Once upon a time, another pastor told me about a new fundraising scheme being used at the church she served: the "Lord's Lottery." Everyone was encouraged to use envelopes for their offering and put their name on it. At the end of the service, they would draw an envelope and the person would get back twice as much as they had put in. "Offerings are up!" the pastor declared proudly.
Well, yes, offerings may well have been up, but Christian stewardship was certainly down. Other things that were up: self-centered-ness, greed, the list goes on and on. Christian stewardship is about selflessness and putting God at the center of your life and relying on him, first, last and always. (Among other things.)
For exactly these same reasons, the Arizona voting lottery is a bad idea. We should all be participating in the electoral process, but if you do not feel obliged, you should not be bribed or otherwise taking part out of some sort of selfish motive. Enlightened self interest is one thing; an appeal to greed is another.
And yes, I know that some politicians make promises that are more or less than exactly that. But that is different than the government doing it.
I used to worry about low voter turnouts. I do not any more. I want an educated, interested electorate showing up to vote. If you will not take the time and interest to educate yourself, it is ok. Just stay home.