# Arrow's theorem, strategic thinking and utilitarianism

I have this problem as part of a course on Decision Theory, and was not sure about question a (4th condition of Arrow's theorem) and question dii (utilitarianism). I have provided the whole question to give context

6 people have 5 options on where to go on holiday. These are: $D, L, R, N$ and $S$

They decide on the following voting scheme: to divide the $5$ options into these $3$ categories:

Category 1: options they like

Category 2: options they find acceptable

Category 3: options they dislike

Options in category $1$ get $2$ points, options in category $2$ get $1$ points and options in category $3$ get $-1$ points

Group preferences are decided by summing individual scores for each of $D, L, R, N$ and $S$

If this does not produce a winning option, with the same maximum sum of scores, they agree to settle on option $D$

Question (a): Discuss this from the perspective of Arrow's theorem Will check whether or not the $4$ conditions of Arrow's condition are satisfied: Unrestricted domain, no dictatorship, Pareto's condition and independence of irrelevant alternatives.

1. Unrestricted domain: yes, the result is a total order.
2. No dictatorship: It does not correspond to the preferences of any individual but the group of a whole.
3. Pareto's condition: Yes, if everyone prefers $X$ over $Y$ then the group will prefer $X$ and $Y$
4. Independence of irrelevant alternatives: I am not sure. I think it holds because removing two options would not affect the preferences of two others, since they are purely ranked on a point-based system.

So I believe all axioms are satisfied but I could be wrong.

Question (b): Would this encourage strategic thinking?

In some instances, yes, for example if someone had $D$ as their second-favorite choice and knew that their favorite choice would never win, they could present $D$ as their favorite option, giving it a higher maximum score, at the expense of their true preference which would never win.

Question (c): Could Harsanyi's theory of utilitarianism be used as an alternative way to combine preferences?

Harsanyi's theory of utilitarianism:

If a planner obeys anonymity and strong Pareto principle then the function $f$ must be $f(U_1, U_2, ..., U_m)=U_1+U_2+...+U_m$

Anonymity: It does not matter which utility corresponds to which citizen.

Strong Pareto principle: If each individual is indifferent between two options then so is the planner. If no citizen prefers $x$ to $y$ and some prefer $y$ to $x$ then the planner will prefer $y$ to $x$

So the planner acts a sum utilitarian.

This could be used an alternative way, if each person ranks his least preferred option $0$ and most preferred as $1$. Then the option with highest summed utility will be chosen.

Question (di): Further option: any option that is put into category 3 (options they dislike) by anyone should be disregarded. Is this sensible from the perspective of Arrow's theory?

I think this will not satisfy "no dictator" axiom, since an individual could be decisive by putting everything but his favorite option into category 3, immediately leaving his most preferred option as his only choice.

Question (dii): Is it sensible from the perspective of Harsanyi's theory?

You know that the method in (a) cannot satisfy all four conditions: that would violate Arrow’s theorem. You are correct in thinking that it does not violate independence of irrelevant alternatives: the score for each option is independent is not affected by the presence or absence of any other option, so removing a losing option does not change the ranking of the remaining options. The condition that is violated is Pareto optimality. It’s possible that everyone likes both $D$ and $L$ but prefers $L$ to $D$, in which case $D$ and $L$ both get the maximum possible $12$ points, but the method selects $D$, not $L$.
In (b), note that the voters are not actually ranking the destinations: they’re categorizing them. Say that a voter’s first choice is $L$, but she’s sure that it won’t win, and $D$ is her second choice. She should still put $L$ in Category $1$, so that it gets at least $2$ points. Putting $D$ in Category $1$ isn’t strategic voting on her part unless $L$ is the only destination that she actually likes, and $D$ is merely acceptable; in that case moving $D$ up to Category $1$ to give it an extra point is both strategic voting and probably sensible. In other words, strategic voting may be worthwhile, but it’s not so obviously worthwhile as you may have been thinking.