I am wondering what the relationship is between skewness, kurtosis and mean variance efficiency is.
Is it correct that particular investors are willing to give up mean variance efficiency in return for greater probability of positive returns. Does this mean that a investor trades a lower average return for bearing more exposure towards positively skewed return distributions and a higher kurtosis? Similar to a lottery type pay-off function? Or do they accept a lower risk adjusted return instead of average return? Holding other factors constant such as risk aversion.
Mean-variance efficiency generally only considers the first two moments. Skew and kurtosis are outside of it, and Sharpe is commonly criticized for not accounting for skew and kurtosis (leading to metrics like Omega as a replacement).
Regarding your specific question, it depends entirely on what risks a given investor/trader wants to take. One person may prefer a return stream with smaller average returns but a small chance of a large payoff as a result of skew and/or kurtosis. Another may prefer small steady profits with a small risk of large losses (eg, selling deep OTM puts). All depends on which risks you're willing to take on.
Some people have loss aversion. Their satisfaction from gaining \$1 is significantly smaller than their pain from losing \$1. People with this condition may give up potential gains to avoid small potential losses.