed and group-structured populations.Proc. Natl. Acad. Sci.121, e2406885121 (2024)

    work page 2024

  30. [30]

    & Hilbe, C

    Murase, Y . & Hilbe, C. Indirect reciprocity under opinion synchronization.Proc. Natl. Acad. Sci.121, e2418364121 (2024)

    work page 2024

  31. [31]

    & Hilbe, C

    Murase, Y . & Hilbe, C. Indirect reciprocity with stochastic and dual reputation updates.PLOS Comput. Biol.19, e1011271 (2023)

    work page 2023

  32. [32]

    & Nowak, M

    Schmid, L., Chatterjee, K., Hilbe, C. & Nowak, M. A. A unified framework of direct and indirect reciprocity.Nat. Hum. Behav.5, 1292 (2021). 11/16

    work page 2021

  33. [33]

    Costly punishment sustains indirect reciprocity under low defection detectability.J

    Murase, Y . Costly punishment sustains indirect reciprocity under low defection detectability.J. Theor. Biol.600, 112043 (2025)

    work page 2025

  34. [34]

    E., Hilbe, C

    Glynatsi, N. E., Hilbe, C. & Murase, Y . Exact conditions for evolutionary stability in indirect reciprocity under noise (2025). ArXiv:2509.08006 [q-bio.PE], 2509.08006

    work page arXiv 2025

  35. [35]

    & Ohtsuki, H

    Masuda, N. & Ohtsuki, H. Tag-based indirect reciprocity by incomplete social information.Proc. Royal Soc. B: Biol. Sci. 274, 689–695 (2007)

    work page 2007

  36. [36]

    & Masuda, N

    Nakamura, M. & Masuda, N. Indirect reciprocity under incomplete observation.PLoS Comput. Biol.7, e1002113 (2011). 36.Suzuki, S. & Kimura, H. Indirect reciprocity is sensitive to costs of information transfer.Sci. reports3, 1435 (2013)

    work page 2011

  37. [37]

    Yoeli, E., Hoffman, M., Rand, D. G. & Nowak, M. A. Powering up with indirect reciprocity in a large-scale field experiment. Proc. Natl. Acad. Sci.110, 10424–10429 (2013)

    work page 2013

  38. [38]

    & Richerson, P

    Boyd, R. & Richerson, P. J. Punishment allows the evolution of cooperation (or anything else) in sizable groups.Ethol. sociobiology13, 171–195 (1992)

    work page 1992

  39. [39]

    & Boyd, R

    Henrich, J. & Boyd, R. Why people punish defectors: Weak conformist transmission can stabilize costly enforcement of norms in cooperative dilemmas.J. theoretical biology208, 79–89 (2001)

    work page 2001

  40. [40]

    & Richerson, P

    Boyd, R., Gintis, H., Bowles, S. & Richerson, P. J. The evolution of altruistic punishment.Proc. Natl. Acad. Sci.100, 3531–3535 (2003)

    work page 2003

  41. [41]

    & Sigmund, K

    Brandt, H., Hauert, C. & Sigmund, K. Punishing and abstaining for public goods.Proc. Natl. Acad. Sci. United States Am. 103, 495 (2006)

    work page 2006

  42. [42]

    & Boyd, R

    Mathew, S. & Boyd, R. Punishment sustains large-scale cooperation in prestate warfare.Proc. Natl. Acad. Sci.108, 11375–11380 (2011)

    work page 2011

  43. [43]

    Reciprocity: Weak or strong? what punishment experiments do (and do not) demonstrate.Behav

    Guala, F. Reciprocity: Weak or strong? what punishment experiments do (and do not) demonstrate.Behav. brain sciences 35, 1–15 (2012). 44.Raihani, N. J. & Bshary, R. The reputation of punishers.Trends ecology & evolution30, 98–103 (2015)

    work page 2012

  44. [44]

    Raihani, N. J. & Bshary, R. Third-party punishers are rewarded, but third-party helpers even more so.Evolution69, 993–1003 (2015). 46.Raihani, N. J. & Bshary, R. Punishment: one tool, many uses.Evol. Hum. Sci.1, e12 (2019)

    work page 2015

  45. [45]

    & Baek, S

    Murase, Y ., Kim, M. & Baek, S. K. Social norms in indirect reciprocity with ternary reputations.Sci. Reports12, 455 (2022). 48.Tanabe, S., Suzuki, H. & Masuda, N. Indirect reciprocity with trinary reputations.J. Theor . Biol.317, 338–347 (2013)

    work page 2022

  46. [46]

    & Baek, S

    Lee, S., Murase, Y . & Baek, S. K. Local stability of cooperation in a continuous model of indirect reciprocity.Sci. Reports 11, 14225 (2021)

    work page 2021

  47. [47]

    & Baek, S

    Lee, S., Murase, Y . & Baek, S. K. A second-order stability analysis for the continuous model of indirect reciprocity.J. Theor. Biol.548, 111202 (2022). Acknowledgement The authors would like to thank C. Hilbe and S. K. Baek for insightful discussions and feedback on this work. HK acknowledges support by Basic Science Research Program through the National...

    work page 2022