Cost of institutional incentives for promoting cooperation in $2\times2$ games and collective risk games

C.M. Durbac; M.H. Duong; T.A. Han

REVIEW 4 major objections 5 minor 40 references

Cost of institutional incentives for promoting cooperation in $2\times2$ games and collective risk games

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper derives closed-form asymptotic costs for institutional incentives to promote cooperation, showing the expected reward cost is $N^2\theta H_N$ under neutral drift and either $N^2\theta/(N-1)$ or $N^2\theta$ under strong selection.

desk verdict Core asymptotic lemmas for incentive costs in composition-dependent games are correct and new; the finite-beta cost formula and 'phase transition' claims rest on an unvalidated starting-state assumption. read the letter →

arxiv 2506.02968 v2 pith:DEEGUIPC submitted 2025-06-03 q-bio.PE physics.soc-ph

M.H. Duong , C.M. Durbac , T.A. Han This is my paper

classification q-bio.PEphysics.soc-ph MSC 91A2292D1560J10

keywords institutionalincentivesFermistrategyupdatefinitepopulations2x2gamescollectiveriskgameincentivecostfunctionneutraldriftstrongselection

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 asks what it costs an external institution to steer a finite, well-mixed population toward cooperation, and it derives what that expected cost approaches in the two tractable limits of evolutionary dynamics. Under neutral drift ($\beta\to 0$), the expected reward-incentive cost converges to $N^2\theta H_N$ for every general $2\times2$ game and for the collective risk game, independent of the payoff structure. Under strong selection ($\beta\to\infty$), it converges to $N^2\theta/(N-1)$ when the incentive is too weak to overcome the defector's advantage in all states, and to $N^2\theta$ when it is strong enough, with analogous closed forms for punishment and hybrid incentives. These formulas matter because they give an institution an explicit budget: for extreme selection intensities, the cost is a simple function of population size and per-capita incentive, with no need to invert the Markov chain's fundamental matrix. The paper's numerical work shows that between these extremes the cost curves are not monotone, exhibiting phase transitions whose location depends on the game and, for the collective risk game, on the probability of collective loss.

What carries the argument

The carrying object is the absorbing Markov chain on states $S_0,\dots,S_N$ recording the number of cooperators, with tridiagonal transition matrix $W$ whose off-diagonal entries are Fermi probabilities $a_i=(1+e^{-\beta(\delta_i+\theta)})^{-1}$ and $c_i=(1+e^{\beta(\delta_i+\theta)})^{-1}$. The expected cost is written as $E(\theta)=N^2\sum_{j=1}^{N-1} \theta_j/(j(N-j))\,(f_D W^{-1}_{1,j}+f_C W^{-1}_{N-1,j})$, with starting weights $f_D=1/(1+r)$ and $f_C=r/(1+r)$, $r=e^{\beta(N-1)(\Delta+\theta)}$. The asymptotics come from simplifying $W$ before inversion: as $\beta\to 0$ it becomes the Cartan matrix, whose inverse is $2[\min(i,j)-ij/N]$, and as $\beta\to\infty$ it becomes a bidiagonal matrix whose inverse is the cumulative-sum matrix, supplied by the inverse formula for tridiagonal Toeplitz matrices (Lemma 1).

What would settle it

Run a forward simulation of the Fermi process for a finite population, say $N=10$, with a fixed reward incentive $\theta$ and starting from the all-defector state $S_0$ as the model section describes, record the total reward cost until absorption, and compare the mean with Eq (6) evaluated at $f_D=1/(1+r)$, $f_C=r/(1+r)$, $r=e^{\beta(N-1)(\Delta+\theta)}$; a systematic mismatch at finite $\beta$ would show the transient-start weights are the wrong entry distribution, while convergence to $N^2\theta H_N$ as $\beta\to 0$ would confirm Lemma 2.

Watch

Extended reading notes

Core claim

The central claim is that the expected institutional cost $E(\theta)$ for reward, punishment, and hybrid incentive schemes has well-defined closed-form limits even though the underlying $W$ matrix cannot be inverted in closed form for finite $\beta$. Lemma 2 states $\lim_{\beta\to 0} E_r(\theta) = N^2\theta H_N$, where $H_N=\sum_{i=1}^{N-1} 1/i$, for both the general $2\times2$ game and the collective risk game. Lemma 3 states, for reward incentives, $\lim_{\beta\to\infty} E_r(\theta) = N^2\theta/(N-1)$ when $\delta_j+\theta<0$ for all $j$, and $\lim_{\beta\to\infty} E_r(\theta) = N^2\theta$ when $\delta_j+\theta>0$ for all $j$; the appendix derives the corresponding punishment and hybrid factors. The proofs avoid inverting the $\beta$-dependent transition matrix by taking the limit of $W$ first, so that the neutral limit becomes the Cartan matrix with a known inverse and the strong-selection limit becomes a triangular matrix whose inverse is a cumulative-sum matrix.

Load-bearing premise

The load-bearing premise is the entry rule used to compute expected cost: the formulas place the population in the transient state $S_1$ (one cooperator) with probability $f_D$ and in $S_{N-1}$ (one defector) with probability $f_C$, even though the model section describes starting from the homogeneous absorbing states $S_0$ or $S_N$; if the intended entry distribution is different, the finite-$\beta$ cost curves, including the reported phase transitions, would change.

Editorial extensions

If this is right

If the limit formulas are correct, an institution can compute the asymptotic reward-incentive budget from the population size, the per-capita incentive, and the harmonic number alone, with no inversion of the transition matrix.
Under neutral drift the payoff structure drops out entirely: the expected reward cost is $N^2\theta H_N$ for every general $2\times2$ game and for the collective risk game, so game-agnostic budgeting is possible in the weak-selection regime.
Under strong selection the limiting cost is governed by the sign of $\delta_j+\theta$: it is $N^2\theta/(N-1)$ when defection still pays everywhere and $N^2\theta$ when cooperation pays in every state, with the analogous dichotomy for punishment and hybrid incentives in the appendix.
The phase transitions in the Prisoner's Dilemma and collective risk game imply that the optimal per-capita incentive $\theta^*$ is not a smooth function of selection intensity, so an institution cannot safely extrapolate a weak-selection budget to strong selection.
In the Stag Hunt game the cost rises to a peak and then declines to a plateau, indicating that once a cooperation threshold is crossed, additional incentive spending buys little extra cooperation.

Reading between the lines

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

A natural test of the paper's finite-$\beta$ cost definition is to recompute $E(\theta)$ with the population starting directly from the absorbing homogeneous states $S_0$ and $S_N$; because the paper's transient-state weighting is not fully specified, the reported phase-transition curves would likely shift even though the two asymptotic limits, which are insensitive to the starting weights, would
The sign-condition dichotomy in Lemma 3 should extend beyond $2\times2$ games to other $n$-player dilemmas whose average payoff difference $\delta_j$ keeps a uniform sign, making the strong-selection budget a general two-regime law rather than a special fact about the games studied here.
Since the neutral limit is game-independent, the question of which social dilemma is most expensive to incentivise is decided by the finite-$\beta$ corrections, so computing the first-order term in $\beta$ around $N^2\theta H_N$ would identify where game structure first enters the budget.
In the collective risk game, the amplification of the phase transition with the loss probability $r$ suggests a policy-relevant prediction: the per-capita incentive needed to sustain cooperation should depend nonmonotonically on perceived risk, with the sharpest transitions at high risk.

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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

Colleague, here's my read.

The real new content is analytical: closed-form β→0 and β→∞ limits of the expected institutional cost for reward, punishment, and hybrid incentives in general 2×2 games (composition-dependent payoff gaps) and the collective risk game. The neutral drift result N^2 θ H_N matches prior work, as they acknowledge; the strong-selection limits N^2 θ/(N−1) and N^2 θ are new, and I checked the algebra in Lemma 3—it's sound under their sign assumptions. This is a genuine extension of the Donation Game/PGG analyses in their earlier papers. The method of taking limits on the transition matrix before inversion is a good trick and they use it consistently. Credit where due: the core lemmas are transparently derived and the asymptotic statements are correct.

Now the soft spots. The expected cost E(θ) for finite β, Eq. (6), is load-bearing for every numerical curve, every 'phase transition' claim, and the optimization in §4.3. But the starting-state distribution is introduced heuristically: the text first says the chain starts at absorbing states S0 or SN, then uses transient states S1 and S_{N−1} as starting points, with weights fD and fC derived from a fixation-probability ratio. A fixation ratio doesn't by itself specify an entry distribution over transient states; there's a missing mutation-selection entry model. The asymptotic limits don't care because fD,fC just go to 1/2 or 0/1, but the finite-β curves absolutely depend on this choice. So the phase transitions and optimization results should be treated as conditional on an unstated modeling assumption. Also, the phase transition language is qualitative—no criteria, no error bars, and no code or data to check the figures.

The paper would benefit from acknowledging this and either providing a derivation of the entry rule or framing the numerical results as illustrative for that specific rule. The analytical contribution stands on its own.

For whom: people working on institutional incentives in evolutionary game theory will want the asymptotic formulas. It deserves a serious referee—the math core is worth engaging—but the referee should push on the finite-beta starting state before the numerical claims are accepted.

My recommendation: send to peer review, with a request for revision that either justifies the starting-state distribution or clearly marks the finite-β results as model-dependent.

Referee Report

4 major / 5 minor

Summary. The paper studies the expected cost of institutional reward, punishment, and hybrid incentives in finite, well-mixed populations playing general 2x2 games and the Collective Risk Game, under Fermi strategy update. It derives a Markov-chain expression for the expected incentive cost, obtains asymptotic limits for neutral drift (beta -> 0) and strong selection (beta -> infinity), and reports numerical simulations that show non-monotonic cost curves, which the authors call phase transitions, together with an optimization procedure. The neutral-drift limits and the strong-selection limits for reward and punishment incentives are algebraically consistent under the stated uniform-sign assumptions, but several load-bearing steps in the derivation of the finite-beta cost formula and in the hybrid-incentive asymptotic results need correction.

Significance. The contribution is a useful extension of the authors' earlier work on the Donation Game and Public Goods Game to games in which the cooperator-defector payoff difference depends on the population composition. The limiting formulas are explicit, contain no fitted constants, and follow from the model's transition matrix by standard matrix inversion, which are verifiable strengths. If the finite-beta model is properly justified, the paper provides a practical budgeting rule for institutions and a benchmark for numerical studies. However, the current text contains a sign error in the derivation of the cooperation frequency, an under-specified starting-state distribution in the finite-beta cost formula, and incorrect general-parameter hybrid strong-selection limits; these issues must be fixed before the results can be relied upon.

major comments (4)

[Section 2.3, Eq. (4)] The displayed derivation of the cooperation frequency contains reciprocal errors. For the transition probabilities in Eq. (2), u_{i,i-1}/u_{i,i+1} = (1+e^{-beta(delta_i+theta)})/(1+e^{beta(delta_i+theta)}) = e^{-beta(delta_i+theta)}, whereas the paper writes (1+e^{beta(delta_i+theta)})/(1+e^{-beta(delta_i+theta)}) and concludes that rho_D,C/rho_C,D = e^{beta sum(delta_i+theta)}. The standard result is rho_D,C/rho_C,D = prod_i u_{i,i+1}/u_{i,i-1} = e^{beta sum(delta_i+theta)}, so the final expression is correct but the displayed first equality and the explicit formulas for rho_D,C and rho_C,D are inverted. Please correct the orientation of the ratio and the fixation-probability definitions so that the derivation is self-consistent.
[Section 2.3, Eq. (6)] The expected cost formula weights the fundamental-matrix entries n_{1,j} and n_{N-1,j} by f_D and f_C, described as the long-run probabilities of starting at the absorbing states S0 and SN. The text does not derive these weights from a stated mutation-selection model. A fixation-probability ratio determines the stationary distribution over the absorbing states in a rare-mutation limit, but the entry distribution into the transient chain (S1 versus S_{N-1}) also requires specifying the mutation process, for example symmetric mutation rates and exactly one mutant per invasion attempt. As written, Eq. (6) is an additional modelling assumption, and all finite-beta cost curves in Figures 3-10 and the claimed phase transitions in Section 4 depend on it. Please state the rare-mutation model explicitly, or treat Eq. (6) as an assumption and provide a sensitivity check for the entry distribution.
[Appendix, Lemma 5 (hybrid incentives)] The hybrid-incentive strong-selection limits are not correct for general a and b. From Eq. (12) and the limiting inverse of W, part 1 (delta_j+theta<0 for all j) has only the j=1 term contributing, giving lim E_mix = N^2 theta/(N-1) * min(1/a, (N-1)/b); part 2 has only the j=N-1 term contributing, giving lim E_mix = N^2 theta/(N-1) * min((N-1)/a, 1/b). The stated formulas N^2 theta/(a(N-1)) and N^2 theta/(b(N-1)) hold only when the corresponding minimum is attained by the reward or punishment branch, a condition that is neither stated nor implied. Please correct the lemma and its proof, or add the missing conditions on a and b.
[Section 4 and Discussion] The term 'phase transition' is used for the non-monotonic dependence of E(theta) on beta or theta, but no quantitative criterion is given: there is no definition of the transition, no critical value, no scaling with N, and no order parameter. The claims that the cost functions 'exhibit a phase transition' are supported only by visual inspection of the plotted curves. If the phase transition is to be a main result, it needs a precise definition and quantitative characterization; otherwise the language should be softened to 'non-monotonic behaviour'.

minor comments (5)

[Abstract and Section 2.3] There are typos: 'functons' should be 'functions', and 'cost of inference over all generations' should be 'cost of incentives over all generations'.
[Section 2.3, Eq. (4)] In the sentence preceding Eq. (4), 'ui,i-1 and ui,i-1' should read 'ui,i-1 and ui,i+1'.
[Figures 2 and A3/A4] The caption of Figure 2 says the strong-selection limit is 'in accordance to Lemma 2', but it should refer to Lemma 3; the same issue occurs in the captions of Figures A3 and A4.
[Table 1] The abbreviations in the first column, such as 'PD (D)' and 'PD (C)', are not explained in the caption; a brief explanation of the notation (defective/cooperative versions) would improve readability.
[Section 4.3, Algorithm 1] The pseudocode lists theta as an input to CostFunction, but the transition matrix W depends on theta through the payoffs; please clarify that the matrix is constructed for the current theta value being evaluated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the asymptotic cost limits follow from the paper's own transition matrix via standard matrix inversions, and the central claims do not reduce to a fit or to a self-citation chain.

full rationale

The paper's central analytical results, Lemmas 2 through 5, are derived from the explicit transition matrix of the absorbing Markov chain, using the standard inverse of the tridiagonal fundamental matrix (Lemma 1, citing Huang and McColl [17]) and the known inverse of the Cartan matrix ([39]). The limiting formulas contain no fitted constants and are not defined in terms of the quantities they purport to predict. The finite-beta cost expression in Eq. (6) does rely on a heuristic entry distribution: the paper weights transient starts at S1 and S_{N-1} by f_D and f_C, which are the equilibrium frequencies of the two absorbing homogenous states. This is a modelling assumption, not a circular step; it is not itself derived from the cost function, and changing it would change the numerical curves, but the curves are not equivalent to the assumed weights by construction. Similarly, the phase-transition observations are numerical consequences of the stated model, not fitted inputs renamed as predictions. The self-citations to [5, 6, 7, 14] supply the incentive framework and earlier related results, but the new asymptotic limits are proven from the model equations in this paper, so those citations are not load-bearing in a circular sense. The paper also openly acknowledges limitations and intractability of a fully rigorous finite-beta optimisation, which further supports the assessment that the claimed limits are genuine derived results rather than restatements of assumptions.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The model relies on standard evolutionary game theory assumptions (Fermi update, absorbing Markov chain, fixation probability ratio as cooperation frequency). The load-bearing ad hoc assumption is the starting-state distribution: expected cost is computed as f_D n_{1,j} + f_C n_{N−1,j}, with f_D, f_C from the fixation ratio r. This is not derived from a well-defined mutation process. For the strong selection limits, the paper must assume δ_j+θ has a uniform sign across all j. No invented entities are introduced. The free parameters are the incentive cost θ, the incentive efficiencies a and b (normalized to 1), and the target cooperation level ω.

free parameters (3)

θ (per-capita incentive cost) = varied; optimization variable
Central control parameter. The strong selection limits require θ to be uniformly small or large relative to all δ_j.
a, b (incentive delivery efficiencies) = 1 (normalized)
Set to 1 for simplicity in Sections 3 and Appendix; the hybrid formulas retain factors 1/a and 1/b.
ω (target cooperation fraction) = user choice
Appears in the constraint θ ≥ θ0(ω); the institution chooses the desired cooperation level.

assumptions (7)

domain assumption Population dynamics follows Fermi's strategy update rule with intensity of selection β.
Section 2.1 defines the update probability P_{X,Y} = (1+e^{-β(f_Y-f_X)})^{-1}; all subsequent transition probabilities use this.
domain assumption Small mutation rate justifies an absorbing Markov chain on the number of cooperators with absorbing states S0 and SN.
Section 2.1 states the dynamics is modelled as an absorbing Markov chain of N+1 states following the timeline from mutation appearance to fixation or extinction.
domain assumption The institution operates a full-incentive scheme with complete information about population composition.
Section 2.3 assumes every player receives incentives in each generation and the number j of cooperators is known; Discussion lists this as a limitation.
ad hoc to paper Expected cost is computed with starting states S1 and S_{N−1} weighted by f_D = 1/(1+r) and f_C = r/(1+r), with r = e^{β(N−1)(Δ+θ)}.
Section 2.3, around Eq (6): 'The population will start, in the long run, at i=0 (i=N) with probability ... f_D (f_C)' followed by 'the expected number of visits at state Si will be f_D n_{1,i} + f_C n_{N−1,i}'. This mixes absorbing and transient starting states and is not derived from a microscopic mutation model.
standard math Cartan matrix inverse formula (Wbar^{-1})_{i,j} = 2[min(i,j) - ij/N].
Used in Lemma 2 proof, cited to [39]; accepted linear algebra result.
standard math Analytical inversion of tridiagonal Toeplitz matrices (Lemma 1).
Used in Lemma 3 proof, cited to [17]; accepted linear algebra result.
ad hoc to paper In the strong selection limits, θ is assumed small (δ_j+θ<0 for all j) or large (δ_j+θ>0 for all j) so that the sign is uniform across all transient states.
Lemma 3 and Appendix state these restrictions; they exclude mixed-sign regimes where the limiting dynamics is more complex.

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Cite this review

Pith. "Pith review of Cost of institutional incentives for promoting cooperation in $2\times2$ games and collective risk games." pith.science (2026). https://pith.science/paper/DEEGUIPC

@misc{pith2026250602968,
  author       = {Pith},
  title        = {Pith review of: Cost of institutional incentives for promoting cooperation in $2\times2$ games and collective risk games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DEEGUIPC}},
  note         = {Machine review of arXiv:2506.02968}
}

abstract

Prosocial behaviours have been extensively studied across multiple disciplines. Cooperation, requiring a personal cost for collective benefits, is widespread in nature and human society, having been explained through mechanisms such as kin selection, direct and indirect reciprocity, and network reciprocity. Institutional incentives, which reward cooperation and punish anti-social behaviour, offer a promising approach to fostering cooperation in groups of self-interested individuals. Focusing on general $2\times2$ games and the collective risk game (which is a fundamental model for climate action), we analyse the associated cost of providing incentives under evolutionary dynamics governed by Fermi's rule, exploring the asymptotic behaviour of the incentive cost functons in the limits of neutral drift and strong selection. We also implement numerical simulations to study how parameters such as the intensity of selection affect the behaviour of the aforementioned cost functions.

Figures

Figures reproduced from arXiv: 2506.02968 by the authors.

**Figure 2.** The dashed blue line represents the strong selection limit of the reward cost function. [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

**Figure 3.** Comparison of the reward cost function Er(θ) for the General 2 × 2 Games: Prisoner’s Dilemma (R = 2, S = 0, T = 4, P = 1), Hawk and Dove (R = 2, S = 1, T = 3, P = 0), and Stag Hunt (R = 3, S = 0, T = 2, P = 1), shown in this order. Each figure illustrates the behaviour of Er(θ) for various values of the intensity of selection β, with population size N = 10. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Comparison of the punishment cost function [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]

**Figure 5.** Figure 5: Comparison of the hybrid cost function Emix(θ) for the General 2×2 Games: Prisoner’s Dilemma (R = 2, S = 0, T = 4, P = 1), Hawk and Dove (R = 2, S = 1, T = 3, P = 0), and Stag Hunt (R = 3, S = 0, T = 2, P = 1), in that order. The cost function is evaluated across vario…

**Figure 6.** Figure 6: Reward, punishment, and hybrid cost functions under varying values of the intensity of [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 7.** Figure 7: Comparison of the cost functions for reward ( [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

**Figure 8.** Figure 8: Behaviour of the reward cost function Er(θ) in the Collective Risk Game for varying intensities of selection β and risk probabilities r = 0.01, 0.3, 0.7, and 0.9 (in this order). Parameters are fixed at B = 1, c = 0.1, n = 5, m = 2, and N = 20 [PITH_FULL_IMAGE:figures…

**Figure 9.** Figure 9: Behaviour of the punishment cost function [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]

**Figure 10.** Figure 10: Behaviour of the hybrid cost function Emix(θ) in the Collective Risk Game for varying intensities of selection β and risk probabilities r = 0.01, 0.3, 0.7, and 0.9 (in this order). Parameters are fixed at B = 1, c = 0.1, n = 5, m = 2, and N = 20. The Collective Risk G…

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