  We show a near optimal direct-sum theorem for the two-party randomized
communication complexity. Let $f\subseteq X \times Y\times Z$ be a relation,
$\varepsilon> 0$ and $k$ be an integer. We show,
$$\mathrm{R}^{\mathrm{pub}}_\varepsilon(f^k) \cdot
\log(\mathrm{R}^{\mathrm{pub}}_\varepsilon(f^k)) \ge \Omega(k \cdot
\mathrm{R}^{\mathrm{pub}}_\varepsilon(f)) \enspace,$$ where $f^k= f \times
\ldots \times f$ ($k$-times) and $\mathrm{R}^{\mathrm{pub}}_\varepsilon(\cdot)$
represents the public-coin randomized communication complexity with worst-case
error $\varepsilon$. Given a protocol $\mathcal{P}$ for $f^k$ with
communication cost $c \cdot k$ and worst-case error $\varepsilon$, we exhibit a
protocol $\mathcal{Q}$ for $f$ with external-information-cost $O(c)$ and
worst-error $\varepsilon$. We then use a message compression protocol due to
Barak, Braverman, Chen and Rao [2013] for simulating $\mathcal{Q}$ with
communication $O(c \cdot \log(c\cdot k))$ to arrive at our result. To show this
reduction we show some new chain-rules for capacity, the maximum information
that can be transmitted by a communication channel. We use the powerful concept
of Nash-Equilibrium in game-theory, and its existence in suitably defined
games, to arrive at the chain-rules for capacity. These chain-rules are of
independent interest.
