AN APPLICATION OF SOCIAL CHOICE THEORY TO ORGANIZATIONAL DECISION MAKING

STRUCTURE-INDUCED (DIS)EQUILIBRIUM:

AN APPLICATION OF SOCIAL CHOICE THEORY TO ORGANIZATIONAL DECISION MAKING

By

Adrian Van Deemen

Radboud University Nijmegen Institute for Management Research P.O. Box 9108

6500 HK Nijmegen The Netherlands

Tel. ++ 31 (0)24 3613076

E-mail: a.vandeemen@fm.ru.nl

Preliminary Version 1.2

ABSTRACT

In this paper, an extension of the Condorcet paradox is presented that may be relevant for decision-making in organizations. The extension deals with democratic organizational structures of collective decision making units. Each decision unit aggregates a profile of complete and transitive individual preferences into a social preference by using the majority rule. The majority social preferences are aggregated in (fewer) decision-making units at a higher organizational level and so on until the top level unit is reached. In the paper it is assumed that in the end there is only one top level unit. The aggregated outcome of this unit is considered to be the organizational decision.

We first show that the way of structuring the organization of decision making units may lead to cyclical majority preferences at the top level and hence to the absence of a majority social choice at that level. The basic point here is that this Condorcet paradox at the top level is invoked purely on structural grounds. It is the structure that causes the decision deadlock.

We also show the reverse possibility, namely that top level majority social choices may exist in hierarchical structures in spite of the fact that local cycles at several levels in the hierarchy exist. The structure induces a global equilibrium in spite of the fact that local disequilibria (local Condorcet paradoxes) exist.

Subsequently, we briefly discuss the consequences of our results for the well-known structure-induced equilibrium concept of Shepsle (1979) and for the Garbage Can organizational decision making model (Cohen, March and Olsen (1972).

1. I

The Arrow Impossibility Theorem (cf. Arrow 1963) and the related Condorcet paradox (cf.

Black 1957) have been studied extensively in social choice theory. As far as we know, however, the research after the Condorcet paradox and majority disequilibrium is, with some exceptions, oriented towards stand-alone collective decision making units, that is, to units of individuals with preferences which are not embedded in organizational or hierarchical structures of decision making. This makes the research after both Arrow’s theorem and the Condorcet paradox less relevant for studies of organizational decision making.

In this paper, social choice theory is applied to organizational choice. An extension of the Condorcet paradox is presented and studied that may be relevant for decision-making in hierarchical organizations and which may open a new line of research with respect to the Condorcet paradox. The extensions deal with hierarchical structures of collective decision making units. Each decision unit aggregates a profile of complete and transitive individual preferences into a social preference by using the majority rule. These majority social preferences are in their turn aggregated in (fewer) decision-making units at a higher organizational level and so on until the top level unit is reached. In this paper it is always assumed that in the end there is only one top level unit. The aggregated outcome of this unit is considered to be the organizational outcome. The paper exclusively focuses on democratic organization structures. The higher-level decision units, including the top level unit, are all supposed to use the majority rule.

The basic aim is to show the effects of organizational structure on majority decision making.

We first show that the way of structuring the network of decision making units may lead to

choice at that level. The basic point hereby is that this top level Condorcet paradox is invoked purely on structural grounds. It is only the structure that causes the decision deadlock. If the individual preferences of the same units are joined into an unstructured single decision making unit, a top level majority social choice exists and hence no paradox appears.

We also show the reverse possibility, namely that top level majority social choices may exist in hierarchical structures in spite of the fact that local cycles at several levels in the hierarchy exist. The structure, so to say, corrects the local cycles. It induces a global equilibrium in spite of the fact that local disequilibria (local Condorcet paradoxes) exist. Finally, we show that by imposing or designing a decision structure, Condorcet paradoxes can be solved

Murakami (1966, 1970), Pattanaik (1970), Fine (1972) and Fishburn (1973) study what they call representative democracies or representative systems based on majority voting operators.

Our work is highly inspired by these studies. However, these studies are mainly oriented towards finding characterizations of representative systems or multi-stage majority decision procedures. In contrast, this paper wants to elucidate the effects of structure on social choice making. Moreover, we do not wish to restrict ourselves to the interpretation of representative systems only. In our view the general concept of organizational structure is more appropriate.

This implies that we see a system of representative democracy as an organizational structure.

The theoretical structure outlined here also may be viewed as a system of committees or as a coalition of coalitions as in the theory of the firm (Cyert & March 1992).

Shepsle (1979, 1986) introduces the idea of structure-induced equilibrium in spatial voting games. Starting from a multidimensional policy space as the choice space, he divides this

system each of which has jurisdiction over a subspace. Then he shows that such a jurisdiction structure may induce majority equilibria. For example if the space is n-dimensional Euclidian, then covering this space by n committees each with a jurisdiction over one dimension, Black’s median voter theorem (Black 1957) guarantees the existence of a majority decision for each dimension. Again our work is much inspired by Shepsle’s work. However, we are interested in the effects of (hierarchical) structures on majority decision making. Hereby we will explicitly investigate the effects of structure on majority (dis)equilibria without using jurisdiction compartments of the choice space.

The paper is organized as follows. In Section 2 we present some necessary concepts and notation. Moreover, a ‘language’ is presented to describe and analyze organizational decision making in social choice theoretic terms. In the third section it is shown how organizational decision structures may induce majority disequilibria. Section 4 shows that for a fixed input preference profile any alternative can be a collective decision by varying the structure. Hence, it shows the structural effects on social choice. In Section 5 we study how local disequilibria (Condorcet paradoxes) may be covered by the overall organization structure. In Section 6 it is shown that Condorcet paradoxes can be solved by imposing a decision structure on the set of individual preferences. We finish with a concluding section.

2. P

Let N be the set of individuals or players indexed by i = 1, …,n. We assume n to be nonempty and finite unless stated otherwise. Nonempty subsets of N are called decision units or coalitions.

Let X be a nonempty set of alternatives denoted by x, y. Let Ω be a nonempty class of nonempty subsets of X. Elements of Ω are called feasible subsets or agendas of X. As usual, P(X) denotes the power set of X.

A preference on X is a binary relation on X. Preferences are denoted by R. Further, xRy stands for (x, y) ∈ R. The asymmetric part of a preference R, denoted by P, and its symmetric part denoted by I are defined as:

1. xPy if xRy and not yRx;

2. xIy if xRy and yRx.

xRy can be interpreted as ‘x is at least as good as y’, xPy as ‘x is better than y’, and xIy as ‘x is as good as y’. For convenience, we write xy for xPy and (xy) for xIy.

A preference R is said to be

- reflexive if for all x ∈ X: xRx;

- complete if for all x, y ∈ X: xRy or yRx;

- anti-symmetric if for all x, y ∈ X: xRy and yRx implies x = y;

- transitive if for all x, y, z ∈ X: if xRy and yRz, then xRz;

- quasi-transitive if for all x, y, z ∈ X: if xPy and yPz, then xPz;

- intransitive if it is not transitive;

- acyclic if there are no x1, x2, x3 , … , xk ∈ X such that x1Px2, x2Px3 , …. , xk-1Pxk and xkPx1;

- cyclic if it is not acyclic.

R is a weak preference ordering (WPO) if it is reflexive, complete and transitive; it is a linear preference ordering (LPO) if it is reflexive, complete, anti-symmetric and transitive. The set of WPOs is denoted by O(X); the set of LPOs by L(X). Of course, L(X) ⊂ O(X).

A preference profile for a decision unit or coalition S is a mapping from S into O(X). Notation PS. The preference assigned to player i, when the context is clear, is denoted by Ri. Thus, a preference profile is an s-tuple of WPOs, one and only one for each player in S. Hereby, s =

#S (the number of elements in S). A group preference function assigns to each PS a preference relation.

Consider a preference R and a nonempty subset Y ⊆ X. An alternative x ∈ Y is said to be R- best if xRy for all y ∈ Y. The set of R-best elements of Y is called the choice set of Y. Note that if x is best in Y, then there is no y Y such that yPz. We know that there is a nonempty choice set for every nonempty Y ⊆ X if and only if R is reflexive, complete and acyclic. Sufficient conditions are reflexivity, completeness and transitivity.

Consider an S ⊆ N. Let s(x, y) = #{i ∈ S: xRiy}. For any profile PS we define

xmy := s(x, y) ≥ s(y, x)

for every x, y ∈ X. m is called the majority rule. Clearly, the majority rule is a group

preference function. The preference generated by m is called the majority relation. A majority decision, majority social choice or majority equilibrium for a preference profile PS is an m- best alternative given that profile. A set of m-best alternatives is called a Condorcet set.

As is well known, a majority decision need not exist. The exemplary illustrations are the following profiles:

x y z z x y y z x

or

x z y z y x y x z

The first yields the cyclic majority relation xyzx. This is a so-called forward cycle. The second profile yields the cyclic majority relation xzyx, which is a so-called backward cycle. We say a Condorcet paradox or majority disequilibrium occurs for a profile when there is no majority decision for that profile.

Now, we develop a language to study organizational decision making in social choice theoretic terms. The language is based upon the computer language Lisp (Slade 1998, Seibel 2005). We mainly will work with list structures and operations on (parts of) list structures. In the sequel we will use prefix notation to denote decision structures. The list (m P_S) means that the majority rule is applied to the profile PS = (R1, …, Rs) of decision unit, committee, or coalition S. The majority rule (or any other group preference rule) is always placed at the first

position in a list. When the context is clear, we label the decision units with S1, S2, …, Sk and denote the unit profiles accordingly with P1, P2, … Pk.

We say a list is nested if it contains other lists. Consider, as an example, the list

(m (m P1) (m P2) (m P3)). The big list contains four elements of which three are lists as well.

The first element in the list is the majority rule. It operates upon the results of the second, third and fourth element in the list. The first element in the other lists also is the majority rule.

This is an example of a democratic organizational decision structure. More general, an organizational decision structure is a nested list of list. The top level of the structure is the big list directly containing the elements (which may be lists). With ‘directly’ is meant that it is not contained in another list. The very first operator in the list therefore is the operator of the top level unit. A democratic organizational decision structure is a decision structure in which only the majority rule is used as a decision making mechanism. In such organizations, policy formation proceeds only by means of majority decision making.

Decision structures are processed by starting at the lowest level decision units. They are elaborated from the lowest level list toward the top level list, thus from inside out. That is, we use a bottom-up approach. Note that we can describe any organizational decision structure we wish. Also note that it is quite possible to use other group decision rules than the majority rule, e.g. the Borda rule or the Nash bargaining rule (when preferences are measurable at the interval level). In addition, we can introduce non-decision operators that work on parts of structures like the union of profiles or decision units. In this paper we focus on democratic structures.

3. S

-

In his section we present a simple example of a structure that induces a Condorcet paradox.

Consider the following hierarchical structure consisting of four decision making units.

Examples for this kind of structural arrangements abound: a university board (as top level unit) and three faculty management teams; a faculty board and its department MTs; a parliamentary committee system; a board of executive directors and the MTs of three

divisions in a company; an arrangement of bureaus in a governmental department; and so on.

More specific, consider three decision-making units with profiles

P1 = (xyz, zyx, yzx) P2 = (zyx, xyz, yzx) P3 = (yxz, xzy, xyz)

and the tree structure (m (m P1) (m P2) (m P3)). Applying the majority rule yields yzx for S1, zxy for S2 and xyz for S3. Taking these outcomes together constitutes a Condorcet profile.

Feeding this profile into the aggregating top level decision unit yields the forward majority cycle xyzx. Hence, no overall majority decision exists; we have an instance of the Condorcet paradox. See Figure 1.

Figure 1

xyz zyx yzx zyx xyz zxy yxz xzy xyz

S₁

S₂

S₃

S₄ yzx

xyz

zxy xyzx

no decision

Now, joining profiles P1, P2 and P3 into a single unit we have

3: xyz 1: xzy 1: yzx 1: yxz 2: zyx 1: zxy

This joint profile yields the (transitive and complete) majority relation xyz and, hence, x is the Condorcet social choice. Since the difference between the two concerned sets of individual preferences is the hierarchical structure, it is only this structure that causes the difference in

outcome, in particular the disequilibrium. For this reason we call this kind of disequilibrium a structure-induced disequilibrium.

We have shown that (m (m P1) (m P2) (m P3)) ≠ (m (∪ P1 P2 P3)). Here, ∪ stands for union.

Obviously, the presented model is simple. In reality, organizational structures are far more complex. It is surely possible to complicate our simple model, for example by introducing additional decision units, by allowing amendment procedures for the top level unit or by feeding in preferences of the members of the top level unit. However, the simple form suffices to show the influence of structure on the making of an organizational social choice.

Clearly, the top level decision unit has a problem. It may circumvent the decision deadlock by amending the preferences of the other lower level decision units or by bringing in some additional preferences. However, this may lead to other problems. For example, suppose the top level unit consists of two members who bring in, besides the aggregated result of the lower units, the preferences zxy and yzx. As can be easily seen, this leaves the cycle xyzx in tact. The same is true when they bring in the preferences yzx and xyz or the preferences zxy and xyz. When they both bring in one of the six linear orderings over {x, y, z}, then this ordering will be the collective preference and hence a majority decision will exists. However, in this case a majority in two lower level decision units can be found that prefer some

alternative to the alternative chosen at the top level. For example, if both bring in the preference xyz, then this will be the collective majority relation and x will be the majority decision. But a majority in unit S1 prefer y to x and a majority in unit S2 prefers z to x.

Note that the model is invariant under assigning a certain number of participants in the different decision units. For example assigning in

- P1 xyz to 3 individuals, zyx to 2, and yzx to 2;

- P2 zyx to 2 individuals, xyz to 3, and zxy to 4;

- P3 yxz to 3 individuals, xzy to 3, and xyz to 4 does not make any difference in the model results.

Furthermore, the model is invariant under assigning certain weights to the several decision units. For instance, if we assign the numbers 2 to S1, 3 to S2 and 3 to S4, the model results remain the same.

4. D

,

In this section we show that different structures with the same preference profile for the decision units may yield different social choices. Hence, we fix the individual preferences fed into the lower level decision units but vary the structure by introducing another level of. units.

First consider the following structure (Figure 2)

Figure 2

xyz zyx yzx zyx xyz zxy yxz xzy xyz

S

S

S

yzx

zxy S

xy, yz, (zx)

x is the social choice

S

xyz

zx, (xy), (yz)

Remember that alternatives between parentheses mean indifference. The formal notation of this structure is (m (m (m P1) (m P2)) (m P3)). In this structure, the collective decisions of units P1 and P2 are fed in into a fourth unit P4 which aggregates it without amendment into the intransitive collective preference zx, (xy) and (yz). This preference together with the result of unit P3 is fed into the top level unit which aggregates it without amendment into the intransitive but acyclic social preference xy, yz, (zx). Since x is better than y and is at least as good as z, x is the majority social choice.

Can we find a structure which makes, with the same input profile for the three lower units, y or z a social choice? The answer is affirmative.

First, consider the structure in which now the collective decisions of S2 and S3 are fed into

S3 to S4 instead of from S1 and S2. Subsequently, the result of S4 and S1 are aggregated by the top level unit without amendment. See Figure 3. The formal decision structure now is (m (m (m P2) (m P3)) (m P1))

Figure 3

xyz zyx yzx zyx xyz zxy yxz xzy xyz

S

S

S

zxy

xyz S

yz, zx, (xy)

y is the social choice

S

yzx

xy, (zx), (yz)

The overall collective preference for this structure is the acyclic but intransitive relation yz, zx and (xy). Since y is strictly better than z and as least as good as x, y is the majority decision.

However, the only difference between Figure 2 and Figure 3 is the decision structure. Hence, only this difference in structure can account for the difference in the collective decision.

Finally consider the decision structure in which the results of S1 and S3 are fed in into the fourth unit. Again this unit S4 aggregates these results without using an amendment procedure or by bringing in additional preferences. The formal structure is

Figure 4

zyx xyz zxy xyz zyx yzx yxz xzy xyz

S

S

S

yzx

xyz S

zx, xy, (yz)

z is the social choice

S

zxy

yz, (xy), (zx)

The intransitive collective majority relation yz, (xy) and (zx) of P4 and the transitive relation zxy of unit P2 are fed in into the top level unit which yields zx, xy and (yz). Now, the majority decision is z. Again note that the input preferences are the same as in the previous figures.

Clearly, only the difference in structure accounts for the selection of z as the majority social choice.

Looking at Figures 1, 2, 3, and 4, we see four different structures with the same input preference profiles each yielding a different result:

(m (m P1) (m P2) (m P3)) no majority decision

(m (m (m P1) (m P2)) (m P3)) x is the majority decision (m (m (m P2) (m P3)) (m P1)) y is the majority decision (m (m (m P1) (m P3)) (m P2)) z is the majority decision

Each structure yields a different social choice. Moreover, for each alternative a ϵ{x, y, z} it is possible to construct a decision structure over a fixed set of individual preferences over {x, y, z} such that a is a majority decision. And in addition it is possible to design a structure that implements a decision deadlock. This shows that it is possible to enforce a certain social choice or to create a decision deadlock by redesigning an organizational structure.

Clearly, the analysis above is close to implementation theory as developed in social choice and game theory (Jackson 2001, Maskin & Sjöström 2002). However, in this paper the emphasis is not on implementing a social choice correspondence by some mechanism (game form), but rather to design an organizational structure or organization scheme for a set of group preference functions. The problem is how to design an organizational structure in order to arrive at a desirable social choice

5. F

The first structure (cf. Figure 1) induces a disequilibrium in spite of the fact that each lowest level decision unit produces a ‘nice’ and consistent collective preference. In this section we show the reversed phenomenon, namely, that local disequilibria, i.e. Condorcet paradoxes, yielded by separate lower level decision units, can be turned into a global equilibrium by a decision structure.

Consider three decision units with profiles:

P1 = (xyz, zxy, yzx) P2 = (xzy, zyx, yxz)

Figure 5

xyz zxy yzx xzy zyx yxz yxz xzy xyz

S₁

S₂

S₃

S₄ zxyz

xyz

xzyx xyz

x is the social choice

S1 and S2 yield, respectively, the cycles zxyz and xzyx. So we have two local disequilibria. S3 produces the collective preference xyz. Feeding these three collective relations into the top level unit yields the transitive majority relation xyz. Hence x is the majority decision. In spite of the fact that we have two cyclic collective relations from which no decision can be derived, the global collective relation is transitive and yields a clear decision.

Now consider the case in which P1 and P2 remain the same as in the previous but P3 = (xyz, zyx, yzx). Using the same organizational structure now gives yzx as the collective preference and hence y is the final decision. If we take P3 = (zyx, xyz, zxy), then zxy is the overall organizational preference and hence z is the social choice. In all the three cases, the third unit P3 determines the organizational choice. P1 and P2 produce respectively a forward cycle and a backward cycle, which cancel each other out in the structure. In addition, looking at the unstructured individual preference set in the first case

x y z z x y y z x x z y z y x y x z y x z x z y x y z

we see that x is also the social choice. In the second case also y remains the collective decision and in the third z. Hence, it appears as if structure doesn’t matter in these cases.

6. S

C

P

In this section we show how to solve Condorcet paradoxes by imposing an organizational structure on the set of individuals. Consider the following set of preferences:

3: x y z 2: z y x 2: y z x 2: z x y

Applying the majority rule yields the cyclic majority relation xyzx. Hence, there is no majority decision. We have an instance of the Condorcet Paradox. To solve this paradox, we design a

decision structure with five decision units. See Figure 6. Let the lowest level units have the following preference profiles:

P1 = (xyz, zyx, yzx) P2 = (zyx, xyz, zxy) P3 = (xyz, zxy, yzx)

The overall decision structure is (m (m (m P1) (m P3)) (m P2)). Since (m (m P1) (m P3)) = yz, zx, (xy) and (m P2) = zxy, the overall majority relation is (yz), zx and xy. Hence z is the

majority decision and the paradox has disappeared. Note that (m P3) produces the forward cycle zxyz.

Notice that also this case shows that an overall majority decision can exist along with local cycles. Further notice that z is the majority decision for unit S2.

Figure 6

zyx xyz zxy xyz zyx yzx xyz zxy yzx

S₂

S₁

S₃

yzx

zxyz S₄

zx, xy, (yz)

z is the social choice

S₅ zxy

yz, zx, (xy)

the intransitive relation zx, xy and (yz), and since (m P1) is yzx, this structure yields yz, zx and (xy). Hence, now y is the majority decision, which also happens to be the majority decision of unit S1. So, slightly redesigning the decision structure yields another social choice. Again, the Condorcet paradox has disappeared, which again shows that imposing or (re)designing a decision structure helps to solve it.

Note that the structure (m (m (m P1) (m P2)) (m P3)) does not solve the paradox. It yields the forward cycle zxyz, which happens to be the same cycle as produced by S3.

We study the exemplary Condorcet paradox profile

x y z z x y y z x.

- Consider the structure (m (m xyz zxy) (m yzx)). Clearly, this yields yz, zx and (xy).

Hence, y is the majority decision for this structure.

- Now consider the structure (m (m xyz) (m zxy yzx)). The result now is xy, yz and (xz) and hence x is the structural majority decision.

- Subsequently, when using the structure (m (m xyz, yzx) (m zxy)), the result is zx, (zy) and xy. Therefore z is now the structural majority decision.

So we found three decision structures that solve the paradox. Moreover, for each a ∈{x, y, z}

there exists an organizational structure such that a is the social choice.

More general, we have the following results.

Let ∏ = { π| π : N → O(X)} where O(X) = {R| R is a weak ordering on X}. A π ∈ ∏ for which no majority winner exist is called a Condorcet paradox.

THEOREM 1

For any a ∈ X = {x, y, z} and any Condorcet paradox given X there exists a structure that solves for a

Note that this theorem deals with Condorcet paradoxes with weak orderings. The next theorem shows that any Condorcet paradox can be solved:

THEOREM 2

For any Condorcet paradox there is an organization decision structure that solves the paradox.

The proof of both theorems is in the appendix.

There exists a long discussion about the relevance of the Condorcet paradox and the empirical occurrence of majority cycles. Probability calculations and analytics in the case of spatial voting games show that the paradox should occur frequently. However, as Gerhlein formulates it (Gehrlein, 2006, p.58):

“Numerous empirical studies have been conducted to determine if Condorcet’s Paradox is ever observed in actual elections. After surveying these studies, we must conclude that the evidence does not suggest that the phenomenon is widespread in voting situations.”

Gehrlein does not discuss experimental evidence in the context of spatial voting games.

However, the conclusion is the same: there is no (experimental) evidence of (global) cycling in the case of empty cores. See e.g. Fiorina & Plott 1978, McKelvey & Ordeshook (1990) and Bianco et al. (2006).

Clearly, there are many answers to the question “why there are no cycles” or “why there is so much stability”. One answer according to Theorem 2 is organizational structure. In the social and political world, almost everything is organized. Organizational structures abound in social reality. And, as shown above, organizations may have structures that solve for Condorcet paradoxes. That is, democratic decision making may be organized in such a way that cycles are avoided.

7.CONCLUSION AND DISCUSSION

The above analysis clearly shows that structure matters. Condorcet paradoxes can be solved by imposing a decision structure on the set of individual preferences that constitute a paradox.

Moreover, we have shown that different structures may yield different majority decisions for a fixed preference set. This raises questions about the (re)design of organizational decision structures.

Our work is in the line of the work of Shepsle (1979, 1986). We also raise the question whether majority disequilibria can be avoided by means of structures or institutions. And also our answer is affirmative. However, we also show that structures may induce majority

disequilibria. Moreover, we wished to show the effects of structure on majority decision

Our structural results have clear implications for organizational decision making models, e.g.

the Garbage Can Model of Organizational Choice (Cohen, March and Olsen (1972). One of the key issues in this model is the concept of problematic preferences. Unfortunately, this concept is nowhere elaborated in the Cohen et al. paper or in subsequent work on this model.

However, if we interpret this as intransitive preferences, then our analysis makes clear that organizational garbage cans may produce problematic preferences, but in spite of this, organizational decision making may produce nice and consistent results. Conversely, within organizations the decision units may produce non-problematic preferences but precisely because of the organizational structure, the overall decision making process may be jeopardized. In Bendor et al. (2001) the garbage can model is critically assessed. They propose to combine the model with rational choice analytic approaches in order to innovate and develop the model into a positive theory of organizational decision masking. In a sense, this paper is a first trial in that direction.

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