Surmounting Arrow s Impossibility Theorem+ via Revolutionary Logicist AI
Abstract
Arrow’s stunning “Impossibility Theorem” (AIT) roughly says that, without a “dictator” who holds sway, it’s mathematically impossible for a group of agents in that group to have their individual preferences aggregated to yield preferences for the group as a whole. An immediate and equally stunning corollary of AIT is that a “meta” agent cannot make a decision based on input from an advisory group composed of agents, where that input is an aggregation of the preferences of the individuals in the group. This corollary leaves for instance a commander high and dry, as a matter of mathematics. Yet, there is an escape route. We see it, and wish to bring it into clear focus, so that this route can be taken. Our providing this route is tantamount to giving the sponsor of our work an AI system (A) that can be used by human decision-makers to surmount AIT. But the beauty of the situation is that any system powerful and innovative enough to enable a human decisionmaker to surmount AIT will be all the more effective when decision-making challenges are less severe than those posed by Arrow’s result (and by the family AIT+ that includes broader versions of Arrow’s original theorem; this family is discussed below in §5). Let’s be a bit more precise. G will be our group of human and/or machine agents ai, each of which have individual preferences pref i to be aggregated to yield preferences for G as a whole, i.e. pref G. AIT says that if the aggregation must obey certain seemingly inviolable principles, aggregation is impossible. It may be illuminating to view this distressing theorem from the point of view of an agent C outside and above the group G, as follows. Suppose human Carl = C is such an agent: Carl wishes to make a decision on the basis of what he learns from G; and what he learns from G is to be the systematic aggregation of the advice the individual agents ai in G have issued. Arrow’s theorem tells us that Carl can’t do this! Of course, Carl can dictatorially and brashly cast aside input from G and decide solely on his own, but then what would be the point of having G on hand in the first place? In both democracy as a whole (where Carl personifies a voting system and G the electorate), and in military decision-making specifically (where Carl is a commander and G is composed, in modern conflict, of humans and machines), casting aside G isn’t a live, or wise, option. So, AIT/AIT+ constitutes a severe and sobering barrier to group decision-making, and to meta-decision-making to be made on the basis of what individuals in an advisory group have said. Yet, again, confronted with AIT and its distressing consequences, we will invent and specify formal theory that enables engineering of the AI system A that, in collaboration with humans, enables this barrier to be surmounted.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Aug 20, 2019
- Source ID
- N000141912558
Entities
People
- Selmer Bringsjord
Organizations
- Office of Naval Research
- Rensselaer Polytechnic Institute
- United States Navy