= Solution
In the first tableau the basis is $(x_2,s_5,x_1)$, giving
$$
x=(1/9,2/9,0),\qquad s=(0,2/3,0).
$$
The second basis is $(r_1,y_6,r_3)$, giving $y=(0,0,1/4)$ and $r=(3/4,0,1/4)$. This pair is not yet a <Nash equilibrium>: label $1$ is missing and label $4$ is duplicated.
Resolve the duplicate by bringing $y_4$ into the second tableau. Its column is $(3/4,1/4,9/4)^T$, so the <simplex ratio test> gives
$$
\min\left\{\frac{3/4}{3/4},\frac{1/4}{1/4},\frac{1/4}{9/4}\right\}=\frac19.
$$
Variable $r_3$ leaves, giving
$$
y=(1/9,0,2/9),\qquad r=(2/3,0,0).
$$
Now label $3$ is duplicated. Bring $x_3$ into the first tableau; its column is $(0,3,1)^T$. The positive-entry ratios are $(2/3)/3=2/9$ and $(1/9)/1=1/9$. Thus $x_1$ leaves, giving
$$
x=(0,2/9,1/9),\qquad s=(0,1/3,0).
$$
All labels are now present and $x_ir_i=y_js_j=0$. Each unnormalized strategy has total mass $1/3$, so the <Lemke-Howson algorithm> produces
$$
\boxed{x'=(0,2/3,1/3),\qquad y'=(1/3,0,2/3).}
$$
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