According to the formula for pricing binary options with a volatility skew, it appears that the value of the binary option for a given strike gets lower, the higher the volatility skew at that strike. Why is this, intutively? I (think I) understand the derivation, but intuitively it seems like a positive volatility skew should make far OTM binary call options more expensive, not cheaper, because it should create "fatter tails" in the implied PDF of the price of the underlying at that expiry.
Just on pure intuition (close to a guess); the binary does not pay you for the fat tail. All you want is high volatility locally around the strike to add to the chances for the option to expire in the money. Just narrowly in the money counts as much as far out on the tail.
Edit: The put spread in the other answer above could provide a good starting point to deepen the intuition. It buys volatility nearer the current underlying price and sells it back farther out on the tail. Narrow it down to (an out of the money, OTM) binary: it is long volatilty OTM and short volatility (vola) in the money all (ITM) the way out on the tail. The fat-tail distribution has relatively low vola when we are OTM and want vola to put us ITM. It has high vola when we are ITM and want low vola to stay there.
The takeaway here I guess is not to look at the expected return (P-probability): even though the chances for the very far OTM binary to expire ITM may increase with the fatness of the tail, its fair price will not! The fair price is what you would pay to break-even when you carry the option risklessly with a dynamic delta-hedge in the underlying until expiry (risk neutral Q-probability).
And where you are OTM you are long vola and long gamma and reap high rebalancing profits when vol is high. Where you are ITM and out on the tail it is the reverse, you lose in rebalancing when the large high vola moves producing the fatness even further out occur. Forget the (non-risk adjusted P-probability) expected return and look at how the vola skew matches where you will be long and short vola and gamma!