= Solution
The usual residual-deviance chi-squared calibration is unreliable because random effects are estimated and integrated out, changing both the effective degrees of freedom and the null distribution. Use a <parametric bootstrap>: fit the reported GLMM; compute an observed dispersion statistic such as the Pearson statistic or its ratio to nominal residual degrees of freedom; for each bootstrap replicate draw ten player effects from $N(0,0.1168)$, simulate all 60 Poisson responses from the fitted conditional means, refit the same GLMM, and recompute the statistic. With $B$ replicates, estimate the upper-tail p-value by
$$
\widehat p=\frac{1+\#\{T_b\geq T_{\mathrm{obs}}\}}{B+1}.
$$
Reject at $5\%$ when $\widehat p<0.05$, equivalently when $T_{\mathrm{obs}}$ exceeds the empirical $95$th percentile of the bootstrap null distribution.
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