Chris's new book is fantastic. It's clearly written, compelling, and highly original. Highly recommended.
Both Chris and I are skeptics that there is any duty to vote; indeed, we both argue that most people have a duty not to vote, though we offer different reasons.
Here, though, let me lay out what it would take to show there is a duty to vote. I want to help you all when you write your responses to Chris.
1. Overcome the particularity problem: It's not enough to point to some general duty, G, which you argue people possess, and then to point out that voting is a way of satisfying G. It's not even enough to show that voting is the best way to satisfy G. You need to show voting is the obligatory way of satisfying G.
For instance, suppose someone says, "You should exercise civic virtue, therefore vote!" The problem--which should be obvious but for some reason isn't--is that there are lots of ways of exhibiting civic virtue other than by voting. Indeed, voting isn't even especially good.
Even if voting were the best, that wouldn't obviously make it obligatory. Suppose you have a duty to do productive work, as Stuart White and Lawrence Becker argue. Maybe the best job you could take, given your talents, is medical doctor. But that doesn't mean you are obligated to be a medical doctor; you could be free and permitted to do something less optimal, such as being a philosophy professor.
1.1. Corollary: No bullshit when dealing with particularity: This may sound silly, but it's not. You can't just point to a special feature of voting and then declare the debate over. The special feature must plausibly explain why voting is special. I've refereed papers which say that voting is special because we do it together as a group. Sure, but why does that make it obligatory? No answer. You might as well say voting is special because it's the only major political act that rhymes with "boating" and "gloating".
1.2. A heuristic: If you offer R as a reason to vote, ask yourself, does R imply that something other than voting is obligatory? Does R pick out some other action as even better than voting?
2. You need to deal with bad voters and bad voting: Most defenders of the duty to vote think there is merely a general duty to vote some way, not a specific duty to vote well, for the better side. But then your argument for a duty to vote nevertheless needs to explain, carefully, why the voters for the worse parties or worse candidates aren't doing something morally bad and wrong.
For instance, it's an act of civic virtue if I cast a magic spell which prevent cops from exercising undue force. It's not an act of civic virtue if I instead train cops to get away with using undue force. I avoid free riding when I pay taxes for genuine public goods. I don't avoid free riding by burning the cash that would go toward them.
Many of the supposed arguments for a general duty to vote seem to imply that you must vote well, not merely that you must vote.
3. You need to follow the consequences of your argument. For instance, as Brennan and Lomasky have pointed out, one popular argument for a duty to vote leads to absurd conclusions. "If no one voted, it'd be a disaster, therefore you should vote." First, it wouldn't be a disaster, but secondly, notice the argument works even better for farming: "If no one farmed, we would all starve to death, therefore you should be a farmer." Dumb, right?
Some people argue that high levels of voting produce good consequences. (Probably false but OK.) Those same people almost never say apply this argument to other areas of life. They don't say, "There are very good consequences if everyone does [fill in the blank], so therefore it's obligatory to [fill in the blank]."
Does your argument for voting have implications you are not prepared to endorse? Think it through. After all, if you don't, Chris or I will, and you don't want to hand us free publications criticizing you, do you?
4. Don't magically assume that voters know the right way to vote and that voting well is easy. It isn't. Read up on voter psychology, which says most people just follow what others do and for most people, politics is not about policy. Look at what political hacks even supposedly smart, critical thinking philosophers are. (How many philosophers just parrot whatever the Democrats say today? Note that I wouldn't be surprised if all 3 Republican philosophers in the academy act the same with regard to the Republicans.) Consider how hard it is to be informed, not merely of the particulars, but of the social science needed to understand causation. Consider how difficult it is to predict what politicians will do if elected. (Did you expect Trump to be such a dove compared to Bush II? Did you expect Obama to use so many drone strikes? Why?) Consider how hard it is to find reliable thought leaders to follow.
Of course, it's pretty common for philosophers to say that in the US, it is uniquely easy to vote well: Just vote Democrat all the time. Obviously the Republicans are worse so voting well is easy.
5. Don't straw man liberals with half-baked arguments. I'm looking at you, communitarians! So many papers say, "Liberals emphasis rights but forget that there are duties too! Therefore, citizens should vote!" Not only is this dumb because by definition a claim right implies that others have duties, but it's not even an argument. Fine, citizens have duties too. Which ones do they have and why?
6. For political theorists specifically: Quoting an older theorist stating they believe there is a duty to vote is not an argument.
7. Also for political theorists: Saying something like, "A duty to vote is implicit in our democratic culture" is not an argument for a duty to vote. Is this implicit assumption correct?