Social Welfare: Fairness and Fairness Mechanisms

Last updated Aug 29, 2005.

In this section, we will try and answer the question of how we can fairly combine the preferences of a group of individuals into a choice that is fair.

Part 1: Framework

Concept 1: Review of the multi-agent choice problem

Recall how we set up the multi-agent choice problem at the start of the semester. We defined a multi-agent choice problem as one where the consequences experienced by each agent are a function of the choices (and possibly the state of nature) of all the agents. If the game is fully cooperative and if everybody shares the same information, then this problem is just a single agent problem with a vector of choices. However, if goals differ or if agents have different information, then the problem gets interesting. This is illustrated in the figure where two agents make choices, a consequence results, and then agents evaluate this consequence in terms of the agent's goals. Performing this evaluation for all possible consequences allows an agent to create an ordering over this set. For interesting games, this ordering differs between agents (either because of partial information or differing goals).

Concept 2: The Social Welfare problem: Generating consequences from individual choices

The social welfare problem is typically thought of in a canonical form of the general problem outlined in the above figure. In this canonization, all consequences are represented by a set of choices. This statement probably doesn't make sense, but I'll use some examples to illustrate what I mean.

Example 1: When we talked about the prisoner's dilemma, we discussed how the payoff matrix was created. We did this by thinking about how each player's actions would produce a consequence, and then we looked at how each player evaluated that consequence. For example, if player 1 defects (talks to the police) and player 2 cooperates (remains silent) then the following consequence is produced: player 1 gets a very short prison sentence and player 2 gets a very long prison sentence. For every action pair, we can identify what consequence is produced. Each player then forms a utility function by assigning a numerical value to the consequence. If player 1 defects and player 2 cooperates, then player 1 creates a utility function for the resulting consequence by setting the utility equal to the number of years he or she will spend in prison as the numerical utility value u₁(D,C)=1 meaning player 1 will spend one year in jail. Similarly, player 2 creates a utility function in the same manner yielding u₂(D,C)=4 meaning player 2 will spend four years in jail.

In this example, notice how the consequence included a dimension for what happened to player 1 and another dimension for what happened to player 2. The utilities assigned by each player ignored one of these dimensions so that player 1 only payed attention to its dimension and similarly for player 2. These consequence structures can get much more complicated than this. Consider the following example.

Example 2: In the repeated play prisoner's dilemma, the consequences of an action include the payoff for the current round and the set of possible future payoffs for subsequent rounds. For example, when an Always Defect agent plays against a Tit fot Tat agent, the choice to defect on the first round not only yields the temptation payoff on that round, it also guarantees that the Always Defect agent will get the punishment payoff on the next round. This is because the consequence of the Always Defect agent consisted not only in the immediate payoff, but also in a change to the internal state of the Tit for Tat agent that caused this agent to act differently in future rounds of play. Similarly, the choice to play a Tit for Tat strategy has a consequence structure that includes the payoff for the first round and for future possible rounds. This consequence structure also includes the likely survivability in a society of agents meaning that the choice to be a Tit for Tat-er may mean that future societies are more compatible with this kind of play. In summary, consequences can include different dimensions across agents and different dimensions across time.

Usually (in the stuff that I've read) social welfare of the type addressed by Arrow's impossibility theorem frames the social choice problem in something of a canonical form. In this canonical form, all consequences are represented by an associated choice. Let me illustrate by an example.

Example 3: Electing the President of the United States is a good example of the canonization usually associated with social choice theory. Typically, there are between three and fifteen candidates for president (with only two or three having any real chance of winning). The choices available to voters are precisely the candidates, but each candidate is actually a representation of a potentially very complicated set of consequences. For example, in the 2000 presidential election, let's restrict attention to the two most popular choices, Al Gore and George Bush. A vote for either candidate included the normal consequences on issues such abortion rights, national security, labor, etc., but also included more subtle consequences such as whether Supreme Court appointees (and the next twenty years of Supreme Court decisions) would have liberal leanings or conservative leanings.

In summary, the social choice problem is typified by a set of discrete options available to each agent in the system. Each agent will specify a preference pattern or a utility function over this set of options, and then a social choice mechanism will be invoked that transforms these patterns/functions into a choice of one of the options.

Concept 3: Combining choices, types, and mechanisms

There are a zillion different ways of combining preferences and utilities. The conventional voting procedure used in presidential elections translates each voters preference pattern into a single choice, and then counts all of the choices. This is not the only procedure that could be used, however. Elections could be decided instead by giving each voter 10 points and then letting them distribute these points among the various candidates; the candidate with the highest number of total points would be the winner.

The process of designing a system whereby agents can express their preferences/utilities and then transforming what the agent expresses is called mechanism design. In other words, somebody has to design a mechanism by which individual's preferences are translated into a choice among the options. Arrow's impossibility theorem essentially states that there is no fair way to combine the preference patterns of each agent into a social choice. In other words, there is no fair social choice mechanism. We will prove this by formalizing what we mean by fair and then showing that all of the dimensions of fairness cannot be simultaneously satisfied.

It is interesting to note that since no social choice mechanism is fair, in the sense of Arrow, the process of mechanism design is essentially equivalent to "picking a poison" --- one dimension of inequity must be admitted, and the decision of the mechanism designer is to choose the least unpalatable dimension. This difficulty is confounded by the fact that once the mechanism is chosen, agents may not tell the truth about how they really feel if they think lying will help their real preference pattern be selected by the mechanism. For example, consider the 10 point voting system that I described above. Suppose that a voter feels that Bush deserves six points and Gore deserves four, but this voter also believes that, on the average, other voters will give Bush and Gore five points each. This voter might want the choice to reflect his or her preference pattern so he or she may lie and give Bush all ten points. This doesn't reflect his or her true preferences, but in order to manipulate the mechanism the voter may lie. A type is what the voter chooses to give as input to the social choice mechanism; sometimes a type is the voter's true preference pattern and sometimes a type is some distortion of this. When we talk about different social choice functions, we will revisit the idea of a type again.

Part 2: Preferences and Utilities

Since we already talked about this in a previous lecture, you can skip this if you want. I've chosen to keep it in the notes because (a) the notes evolve from semester to semester and so historical impetus keeps them here and (b) these concepts are important to review at this time in the semester. If you haven't masterd this material, please take the time to do so now.

Concept 1: Preferences and preference patterns

Consider a set of choices {c₁, c₂, . . . , c_n}. (Note how I use c's to represent choices and c's to represent consequences. This is OK given that we are using the canonization described above wherein the choices encode consequences. In other words, since the choices are representations of consequences, using the same notation is no problem.) A preference pattern is a relation between pairs of choices such that every set of choices can be compared to every other. (I will use the greater than symbol to represent the notion of preferred to, but note that this is not typically done. I'm stuck with > because I cannot figure out how to get netscape to give me the real symbol that I want. I'll show you this symbol in class and I want you to use it on exams and in our discussions.) A preference pattern is a total ordering that is transitive. This means that the following axioms hold:

"c_i, c_jeither c_i>c_j, c_i<c_j, or c_i~c_j. In other words, for all choices I prefer one to the other (>) or I'm indifferent (~) between them. The greater than or equal to symbol (written >~ herein) is read as at least as preferred as so that writing c_i>~c_j is read c_i is at least as preferred as c_j. This property is known as total ordering.
c_i>c_j and c_j>c_k implies c_i>c_k. This is known as transitivity.

Concept 2: Utilities and the principle of maximum expected utility

What we want to do is take a total preference ordering over a set of consequences and turn it into a numerical function. The way that we will do this is by realizing that preferences only tell us that we like one choice more than another. In reality, we may have strengths of preference meaning that we like choice a much more than choice b, and choice b just a little more than choice c. There are a set of axioms that allow us to encode how strong our preferences are in an almost unique utility function. The assumption behind turning these axioms into a utility function is that we will maximize this utility function. More explicitly, we will maximize the expected value of this utility function. The key to finding such a utility function is using lotteries.

Concept 3: Lotteries and preferences over lotteries

A lottery is simply a probabilistic outcome. Most concretely, when you buy a lottery ticket, you are buying a probabilistic outcome --- with a small probability you can win a huge prize, and with a large probability you win nothing. Formally, we will adopt Russell and Norvig's notation in Artificial Intelligence: A Modern Approach and write a lottery as follows: [p,A; 1-p,B]. This notation is read as "With probability p I get outcome A and with probability 1-p I get outcome B. Concretely put, [p,A; 1-p,B] is a lottery ticket with a probability p of winning A (the jackpot) and probability 1-p of winning B (nothing). Constructing utility functions is based on finding probabilities such that the human is indifferent between a sure outcome and a lottery ticket. We will now formalize this notion using the axioms explained in Russel and Norvig. I will quote them directly in what follows, except that I will use >, >~, and ~ instead of the correct shape for preference.

Concept 4: The axioms of utility theory

Total ordering. See concept 1 above.
Transitivity. See concept 1 above.
Continuity. If some state B is between A and C in preference, then there is some probability p for which the rational agent will be indifferent between getting B for sure and the lottery that yields A with probability p and C with probability 1-p.

there exists a

Substitutability: If an agent is indifferent between two lotteries, A and B, then the agent is indifferent between two more complex lotteries that are the same except that B is substituted for A in one of them. This holds regardless of the probabilities and the other outcome(s) in the lotteries.

A~B ==> [p,A; 1-p,C] ~ [p,B; 1-p,C]

Monotonicity: Suppose that there are two lotteries that have the same two outcomes, A and B. If an agent prefers A to B, then the agent must prefer the lottery that has a higher probability for A (and vice versa).

This is kind of a funny one, so I'm going to comment on it here. The form of this statement is that the left side (A is more preferred than B) implies an equivalence. This equivalence states that the existence of a probability p and q such that p is no less than q is equivalent to saying that the lottery with odds equal to p is at least as preferred as the lottery with odds equal to q.

Decomposability: Compound lotteries can be reduced to simpler ones using the laws of probability. This has been called the "no fun in gambling" rule because it says that an agent should not prefer one lottery just because it has more choice points than another.

Again, this is kind of funny. The left side of the equivalence is a lottery consisting of an outcome A and another outcome which is a lottery [q,B; 1-q,C]. This is a bit like buying a lottery ticket where with probability p you win $5, and with probability 1-p you get another lottery ticket for different outcomes with different odds. The right side of the equation is a compound lottery ticket with odds p of winning A, odds (1-p)q of winning B, and odds (1-p)(1-q) of winning C.

At this point, we are no longer quoting from Russell and Norvig. In the next concept, we will show how to use these axioms to construct a utility function.

Concept 5: Building a utility from preferences over lotteries

Given these axioms, can we build a utility function? The answer is, of course, yes. In fact, there is a theorem that says we can. The following is a statement of this theorem. It is partially taken from Russell and Norvig, and partially stated in my own terms.

Theorem. If an agent's preferences obey the axioms of utility, then there exists a real-valued function U such that
u(A)>u(B) <==> A>B,
u(A)=u(B) <==> A~B.

How would we prove this? By constructing such a function. Assuming that we have a discrete set of choices (like in the canonization of the social welfare problem), here's how.

Find the most preferred and least preferred outcome. We will call these A and Z, respectively. We know that a most preferred and least preferred outcome exist because the set of outcomes is ordered.
Make up a number u(A) and u(Z) that will represent the utility of the most preferred and least preferred outcome. If we are indifferent between A and Z (i.e., A~Z) then set u(A)=u(Z). Otherwise, we can choose any number we want so long as u(A)>u(Z).
Choose any other option. Let's call it B. If B~A then set u(B)=u(A). If B~Z then set u(B)=u(Z). Otherwise, use continuity to find the value of p such that [p,A; 1-p,Z]~B, and then set u(B)=pu(A)+(1-p)u(Z).
Repeat for all options.

The third step is really the key one because it is where we transform our preference A>B>Z into a probability that we can use to create the utility. You should probably know that scientists have used this technique for figuring out what people's utilities are, but that the resulting utilities don't match what people actually choose; this means that utilities are nice for doing designs, but they probably don't (in their formal form) represent how people make decisions.

Can you prove the theorem now? You should probably try since I've asked other students to do this proof on an exam.

Concept 6: Uniqueness: positive affine transformations

The philosophy behind these axioms, the theorem, and the algorithm is that the utility will be maximized in some way: Which outcome should I chose? The one that maximizes utility. What if the outcome is actually probabilistic? Then I maximize expected utility. Thus, constructing a utility function assumes that this utility function will be used to make a choice as follows:

c^* = arg max_c u(c),
a^* = arg max_a Sum_s p(s) u(a,s) = arg max_a Sum_s p(s) u(c(a,s))

The first equation says that the choice I make is the one that has highest utility. The second equation says that the optimal action to choose, a^*, is the one that, on average across a set of probabilistic states, produces consequences with high utility. I realize that this equation is a bit difficult to discern, but I believe it is worth your time to really figure out what it means. The key is that the outcomes are consequences, and that these consequences are functions of actions and states.

Shifting focus a bit, look at the algorithm we used to find the utility function. In step 2, we just made up numbers for u(A) and u(Z). Since we are only going to take the argument that maximizes the resulting function, it doesn't really matter what the actual values of u() are as long as they obey the axioms of preference. This does, however, mean that the utility function is not unique. In terms of finding the argument that maximizes the utility (or expected utility), if u() satisfies the axioms than so does au+b where a is greater than zero and where b can be anything. Another way of saying this is that the utility function is unique only up to a positive affine transformation. The transformatin is positive since a is greater than zero and it is affine since b can be anything. Let's check and see if we get the same maximal solution for u and au+b.

c^* = arg max_c u(c) = arg a[max_c u(c)] = arg max_c[a u(c)] = arg max_c[a u(c) +b].

Thus, we get the same solution even if the utility is unique only up to a postivie affine transformation. Unfortunately, this nonuniqueness causes problems when get into the world of multi-agent choice.

Part 3: Social Welfare Concepts and Problems

Concept 1: Combining preference patterns

Suppose that you are given the job of finding a mechanism that takes the preference patterns of multiple agents and combines them fairly. Let {P₁,P₂, . . ., P_n} denote the preference patterns for agents 1, 2, . . . n. Your job is to come up with a way to combine these preference patterns into a preference pattern for society. Let society's preference pattern be denoted P (with no subscript), and let Pset denote the set of all possible preference patterns for a given set of choices. In other words, your job is to find a function that does the following:

f: Pset₁ x Pset₂ x . . . Pset_n x --> Pset,
P = f(P₁,P₂, . . ., P_n).

It is helpful to give some examples of just what I mean here. Suppose that we have a problem with two agents and three choices. Denote these choices A, B, and C. Assuming, for simplicity, that the agents will not be indifferent between any two options, there are several types of preferences patterns that an agent can have. These possible preference patters include A>B>C, and A>C>B and so on. Thus,

Pset = { A>B>C, A>C>B, B>A>C, B>C>A, C>A>B, C>B>A }.

No matter which set of preference patterns are held by the two agents, the function f must be able to take these and return a preference pattern for society.

What is the domain of f? For the two agent, three choice problem the domain is Pset x Pset (the cross product of the two sets). What is the range of f? It is Pset.

Concept 2: Arrow's impossibility theorem: a foreshadowing

Arrow's impossibility theorem states that there is NO function f that "fairly" combines these preferences. This means that if we are going to use preferences as the things that we use to make social choices, we are going to have to compromise a bit. We will talk about some different kinds of compromises.

Concept 3: Strengths of preference: overcoming the impossible

If we are willing to use something besides preferences, we may be able to circumvent Arrow's impossibility. One way to do this is to use the information about strength's of preferences encoded as part of a utility function. This allows us to let those who feel most strongly about an issue have a greater say in the social outcome. For example, we can make a social choice by summing up all of the individual agent's utilties to create a social welfare utility

u(c) = Sum_k u_k(c).

This social welfare utility now encodes the preferences of the agents.

However...

Concept 4: Interpersonal comparison of utility: uniqueness and types revisited

Utilities are unique only up to a positive affine transformation. This means that I can manipulate the system to mimic my preferences by choosing a really big a that causes my utilities to dwarf the sum total of everybody else's utilities. In other words, if everybody else's utilities are in the range from 0 to 10, and if I know that there are only 10 agents, then an a=100,000 means

u(c) = S_k u_k(c) = u_i(c) + S_{k
!= i} u_k(c)
u'(c) = au_i(c) + S_{k
!= i} u_k(c) = 100,000 u_i(c) + S_{k != i} u_k(c) which approximately equals 100,000 u_i(c).

If I am player i, then I can manipulate the system by choosing a representation of my utilities which is out of scale with everybody elses. This problem, caused by the nonuniqueness of utilities, is called the problem of interpersonal comparison of utilities. In essence, it is a problem of finding a common currency for expressing utilities that make my utilities comparable to yours. You will get to play around in a lab where you use different social choice functions that are based on preferences and on utilities.