Beyond Voting
~ we might escape strategic votes, in the land of machine intelligence ~
TL;DR — Voting seemed cursed since the 70’s, due to Gibbard’s Theorem, that any voting method is unfair in one way or another: dictatorial, strategic, or it gives only two choices. Well! As far as I can tell, Gibbard only seems to apply to ranked lists… while Machine Intelligence research shows that our algorithms are more competent when given vectors in a ‘high-dimensional latent space’ instead. Gibbard’s Theorem does NOT apply to such a data-type, and so that region of algorithms MIGHT hold a strategy-proof and collegiate voting system! I sketch the arguments for such a method, below.
Gibbard’s Bane
In 1973, Allan Gibbard expanded upon the intuitions of fellow researcher William Vickrey — to use Arrow’s Impossibility Theorem to prove that no voting system escaped strategic votes, unless only one or two controlled all. So much for a ‘more perfect union’!
And there we have sat, for 48 years. Yet, look at Gibbard’s phrase, explaining what must come first for a ‘trick’ ballot to yield better results for the person casting it: “Both a constitution and a voting scheme take preference n-tuples as arguments” (p.592, 3rd parag. emph. mine) “Preference n-tuples” are ranked lists; your first pick down to your least favorite.
And, let’s be clear — whenever you have ONLY this preference-list, then yes, Gibbard says there is no fair vote. (Choose Among: Strategic, Dictatorial, or Two-Options Only) Yet, that formulation of the problem cannot make any claim upon the case where each person’s preferences are expressed as a vector in a high-dimensional latent space (the kinds used by Variational Auto-Encoders and Transformers, those new-fangled A.I.).
Sure, that high-dimensional vector can be converted down to a preference-list, yet that conversion throws-away almost all the important information! When we want an algorithm to perform well, we can’t declare failure after denying it sufficient information. Consider the real-world examples in proven A.I. applications: A Variational Auto-Encoder takes an image (for instance) and converts that into a ‘latent-space vector’… which it can then convert back into the original image, successfully. Given all the information it needed, it was able to function properly. That ‘latent-space vector’ was holding-onto all the important information, and THAT is why the VAE could reconstruct the input. However, you can ask any ML researcher: “If we convert those latent-space vectors into scalars, to form an ordered list, will we still be able to reconstruct the input?” No, they will explain, because you threw-out all the important data when you converted a vector into a scalar.
Similarly, Transformers rely entirely upon their key and query vectors’ orientation. If you took the scalar length of each, or any distance metric, then they become meaningless. No, we cannot expect our best algorithms to fail with the best data, just because Gibbard’s algorithms fail when he gave them collapsed data. We simply do not know either way, at this moment.
Hints?
There are some vague intuitions to believe that Gibbard’s Theorem won’t hold for latent-space vectors and machine intelligence: with loss-minimization, such spaces are provably convex, doomed to succeed. A formulation of ‘minimal regret’ would seem to follow the same constraints. More importantly, consider what happens to someone’s ‘vote’ when they are strategic, and how that is portrayed and dealt-with in a latent-space:
Suppose Voter A wants Bernie Sanders to win, using a new voting system we decide to try. Yet, Voter A’s second-favorite, Kucinich, is more likely to win! So, to be strategic, Voter A places Kucinich at the bottom of their ballot, ‘claiming’ falsely that he is their least favorite. And this lie is enough to bring Kucinich down, letting Bernie win. A strategy!
Yet, if voters’ preferences are projected onto a high-dimensional latent-space, then they are naturally ‘clustered’ by similarity. So, that one Voter A, with Bernie at the top of their ballot and Kucinich at the very bottom (below Ted Cruz?!?) would be placed OUTSIDE all the normal clusters! “Weirdo. Obviously some kind of fraud or delusion,” say the clusters. And, if the method for settling on an elected choice tends toward a broad ‘mode’ among those clusters, then that ‘strategic’ vote is given LESS weight in the outcome.
That ‘mode-seeking’ is the key insight for how to design a cluster-based method which might prevent strategic voting, in its entirety: “the best outcome for you will happen when you are honest.” Because strategic votes are idiosyncratic, they are first to be ignored.
How do we find those clusters, from the voters? Machine Intelligence, again — it can read detailed interviews and longitudinal surveys from each person, encrypted homomorphic records of their voting history, aggregations of concerns and interests. Put all that together to form the ‘profile’ of each voter, which the Variational Auto-Encoder places somewhere in its cupboard of clustered vectors. If someone wants to ‘vote strategically’, then they must plan long in advance to fake their entire history up to that point, if only to fool Kucinich ONCE. That sounds unlikely.
So, there are compelling reasons to check if a voting system might succeed by using latent-space vectors, despite the fact that Gibbard proved all preference-orderings will fail. A “failure of algorithms upon the preference-ordering” does NOT imply a “failure of algorithms upon the latent-space vectors.” A new vista is open to us.
