Definition. An interior automation game [ftip-00OB]
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    Fix an integer \(N>1\), \(L,k>0\), and \(0<\ell <s<k+\ell / N\). Firm \(i\) chooses \(\alpha _i\in [0,1]\), average automation is \(\bar \alpha =N^{-1}\sum _i\alpha _i\), and its profit is

    \[\pi _i=\Pi _0+L\left (s\alpha _i-\ell \bar \alpha -\frac {k}{2}\alpha _i^2\right ).\]

    The saving \(s\) and demand leakage \(\ell \) are fixed parameters. In the seed model \(s=w-c\) and \(\ell =\lambda (1-\eta )w\), with wage \(w\), automation cost \(c\), spending propensity \(\lambda \) and replacement-income fraction \(\eta \). These are static parameters; they do not specify training expenditure or a law of economic development.

    Strict concavity gives the unique Nash choice and the unique joint-profit maximizing choice, both interior:

    \[\alpha ^{\rm NE}=\frac {s-\ell / N}{k},\qquad \alpha ^{\rm CO}=\frac {s-\ell }{k}.\]

    Proof. 

      Differentiate individual profit in \(\alpha _i\) to obtain \(L(s-\ell / N-k\alpha _i)\). Differentiate total profit in each choice to obtain \(L(s-\ell -k\alpha _i)\). The stated inequalities put both stationary choices strictly between zero and one; negative second derivatives establish the maxima.