Draw a picture. For each scenario, there are obvious circumstances that the payoff for each would be better.
For the N day option, the payoff would be better if there was a slow gradual decline in price and a slow gradual increase over the same period, such that the final difference in the price of the underlying was largely unchanged. For multiple options issued over that period, at expiration, there would be several options that would need to pay off, so that would be worse.
In the event of quick downward and upward movements, I'm unable to say which would be better. The multiple options would benefit from increased premium from higher implied volatilities, but will be further out of the money.
In the event of a slow decreasing or increasing underlying price, the multiple options would benefit from strike prices that are struck at different levels and therefore, the overall volatility of the strategy would be lower than just selling one option.
You will find that if you look at these strategies using historical prices, the multiple options will have a lower return, but the decrease in volatility will push the information ratio of the strategy higher than the individual sale. Given you are selling options, and depending on your cash coverage requirements, this could mean that, for the same volatility, you can put on a larger position in the second scenario and have a larger return than the first.