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.
- Unrestricted domain: yes, the result is a total order.
- No dictatorship: It does not correspond to the preferences of any individual but the group of a whole.
- Pareto's condition: Yes, if everyone prefers $X$ over $Y$ then the group will prefer $X$ and $Y$
- 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?
I am not sure about this from the utilitarian perspective...
Thank you for your help!