maximum approval
i’ve recently written specifically on two common fallacies in the anti-utilitarian vein: majority rule and maximin. but recently i saw the twitter user BringBackLevity endorsing another commonly cited, tho similarly debunked, alternative to utilitarianism. he proposes that we should maximize the number of people who “approve” of a candidate.
there are two critical flaws with this social welfare function. first, it creates arbitrarily large biases for a single individual. imagine candidate x makes a million voters ecstatic, and one voter infinitesimally below their “approval” threshold, while candidate y is exactly at every voter’s approval threshold. levity’s proposal is that we must elect y, thus making a million voters arbitrarily worse off, for the sake of making a single voter infinitesimally better off. we can also do this in reverse. imagine candidate x puts all voters infinitesimally below their approval threshold, while candidate y is approved by one voter, and makes a million voters arbitrarily worse off (e.g. he tortures or even kills them). again, levity’s proposal mandates we elect y, making a million voters arbitrarily worse off in order to make a single voter infinitesimally better off.
but beyond this, levity’s conception contains another unstated misconception: it conflates a voting method with a social welfare function. a voting method is a specific procedure for casting ballots and determining a winner, while a social welfare function is a formula for aggregating preferences. the “approval” a voter indicates on a ballot is not a preference, it’s an action taken by a voter, which is a function of preferences as well as normalization error and strategy. for example, you may hate both w and z, but if they’re the frontrunners, your best strategy is to approve whichever of them you hate less. this doesn’t mean either of them has met your “approval” threshold utility.
reductio ad absurdum
many armchair voting theorists are ignorant of the basic facet of logic that you can actually logically prove things about subjective statements. this seems counterintuitive to a lot of people, but you can do this if the statements lead to “absurdities” — logical self-contradictions. this technique is literally called reductio ad absurdum — latin for “reduction to an absurdity”, and is the basis for our modern mathematically proven understanding that the social welfare function must be the simple sum of the individual utilities of the group members. but BringBackLevity was unaware of this, and made the common intuitive fallacy, saying, “Preferences are subjective. They can’t be ‘mathematically proven.’”
with that in mind, let us return to our two election scenarios featuring candidates x and y. now levity might argue that he’s okay with making an arbitrary number of voters arbitrarily worse off in exchange for making a single voter better off by an arbitrarily small amount — in which case he’s not actually caught in a self-contradiction per se. but his proposal still seems absurd to virtually anyone you ask. you can simply ask, “would you rather have x or y in either of these scenarios, given you’re a random voter?” the cost-benefit calculus is simple. if x wins, there’s a massive 1,000,000 in 1,000,001 chance you’ll be radically happier than if y wins — and a vanishingly small 1 in 1,000,001 chance you’ll be a tiny bit less happy. we know how most people will answer this, because we see people making decisions like this in the market every day. people may take a smaller chance of a higher payoff, or a higher chance of a smaller payoff — but no one takes the higher chance of a smaller payoff.
there’s a deeper practical issue at play in this expected payoff framing, which gets to the underpinnings of choice. it’s critical to understand that the entire point of decision making is to maximize one’s expected utility, and that an election is just a decision made by more than one person. so the same principles of individual decision making apply: voters logically want the voting method that maximizes their expected satisfaction, given uncertainty about how future elections will go. this concept was articulated by the economist john harsanyi:
Let uk(e) denote the utility (according to individual k) of an event e. We want to investigate how to aggregate the individual utilities into a “social utility” saying “how good e is for all of society.” For what reason should we claim that social utility is just the average of individual utilities?
J.C.Harsanyi, in a 2-page article involving no mathematics whatever [J.Political Economy 61,5 (1953) 434–435], came up with the following nice idea: “Optimizing social welfare” means “picking the state of the world all individuals would prefer if they were in a state of uncertainty about their identity.” I.e. if you are equally likely to be anybody, then your expected utility is the summed utility in the world divided by the number of people in it — i.e. average utility. Then by the linear-lottery property (Lin) of von Neumann utility, it follows that social utility is averaging.
but all of this rich established body of research is lost on levity, who is blissfully unaware of it. even after these issues were pointed out to him, he again made the error of confusing social welfare functions and voting methods:
One perspective on the best way to run a society is to have each person state what they want and how much, add up how much everyone wants each thing, and go with the top option.
Another perspective is that we should do that, but then rerun it with the top two.
here he’s describing voting methods. but the point of contention is what is the right social welfare function. once you have the correct social welfare function, you measure the performance of a voting method by averaging its social utility efficiency using that function.
