Forward relative performance processes with CRRA wealth utility and separable time-space dependence necessarily induce matching CRRA consumption utility, yielding closed-form Nash equilibria for n-player and mean-field games with consumption.
An exact connection between two solvable SDEs and a nonlinear utility stochastic PDE.SIAM Journal on Financial Mathematics, 4(1):697–736
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Optimal investment and consumption under forward utilities with relative performance concerns
Forward relative performance processes with CRRA wealth utility and separable time-space dependence necessarily induce matching CRRA consumption utility, yielding closed-form Nash equilibria for n-player and mean-field games with consumption.