1. CMPE539 SOCIAL CHOICE
Dr. ADNAN ACAN

2. SOCIAL CHOICE
In This chapter we want to examine the “designer perspective”: we ask what rules should be put in place by the authority (the “designer”) orchestrating a set of agents.
A simple example of the designer perspective is voting. How should a central authority pool the preferences of different agents so as to best reflect the wishes of the population as a whole?

3. SOCIAL CHOICE
Voting is only a special case of the general class of social choice problems.
Social choice is a motivational but nonstrategic theory - agents have preferences, but they do not try to camouflage them in order to manipulate the outcome (of voting, for example) to their personal advantage.
Example: plurality voting - ask each agent to vote for his favorite preference and then pick the one that received the largest number of votes. This is called the plurality voting method.

4. SOCIAL CHOICE
While quite standard, plurality voting is not without problems:
For one thing, we need to select a tie-breaking rule (e.g., we could select the candidate ranked first alphabetically). A more disciplined way is to hold a runoff election among the candidates tied at the top.
Even absent a tie, however, the method is vulnerable to the criticism that it does not meet the Condorcet condition: This condition states that if there exists a candidate x such that for all other candidates y at least half the voters prefer x to y, then x must be chosen.

5. SOCIAL CHOICE
Let’s consider the following practical case: There are three voters X, Y, and Z who vote for three preferences a, b, and c. Let ab denote that outcome a is preferred to outcome b. Asssume that the voters ordered their preferences as follows:
X: ab c
Y: bc a
Z: cb a
Standard plurality would declare a tie between all three candidates and would choose a. However, the Condorcet condition would choose b, since two of the three voters prefer b to a, and likewise prefer b to c.

6. SOCIAL CHOICE
Based on this example the Condorcet rule might seem unproblematic (and actually useful since it breaks the tie without resorting to an arbitrary choice such as alphabetical ordering), but now consider a similar example in which the preferences are as follows.
X: a ≻ b ≻ c
Y: b ≻ c ≻ a
Z: c ≻ a ≻ b
In this case the Condorcet condition does not tell us what to do, illustrating the fact that it does not tell us how to aggregate arbitrary sets of preferences.
Hence, we have seen that social choice is not a straightforward matter. In order to study it precisely, we must establish a formal model.

7. SOCIAL CHOICE
A formal model: Let N = {1, 2, , n} denote a set of agents, and let O denote a finite set of outcomes (or alternatives, or candidates). Let’s denote the proposition that «agent i weakly prefers outcome o1 to outcome o2» by o1 i o2. The notation
o1 ≻i o2 is used to capture strict preference, (shorthand for
o1 i o2 and not o2 i o1) the notation o1 ∼i o2 is used to capture
indifference (shorthand for o1 i o2 and o2 i o1).

8. SOCIAL CHOICE
Because preferences are transitive, an agent’s preference relation induces a preference ordering, a (nonstrict) total ordering on O. Let preference L- be the set of nonstrict total orders; we will understand each agent’s preference ordering as an element of L-.
We denote an element of L- using the same symbol we used for the relational operator: i ∈ L-.
Likewise, we define a preference profile [i] ∈ L-n as a tuple giving a preference ordering for each agent.

9. SOCIAL CHOICE
We have altready seen that preference orderings and utility functions are tightly related. We can define an ordering i ∈ L- in terms of a given utility function ui : O → R for an agent i by requiring that o1 is weakly preferred to o2 if and only if ui(o1) ≥ ui(o2).
We define two kinds of social functions:
Social choice functions simply select one of the alternatives:
Definition (Social choice function): A social choice function (over N and O) is a function C : L-n → O.

10. SOCIAL CHOICE
A social choice correspondence differs from a social choice function only in that it can return a set of candidates, instead of just a single one.
Definition (Social choice correspondence) : A social choice correspondence (over N and O) is a function C : L-n → 2O.

11. SOCIAL CHOICE
In the previous example there were three agents (X, Y and Z) and three possible outcomes (a, b, c). The social choice correspondence defined by plurality voting of course picks the subset of candidates with the most votes; in this example either the subset must be the singleton consisting of one of the candidates or else it must include all candidates.
X: ab c
Y: bc a
Z: cb a
Plurality is turned into a social choice function by any deterministic tie-breaking rule (e.g., alphabetical).

12. SOCIAL CHOICE
Let #(oi ≻ oj) denote the number of agents who prefer outcome oi to outcome oj under preference profile [i] ∈ L-n. We can now give a formal statement of the Condorcet condition.
Definition (Condorcet winner) An outcome o ∈ O is a Condorcet winner if ∀o′ ∈ O, #(o ≻ o′) ≥ #(o′ ≻ o).
A social choice function satisfies the Condorcet condition if it always picks a winner when one exists.
We saw earlier that for some sets of preferences there does not exist a Condorcet winner. (Indeed, under reasonable conditions the probability that there will exist a Condorcet winner approaches zero as the number of candidates approaches infinity.) Thus, the Condorcet condition does not always tell us anything about which outcome to choose.

13. SOCIAL CHOICE
An alternative is to find a rule that identifies a set of outcomes among which we can choose.
Definition (Smith set) The Smith set is the smallest set S ⊆ O having the property that ∀o ∈ S, ∀o′  S, #(o ≻ o′) ≥ #(o′ ≻ o).
That is, every outcome in the Smith set is preferred by at least half of the agents to every outcome outside the set. This set always exists. When there is a Condorcet winner then that candidate is also the only member of the Smith set; otherwise, the Smith set is the set of candidates who participate in a “stalemate” (or “top cycle”).

14. SOCIAL CHOICE
The other important flavor of social function is the social welfare function. These are similar to social choice functions, but produce richer objects, total orderings on the set of alternatives.
Definition (Social welfare function) A social welfare function (over N and O) is a function W : L-n → L-.

15. VOTING
Voting methods
The most standard class of voting methods is called nonranking voting, in which each agent votes for one of the candidates. we have already discussed plurality voting.
Definition (Plurality voting) Each voter casts a single vote. The candidate with the most votes is selected.
As discussed earlier, ties must be broken according to a tie-breaking rule (e.g., based on a lexicographic ordering of the candidates; through a runoff election between the first-place candidates, etc.).

16. VOTING
Plurality voting gives each voter a very limited way of expressing his preferences. Various other rules are more generous in this regard. Let’s consider cumulativevoting
Definition (Cumulative voting) Each voter is given k votes, which can be cast arbitrarily (e.g., several votes could be cast for one candidate, with the remainder of the votes being distributed across other candidates). The candidate with the most votes is selected.

17. VOTING
Definition (Approval voting) Each voter can cast a single vote for as many of the candidates as he wishes; the candidate with the most votes is selected.
Although it is more expressive than plurality, approval voting still fails to allow voters to express their full preference orderings.
Voting methods that allow voters to express their full preference orderings are called ranking voting methods. Among them, one of the best known is plurality with elimination; for example, this method is used for some political elections.
When preference orderings are elicited from agents before any elimination has occurred, the method is also known as instant runoff.

18. VOTING
Definition (Plurality with elimination) Each voter casts a single vote for their most-preferred candidate. The candidate with the fewest votes is eliminated. Each voter who cast a vote for the eliminated candidate casts a new vote for the candidate he most prefers among the candidates that have not been eliminated. This process is repeated until only one candidate remains.

19. VOTING
Definition (Borda voting): Each voter submits a full ordering on the candidates. This ordering contributes points to each candidate; if there are n candidates, it contributes n−1 points to the highest ranked candidate, n−2 points to the second highest, and so on; it contributes no points to the lowest ranked Candidate. The winners are those whose total sum of points from all the voters is maximal.

20. VOTING
EXAMPLE: The mayor of Smallville is being chosen in an election using the Borda Count Method. The four candidates are Paul (the town barber), Rita (head of the town council), Sarah (Superintendent of Education), and Tim (former District Attorney). 500 registered voters cast their preference ballots. The results are summarized in the preference schedule below.

21. VOTING
The calculation of total votes based on the Borda count method is as follows: Paul: 3*130 +2*0 +1*250 = =640 Rita: 3*0 +2*500 +1*0 =1000 Sarah: 3*150 +2*0 +1*250= =700 Tim: 3*220 +2*0 +1*0 =660 The winner is Rita.

22. VOTING
Nanson’s method is a variant of Borda Count that eliminates the candidate with the lowest Borda score, recomputes the remaining candidates’ scores, and repeats. This method has the property that it always chooses a member of the Condorcet set if it is nonempty, and otherwise chooses a member of the Smith set.
Definition (Pairwise elimination) In advance, voters are given a schedule for the order in which pairs of candidates will be compared. Given two candidates (and based on each voter’s preference ordering) determine the candidate that each voter prefers. The candidate who is preferred by a minority of voters is eliminated, and the next pair of noneliminated candidates in the schedule is considered. Continue until only one candidate remains.

23. VOTING EXAMPLE: Suppose we have the following situation:
If we were to distribute preferences, we would first distribute 40,000 to A, 25,000 to B and 35,000 to C. Then B would be eliminated, and B‘s votes would be distributed to A and C: 15,000 to A and 10,000 to C, thus giving A the win with 55,000 votes to C‘s 45,000.

24. VOTING
But this, it can be argued that, is not really fair. If we examine the above ballot distribution, we see that a total of 55,000 voters have given B a higher preference than A, and a total of 55,000 voters have given B a higher preference than C. This can be more easily seen by a slight rewriting of the above ballots: Since B is preferred to either A or C by a majority of voters, the win should really go to B.

25. VOTING
In a case, where one candidate is preferred to all others individually by a majority of voters, that candidate is called a Condorcet winner. A Condorcet winner does not exist for all voting situations, but if there is a Condorcet winner, it seems reasonable to elect that particular candidate. Any voting system which chooses a Condorcet winner, if there is one, is called a Condorcet system.

26. VOTING
One problem with the ranking “X is preferred to Y by a majority of voters” is that it is not transitive. For example: Here A is preferred to B by 67% , B is preferred to C by 66% of voters, and C is preferred to A by 67% of voters. This is an example of a voting cycle. In such a case there is no Condorcet winner, and so, if a Condorcet method is being used to select the winner, some method must be used to “break” such cycles.

27. VOTING
Borda count takes into account all preferences. Each candidate is assigned a numerical value based on the number of preferences received. If there are n candidates, each first preference counts (n-1) and each second preference counts (n-2), down to preference n which counts 0. In general, a preference i will count (n-i).
For example, in the situation given above, candidate A has received 40,000 first preferences, 20,000 second preferences and 40,000 third preferences, so the Borda value is
40.000* * *0=
The values for candidates B and C are and respectively. The highest value is that of B, who wins.

28. VOTING
Even if a Condorcet winner exists, that winner is not necessarily chosen by the Borda count. A simple example is provided by the following voting situation: The Borda counts values are: A: 2*30 = 60 B: 2*20+1*30 = 70 C: 1*20 = 20 Winner is B. However, on inspection there is a Condorcet winner in this case, and it is candidate A .

29. VOTING
Nanson’s Method: The Borda count is by no means perfect as a general method for establishing a majority choice. If we consider the situation:
We see that candidate A has a clear majority on first preferences alone, and so should win. If, however, we calculate the rank-order values, we obtain:

30. VOTING which would give the win to B.
Nanson’s method is designed to be an all purpose method that will deal with situations like this.

31. VOTING
To begin, the Borda counts of the candidates are calculated, and the average (the arithmetic mean) found. Any candidate whose value is less than the average is eliminated. In the situation described above, the average is
( )/3=
and so would be eliminated.
The remaining candidates are then ranked as before, and the new Borda values calculated. We continue eliminating candidates whose values fall below the average, and ranking the remaining candidates, until one candidate is left.

32. VOTING
After C has been eliminated, we rank A and B, writing 1 for a high preference and 2 for a lower preference:
Note that A and B have retained their orders with respect to each other – only the numbers used to indicate preferences have changed. The new rank-orders are 63,000 for A and 37,000 for B. Thus B is eliminated, and the winner is A .

33. VOTING
There is another version of Nanson’s method, where instead of eliminating all candidates whose rank-order values are less than the average, only the candidate with the lowest rank-order value is eliminated at any stage – this is called Baldwin’s method.
Both methods are Condorcet methods, in that they will always choose a Condorcet winner if one exists.

34. MECHANISM DESIGN
Mechanism design is a strategic version of social choice theory, which adds the assumption that agents will behave so as to maximize their individual payoffs.
Example : Strategic voting
Will, Liam, and Vic are sweet souls who always tell you their true preferences. But Ray is always figuring things out and so he goes through the following reasoning process.

35. MECHANISM DESIGN
He prefers the most deskbound activity possible (hence his
preference ordering). But he knows his friends well, an in particular he knows which activity each of them will vote for.
He thus knows that if he votes for his true passion— (c) — he will end up (b) winning. So he votes for (a), ensuring that this indeed is the outcome.
Is there anything you can do to prevent such manipulation by little Ray?

36. MECHANISM DESIGN
This is where mechanism design, or implementation theory, comes in. Mechanism design is sometimes called “inverse game theory.”
We framed game theory as: Given an interaction among a set of agents, how do we predict or prescribe the course of action of the various agents participating in the interaction?
In mechanism design, rather than investigate a given strategic interaction, we start with certain desired behaviors on the part of agents and ask what strategic interaction among these agents might give rise to these behaviors.

37. MECHANISM DESIGN
We will assume unknown individual preferences, and ask whether we can design a game such that, no matter what the secret preferences of the agents actually are, the equilibrium of the game is guaranteed to have a certain desired property or set of properties.
Example: buying a shortest path: Consider the transportation network:

38. MECHANISM DESIGN
The number next to a given edge is the cost of transporting along that edge, but these costs are the private information of the various shippers that own each edge. The task here is to find the shortest (least-cost) path from S to T; this is hard because the
shippers may lie about their costs.
Your one advantage is that you know that they are interested in maximizing their revenue. How can you use that knowledge to
extract from them the information needed to compute the desired path?

39. MECHANISM DESIGN Mechanism design with unrestricted preferences
Definition (Bayesian game setting): A Bayesian game setting is a tuple (N,O,, p, u), where
• N is a finite set of n agents;
• O is a set of outcomes;
•  = 1 × · · · × n is a set of possible joint type vectors;
• p is a (common-prior) probability distribution on  ; and
• u = (u1, , un), where ui : O ×  → R is the utility function for each player i.
Given a Bayesian game setting, we can define a mechanism.

40. MECHANISM DESIGN Mechanism design with unrestricted preferences
Definition (Mechanism) A mechanism (for a Bayesian game setting (N,O, , p, u)) is a pair (A,M), where
A = A1 × · · · × An, where Ai is the set of actions available to agent i ∈ N; and
• M : A → (O) maps each action profile to a distribution over outcomes.
A mechanism is deterministic if for every a ∈ A, there exists o ∈ O such that M(a)(o) = 1; in this case we write simply M(a) = o.