A bstract A flavour-tagged decay-time-dependent amplitude analysis of B s 0 → ( K + π − )( K − π + ) decays is presented in the K ± π ∓ mass range from 750 to 1600MeV/ c 2 . The analysis uses pp collision data collected with the LHCb detector at centre-of-mass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3 . 0 fb −1 . Several quasi-two-body decay modes are considered, corresponding to K ± π ∓ 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 $$ {B}_s^0\to {K}^{\ast }(892){}^{\circ}{\overline{K}}^{\ast }{(892)}^0 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mi>s</mml:mi> <mml:mn>0</mml:mn> </mml:msubsup> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mfenced> <mml:mn>892</mml:mn> </mml:mfenced> <mml:mi>°</mml:mi> <mml:msup> <mml:mover> <mml:mi>K</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>∗</mml:mo> </mml:msup> <mml:msup> <mml:mfenced> <mml:mn>892</mml:mn> </mml:mfenced> <mml:mn>0</mml:mn> </mml:msup> </mml:math> decay is measured as f L = 0 . 208 ± 0 . 032 ± 0 . 046, where the first uncertainty is statistical and the second is systematic. The first measurement of the mixing-induced CP -violating phase, $$ {\phi}_s^{d\overline{d}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ϕ</mml:mi> <mml:mi>s</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mover> <mml:mi>d</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:msubsup> </mml:math> , in $$ b\to d\overline{d}s $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>b</mml:mi> <mml:mo>→</mml:mo> <mml:mi>d</mml:mi> <mml:mover> <mml:mi>d</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>s</mml:mi> </mml:math> transitions is performed, yielding a value of $$ {\phi}_s^{d\overline{d}}=-0.10\pm 0.13\left(\mathrm{stat}\right)\pm 0.14 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ϕ</mml:mi> <mml:mi>s</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mover> <mml:mi>d</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mn>0.10</mml:mn> <mml:mo>±</mml:mo> <mml:mn>0.13</mml:mn> <mml:mfenced> <mml:mi>stat</mml:mi> </mml:mfenced> <mml:mo>±</mml:mo> <mml:mn>0.14</mml:mn> </mml:math> (syst) rad.