Tuesday, June 17, 2014
Arrow's Theorem and Indecision
Whether or not one can demonstrate a link between Arrow's theorem and procrastination, the possibility seems intriguing. In essence (to be elaborated below), Arrow's theorem declares the impossibility of arriving at a satisfactory consensus when there are more than two options and more than two deciders, and a definition of "satisfactory" that most might find reasonable and even minimal. This could lead to apparent procrastination if one considers the actor to have more than two possible activity options and more than two purposes that same actor wishes to achieve; the actor may be unable to reconcile the competing purposes and decide what it would be best to do now. This effect could have consequences for design of personal time management systems.
Arrow's (impossibility) TheoremAlso known as "Arrow's paradox," the theorem can be expressed in several ways; the Wikipedia article "Arrow's Impossibility Theorem" provides both informal and formal definitions. In essence (I hope I'm getting this right) it says that when there are three or more options to rank, and three or more ways to rank them (multiple voters, or multiple criteria for deciding the ranking), it is not possible to be sure a method for computing a consensus will respect all three of the following desiderata:
- unanimity prevails: if an option is ranked top in all rankings, it will be ranked top in the consensus.
- insensitivity to irrelevant alternatives: removing an option other than the winner should not change the consensus result.
- no dictators: there is no individual (source of a ranking) whose ranking will always match the consensus ranking.
A Sort of "Perfection Paralysis"If we cast this in terms of a personal decision about what to do next, any rule we might apply can only satisfy two of these three constraints. If we choose to protect the first and third (unanimity and non-dictatorship) we will be subject to unstable choices. This would lead to a situation similar to "perfection paralysis"; perfection paralysis has procrastinators postponing the risk of failure, this conundrum has procrastinators postponing the risk of pursuing the wrong activity.
For instance, we might choose to weigh importance, urgency, and likelihood of a quick success as the criteria for our possible activities, to name just three, and rank activities on those criteria. If something is at the top on all three criteria (a stop in the restroom, for instance) the unanimity rule says we should do that first. But we should not always pick the most urgent (nor always the most important or the surest to deliver a quick success), our other ranking criteria must matter. If our decision rule respects those two constraints, then eliminating even low ranked options is liable to destabilize our consensus rankings of the remaining options.
Consequences and SolutionsThe consequences of this are well known and studied in social welfare functions and social choice theory, particularly voting systems design. The Wikipedia article discusses various was of attempting to change the decision problem to avoid using the problem, such as not using a ranking system, or not having more than two choices.
An example of this latter approach, and its flaws, can be found in the French presidential election system. After a first round, if no candidate has a majority, only the top two candidates compete in a run-off election. The result is always a clear majority for the elected candidate. However, there can be a "Condorcet candidate" (this occurred in 2007), one who would beat either of the top two candidates in a run-off, but who places third (or lower) in the first round and so cannot be elected.
In personal time and agenda management, similar tactics may be introduced to make decisions possible. For instance, it is common to manage "work" and "life" separately, so that the criteria operable are limited to those of one or the other sphere. We don't decide whether a meeting is more urgent and important than going to the opera, we decide whether or not it fits (is more urgent and important than other work activities) in the time we reserve for work, and we decide whether or not we have time to go to the opera after work. However, this may lead to poorly considered decisions when we should be applying more criteria because more aspects matter to us. "Life" may be self, health, spouse, parents, children, and more: should each be considered separately, or should we plan our attention and efforts for each globally, as globally as we can? How can we?
NotesIt is not clear that people are generally aware of the problem, but who knows whether they are? It has an influence on our inability to manage trade-off decisions when more than one criterion is involved.
As I wrote this note, my purpose--and plan for it--evolved. I wanted to note, before I forgot, the likelihood that Arrow's paradox has consequences for agenda and to-do list management, but maybe not yet develop that aspect analytically. I wanted to note my questions about whether there is a general procedure similar to the one used for NP problems, to show that a decision problem is Arrow-ranking-paradoxical. While I was at it, the note should be of interest and use to others.