  The Hubble tension between the $\Lambda$CDM-model-dependent prediction of the
current expansion rate $H_0$ using Planck data and direct, model-independent
measurements in the local universe from the SH0ES collaboration disagree at
$>3.5\sigma$. Moreover, there exists a milder $\sim 2\sigma$ tension between
similar predictions for the amplitude $S_8$ of matter fluctuations and its
measurement in the local universe. As explanations relying on unresolved
systematics have not been found, theorists have been exploring explanations for
these anomalies that modify the cosmological model, altering
early-universe-based predictions for these parameters. However, new
cosmological models that attempt to resolve one tension often worsen the other.
In this paper, we investigate a decaying dark matter (DDM) model as a solution
to both tensions simultaneously. Here, a fraction of dark matter density decays
into dark radiation. The decay rate $\Gamma$ is proportional to the Hubble rate
$H$ through the constant $\alpha_{\rm dr}$, the only additional parameter of
this model. Then, this model deviates most from $\Lambda$CDM in the early
universe, with $\alpha_{\rm dr}$ being positively correlated with $H_0$ and
negatively with $S_8$. Hence, increasing $\alpha_{\rm dr}$ (and allowing dark
matter to decay in this way) can then diminish both tensions simultaneously.
When only considering Planck CMB data and the local SH0ES prior on $H_0$, $\sim
1$\% dark matter decays, decreasing the $S_8$ tension to $0.3\sigma$ and
increasing the best-fit $H_0$ by $1.6$ km/s/Mpc. However, the addition of
intermediate-redshift data (the JLA supernova dataset and baryon acoustic
oscillation data) weakens the effectiveness of this model. Only $\sim 0.5$\% of
the dark matter decays bringing the $S_8$ tension back up to $\sim 1.5 \sigma$
and the increase in the best-fit $H_0$ down to $0.4$ km/s/Mpc.
