{
  "format_version": 3,
  "claim_natural": "Consider a sector with \\(N \\geq 2\\) symmetric firms, each endowed with L task-positions. Each firm i chooses an automation rate \\(\\alpha_i\\) in [0,1], paying wage w per human-staffed task and cost c per automated task, with integration friction cost \\((k/2) \\cdot L \\cdot \\alpha_i^2\\) where \\(k > 0\\). Aggregate sectoral demand is \\(D = A + \\lambda \\cdot w \\cdot L \\cdot [N - (1-\\eta) \\sum_j \\alpha_j]\\), where \\(A > 0\\) is exogenous demand, \\(\\lambda \\in (0,1]\\) is workers' marginal propensity to consume from wages, and \\(\\eta \\in [0,1)\\) is the fraction of displaced wage income recovered through reemployment. Each firm's revenue is \\(D/N\\) (equal market shares). Define \\(s = w - c > 0\\) and \\(\\ell = \\lambda(1-\\eta)w > 0\\). Each firm i maximizes \\(\\pi_i = D/N - wL(1-\\alpha_i) - cL\\alpha_i - (k/2)L\\alpha_i^2\\). The Nash equilibrium automation rate is \\(\\alpha^{NE} = (s - \\ell/N)/k\\). The cooperative optimum is \\(\\alpha^{CO} = (s - \\ell)/k\\). The difference \\(\\alpha^{NE} - \\alpha^{CO} = \\ell \\cdot (1 - 1/N)/k\\) is strictly positive.",
  "claim_formal": {
    "subject": "Symmetric N-firm automation game with demand externality",
    "property": "All three equilibrium results hold",
    "operator": "==",
    "operator_note": "The claim asserts three algebraic identities derived from first-order conditions of the given model: (1) the symmetric Nash equilibrium rate alpha^NE = (s - ell/N)/k, (2) the cooperative (joint) optimum alpha^CO = (s - ell)/k, and (3) the gap alpha^NE - alpha^CO = ell*(1 - 1/N)/k > 0. These are interior solutions from unconstrained FOCs; the claim implicitly assumes parameters place the optima in (0,1). Strict positivity of the gap follows from ell > 0 and N >= 2.",
    "threshold": true,
    "is_time_sensitive": false
  },
  "evidence": {
    "A1": {
      "type": "computed",
      "label": "Nash equilibrium rate alpha^NE = (s - ell/N)/k via FOC",
      "sub_claim": null,
      "method": "SymPy symbolic differentiation of pi_i w.r.t. alpha_i; solve FOC; verify result equals (w-c-lambda(1-eta)w/N)/k",
      "result": "Confirmed: residual = 0",
      "depends_on": []
    },
    "A2": {
      "type": "computed",
      "label": "Cooperative optimum alpha^CO = (s - ell)/k via joint FOC",
      "sub_claim": null,
      "method": "SymPy symbolic differentiation of total profit Pi w.r.t. alpha at symmetric profile; solve FOC; verify result equals (w-c-lambda(1-eta)w)/k",
      "result": "Confirmed: residual = 0",
      "depends_on": []
    },
    "A3": {
      "type": "computed",
      "label": "Gap alpha^NE - alpha^CO = ell*(1 - 1/N)/k > 0",
      "sub_claim": null,
      "method": "SymPy symbolic subtraction alpha^NE - alpha^CO; verify equals lambda(1-eta)w(1-1/N)/k; positivity from parameter assumptions",
      "result": "Confirmed: gap formula correct, strictly positive for N >= 2",
      "depends_on": [
        "A1",
        "A2"
      ]
    }
  },
  "cross_checks": [
    {
      "description": "Numerical spot-check at N=5, w=1, c=0.5, k=1, lambda=0.5, eta=0.4 (s=0.5, ell=0.3)",
      "fact_ids": [
        "A1",
        "A2",
        "A3"
      ],
      "agreement": true,
      "values_compared": [
        "NE FOC residual = 0.00e+00",
        "CO FOC residual = 0.00e+00",
        "Gap direct = 0.24, gap formula = 0.24"
      ]
    },
    {
      "description": "Second-order conditions verified symbolically (strict concavity of individual and joint profit)",
      "fact_ids": [
        "A1",
        "A2"
      ],
      "agreement": true,
      "values_compared": [
        "d^2 pi_i/dalpha_i^2 = -kL: True",
        "d^2 Pi/dalpha^2 = -NkL: True"
      ]
    }
  ],
  "adversarial_checks": [
    {
      "question": "Does the FOC yield a maximum (not minimum or saddle)?",
      "verification_performed": "Computed second-order conditions symbolically with SymPy. d^2 pi_i / d alpha_i^2 = -kL < 0 (k > 0, L > 0): strict concavity. d^2 Pi / d alpha^2 = -NkL < 0: joint problem also strictly concave. Both SOCs confirmed computationally.",
      "finding": "SOC confirmed: both individual and joint profit are strictly concave, so FOC solutions are global maxima.",
      "breaks_proof": false
    },
    {
      "question": "Is the symmetric Nash equilibrium unique?",
      "verification_performed": "Examined the best-response function. The FOC solution alpha_i* = (s - ell/N)/k does not depend on rivals' strategies, making it a dominant strategy. Strict concavity ensures uniqueness of each firm's best response.",
      "finding": "The NE is unique: alpha_i* is a dominant strategy, independent of rivals' choices. No asymmetric equilibria exist.",
      "breaks_proof": false
    },
    {
      "question": "Could the interior solutions lie outside [0,1]?",
      "verification_performed": "Checked parameter conditions for interiority. alpha^NE in (0,1) requires ell/N < s < k + ell/N. alpha^CO in (0,1) requires ell < s < k + ell. The claim derives unconstrained FOC solutions; interiority is a parameter assumption noted in operator_note.",
      "finding": "Formulas are correct as interior FOC solutions. Whether they fall in [0,1] depends on parameter magnitudes, which is an implicit assumption of the claim.",
      "breaks_proof": false
    },
    {
      "question": "Can aggregate demand D become negative?",
      "verification_performed": "At maximum automation (all alpha_j = 1): D = A + lambda*w*L*N*eta >= A > 0. D is linear and decreasing in each alpha_j, so it is minimized at full automation. Since D > 0 even there, D > 0 always.",
      "finding": "D >= A > 0 for all feasible profiles. Model is well-specified.",
      "breaks_proof": false
    }
  ],
  "verdict": {
    "value": "PROVED",
    "qualified": false,
    "qualifier": null,
    "reason": null
  },
  "key_results": {
    "A1_NE_verified": true,
    "A2_CO_verified": true,
    "A3_gap_verified": true,
    "numerical_crosscheck_passed": true,
    "SOC_verified": true
  },
  "generator": {
    "name": "proof-engine",
    "version": "1.18.0",
    "repo": "https://github.com/yaniv-golan/proof-engine",
    "generated_at": "2026-04-16"
  },
  "proof_py_url": "/proofs/consider-a-sector-with-n-2-symmetric-firms-each-endowed-with-l-task-positions/proof.py",
  "citation": {
    "doi": null,
    "concept_doi": null,
    "url": "https://proofengine.info/proofs/consider-a-sector-with-n-2-symmetric-firms-each-endowed-with-l-task-positions/",
    "author": "Proof Engine",
    "cite_bib_url": "/proofs/consider-a-sector-with-n-2-symmetric-firms-each-endowed-with-l-task-positions/cite.bib",
    "cite_ris_url": "/proofs/consider-a-sector-with-n-2-symmetric-firms-each-endowed-with-l-task-positions/cite.ris"
  },
  "depends_on": []
}