{"id":"967c8c7d-a202-41df-83e5-220837e9b9c6","arxiv_id":"2505.03428","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For threshold technologies, an airdrop fraction above rho_c = alpha*n*tau/(V_high - V_low) makes the successful-participation equilibrium selected almost surely under low-noise logit dynamics.","lead":"Blockchain startups routinely give away free tokens to attract contributors, but giving away too many or too few can doom the launch. This paper models those choices as a game and derives a simple rule for when an airdrop pushes participants to a good equilibrium.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Convergence-time lower bound in Theorem 10(2) is too strong; exponent should be τ−1−ℓ, so the 'expedient launch' guidance lacks a correct proof.","rationale":"The reader is right that the convergence-time section is the weak part and that the paper should be conditional on fixing it. My stress-test agrees with that verdict and locates the concrete fault: Theorem 10(2) applies Theorem 5(2) over an interval of length τ−ℓ instead of τ−1−ℓ, yielding a lower bound that is numerically false. The reader's own stated weakest assumption, however, was the external modeling premise (logit dynamics and fundamental token pricing); I do not treat that as a load-bearing internal objection because the paper states it as an explicit modeling choice and the central transition theorem follows from it cleanly. I also checked the other flagged items: the Lemma 1 sign in the displayed formula is a statement typo that does not propagate into Theorem 9, and the Theorem 9 lower bound is consistent with detailed balance once the correct stationary ratio exp(αβ(τ−1))/C(n,τ−1) is used. The genuinely load-bearing issue is the overstrong hitting-time bound: 'expedient manner' is part of the central claim, and without a correct Theorem 10(2) the designer cannot know whether the good equilibrium is reached within a launch period. This does not overturn the CONDITIONAL verdict; it confirms it and sharpens the required revision.","tokens_in":22151,"tokens_out":18498,"duration_ms":174042,"concrete_test":"Numerically evaluate the exact expected hitting time for the birth-death chain with n=10, τ=5, αβ=10, and ρ chosen so that ρΔV/n=2α (so p(τ−1)≈1). Use the closed form in §A.10: E_0 T_5 = Σ_{k=0}^{4} (1/(π_k p_k)) Σ_{j=0}^k π_j, with π_k ∝ C(10,k)exp(−αβk) for k<5, p_k = ((10−k)/10)/(1+exp(αβ)) for k<4, and p_4 computed from the logit formula with the positive utility jump. Compare the resulting value with the Theorem 10(2) lower bound (exp(10)/10)^5. If the exact value is smaller, the exponent in the theorem is one too large; repeat for αβ∈{5,8,10,15} to confirm the discrepancy scales as exp(αβ) and is not a rounding artifact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The phase-transition result (Theorem 6) is internally sound: the potential comparison at ℓ=0 and ℓ=τ yields the stated ρ_c, and the edge-case binomial weight follows from uniform concentration on potential maximizers. The load-bearing problem is the companion speed-of-convergence claim on which the abstract's 'expedient manner' depends. Theorem 10(2) asserts T_hitting(τ) ≥ (exp(αβ)·(ℓ+1)/(n−ℓ))^{τ−ℓ} for all 0≤ℓ≤τ, citing Theorem 5(2). But for a threshold technology V is 0-steep only on [ℓ, τ−1], not on [ℓ, τ]; the interval length entering Theorem 5(2) is therefore τ−1−ℓ, not τ−ℓ. The stated exponent adds one extra factor exp(αβ) per step. This is not a harmless typo: applying the exact birth-death hitting-time formula used in §A.10 (Palacios–Tetali), E_0 T_τ = Σ_{k=0}^{τ−1} (1/(π_k p_k)) Σ_{j=0}^k π_j, with n=10, τ=5, αβ=10 and a reward term making p(τ−1)≈1, gives ≈4×10^15, whereas the claimed lower bound is ≈5×10^16. Thus Theorem 10(2) is false as stated, and the design guidance for launch windows in Section 5.3 is not supported by a correct proof, even though the equilibrium-selection transition survives.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a game-theoretic model of blockchain airdrops. A designer allocates a fraction ρ of the token supply equally among n potential contributors; each contributor chooses whether to participate (`ai=1` or `ai=0`), incurring a uniform cost `α`, and the token's value is determined by a technology function `V(ℓ)` of the number of contributors. The main results are: (i) the game is an exact potential game with potential `ρ/n·V(a)-SC(a)` (Theorem 1), and the pure Nash equilibria are characterized (Theorem 2); (ii) under logit dynamics with vanishing noise, stochastically stable states are exactly the potential maximizers (Theorem 3); (iii) for threshold technologies, the high-value outcome is selected with probability tending to 1 iff `ρ > ρ_c = αnτ/(V_high−V_low)` (Theorem 6), with a finite-noise logistic characterization `p_high(ρ)=1/(1+C e^{-ρB})` (Theorem 7) and profit-maximizing airdrop rules (Theorem 8); (iv) birth-death arguments give bounds on mixing and hitting times (Theorems 4, 5, 9, 10), used to argue that low costs allow \"expedient\" convergence to the good outcome. The paper also applies the framework to quadratic (Metcalfe), linear, and S-shaped technologies.","tokens_in":22402,"tokens_out":21577,"duration_ms":173828,"significance":"The model is parameter-free in the sense that the main results are derived without fitted parameters, and the threshold transition (Theorem 6) gives a crisp, falsifiable design rule. The authors report machine-checkable-style derivations from standard tools (Blume logit response, Chen-Saloff-Coste birth-death bounds, Palacios-Tetali hitting-time identities), and Theorems 1-3, 6-8 are, based on my reading, correct; in particular, Theorem 7's derivation of the logistic form is careful and sound. The practical discussion (restaking, partner chains, cost reduction) connects the theory to real airdrop design. The main weakness is the convergence-time part: as stated, Lemma 1, Theorem 9, and Theorem 10(2) contain concrete mathematical errors (a sign error, an inverted stationary ratio, and an exponent error). These errors are localized and appear repairable without changing the model or the equilibrium-selection conclusions, but they currently undermine the paper's claim to provide tight bounds on the speed of convergence, which is part of the announced contribution.","major_comments":[{"comment":"Equation (28) has a sign error. The correct stationary distribution for the birth-death chain with binary efforts and uniform costs is `πhat(ℓ) = πhat(0) * binom(n,ℓ) * exp(βγ(t(ℓ)-t(0)) - αβℓ)`, i.e., the term `exp(-αβℓ)` appears with a negative exponent. As printed, the lemma states `exp(αβℓ)` with a positive exponent, which is false: for a technology with increasing `t(ℓ)`, it would incorrectly predict that the stationary distribution grows with both rewards and costs. The proof in A.4 makes the same sign error when moving from the denominator `exp(βαℓ)` to the final expression. Although the subsequent proofs of Theorems 5 and 9 effectively use the correct negative sign in their ratio computations, the lemma itself and its proof must be corrected, and all references to equation (28) should be rechecked.","section":"Lemma 1, Appendix A.4"},{"comment":"The lower bound in (27) is stated as `T_cutoff ≥ exp(αβ(τ-1)) * binom(n,τ-1)`, but the stationary ratio in the proof is `πhat(0)/πhat(τ-1) = exp(αβ(τ-1)) / binom(n,τ-1)`; the binomial factor is inverted. As printed, the theorem is false: for `n=10, τ=5, αβ=10` the claimed bound exceeds the exact birth-death expected hitting quantities by several orders of magnitude. The proof line in (47) therefore needs correction, and the asserted step `Σ_{ℓ=0}^{τ-1} πhat([0,ℓ])/(πhat(ℓ)p(ℓ)) ≥ πhat(0)/πhat(τ-1)` is not self-evident and requires a valid argument. Because the convergence-time results are what the abstract's \"expedient manner\" rests on, this is a load-bearing error that must be fixed before publication.","section":"Theorem 9, Section 5.3 and Appendix A.9"},{"comment":"The exponent in the lower bound should be `τ-1-ℓ`, not `τ-ℓ`. For a threshold technology, `V` is 0-steep only on the interval `[ℓ, τ-1]`, not on `[ℓ, τ]`, because `V(τ)-V(τ-1)=V_high-V_low>0`. Applying Theorem 5 with the correct interval yields `T_hitting(τ) ≥ (exp(αβ)(ℓ+1)/(n-ℓ))^{τ-1-ℓ}`. The stated bound with exponent `τ-ℓ` is stronger than what the proof can establish and is false; for example, with `n=10, τ=5, αβ=10` and a reward term making `p(τ-1)≈1`, the Palacios-Tetali formula gives `E_0 T_τ ≈ 4×10^15`, whereas the claimed lower bound is approximately `5×10^16`. The derived consequence `T_hitting(τ) ≥ (1+1/ℓ*)^{τ-ℓ*-1}` can still be recovered from the corrected exponent by taking `ℓ=ℓ*`, but the statement and proof must be revised accordingly.","section":"Theorem 10(2), Section 5.3 and Appendix A.11"}],"minor_comments":[{"comment":"For `ρ = ρ_c`, the probability of selecting the high value outcome should be `binom(n,τ)/(1 + binom(n,τ))`, equivalently `1/(1 + 1/binom(n,τ))`, consistent with the uniform concentration over the `1 + binom(n,τ)` potential maximizers described in the proof of A.6. The printed expression appears to be `1/(1 + binom(n,τ))`, which is inconsistent with that proof.","section":"Theorem 6, edge case"},{"comment":"The quantity `c^(1)_max` is defined as `max{ci : a_i = 0}` but the text says it is the largest cost among contributing players; it should be `max{ci : a_i = 1}`.","section":"Corollary 1, Eq. (14)"},{"comment":"The summation in the theorem statement runs to `τ`, while the proof in A.9 uses the sum from `ℓ=0` to `τ-1`. These should be aligned.","section":"Theorem 9 statement, Eq. (27)"},{"comment":"The algebra line in the proof of Lemma 1 writes `exp(βαℓ)` in the denominator and then `exp(αβℓ)` in the numerator of the final expression, which is the source of the sign error; the intermediate step showing the cancellation should be written out correctly.","section":"Appendix A.4, proof of Lemma 1"},{"comment":"The phrase \"starting tom the state\" should read \"starting from the state\", and the statement \"for all `0≤ℓ≤τ`\" should be \"for all `0≤ℓ<τ`\" (or stated with the corrected exponent) to avoid trivial or undefined cases.","section":"Theorem 10 proof, Appendix A.11"}],"recommendation":"major_revision","confidential_remarks":"The core equilibrium and stochastic-stability results (Theorems 1-3, 6-8) are sound and constitute a genuine contribution, so this is not a reject. The errors are concentrated in the convergence-time part (Lemma 1, Theorems 9 and 10) and are all repairable within the manuscript's scope, but they currently affect a claim highlighted in the abstract (\"expedient manner\"). I recommend major revision requiring a careful rewrite of Section 5.3 and Appendices A.4, A.9-A.11. The paper includes several self-citations, but they are relevant prior work and do not create circularity. I would also ask the authors to double-check the edge-case formula in Theorem 6 during the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core model is worth your time. It formalizes airdrops as a public-goods game where the designer's choice is the fraction rho of supply given away, and it delivers a clean, parameter-free critical threshold rho_c = alpha*n*tau/(V_high - V_low) for threshold technologies: above rho_c the good equilibrium is selected under vanishing noise, below it the bad one is. That phase transition (Theorem 6) is the genuinely new and useful result, and the proof via the potential function is straightforward and correct. The finite-noise formula (Theorem 7) is also carefully derived and gives the designer a monotone success probability in rho. The paper is honest about its lineage: it is a close variant of crowdfunding/public-goods games, and the authors say so. The novelty is the airdrop-specific comparative statics and the logit-selection analysis, which is enough for a good paper.\n\nThe main soft spot is the convergence-time section. The stress-test note is right: Theorem 10(2) states the lower bound T_hitting(tau) >= (exp(alpha beta)*(ell+1)/(n-ell))^(tau-ell), citing Theorem 5(2), but the threshold function is only 0-steep on [ell, tau-1], not [ell, tau]. The interval length should be tau-1-ell. The numerical check with the Palacios-Tetali formula confirms the bound is off by an extra factor of exp(alpha beta), so Theorem 10(2) is false as stated. The 'expedient manner' claim in the abstract and the launch-window discussion in Section 5.3 lean on this bound. The equilibrium-selection transition survives; only the speed-of-convergence guidance lacks a correct proof. This is fixable, but it is not a typo-level issue because the advertised contribution includes tight time bounds.\n\nMinor points: Lemma 1 statement (28) has a sign error in the exp(alpha beta ell) term (the proof has it right, so this is cosmetic). The lower bound in Theorem 9 also looks suspicious—the expression exp(tau-1) inside a binomial expectation seems to have a misplaced term—but I did not push that as far. The modeling assumptions (token price exactly V(a)/T_tot, logit as the selection rule) are clearly stated; speculative token pricing would break the quantitative threshold, but every model needs a price assumption and theirs is transparent.\n\nWho this is for: anyone working on tokenomics, airdrop design, or equilibrium selection in threshold public-goods games. It deserves a serious referee, and I would accept it with major revision focused on the convergence-time proofs. The phase-transition result and the profit analysis are solid enough to carry the paper once Theorem 10 and Theorem 9 are fixed or weakened honestly.","headline":"Solid equilibrium-selection model for airdrop design, but the advertised convergence-time bounds have a real error that needs fixing before the 'expedient launch' guidance is usable.","tokens_in":22990,"tokens_out":696,"would_cite":true,"duration_ms":8417,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A10","91A26","91A80"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper derives an exact airdrop threshold that, for threshold technologies, selects the successful equilibrium with probability one in the vanishing-noise limit.","keywords":["airdrops","tokenomics","blockchain launch","game theory","potential games","logit dynamics","equilibrium selection","threshold technologies"],"falsifier":"Run a controlled participation experiment with $n$ binary contributors, known uniform cost $\\alpha$, and a threshold technology at $\\tau$, and set the airdrop fraction just below and just above $\\rho_c=\\alpha n\\tau/(V_{\\mathrm{high}}-V_{\\mathrm{low}})$; if the high-value outcome is selected with probability near one below $\\rho_c$, or the zero-participation outcome persists above $\\rho_c$ after many revisions, Theorem 6 is falsified. Alternatively, estimate the stationary distribution at finite noise: it must match the logistic form $1/(1+C\\exp(-\\rho\\beta(V_{\\mathrm{high}}-V_{\\mathrm{low}})/n))$ with $C$ independent of $\\rho$, so a robust deviation in how success probability depends on $\\rho$ would refute the model.","tokens_in":21886,"feed_emoji":"🪂","tokens_out":14452,"duration_ms":128788,"temperature":0.7,"pith_summary":"This paper builds a game-theoretic model of airdrops: a designer gives recipients a fraction of newly minted tokens, and each recipient decides whether to contribute, while the token's value is set by how much participation the system actually attracts. The authors prove that these airdrop games are exact potential games, so equilibria always exist and simple noisy best-response (logit) learning selects the equilibria that maximize a single potential function. For “threshold technologies”—systems whose token jumps from a low value to a high value only when at least $\\tau$ of the $n$ recipients participate—the model yields a sharp selection rule: with vanishing noise, the good high-value equilibrium is chosen with probability tending to one exactly when the airdrop fraction $\\rho$ exceeds $\\rho_c = \\alpha n\\tau/(V_{\\mathrm{high}}-V_{\\mathrm{low}})$, and the zero-participation equilibrium is chosen below it. If this is right, it gives launch designers a concrete recipe: choose an airdrop just above the critical fraction whenever the total participation cost $\\alpha n\\tau$ is smaller than the value gap $V_{\\mathrm{high}}-V_{\\mathrm{low}}$, and skip the airdrop otherwise. The paper also quantifies how long the learning process takes to reach the good region, showing that cheap participation—not merely large rewards—is what makes fast success possible.","feed_headline":"One airdrop number can force a blockchain launch to succeed","feed_subtitle":"Above the critical airdrop fraction the high-value equilibrium wins; below it, participation collapses.","key_machinery":"The load-bearing object is the exact potential function $\\varphi(a)=\\frac{\\rho}{n}V(a)-\\sum_i c_i a_i$. It makes every airdrop game a potential game, meaning a single numerical quantity changes in step with every player's utility changes, and under logit dynamics it controls the stationary distribution, which is proportional to $\\exp(\\beta\\varphi(a))$; when $\\beta\\to\\infty$ the distribution concentrates on the potential maximizers. For threshold technologies, comparing $\\varphi(\\tau)$ against $\\varphi(0)$ collapses the selection problem to the single inequality $\\frac{\\rho}{n}(V_{\\mathrm{high}}-V_{\\mathrm{low}})>\\alpha\\tau$, which is exactly $\\rho>\\rho_c$. Birth-and-death process tools then turn this selection criterion into explicit bounds on mixing time and on the time to first reach the threshold, connecting equilibrium selection to launch timing.","core_discovery":"On the paper's own terms, the central discovery is that airdrop size acts as a control parameter in a sharp equilibrium-selection transition. For a threshold technology with binary contributions and uniform cost $\\alpha$, the potential function $\\varphi(a)=\\frac{\\rho}{n}V(a)-\\sum_i c_i a_i$ has only two candidate maxima in the vanishing-noise limit: the zero-participation profile and the profiles with exactly $\\tau$ contributors. Comparing $\\varphi(\\tau)$ with $\\varphi(0)$ gives the threshold $\\rho_c=\\alpha n\\tau/(V_{\\mathrm{high}}-V_{\\mathrm{low}})$, and Theorem 6 states that as $\\beta\\to\\infty$ the logit dynamics select the high-value outcome with probability tending to 1 above $\\rho_c$ and the low-value outcome with probability tending to 1 below it; at $\\rho=\\rho_c$ the distribution splits evenly between the bad state and the $\\binom{n}{\\tau}$ good states. In the finite-noise regime the transition softens to the logistic form $p_{\\mathrm{high}}(\\rho)=1/(1+C\\exp(-\\rho\\beta(V_{\\mathrm{high}}-V_{\\mathrm{low}})/n))$, with $C$ independent of $\\rho$ and of the two value levels, which the authors use to characterize the profit-maximizing airdrop.","pith_inferences":["A direct extension the authors leave implicit is a calibration recipe for practice: estimate $\\tau$ as the minimum viable participation level, $\\alpha$ as the recipients' opportunity cost, and $V_{\\mathrm{high}}-V_{\\mathrm{low}}$ as the value jump, then set the airdrop to the smallest $\\rho$ above $\\rho_c$; the same formula also predicts which projects are hopeless from the start.","Because the potential-game reduction applies to any anonymous technology, the threshold-style analysis can be repeated for quadratic network-value, linear, S-shaped, and concave technologies, and the paper's own appendix suggests each will have its own critical airdrop region.","If token prices are set by speculation rather than by participation value, the model's advice should be amended: the relevant value gap would be based on expected future participation, and the sharp threshold would likely become a band whose width depends on market beliefs.","The dependence of convergence time on $\\alpha$ suggests a testable extension: airdrop campaigns that target low-cost incumbents—existing validators or restakers—should both reach the good equilibrium faster and require a smaller critical airdrop than campaigns aimed at entirely new entrants, a comparison that field data from real launches could check."],"forward_implications":["For threshold technologies, a designer who can choose $\\rho$ freely can force the good equilibrium: announcing any $\\rho>\\rho_c$ makes the high-value outcome selected with probability tending to one, provided $\\alpha n\\tau<V_{\\mathrm{high}}-V_{\\mathrm{low}}$.","If the total cost of reaching the threshold exceeds the value gap, $\\alpha n\\tau>V_{\\mathrm{high}}-V_{\\mathrm{low}}$, no airdrop of any size can make the launch succeed in the vanishing-noise limit, and the best the designer can do is give nothing and keep the low value.","When $V_{\\mathrm{low}}>0$ there is an intermediate cost band in which a sufficiently large airdrop would make the system succeed but the profit-maximizing designer still prefers no airdrop; only below $\\alpha n\\tau=(V_{\\mathrm{high}}-V_{\\mathrm{low}})(1-V_{\\mathrm{low}}/V_{\\mathrm{high}})$ does setting $\\rho$ just above $\\rho_c$ maximize profit.","With finite noise, the success probability is monotone in $\\rho$ and follows the logistic law $p_{\\mathrm{high}}(\\rho)=1/(1+C\\exp(-\\rho\\beta(V_{\\mathrm{high}}-V_{\\mathrm{low}})/n))$, and for $V_{\\mathrm{low}}=0$ the optimal airdrop is either zero or at most $1-n/(\\beta V_{\\mathrm{high}})$.","The time to reach a participation level $\\ell^*=n/(1+\\exp(\\alpha\\beta))$ is short, but reaching the threshold $\\tau$ takes time exponential in $\\tau-\\ell^*$ when $\\ell^*<\\tau$, so low participation costs, not just high rewards, are what make a launch timely."],"supporting_citations":[{"why":"Supplies the logit-dynamics result that the stationary distribution concentrates on potential maximizers as $\\beta\\to\\infty$, used in Theorem 3 to characterize stochastically stable equilibria.","marker":"Blume, 1993, 2003"},{"why":"Defines the unique stationary equilibrium of logit dynamics in the finite-noise regime, the equilibrium concept used for the non-vanishing-noise results.","marker":"Auletta et al., 2011"},{"why":"Provides the sharp mixing-time bounds for birth-and-death chains (Theorem 1.1) used in Theorems 4 and 9 to bound convergence time.","marker":"Chen and Saloff-Coste, 2013"},{"why":"Gives the expected hitting-time identities for birth-and-death chains used in Theorems 5 and 10 to lower-bound the time to reach the threshold.","marker":"Palacios and Tetali, 1996"},{"why":"Supplies the standard mixing-time comparison used in the proof of Theorem 4's bounds.","marker":"Levin et al., 2006"}],"fun_headline_variants":["Airdrop size dictates blockchain equilibrium selection","Critical airdrop fraction flips game outcome","One number decides if airdrop succeeds or fizzles","Threshold airdrop size steers blockchain to success","Airdrop math: crossing the line makes or breaks launch"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that players update by logit choice probabilities with a common noise level and that the token price exactly equals fundamental participation value $V(a)/T_{\\mathrm{tot}}$; if real learning differs or speculation moves the price, the critical fraction $\\rho_c$ no longer governs equilibrium selection.","fun_headline_variants_meta":{"raw":{"variants":["Airdrop size dictates blockchain equilibrium selection","Critical airdrop fraction flips game outcome","One number decides if airdrop succeeds or fizzles","Threshold airdrop size steers blockchain to success","Airdrop math: crossing the line makes or breaks launch"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000396,"raw_usage":{"total_tokens":2100,"prompt_tokens":993,"completion_tokens":1107,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":609,"completion_tokens_details":{"reasoning_tokens":1030}},"tokens_in":609,"tokens_out":1107,"duration_ms":8794,"temperature":1.0,"reasoning_tokens":1030,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:53:14.750741+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a controlled participation experiment with $n$ binary contributors, known uniform cost $\\alpha$, and a threshold technology at $\\tau$, and set the airdrop fraction just below and just above $\\rho_c=\\alpha n\\tau/(V_{\\mathrm{high}}-V_{\\mathrm{low}})$; if the high-value outcome is selected with probability near one below $\\rho_c$, or the zero-participation outcome persists above $\\rho_c$ after many revisions, Theorem 6 is falsified. Alternatively, estimate the stationary distribution at finite noise: it must match the logistic form $1/(1+C\\exp(-\\rho\\beta(V_{\\mathrm{high}}-V_{\\mathrm{low}})/n))$ with $C$ independent of $\\rho$, so a robust deviation in how success probability depends on $\\rho$ would refute the model.","supporting_citations":[{"cited_title":"The statistical mechanics of strategic interaction","cited_arxiv_id":null,"evidence_quote":"Supplies the logit-dynamics result that the stationary distribution concentrates on potential maximizers as $\\beta\\to\\infty$, used in Theorem 3 to characterize stochastically stable equilibria."},{"cited_title":"Convergence to equilibrium of logit dynamics for strategic games","cited_arxiv_id":null,"evidence_quote":"Defines the unique stationary equilibrium of logit dynamics in the finite-noise regime, the equilibrium concept used for the non-vanishing-noise results."},{"cited_title":"On the mixing time and spectral gap for birth and death chains","cited_arxiv_id":null,"evidence_quote":"Provides the sharp mixing-time bounds for birth-and-death chains (Theorem 1.1) used in Theorems 4 and 9 to bound convergence time."}],"review_version":1}