Decision making beyond arrow's "impossibility theorem, " with the analysis of effects of collusion and mutual attraction

In 1951, K. J. Arrow proved that, under certain assumptions, it is impossible to have group decision making rules which satisfy reasonable conditions like symmetry. This Impossibility Theorem is often cited as a proof that reasonable group decision making is impossible. We start our paper by remarking that Arrow’s result only covers the situations when the only information we have about individual preferences is their binary preferences between the alternatives. If we follow the main ideas of modern decision making and game theory and also collect information about the preferences between lotteries (i.e., collect the utility values of different alternatives), then reasonable decision making rules are possible: e.g., Nash’s rule in which we select an alternative for which the product of utilities is the largest possible. We also deal with two related issues: how we can detect individual preferences if all we have is preferences of a subgroup, and how we take into account mutual attraction between participants. I. GROUP DECISION MAKING AND ARROW’S IMPOSSIBILITY THEOREM In 1951, Kenneth J. Arrow published his famous result about group decision making [1], a result that became one of the main reasons for his 1972 Nobel Prize; see also [16], [22], [23], [34]. The problem. Arrow’s result deals with the following setting. A group of n participants P1, . . . , Pn needs to select between one of m alternatives A1, . . . , Am. To find individual preferences, we ask each participant Pi to rank the alternatives Aj from the most desirable to the least desirable: Aj1 Âi Aj2 Âi . . . Âi Ajn . Based on these n rankings, we must form a single group ranking (in the group ranking, equivalence ∼ is allowed). Case of two alternatives is easy. In the simplest case when we have only two alternatives A1 and A2, each participant either prefers A1 or prefers A2. In this case, it is reasonable, for a group: • to prefer A1 if the majority prefers A1, • to prefer A2 if the majority prefers A2, and • to claim A1 and A2 to be of equal quality for the group (denoted A1 ∼ A2) if there is a tie. Case of three or more alternatives is not easy. When we have three or more alternatives, there is no such simple rule; to be more precise, we can still come up with many possible group decision rules, but all these rules will be, in some sense, counter-intuitive. Arrow’s result. Arrow has explicitly formulated several reasonable conditions and showed that no group decision rule can satisfy all these conditions. Arrow’s conditions are very straightforward and very natural. The first is the Pareto condition: that if all participants prefer Aj to Ak, then the group should also prefer Aj to Ak. The second condition is Independence from Irrelevant Alternatives: the group ranking between Aj and Ak should depend only on how participants rank Aj and Ak – and should not depend on how they rank other alternatives. Arrow has shown that every group decision rule which satisfies these two condition is a dictatorship rule – the rule according to which the group accepts the preferences of one of the participants as the group decision and ignores the preferences of all other participants. This clearly violates another reasonable condition of symmetry: that the group decision rules should not depend on the order in which we list the participants. II. BEYOND ARROW’S IMPOSSIBILITY THEOREM: NASH’S BARGAINING SOLUTION It is sometimes claimed that reasonable group decision making is impossible. Arrow’s Impossibility Theorem is often cited as a proof that reasonable group decision making is impossible – e.g., that a perfect voting procedure is impossible; see, e.g., [34]. Arrow’s result is only valid if we have binary (partial) information about individual preferences. We will see that the pessimistic interpretation of Arrow’s result is, well, too pessimistic. Indeed, Arrow’s result assumes that the only information we have about individual preferences is their binary (“yes”“no”) preferences between the alternatives. This information does not fully describe a persons’ preferences: e.g., the same preference A1 Â A2 Â A3 may indicate that a person strongly prefers A1 to A2, and A2 to A3, and it may also indicate that this person strongly prefers A1 to A2, and at the same time, A2 is almost of the same quality as A3. To describe this degree of preference, researchers in decision making use the notion of utility; see, e.g., [22], [23]. What is utility: a reminder. A person’s rational decisions are based on the relative values to the person of different outcomes. In financial applications, the value is usually measured in monetary units such as dollars. However, the same monetary amount may have different values for different people: e.g., a single dollar is likely to have more value to a poor person than to a rich one. In view of this difference, in decision theory, to describe the relative values of different outcomes, researchers use a special utility scale instead of the more traditional monetary scales. There are many different ways to elicit utility from decision makers. A common approach is based on preferences of a decision maker among lotteries. A simple way to define a lottery is as follows. Take a very undesirable outcome A− and a very desirable outcome A, and then consider the lottery A(p) in which we get A with probability p and A− with probability 1 − p (p is given and is usually understood as an “objective” probability). Clearly, the larger p, the more preferable A(p): p < p′ implies A(p) < A(p′). Traditional decision theory is based on assumptions concerning preferences over lotteries. For example, the following two assumptions are usually adopted as axioms: • the comparison amongst lotteries is a linear order – i.e., a person can always select one of the two alternatives, and • the comparison is Archimedean – i.e. if for all ε > 0, an outcome B is better than A(p − ε) and worse than A(p+ε), then it is of the same quality as A(p): B ∼ A(p) (where A ∼ B means that A and B are of the same quality). Because of our selection of A− and A, most reasonable outcomes are better than A− = A(0) and worse than A = A(1). Due to linearity, for every p, either A(p) < B, or B ∼ A(p), or B < A(p). If we define the utility of outcome B as u(B) def = sup{p |A(p) < B}, we have A(u(B)−ε) < B and A(u(B)+ε) > B; thus, due to the Archimedean property, we have A(u(B)) ∼ B. This value u(B) is called the utility of the outcome B. As defined above utility always takes values within the interval [0, 1]. It is also possible to define utility to take values within other intervals. Indeed, note that the numerical value u(B) of the utility depends on the choice of reference outcomes A− and A. If we select a different pair of reference outcomes, then the resulting numerical utility value u′(B) is different. The usual axioms of utility theory guarantee that two utility functions u(B) and u′(B) corresponding to different choices of the reference pair are related by a linear transformation: u′(B) = a · u(B) + b for some real numbers a > 0 and b. By using appropriate values a and b, we can then re-scale utilities to make the scale more convenient (e.g. in financial applications, closer to the monetary scale). Expected utility. Often, we have a “branching” situation involving n incompatible events E1, . . . , En with probabilities p1, . . . , pn such that exactly one of them will occur. E.g. coins can fall heads or tails, dice can show 1 to 6, etc. In such situations, for every n outcomes B1, . . . , Bn, we can form a lottery by assigning outcome Bi if event Ei occurs. If we know the utility ui = u(Bi) of each outcome Bi, and we know the probability pi = P (Ei) of each event Ei, then the utility of the corresponding lottery may be determined as follows. We know the probability pi of each event Ei. Thus, the lottery “Bi if Ei” is equivalent to the lottery in which we get Bi with probability pi. The fact that u(Bi) = ui means that each Bi is equivalent to getting A with probability ui and A− with probability 1 − ui. By replacing each Bi with this new “lottery”, we conclude that the lottery “if Ei then Bi” is equivalent to a two-step lottery in which we: • first select Ei with probability pi, and • then, for each i, select A with probability ui and A− with the probability 1− ui. In this two-step lottery, the probability of getting A is equal to p1·u1+. . .+pn·un (often this is obtained by adding suitable axioms on combination of lotteries, but the meaning should be intuitive here). Thus, by our definition of utility, the utility of the lottery “if Ei then Bi” is equal to u = n ∑