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A flavour-tagged decay-time-dependent amplitude analysis of$B_s^0\to(K^+\pi^-)(K^-\pi^+)$ decays is presented in the $K^{\pm}\pi^{\mp}$mass range from 750 to 1600 MeV$/c^2$. The analysis uses $pp$ collision datacollected with the LHCb detector at centre-of-mass energies of $7$ and $8$ TeV,corresponding to an integrated luminosity of $3.0$ fb$^{-1}$. Severalquasi-two-body decay modes are considered, corresponding to $K^{\pm}\pi^{\mp}$combinations with spin 0, 1 and 2, which are dominated by the $K_0^*(800)^0$and $K_0^*(1430)^0$, the $K^*(892)^0$ and the $K_2^*(1430)^0$ resonances,respectively. The longitudinal polarisation fraction for the $B_s^0\toK^*(892)^0\overline{K}^*(892)^0$ decay is measured as $f_L=0.208 \pm 0.032 \pm0.046$, where the first uncertainty is statistical and the second issystematic. The first measurement of the mixing-induced $CP$-violating phase,$\phi_s^{d\overline{d}}$, in $b\to d\overline{d}s$ transitions is performed,yielding a value of $\phi_s^{d\overline{d}}=-0.10$ $\pm$ $0.13$ (stat) $\pm$$0.14$ (syst) rad.

First measurement of the $CP$-violating phase $\phi_s^{d\overline{d}}$ in $B_s^0\to(K^+\pi^-)(K^-\pi^+)$ decays