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REVIEW 6 major objections 5 minor 80 references

Noncooperative dynamics in election interference

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Pith's one-line read If either player in an election-interference game treats the outcome as all-or-nothing, rational play drives both sides' interference spending up superexponentially as election day approaches.

desk verdict Worth engaging for the arms-race differential game result; the empirical 'captures' claim is in-sample and the tweet-proxy attribution is mismarked, but the theory deserves refereeing. read the letter →

arxiv 1908.02793 v4 pith:ZLBN5H5O submitted 2019-08-07 physics.soc-ph econ.GNq-fin.EC

David Rushing Dewhurst , Christopher M. Danforth , Peter Sheridan Dodds This is my paper · ORCID
classification physics.soc-phecon.GNq-fin.EC MSC 91A2391A8060H3062F15
keywords electioninterferencedifferentialgamesHamilton-Jacobi-Bellmanequationssubgame-perfectNashequilibriumarmsraceBayesianstructuraltimeseriesstochasticoptimalcontrol2016U.S.
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

The paper builds a continuous-time game in which a foreign power (Red) spends effort to steer a two-candidate election toward its preferred candidate, while the target country's domestic agency (Blue) spends effort to cancel that influence. It argues that once either side treats the final result as all-or-nothing, the equilibrium response is for both sides to escalate interference and counter-interference at a superexponential rate near election day. The same framework is taken to data from the 2016 U.S. election, with daily posts from flagged accounts serving as a proxy for Red's effort and aggregated polls as the observed electoral state. The fitted model reproduces the broad temporal shape of the inferred effort and polling dynamics for most of the post-convention campaign window, with the paper explicitly noting a misfit in the first two weeks after the conventions. The arms-race result is presented as a general property of any two-actor strategic interaction with this payoff structure.

What carries the argument

The load-bearing mechanism is the coupled Hamilton-Jacobi-Bellman system (Eqs. 11–12), a pair of nonlinear partial differential equations that describe each player's minimal expected cost as a function of time and the latent poll state. The Nash controls are the negative half-gradients of the value functions, and the quadratic running costs $u_i^2 - \lambda_i u_{\neg i}^2$ give the coupling through the opponent's effort. Discontinuous terminal payoffs, the Heaviside forms, make the terminal control behave like a Dirac mass, which is what drives the superexponential escalation. For the one-sided problem under a credible commitment by the opponent, a logarithmic change of variables linearizes the HJB equation into a backward Kolmogorov equation, and a Feynman-Kac path-integral representation supplies closed-form Laplace approximations for the value function and policy. The inferential apparatus is a Bayesian structural time series model with Gaussian random-walk priors on the latent controls, a logit-normal likelihood for the poll, and a normal likelihood for the normalized daily post counts.

What would settle it

Apply the same estimation pipeline to a second documented election-interference campaign with a hard-line payoff and daily activity data: if the inferred control magnitudes do not grow faster than exponentially in the final weeks before the election, the arms-race claim fails. A complementary check would be to compare the inferred control series against internal records of the operation, showing that post volume was uncorrelated with actual interference spending.

Watch

Extended reading notes

Core claim

The central discovery is that the qualitative shape of the terminal payoff, not its scale, determines whether the game escalates. Red and Blue minimize cost functionals with quadratic running costs $u_i^2 - \lambda_i u_{\neg i}^2$ and terminal costs $\Phi_R(X_T)$ and $\Phi_B(X_T)$; the Nash equilibrium policies are $u_R(t)=-\tfrac12\partial V_R/\partial x$ and $u_B(t)=-\tfrac12\partial V_B/\partial x$, where the value functions solve a coupled pair of Hamilton-Jacobi-Bellman equations. Numerical sweeps over terminal conditions show that replacing a smooth payoff such as $\tanh(x)$ with a discontinuous Heaviside payoff such as $\Theta(x)-\Theta(-x)$ makes the value-function derivatives grow sharply near the terminal time, so both players' control magnitudes and their variances grow superexponentially. The paper's summary statement is that an all-or-nothing mindset by either Red or Blue about the final outcome leads to an arms race that negatively affects both players. In the empirical half, a Bayesian structural time series model infers latent controls and the latent poll from daily post counts and poll aggregates, and the theoretical model's free parameters are tuned to match those inferred series, giving an adequate fit over most of the campaign window.

Load-bearing premise

The empirical part of the paper rests on the premise that the daily count of posts from the flagged accounts measures Red's actual interference effort, so the inferred control series tracks real operations rather than an incidental feature of social-media activity.

Editorial extensions

If this is right

  • If either side's terminal payoff is discontinuous, the equilibrium magnitudes and variances of both sides' interference controls grow superexponentially near the terminal time, so last-minute escalation should be the expected signature of the game.
  • The 2016 fit implies that the observed surge in state-linked account activity is broadly consistent with an optimal control response once the race narrowed to two candidates, rather than with unstructured or random activity.
  • A credible commitment by one side to a fixed strategy reduces the other side's problem to a single-player optimal-control problem with tractable approximations, so announced strategies can be converted into predictions about the opponent's counter-escalation.
  • Because the arms-race property is stated for any strategic interaction of the form of Eqs. 3–5, the qualitative result should transfer to other contests with all-or-nothing final rewards, not only election interference.

Reading between the lines

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

  • A sharper test of the arms-race mechanism would examine another documented interference campaign with a hard-line payoff: if inferred daily effort does not accelerate faster than exponential as the endpoint approaches, the mechanism is not universal.
  • If the daily post series is read as a public signal rather than the true control, the fitted coupling parameters $\lambda_R,\lambda_B$ become interpretable as each side's sensitivity to the other's visible activity, a quantity that could be estimated for other geopolitical contests.
  • The paper's acknowledged misfit in the first two weeks after the conventions suggests a natural extension: a higher-dimensional state that tracks several primary candidates and collapses when the field narrows, which would make the transition itself part of the game.
  • The theory also implies that public social-media activity near an election is a strategic variable, so anomaly detection for influence operations should expect increased volume whenever a state publicly stakes its reputation on a candidate's victory.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

6 major / 5 minor

Summary. The paper develops a continuous-time, two-player, nonzero-sum stochastic differential game of foreign election interference. A latent electoral process X_t is influenced by Red and Blue control policies u_R(t) and u_B(t), with quadratic running costs and a cross-player payoff parameter λ_i, and each player minimizes a terminal cost Φ_i(X_T). The authors derive coupled Hamilton-Jacobi-Bellman equations, solve them numerically for a variety of terminal payoff structures, and report that discontinuous, all-or-nothing terminal conditions lead to superexponential growth of both players' control magnitudes near the election. They also analyze a credible-commitment variant using path-integral and Laplace methods. The paper then confronts the model with 2016 U.S. presidential election polling data and tweet counts from the fivethirtyeight Russian troll dataset, using a Bayesian structural time series model to infer latent controls and then calibrating the game's parameters to those inferred quantities. The abstract claims that the analytical model 'adequately captures many temporal characteristics' of the election and social media activity.

Significance. The theoretical setup is clear, stylized, and potentially useful: the arms-race mechanism, if established rigorously, is a non-obvious qualitative insight about terminal payoff structure in strategic interference games. Strengths include a reproducible simulation codebase, explicit discussion of several model limitations, and analytic reductions for the credible-commitment case. However, the empirical contribution is currently an in-sample calibration rather than a predictive test, the Twitter proxy is attributed to the wrong type of actor, there is an unresolved sign inconsistency between the theory and the BSTS state equation, and two reported sets of fitted parameters disagree. These issues do not necessarily invalidate the theoretical core, but they substantially weaken the empirical claims made in the abstract and conclusions.

major comments (6)
  1. [§II.B.3, §III, Eq. (49)] The empirical claim that Q 'adequately captures' the data is an in-sample calibration, not an independent test. The BSTS model M infers u_R, u_B, and X from the data using random-walk priors, and Q is then fit by minimizing the loss L(θ|Q) against M's posterior means, including the Legendre coefficients of the terminal payoff functions. The paper explicitly states in §II.B.3 that it does not predict any future values. Agreement between Q's credible intervals and M's inferred means is therefore partly produced by the fitting procedure and cannot serve as confirmation of the model. The section should be reframed as calibration or exploration, or supplemented with a genuine out-of-sample or posterior-predictive check.
  2. [§III, footnote [51]] The Twitter data are attributed to 'Russian military intelligence-associated' accounts and the empirical Red player is identified as 'the Russian military foreign intelligence service,' but the cited fivethirtyeight/russian-troll-tweets dataset consists of accounts associated with the Internet Research Agency, a private troll operation, not the GRU/SVR. Since tweet volume is the only direct observable for Red's control policy u_R, misidentifying the actor undermines the mapping from the data to the theoretical Red player. The attribution should be corrected and the implications of the actor mismatch for the empirical conclusions should be discussed.
  3. [§III, Eq. (46); §II.A, Eq. (3)] The BSTS state equation is inconsistent in sign with the theoretical state equation. The theory states dX_t = [u_R(t) + u_B(t)]dt + σ dW_t, while Eq. (46) gives X_t ~ N(X_{t-1} + u_{B,t-1} − u_{R,t-1}, 1). Combined with Eq. (48), where normalized tweet counts are modeled as N(u_R,t, σ_Tweets^2), and with Red's stated objective of favoring candidate A (Clinton), the inferred u_R has the opposite sign from the theoretical control policy unless an explicit reparameterization is introduced. This affects the sign and interpretation of the inferred controls and every fitted quantity that depends on them.
  4. [§II.B.4, Fig. 6] The statement that an all-or-nothing mindset by either player leads to an arms race is presented as a general feature, but the numerical evidence covers only selected terminal payoff functions and a single coupling value (λ_R = λ_B = 3) in Fig. 6, with nine combinations in Appendix A. Discontinuous terminal payoffs clearly produce steeper value-function gradients, but the specific claim of superexponential growth of both players' control magnitudes for arbitrary discontinuous final conditions requires either a proof or a systematic study over a wider class of terminal functions, parameters, and numerical resolutions before it is stated as a general result.
  5. [§III, Fig. 13] The reported fitted parameter values are internally inconsistent. The main text reports (λ_R, λ_B, σ) = (0.1432, 1.7847, 0.7510), while the caption of Fig. 13, for the same K = 10 and η = 0.002, reports (0.849, 0.727, 1.509). No explanation is given for the discrepancy, and it is unclear which parameter set was used to generate the displayed credible intervals. This needs to be reconciled before the empirical results can be assessed.
  6. [§II.B.3, Eqs. (11)–(12)] The inference and equilibrium analysis assume that the coupled HJB system has a unique solution for given final conditions Φ_R and Φ_B, and the manuscript admits that this uniqueness is not proved. Because the posterior in Eq. (21), the interpretation of the numerical solutions as subgame-perfect Nash equilibria, and the subsequent parameter inference all rely on this assumption, this is a load-bearing gap. A proof, a citation to a theorem covering this class of coupled systems, or an explicit statement that all equilibrium and inference results are contingent on uniqueness is needed.
minor comments (5)
  1. [Throughout] Several typographical errors remain, including 'foriegn' in §I, 'connvenience' in §II.A, and an unmatched parenthesis after 'Electoral College' in §II.A.
  2. [Footnote [30]] The Gaussian process citation contains an unresolved placeholder '[ ? ]' that must be completed before publication.
  3. [Fig. 13 caption] The caption's panel B description says 'middle 80% credible intervals of ˆu_R and ˆu_R'; this should presumably read 'of ˆu_R and ˆu_B.'
  4. [Eqs. (18)–(20)] The path-integral representation involving functional Gaussian distributions and the partition function Z is used only formally and is not used in the numerical solution; a brief statement clarifying that these expressions are not employed in the numerical work would reduce confusion.
  5. [§III, Eq. (48)] The tweet count series is normalized and shifted to have a minimum of zero and then modeled with a normal likelihood, discarding its count nature; a brief justification of this choice relative to the Poisson alternative mentioned in the text would be helpful.

Circularity Check

1 steps flagged · score 6.0 of 10

Empirical 'adequately captures' claim is an in-sample fit to the M-inferred latent controls that are then used as the validation target; the theoretical arms-race result is independent and not circular.

  1. fitted input called prediction [Section III (Application), Eq. 49 and Fig. 13; Discussion]
    "After inferring the latent control policies and electoral process, we searched for the parameter values θ = (λR,λB,σ,ΦR,ΦB) of the theoretical model that best explain the observed data and inferred latent variables. ... The 𝓁2 terms in Eq. 49 penalize deviation by Q from the mean of M’s inferred posterior distribution."

    The M-inferred posterior means of u_R, u_B, and X are simultaneously (i) the targets to which Q's parameters—including the Legendre-expanded final payoff functions Φ_R and Φ_B—are fitted by minimizing Eq. 49, and (ii) the benchmark against which Q's output is displayed in Fig. 13 and declared to 'capture' the election interference dynamics. The subsequent claim that 'the observed logit(Zt) is centered in the credible interval' and that the mean control paths 'do lie in these credible intervals' is therefore a report of the optimized loss, not an independent confirmation.

full rationale

The central theoretical result—that an all-or-nothing (discontinuous) terminal payoff for either player drives superexponential growth of both controls—follows from the coupled HJB equations (Eqs. 11-12) and is demonstrated numerically in Fig. 6; no fitted data enter that derivation, so that part of the paper is self-contained. The circularity is located in the empirical validation. The BSTS stage M constructs latent controls u_R and u_B with random-walk priors (Eqs. 44-45), identifies u_R almost directly with normalized tweet counts through Eq. 48, and has no direct observation of u_B. The theoretical model Q is then fit to M's posterior means by minimizing Eq. 49 over θ=(λ_R,λ_B,σ,Φ_R,Φ_B), and the same posterior means are used as the basis for the 'captures' verdict in Fig. 13. That is an in-sample calibration presented as confirmation, not a prediction. Consistent with this, the paper states it does not predict future values, and it provides no hold-out assessment. I also note a non-circularity issue: the main text reports (λ_R,λ_B,σ)=(0.1432,1.7847,0.7510) while the Fig. 13 caption reports (0.849,0.727,1.509) for the same K=10, η=0.002; this inconsistency is a correctness/reproducibility concern rather than a circularity. There are no load-bearing self-citations or imported uniqueness theorems. Overall: theoretical contribution independent (score would be low for that part), but the empirical adequacy claim is forced by construction, giving a partial circularity score of 6.

Assumptions & free parameters 6 free parameters · 6 assumptions · 3 invented entities

The model's central claims rely on standard optimal-control mathematics plus several domain assumptions: zero-drift latent polls, quadratic running costs, uniqueness of the coupled HJB system, and the tweet-volume proxy for Red's control. The free parameters include the coupling constants, volatility, and the Legendre representation of both final payoff functions; their fitted values are inconsistent between text and figure caption.

free parameters (6)
  • lambda_R (Red coupling parameter) = 0.1432 in main text; 0.849 in Fig. 13 caption (discrepancy)
    Controls how much Red values Blue's spending; fit to the M-inferred controls via the loss in Eq. 49.
  • lambda_B (Blue coupling parameter) = 1.7847 in main text; 0.727 in Fig. 13 caption (discrepancy)
    Controls how much Blue values Red's spending; fit to the M-inferred controls via Eq. 49.
  • sigma (latent volatility) = 0.7510 in main text; 1.509 in Fig. 13 caption (discrepancy)
    Volatility of the latent election process; fit to data via the optimization in Sec. III.
  • Legendre coefficients a_{R,k}, a_{B,k}, k=0..10 = not individually reported; 21 values fitted
    Approximate the unknown final payoff functions Phi_R and Phi_B; these are the bulk of the model's 23 free parameters.
  • K (Legendre truncation order) = 10
    Chosen by hand as a compromise between approximation accuracy and overparameterization.
  • eta (loss regularization) = 0.002
    Chosen by hand; penalizes dispersion of Q's generated paths in Eq. 49.
assumptions (6)
  • ad hoc to paper Coupled HJB system (Eqs. 11 and 12) has a unique solution for given final conditions Phi_R and Phi_B.
    Explicitly assumed in Sec. II.B.3; the authors state they have numerical evidence but no proof.
  • domain assumption The latent election process has zero endogenous drift; controls enter additively and affect only the mean, not volatility.
    Eqs. 1 through 3; this supports dX_t = (u_R + u_B) dt + sigma dW_t.
  • domain assumption Running costs reduce to u_i^2 - lambda_i u_neg_i^2 by Taylor expansion under even-function and zero-cost-at-zero assumptions.
    Sec. II.B, cost formulation; the paper justifies the quadratic form with symmetry and no-cost-for-zero-effort.
  • standard math Feynman-Kac and path integral control formulas apply to the transformed HJB equation (Eq. 26).
    Sec. II.C.1; assumes sufficient regularity and integrability for the change of variables and Monte Carlo evaluation.
  • domain assumption Tweet volume, after normalization, is a noisy observation of Red's control u_R.
    Eq. 48; the dataset is also described as Russian military intelligence-associated, which is questionable.
  • ad hoc to paper The BSTS state equation uses -u_R where the theory uses +u_R; this is either a sign reparameterization or a model inconsistency.
    Compare Eq. 46 with Eq. 3; the paper does not discuss the sign flip or reconcile it.
invented entities (3)
  • Latent electoral process X_t independent evidence
    purpose: Unobserved state whose logit maps to observed polls Z_t; Red and Blue push it via their controls.
    Testable through the observed poll series Z_t via the logit-normal likelihood in Eq. 47.
  • Latent Red control policy u_R(t) independent evidence
    purpose: Unobserved level of interference effort; proxied by daily Russian troll tweet counts.
    The tweet time series provides a noisy observable via Eq. 48, though attribution to military intelligence is questionable.
  • Latent Blue control policy u_B(t)
    purpose: Unobserved counter-interference effort by the defending country.
    No direct observable; identified only from polls and model structure, so it is weakly identified.

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Cite this review

Pith. "Pith review of Noncooperative dynamics in election interference." pith.science (2026). https://pith.science/paper/ZLBN5H5O

@misc{pith2026190802793,
  author       = {Pith},
  title        = {Pith review of: Noncooperative dynamics in election interference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZLBN5H5O}},
  note         = {Machine review of arXiv:1908.02793}
}
read the original abstract

Foreign power interference in domestic elections is an existential threat to societies. Manifested through myriad methods from war to words, such interference is a timely example of strategic interaction between economic and political agents. We model this interaction between rational game players as a continuous-time differential game, constructing an analytical model of this competition with a variety of payoff structures. All-or-nothing attitudes by only one player regarding the outcome of the game lead to an arms race in which both countries spend increasing amounts on interference and counter-interference operations. We then confront our model with data pertaining to the Russian interference in the 2016 United States presidential election contest. We introduce and estimate a Bayesian structural time series model of election polls and social media posts by Russian Twitter troll accounts. Our analytical model, while purposefully abstract and simple, adequately captures many temporal characteristics of the election and social media activity. We close with a discussion of our model's shortcomings and suggestions for future research.

Figures

Figures reproduced from arXiv: 1908.02793 by the authors.

Figure 1
Figure 1. FIG. 1. The random walk latent space election model is an accurate approximation to multiple different population candidate [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Example value functions corresponding to the system Eqs. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. We display realizations of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (23 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Example sweeps over the coupling parameters [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Example sweeps over the coupling parameters [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. In the case of strong coupling ( [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Result of the path integral Monte Carlo solution [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. When player [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. If player [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The solution to Eq [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. We approximate the time series components of [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Panel A displays the logit of the observed elec [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. We display credible intervals of latent election process [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. We demonstrate the lack of qualitative changes in the value functions given by solutions to Eqs. [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. We demonstrate the lack of qualitative changes in the value functions given by the solution to Eqs. [PITH_FULL_IMAGE:figures/full_fig_p020_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Parameter sweep over coupling parameters [PITH_FULL_IMAGE:figures/full_fig_p023_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Parameter sweep over coupling parameters [PITH_FULL_IMAGE:figures/full_fig_p024_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Parameter sweep over coupling parameters [PITH_FULL_IMAGE:figures/full_fig_p025_18.png]
Figure 20
Figure 20. Figure 20: FIG. 20. Parameter sweep over coupling parameters [PITH_FULL_IMAGE:figures/full_fig_p026_20.png]
Figure 22
Figure 22. Figure 22: FIG. 22. Parameter sweep over coupling parameters [PITH_FULL_IMAGE:figures/full_fig_p027_22.png]
Figure 24
Figure 24. Figure 24: FIG. 24. Parameter sweep over coupling parameters [PITH_FULL_IMAGE:figures/full_fig_p028_24.png]
Figure 26
Figure 26. Figure 26: FIG. 26. Parameter sweep over coupling parameters [PITH_FULL_IMAGE:figures/full_fig_p029_26.png]
Figure 27
Figure 27. Figure 27: FIG. 27. Parameter sweep over coupling parameters [PITH_FULL_IMAGE:figures/full_fig_p030_27.png]
Figure 29
Figure 29. Figure 29: FIG. 29. Parameter sweep over coupling parameters [PITH_FULL_IMAGE:figures/full_fig_p031_29.png]
Figure 31
Figure 31. Figure 31: FIG. 31. Parameter sweep over coupling parameters [PITH_FULL_IMAGE:figures/full_fig_p032_31.png]
Figure 33
Figure 33. Figure 33: FIG. 33. Parameter sweep over coupling parameters [PITH_FULL_IMAGE:figures/full_fig_p033_33.png]

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Works this paper leans on

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    First-order

    Choice of final conditions Finding optimal play in noncooperative games often requires solving the game backward through time [23–26]. Therefore, we must define final conditions that specify the cost that Red and Blue incur from the actual election result φ(XT ). Red and Blue might have different final conditions because of their qualitatively distinct objec- ...

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    Value functions Applying the dynamic programming principle [17, 18] to Eqs. 3, 4, and 5 leads to a system of coupled Hamilton- Jacobi-Bellman equations for the Red and Blue value functions, −∂VR ∂t = min uR {∂VR ∂x [uR +uB] +u2 R−λRu2 B + σ2 2 ∂2VR ∂x2 } , (7) and −∂VB ∂t = min uB {∂VB ∂x [uR +uB] +u2 B−λBu2 R + σ2 2 ∂2VB ∂x2 } . (8) The dynamic programmi...

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    Inference and prediction The solutions to Eqs. 11 and 12 are functions of the final conditions VR(x,T ) = Φ R(x) and VB(x,T ) = ΦB(x). It is possible to perform both inference and pre- diction at times t < Teven when Φ R(x) and Φ B(x) are not known. To do this, we assume that the system given by Eqs. 11 and 12 has a unique solution given particular final co...

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    Dependence of value functions on parameters We conducted a coarse parameter sweep over λR, λB, ΦR, and ΦB to explore qualitative behavior of this game. We display the results of this parameter sweep for two combinations of final conditions in Figs. 4 and 5. The upper right-hand corner of each panel of the figures dis- plays the final condition of each player...

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.