CSCE 475/875

Handout 8: The Impossibility Theorems

September 29, 2011

(Based on Shoham and Leyton-Brown 2011)

Existence of Social Functions

Is it possible that social functions of desired properties do not exist for some situations?

To simplify the discussions, let us first assume that all agents’ preferences are strict total orderings on the outcomes, rather than nonstrict total orders; denote the set of such orders as , and denote an agent ’s preference ordering as . Denote a preference profile (a tuple giving a preference ordering for each agent) as , and denote agent ’s preferences from preference profile as .

We also redefine social welfare functions to return a strict total ordering over the outcomes, In other words, we assume that no agent is ever indifferent between outcomes and that the social welfare function is similarly decisive.

(Note that that this assumption is not required for the results that follow.)

Finally, let us introduce some new notation. Social welfare functions take preference profiles as input; denote the preference ordering selected by the social welfare function , given preference profile , as When the input ordering is understood from context, we abbreviate our notation for the social

ordering as .

Arrow’s Impossibility Theorem, for Social Welfare Functions …

Definition 9.4.1 (Pareto efficiency (PE)) is Pareto efficient if for any implies that .

That is, when all agents agree on the ordering of two outcomes, the social welfare function must select that ordering. Observe that this definition is effectively the same as strict Pareto efficiency.

Definition 9.4.2 (Independence of irrelevant alternatives (IIA)) is independent of irrelevant alternatives if, for any and any two preference profiles i (o1 ≻′i o2 if and only if implies that if and only if .

That is, the selected ordering between two outcomes should depend only on the relative orderings they are given by the agents. (Note: Think about how this counters ranking voting schemes such as the Borda voting.)

Definition 9.4.3 (Nondictatorship) does not have a dictator if

Nondictatorship means that there does not exist a single agent whose preferences always determine the social ordering. We say that is dictatorial if it fails to satisfy this property.

Surprisingly, it turns out that there exists no social welfare function that satisfies these three properties for all of its possible inputs.

Theorem 9.4.4 (Arrow, 1951) If , any social welfare function that is Pareto efficient and independent of irrelevant alternatives is dictatorial.

What’s the implication of the above theorem? Think about how some agent(s) might be perceived in the environment, and how an agent can make use of this theorem in its reasoning … Does the above theorem tell us that we cannot hope to find a voting scheme that satisfies all of the notions of fairness that we find desirable?

Muller–Satterthwaite’s Impossibility Theorem, for Social Choice Functions …

Definition 9.4.5 (Weak Pareto efficiency) A social choice function is weakly Pareto efficient if, for any preference profile , if there exist a pair of outcomes and such that

This definition prohibits the social choice function from selecting any outcome that is dominated by another alternative for all agents.

The definition implies that the social choice rule must respect agents’ unanimous choices: if outcome is the top choice according to each , then we must have .

Definition 9.4.6 (Monotonicity) is monotonic if, for any and any preference profile with , then for any other preference profile with the property that , it must be that .

Monotonicity says that when a social choice rule selects the outcome for a preference profile , then for any second preference profile in which, for every agent , the set of outcomes to which is preferred under is a weak superset of the set of outcomes to which is preferred under , the social choice rule must also choose outcome . Intuitively, monotonicity means that an outcome must remain the winner whenever the support for it is increased relative to a preference profile under which was already winning.

Definition 9.4.7 (Nondictatorship) is nondictatorial if there does not exist an agent such that always selects the top choice in ’s preference ordering. F

Theorem 9.4.8 (Muller–Satterthwaite, 1977) If , any social choice function that is weakly Pareto efficient and monotonic is dictatorial.

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