Symmetric or skewed? Compare the two books

You're long $4{,}000$ shares of a stock worth $20.00$ whose position P&L has per-share variance $\sigma^2 = 0.04$ dollars$^2$ (so \sigma = \0.20$). You're choosing between two quoting styles for the next interval:

• Symmetric: quote $19.96$ / $20.04$. Expected spread income \60$, but your position stays around$q = +4{,}000$.
• Skewed: quote $19.93$ / $20.01$. Expected spread income \40$(thinner edge on the eager sell side), but the lean is expected to cut your position to about$q = +1{,}500$.

Score each with the risk-adjusted objective $U = E[\text{profit}] - \tfrac{\gamma}{2}\,\sigma^2 q^2$, using risk aversion $\gamma = 0.0005$.

Which strategy wins, and why?