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Collective Choice Theory in Collaborative Computing

This paper presents some fundamental collective choice theory for information system designers, particularly those working in the field of computer-supported cooperative work. This paper is focused on a presentation of Arrow's Possibility and Impossibility theorems which form the fundamental boundary on the efficacy of collective choice: voting and selection procedures. It restates the conditions that Arrow placed on collective choice functions in more rigorous second-order logic, which could be used as a set of test conditions for implementations, and a useful probabilistic result for analyzing votes on issue pairs. It also describes some simple collective choice functions. There is also some discussion of how enterprises should approach putting their resources under collective control: giving an outline of a superstructure of performative agents to carry out this function and what distributing processing technology would be needed.

1 Collective Choice in Information Systems 1.1 Naming Services

Windows NT
If one uses a system then, from time to time, one might receive an event message saying that "The Browser Has Forced an Election..." [MSK99a,MSK99b].

The Internet and the Domain Naming Service DNS
Internet connected systems could not function without the Domain Naming Service and this too relies upon elections: individual system administrators choose when their name-server is to authoritative or not [Wel99,Ricly] and which name-servers it will rely upon.
The difference between the two systems is that Windows NT is designed to manage the naming of relatively small domains and can use a direct election amongst all its naming components, the browsers. This is represented in figure  1, where the master browser is elected by the itself and the other browsers.
The browsers apply the same fitness criteria in choosing their master. There can be no conflict in policy.  The Internet's DNS has to rely upon a loosely co-ordinated database of name servers. This is represented in figure 2. Here the Administrator of each DNS chooses which other DNS it will use to resolve names. In this example, for all names other than their own, the Superior Administrator is chosen as authoritative.

Names
The Administrators need not apply the same fitness criteria in choosing their superior DNS. There may be conflicts in policy.
They can therefore attempt malevolent actions collectively if they so wish, for example: Consider an electronic commerce web site. The user's web browser makes a secure connection to the site, providing a protected channel. If the DNS entry for the server's address was replaced by one indicating an attacker's address, the browser will connect to the malicious site, possibly without the user's knowledge. In this scenario, the DNS spoofer could monitor the traffic over the "secure" connection, since the secure connection would actually be to the spoofer, and forward the transaction data to the real website or process the traffic itself [Wel99]. But even if computer programs have predictable behaviour they may not apply the same fitness criteria in making choices. This may lead to conflicts in policy.

Groups choosing policies
Most people will have come across moderated newsgroups and mailing lists. Potentially these services could be made self-managing and would be simple examples of a computer-supported working environments.

Joining the Group
A policy decision is needed to determine whether or not an individual should be allowed to join a particular group and take the rights and privileges enjoyed by its members. A collective choice, probably by a membership committee, is made based upon the applicant's credentials.
The procedure is usually carried out using a set of recommendations. The individual requesting the rights fills in a form stating his credentials. People are assigned to check the applicant's trustworthiness, qualifications and so forth. If the membership committee is satisfied, someone is instructed to assign the applicant to the group.

Expulsion from the Group
Should the membership committee decide to expel an individual from the group, that, too, would be a collective policy decision.
A situation that could clearly arise is for a number of individuals to infiltrate a group, subvert it by having themselves elected to the membership committe and then expelling all the members of the group who are not sympathetic to the infiltrators. Of course, those individuals might also do this legitimately, if the selection of the membership committee reflected the views of the current membership. What legitimates actions is a wider consensus.
It could be described as a dictatorial Discretionary Access Control scheme. The only latitude that individuals have is to be able to grant or deny access rights to members of their own group or to everyone. The privileged user can undo any access control operations performed by any individual.
To use some better terminology: subjects are entities who may possess access rights and objects are those entities to which subjects have rights to use. An operating system that uses a MAC scheme is represented in figure 3 and one that uses a DAC scheme is represented in figure 4.

MAC Scheme
This is a simple scheme. The administrator classifies all the subjects and the objects and it classifies some subjects above other subjects so that higher subjects can access everything that lower subjects can. This is more sophisticated. The administrator classifies all the subjects and can denote that they belong to certain a Group. Every subject belongs to the group of Everyone.
Individual subjects own some objects and can choose to grant access to • Either: their groups (or groups) • Or: to Everyone Groups do not own anything and neither can the group Everyone. In figure 4, the object A is owned by one of the subjects and has allowed access to the group its owner belongs to and to Everyone. The object B can be accessed by the group, but not by Everyone.
There is degree of autonomny granted to the subjects in that they may classify objects to be accessible to the members of the groups they belong to, but they may not choose which group, or groups, they belong to. Neither may they change ownership of an object they own 1 .

Database Management Systems DBMS
Some DBMSs have more flexible DACs which allow some individuals to be more privileged than others by granting them the right to grant rights. This feature is available in some DBMSs that support the Structured Query Language, SQL, which was based upon System-R, which is descibed by Denning in [Den89].
In effect, this is the same as the DAC scheme for operating systems, see figure 4, but the owner can grant access to subjects other than those in its group and it can grant to others the right to grant access, but not to those to whom access has been explicitly denied by the owner.

General Resource Management Systems
Information processing systems can be thought of as general resource management systems and the Open Distributed Processing ODP standards, in particular the prescriptive model, [ISO95], describe how an enterprise modelling language could be used to state the relationships between resource owners and users as behavioural contracts given in terms of a set of permissions, prohibitions and obligations. This can be seen as a generalization of the MAC and DAC schemes, but the ODP standards are only reference models and each information system should contain some component that embodies its enterprise model. This is a wholly new superstructure to an information processing system: there are some simple class relationships describing the ODP model set out in appendix A, these rather vaguely state the information model for the superstructure. Some mechanisms that could be used for applying policy, implementing the administrators, has already been proposed [Eav99b].
Recently, the Java programming environment has provided a powerful language for expressing permissions [SUN98]. However, it assumes the existence of some agent which would negotiate the contracts between resource owner and user. There have been only a few efforts made to develop arbitrator agents which could generate such contracts. There are some system proposals which attempt to apply abstract behavioural rules in terms of concrete permissions: in papers by Minsky [Min89,Min95]. Minsky's treatment is for information processing systems in general, but a paper by Rabitti et al. [RBKW91] describes authorization generation mechanisms which support a lattice model of authorization policy for an object-oriented database. The innovation of the system is that it generates authorization policy as it operates. Authorization is viewed as having three dimensions: Expression Authorizations specified by users, which are known as explicit and those that are derived by the system as known as implicit.  Bertino et al. attempt to produce mechanical means of generating authorization policy no-one would seriously expect a system to be driven wholly by mechanical recommendation, it would require choices to be made by people and, if that is the case, then a suitable collective choice procedure must be found.

Summary
Information processing systems are not mechanical systems. They represent the interests of people and the agents that comprise an information processing system will require policies that represent people's interests which will be used in formulating behavioural contracts between agents that own resources and the agents that use them.

Issues in Collective Choice Theory
The principal difficulty in collective choice theory is that if a group of people have to choose between more than two issues, there is no choice procedure they can adopt that is not open to abuse. By abuse, it is meant: • Denial of service: some agent can exercise a veto on any policy proposed.
• Enforced service: some agent can force a policy upon others.
An abuse takes the form of insincere behaviour: an agent acts not to fulfil his own interests, but to prevent others from fulfilling theirs.
This simple example, known as the Voting Paradox might help: Scenario 2.1 (Denial of Service by Policy Cycle). Three agents, x, y and z, have to choose between three services: A, B and C. x and y rank sincerely, but z ranks the policies to prevent x and y from reaching a compromise, i.e. policy A, see table 1. Just to clarify terminology: an election is an expression of collective choice and the policy chosen by an election is the outcome of what statisticians might call a trial. In a trial, there are a number of choices available to each voter, or individual, taking part. A policy is usually chosen with regard to an issue; the proposal that a policy should be followed regarding an issue is called a motion.

Two Policy Issues
This could also be described as a a two outcome trials. These arise when there are only two policies which can result: the choice is to accept a policy or not.

Number of Choices is always three
Although there are two policies, there are three choices: one can vote For or Against. One may also be given the explicit right to Abstain, and, by doing so, state that one cannot vote for or against. One may also choose Not to Vote.

Abstentions and Not Voting
Usually in referenda, there is no option to abstain and those who do not vote are considered to have abstained. This assumption is legitimate if one is sure that all individuals who are entitled to vote have been informed that they may do so and have made a decision not to. In most business processes, this would not be the case.
In what follows, it is assumed that all abstentions are explicitly made and that individuals who do not vote have excluded themselves.

Choice Function
Simple majority rule is the usual method for resolving two policy elections, but it by no means the only one. Two policy collective choice is dealt with in some detail in appendix B.
What is required from a choice function is that it is not open to abuse and it is decisive. Two policy issues can not be abused because it can be shown to be the case that the majority have chosen the policy. The only difficulty is resolving ties.

Three policy issues
A three-outcome trial, for example, one of A, B or C must be chosen. The voting procedures described here are explained at greater length in [Saa94] and and there is some documentation on voting methods at [VOT98]. What follows illustrates some of the problems that arise from using them.

Number of Choices can vary (a) Four Choices
One can organize the election so that are four choices: any of A, B or C and to abstain.
(b) Seven (or Eleven) Choices One could also allow voters to express a choice between their first two preferences and to abstain. So a vote might be: (A, B) which implies that A > B > C.
One can also allow voters to state they are indifferent between their first two choices, but prefer them to the third; an example of this kind of vote is: (A = B) > C.
(c) Thirteen (or Seven) Choices Also one can allow voters to express their choices as a ranking over all three policies, in two ways: • Strong ordering no statements of indifference allowed.
• Weak ordering statements of indifference are allowed.
The former allows seven choices of rankings, the latter thirteen.

Choice Functions
(a) Simple Majority Rule This could only be used when the voters are presented with four choices and clearly would not work: • Three voters: a tie can result from a policy cycle and can be used by one voter to deny service, see table 1. • Seven voters: if three vote A, two B and two C, then, even though a majority did not want A, A is chosen.
(b) Single Transferable Vote This could be used when the voters are presented with seven or eleven choices. It suffers from the same problem as the next procedure.
(c) Hare Voting System and Borda Preferendum These two can be used if one presents to the voters thirteen or seven choices; they, and the Single Transferable Vote procedure, all suffer from the same fault, [Dor79,Fis73], which is that voting is affected Policy and Ranking Voter w x y z i 4(3) 3(-) 2(2) 1(1) j 4(3) 3(-) 2(2) 1(1) k 1(1) 2(-) 4(3) 3(2) 9(7) 8(-) 8(7) 5(4) If the voters i, j and k are asked to rank the four policies w, x, y and z, then the order is w > x > y > z, but if asked to choose between w, y and z then (w = y) > z, but it was clear that w was preferred over y. The anomaly being that an irrelevant policy, x, serves to differentiate between relevant ones. It will be shown that there is no satisfactory choice function for more than two policy issues. The next note shows that the number of different orderings for a given number of issues increases dramatically.

n policy issues
The total number of different weak orderings for n policies can be calculated as follows: 1. Generate all the partitions[Ski90, p. 56] of n.
A Mathematica [Ins99] package is available [Eav99a] that performs the calculation. Table 3 lists the total number of different preference orders for up to 6 policies and clearly shows how large the search space becomes.

Sincere and Sophisticated Voting
Sophisticated voting utilizes some strategy whereby a voter does not vote for their first choice to ensure that their least-preferred policy is not chosen. For example, an electorate of seven votes sincerely for three policies x, y and z thus: 4 x, 3 y and 2 z, then x would be chosen. However, the z voters may prefer y to x so their sophisticated vote is for y.
An interesting example of a sophisticated vote is global abstention. If a motion is formulated which requires a choice between A or B, but all voters prefer C which is not proposed, a sophisticated response is a global abstention.
Unfortunately, sophisticated voters enjoy an advantage over sincere voters, but, to do so, they must formulate their own voting policy, which usually requires that they have some information as to the relative strengths of the different coalitions within an electorate and their choice of voting policy would, presumably, be decided by a sincere vote amongst them. One of the attractions of presenting an electorate with a complex agenda-of more than two issuesis that they are less able to formulate strategies amongst themselves, so that complex agendas should elicit more sincere voting, but the likelihood of a policy cycle arising is greater, as is made clear in a later section, §5 and by equation (2). Sincere voting would allow voters' underlying values to more precisely determined, which one would hope, would in the long-run be a more stable basis for decision-making.

Collective Choice Mathematical Model
This is more rigorous presentation of collective choice theory. This following section introduces the notation that will be used to formalize the conditions that are placed on collective choice functions.

Some Notation
Notation 3.1 (Relations, Preferences and Their Ordering: ≻). is a preference ordering over a set of objects in a finite set X = {x 1 , . . . , x n } constructed thus: R is an instance of a class of binary relations between any two objects. To state that x 1 is related to x 2 in some way, one would write: x 1 R x 2 . The particular relation might be any of the following >, ≥, ≤, =. At this stage, R is taken to be transitive and connected.
≻ is a statement of an individual's preference order or preferences over the elements of X. It is a tuple, i.e. a vector, with exactly #X elements, with each element being of the form x i R x j , where R is instantiated to one of the values that the class might take. The ordering is assumed to be consistent for whatever qualities R possesses. There must be at least one statement of preference for each element of X, even if that statement is one of indifference. It is assumed that such a preference ordering is consistent with the qualities of R. The power set of ≻ is X × X Notation 3.2 (Policies and Voters). Some simple set definitions are needed.
X a non-empty set is the universal set of social alternatives, or policies, at least one of which must be chosen by the voters.
X is a subset of the power set P (X) of X; it is a non-empty set of non-empty subsets of X and describes the potential feasible policy sets of X.
Y is an element of X . It is the set of policies that are presented to an electorate for them to vote on: the proposal set.
D is a preference profile of all voters, it will be called a vote, but will contain more than just the voters' preferences on the elements of the proposal set. It contains the preference orders of all the individuals in the society for all alternatives in X. For the n individuals, if individual i is presumed to have a preference order ≻ i , D can be written as the n-tuple (≻ 1 , ≻ 2 , . . . , ≻ n ) of preference orders on X.
P D is the power set of all votes, feasible and infeasible. For a given set of policies and a given set of individuals only a subset of these votes will occur.
(Y, D) is an ordered pair called the situation. It is the feasible set of policies, Y , presented to the electorate, and a vote D.
The important word is feasible. Only some votes will be feasible given the preferences held by voters; therefore only some policy sets will be feasible.
Definition 3.1 (Sincere and Sophisticated). The following function definitions clarify how voters make up their minds and form their preference orders. They are, therefore, purely notional and one or the other is performed by each individual, i. How a preference order is formed is dependent on whether the individual votes sincerely or is sophisticated. If an individual, or population, is voting sincerely, then: If an individual is a sophisticated voter, then a new set of histories is needed: D-and its power set P D -which is the set of all votes that have taken place.
Definition 3.2 (Promotion and Demotion). When stating conditions it is useful to construct votes from other votes. These may be elements of P D that are infeasible.
These functions promote and demote a policy within a vote.
And similarly, demote() would be implemented as follows: 1. Every preference not involving x is unchanged: 2. Everything that involves x is unchanged if x if preferred over something else; or is changed so that x is now preferred over the other policy Along the same lines, two other variants of promote and demote can be defined, which promote or demote for a particular voter on a particular policy and resolve any conflicts.
Definition 3.3 (Collective Choice Function). Is a function that maps each situation to a subset of the feasible subsets for that situation. The collective choice function F yields the choice set of the proposal set.
Typically the choice set will contain only one policy, the one that is preferred over all others. The chosen policy can then be removed from Y and the next policy found. In the event of a tie-if ties are tolerated-the choice set will contain the tied policies.
Remark 3.1 (Quorums). A quorum is usually taken to be the minimum number of voters that can demand that a policy be a legitimate choice. However, it may be that case that there is a quorum, but all votes, bar one, are abstentions and that the choice of that single individual becomes mandatory.
For the time being, this anomaly should be noted, and there will be references in the text to the validity of a policy decision.

Conditions on a Collective Choice Function
These conditions prescribe the behaviour of a collective choice function 2 . These are derived from Arrow's work [Arr63] and have been the subject of considerable debate. This rendering is original and, it is hoped, is more explicit, self-contained and rigorous than that given in Arrow's work. Each condition is expressed as a deduction rule in second-order logic: if the premises are fulfilled then the conclusion is required i.e. it is expected behaviour. The conditions therefore constitute tests for an implementation of a collective choice function.
A more succinct rendering can be found in a paper by Batteau et al. [PB81]. It requires some familiarity with Arrow's conditions and the theory of games 3 , but has the advantage of relating Arrow's conditions to work in games theory and also to requirements on the behaviour of collective choice functions. This latter task has been carried out very successfully by Fishburn [Fis74], but only for two issue collective choice functions. The paper by Batteau et al. only addresses collective choice functions that use strong orderings.
There are five conditions in all. There is a brief description of the meaning of each.
By specifying that individual voters must present orderings that are proper subsets of Y , this eliminates orderings that are completely indifferent or are cyclical, e.g. X = {x, y, z}, ≻ i = ( x > y, y > z, z > x)-so it is a condition on the vote set as well as on the collective choice function. In practice, it would be best to ensure that the orderings in D are well-formed.
The choice set F () cannot be empty and it must be a proper subset of Y . If it is neither of these then there is either global indifference or a policy cycle. ceterus paribus, it will be chosen for society again, or, more formally: Condition 3.3 (Independence). or "independence of irrelevant alternatives" [Arr63, p. 27] requires that the collective choice function return a choice set regardless of any individual's preferences for policies that are not explicitly part of the proposal set. This means that individuals may take on or discard values, or they may change their values regarding other matters, but these changes should not effect those values that have not changed. Formally, this can be expressed thus: This condition on the implementation of the collective choice function is probably unimportant in practice; normally, the input to the collective choice function is D which only contains the preferences on the contents of Y , but as can be deduced from the discussion of the effect of an irrelevant alternative in the Borda preferendum, see table 2, these can affect a preference order.
The following two conditions are more contentious. They are different from the other conditions in that they need not be applicable to all issues and there are two types of tests one can apply.

Condition 3.4 (Non-imposition).
There is no bias in the collective choice function that causes it, on some issues, to yield a choice set that is insensitive to voters' preferences.
The first test is a unilateral test, viz.
That is, it should be possible on a particular set of issues to construct a vote that does not return a particular policy for all votes that select that policy.
The second test is used in the event of a tie between some policies to ensure that the collective choice function does not prefer one policy over the other.
It might not be immediately clear from this formulation but this is an exact statement of the irrelevant alternative anomaly observed in the Borda preferendum, see table 2.
Condition 3.5 (Non-dictatorial). There is no one individual whose choice on some issues is always returned by the collective choice function, a dictator, nor is there any one individual who can reject some policies, a vetoer 4 . (Unfortunately, there is some contention about the use of the term "one individual", see the discussion following.)

Unilateral Tests
These test whether it is possible to overcome a dictator's choice or a vetoer's rejection. The dictator or vetoer is placed in position 1 for convenience.

Tie-Breaking Tests
Subtler tests are those that are applied in the event of a tie.

(a) Dictator
The dictator is preferred in some way. In that, if the dictator changes allegiance, policy changes, but if anyone else it does not.
This definition states that "no one individual" can dictate a vote, which would seem to suggest that individuals can change their minds, but must do so en masse. Arrow requires that a deciding set 5 of voters must change their preferences. How many need to be in that set can only be determined by analyzing a vote. It may appear that simple majority rule does not appear to meet this criteria, since, in a close result, any voter can invert the result, but under simple majority rule on two issues, this can be dismissed, because every voter has exactly the same capability, therefore the simple majority becomes the decisive set.
So, as it stands, this condition is still not accurately expressed, it should state that no subset of voters that is not a deciding set can change the outcome. The problem is that the deciding set cannot be known until the votes have been cast and counted.

Possibility and Impossibility Theorems
Arrow, who originally addressed the problem of the distribution of social welfare, developed these theorems as general statements about a class of functions which seek to combine hierachies of preference relations. Such functions would be of great use in any field where a joint policy must be formulated. Clearly, that includes distributed computing and the "globalization" of local access rules to databases to form lattices of information flow, §1.4 .
There are two results: 1. Possibility theorem for two-policy elections Such a collective choice function does exist for elections which have only a choice between two policies.
2. Impossibility theorem for elections having more than two policies There is, in general, no such collective choice function for elections having more than two policies.
The idea behind the proof of the possibility theorem has already been given in the discussion of deciding sets, condition 3.5, but the same idea is used in the proof of the impossibility theorem.

Conflict Resolution Mechanisms
Arrow's proof for the impossibility theorem consists of analyzing how a collective choice function can choose one preference over another. The notation is as used in §3.1 .
Definition 4.1 (Unanimous Choice). If the voters unanimously agree that one policy is preferred over all others then that policy is chosen.
Note that with this formulation it is possible to be indifferent to x, but one cannot oppose it 6 . For example, if some voter has ≻= (x = y, y > z, x > z) then {x, y} = F ({x, y, z}, (≻)).
Definition 4.2 (Biased Choice). If pairs of voters contradict one another over a policy, one policy is chosen over the other.
Note that this rule can only be applied pair-wise, it cannot be applied to the population as a whole.
This, too, may only be applied pair-wise.

Inadequacy of Conflict Resolution Mechanisms
Unanimity, with abstentions, does not resolve any conflicts. If one attempts to do so using one of the other two choice methods, one or more of the conditions will be breached.

Biased Choice
With two voters there is no majority decision, so the collective choice function must prefer: The first two lead to an imposed policy or indicate a dictatorship, respectively; the latter has been suggested [Zec69], but it is rather arbitrary.

Unresolved Choice
This method allows anyone to act as a vetoer.
One might think that one can improve the biased choice method so that it decides in favour of whichever policy has a simple majority over the other. Unfortunately, if one admits a third voter to make the biased choice decisive, then one also allows that third voter to present a third policy choice; in which case, one is attempting to choose between three policies, which is the problem one is trying to solve.
If, in an attempt to overcome this, one requires that there always be more voters than issues, then on some issues at least one voter will be a dictator, or vetoer 7 . The dictator, or vetoer, is acting as a "Kingmaker". It is possible to stand this dilemma on its head (or feet) and use it as the basis for a collective choice function as in [How89].

Are the Conditions Reasonable?
Hopefully, it should now be clear that it is not possible to construct a collective choice function that satisfies all the conditions given above simultaneously. That said, one can argue that the requirements on the collective choice function's behaviour are too demanding.

Decisiveness
Implicit in the definition of the collective choice function is that it is decisive.

The Non-imposition and Non-dictatorial Conditions
These conditions come in two forms. The first form is unilateral and quite acceptable, although it may even be desirable that one particular voter has an absolute veto over a policy. The other form of the condition is only invoked in the event of a tie, where the requirement is that no policy or voter be preferred over another. This, it has been seen, is the pivotal difference between simple majority rule for a two policy vote and a three (or more) policy vote. In the two policy vote in the event of there only being one vote separating those for and those against, every voter is equally decisive and will be supported by the majority. In a close three (or more) policy vote, some voters can choose the minority position to force a tie.
The issue is, again, the decisiveness of the collective choice function and would so close a result be acceptable.

Independence from Irrelevant Alternatives
The preferendum, see table 2, and Condorcet pairings are both examples of collective choice functions that are non-dictatorial but are not free from the effects of irrelevant alternatives. This is very unfortunate, since an irrelevant alternative is any policy that is ranked lower than the collective winner, but is ranked higher than the collective winner by some voters. This would happen when the voters are assessing the policies with different underlying values which are more abstract.
If one looks at the rankings made in table 2, it is clear that voter k agrees with the others that y > z but disagrees with them regarding the merits of both y and z over both of w and x.

Monotonicity
It can also be argued that monotonicity should be sacrificed to achieve a consensus. There is an attractive suite of collective decision functions called "Kingmaker Trees" [How89]. These do not demonstrate monotonic behaviour, but can be used to obtain a decision in the presence of sophisticated voting.

Are three (or more) policy collective choice functions usable?
Whether the conditions Arrow imposed on collective choice functions are reasonable only arises when the votes are close and that a sophisticated voter would vote in such a way that a policy cycle arises. Three (or more) policy votes can give decisive results and it would be very useful to have them resolve issues: can one therefore quantify how reliable a collective choice is? How likely is it that an election has been subverted by sophisticated voters, given the distribution of votes. Referring again to table 2, if it were possible to compare the two sets of votes, impartially, an administrator would be able to make a better choice of final policy. In which case, it would be better to introduce more irrelevant alternatives to make a more certain choice. Technologically, this is feasible. There are cryptographic algorithms which would allow vote sets to be cast secretly [Sch96] and these could then be assessed by an impartial arbitrator to make the most appropriate choice. The basis for that choice would be probabilistic and an example of a probabilistic criterion that could be employed is given next.

Max-Min Probabilities in Condorcet Pairings
There is an interesting paper by Usiskin [Usi64], which quantifies the probabilities for Condorcet pairings. The paper addresses the "Voting Paradox", but this is slightly misleading, it addresses the organization of votes within committees. It covers the same ground as the seminal works of Black and Farquharson [Bla58,Far69], which describe how committee procedures can be abused, if a policy cycle exists.
In a committee procedure, if there is a policy cycle then for all those voting: • A > B > C has exactly 1 3 of vote • as does C > A > B • and B > C > A If the policies are voted on in pairs, then the order in which they are introduced will determine which is chosen, viz.
The question that Usiskin resolves is how much more popular than one another can they be. In the example above, they all have probability of beating, or of being beaten, of 2 3 . Denote by X i a real-valued random variable that represents the proportion of a simple majority vote received for policy i. The probability of a simple majority vote having the outcome that X i > X j will be: P (X i > X j ). A policy cycle will be revealed if the probabilities for all pairs, P (X 1 > X 2 ), P (X 2 > X 3 ) and so on, for n policies is non-zero and, finally, P (X n > X 1 ) is also non-zero. The maximum minimum value will represent how much more popular one policy can be over another so that a policy cycle might still result.
If one then has at least one policy that beats another by an amount that is greater than this, then there can be no policy cycle.
Theorem 5.1 (Arbitrary Random Variables). The maximum minimum value for the joint probability distribution of a set of n arbitrary random variables is given by: max {min [P (X 1 > X 2 ), . . . , P (X n−1 > X n ), P (X n > X 1 )]} = n − 1 n (2) This is as one would expect, looking at the example given above, where each of A, B and C had a probability of 2 3 of beating the other, if one of these had a greater probability than that, there could be no policy cycle. In the example given above, were there were only three voters, this would mean the voters unanimously agreed on one policy being preferred over at least one other. This may not be the winning policy, it would allow at least one policy to be eliminated, then a two-policy vote can be taken. This result is rather depressing but one can appreciate its intuitive correctness, because it tells us that the more policies there are, the more difficult it is to have one policy beating all others.
This case of arbitrary random variables does not help in understanding the behaviour of sophisticated voters. (The probabilistic events would all be conditioned by at least the previous result, viz. P (X i > X i+1 |X i > X i−1 ) and probably would need to be conditioned by all events.
However, Usiskin does present a result which could be used to interpret sincere voting results. The election results P (X i > X j ) would be based on X i , X j being independent random variables Theorem 5.2 (Independent Random Variables). The maximum minimum value for the joint probability distribution of a set of n independent random variables is given by: max {min [P (X 1 > X 2 ), . . . , P (X n−1 > X n ), P (X n > X 1 )]} = b(n) where b(n + 1) > b(n) and lim This latter result is quite encouraging, because if there is an election where at least one votes has a probability of greater than 3 4 then no policy cycle can exist.
Usiskin also demonstrates a method for formulating the function b(n) and presents some upper and lower bounds.

Some Collective Choice Functions
It was mentioned above, in the discussion of quorums, that a simple majority rule collective choice would be valid even if only one voter expressed a choice and all the others abstained. Simple majority rule is just one of a number of collective choice functions that could be employed. It is worthwhile just listing the collective choice functions. These are only for two policy votes and, because of Arrow's conditions, cannot be extended to three (or more) policy votes, but they are insightful to the acceptability of collective choices. This summary follows Fishburn, [Fis74] and the details are contained in appendix B, but suffice to say that when only two policies, x and y, are under consideration ternary logic can be used with x > y being 1, y < x being −1 and x = y 0. There are some diagrams that illustrate the different types of voting rule and outcomes, figure 10 From the discussion above, §4.1 , a Pareto-optimal collective choice function can also be specified, which is not in the appendix, but is discussed in [ARS98] where it is called "unanimous with abstentions".  Non-minority if 1( D) > n/2 then x > y is the collective choice, see (4). This is a special case of the next type of rule. Absolute special majority is a variant of absolute majority (it is an absolute majority of votes against) and absolute majority is just a variant of specified majority where the complement of the policy is not installed if the required vote count is not reached and, as noted, non-minority rule is a variant of Specified Majority Rule. So all of these can be replaced by that rule.
It is possible to define a Paretian quality which can be added to any voting rule and requires there is no dissenting vote. (Paretian indifference can be thought of as a vote for z (x = y) as opposed to w = (x > y) ∨ (y > x), so it is actually a Paretian vote on a different pair of issues: z and w.) The rules can be ranked in a qualitative order of difficulty of attaining them, see table 4.

Summary
As information systems become more sophisticated they will be used to support human decision-making. The prospect of constructing virtual organizations based on how people interact is attractive: they could potentially be more responsive-Miller describes an evolving information processing organization, [Mil95], which develops its internal structure using a genetic algorithm. There are already some prototypes, [HK96,HH94], which share information based on past usage. This paper acts as a warning that collective decision-making is not something to be taken lightly. Even the most sophisticated voting systems can give rise to erroneous results, §2.2 . Sophisticated voters making policy choices could give rise to systems falling into stasis or being subverted to execute the wrong policies. If a discretionary access control mechanism used to control the release of information from databases were put under the control of collective choice functions and determined access rights based on the criteria proposed by Bertino et al. , §1.4 , it would almost certainly prove to be a vulnerable system.
There is clearly a need for a more sophisticated architecture to deal with access requests which can only be expressed in an enterprise modelling language which would have components similar to that described by ISO in their ODP, [ISO95]. This type of information processing system would need software agents acting on behalf of individuals to ensure that their information is protected. This would necessarily be a probabilistic analysis, based on how trustworthy potential information users are and it may prove expedient to develop systems that are insured against loss or provide degrees of surety, like those proposed by Neumann et al. [LMN97].
These information systems and their users would constitute an economy very much like the everyday commercial world occupied by institutions, corporations and people-only it would be faster, less resource wasteful and, if information system designers integrate the safeguards before they are used, safer.

A ODP Enterprise Entities
These are Booch [Boo94] class diagrams of the relationships that exist between entities in a distributed processing system, or, indeed, any organization as de- Performative agents are also resources. There are three kinds of these: • Administrators

• Arbitrators
• Policy-Makers Administrators and arbitrators have policies they follow, but policy-makers create policy.

Resource Users
These are also agents, but are not performative. They will have their own policies, but they are not explicitly open to arbitration. Resource users may contact administrators prior to using a resource or they may not, it depends on the nature of the resource.
Resource users usually control the policy-makers within communities and this is the case with most societies, since legislative assemblies are usually elected, but this need not be so. Companies are owned by its shareholders who appoint the board of directors, the administrators, but may not use the resources the company makes available. Suffice to say, that, in practice, in most business processes, the resource users have very little influence on the policy-makers.

Administrators, see figure 7
Administrators control sets of resources. Each consumable resource has a behaviour and may use other consumable resources. Since there may be Arbitrators control adminstrators in that they resolve any conflicts that arise between them. They control neither administrators nor resources directly.
6. Policy, see figure 9 Policy is a set of prescriptive statements about the behaviour of a set of resources: usually one sub-set of those resources visà vis another sub-set. There are three kinds of statement: • Permission what one sub-set may do with (or for) the other sub-set.
• Requirement (or Obligation) what one sub-set must do with (or for) the other sub-set. • Prohibition what one sub-set must not do with (or for) the other sub-set.

B Collective Choice Functions
What follows is drawn mostly from [Fis74] (who refers to proofs from his own text [Fis73]: and gives a precise definition of the conditions that a collective How that mapping is achieved is based on the properties of the collective choice function, which decides whether the choice is:

B.1 Two Policies
If each individual has only two policies to choose from, then the policy chosen by the population as a whole will always be one of them, so two policy systems cannot select a set of policies that an individual has not specified.
Definition B.2 (Sets for Two Policies). The sets can be enumerated quite easily for two policies x, y. If a voter prefers x to y, then 1 else if y to x then −1 else 0 signifies indifference-ternary logic.
For the collective choice function over n individuals: Note that the power set of the preferences is written {1, 0, −1} n as shorthand and is the set of all permutations of vectors of length n where each component can take one of three values-#{1, 0, −1} n = 3 n . When a condition is applied to a preference profile, it is either applied with reference to the power set or the vote set: the power set, although large, is denumerable a priori, the vote set is not.

B.2 Egalitarian
Egalitarian 8 collective choice functions treat each voter's vote as identical in effect to every other's.
Definition B.4 (Non-minority Rule). If a collective choice function is Strongly Neutral, Monotonic and Egalitarian and is implemented thus: then the voting system is known as non-minority rule.
Non-minority rule is just one of a class of neutral, monotonic and egalitarian collective choice functions; they differ in effect from the strongly monotonic simple majority rule by having a "dead-band". A geometric insight into the specification of a collective choice function can be given using a unit simplex [Saa94]. There is only one dimension.
The election vector q emanates from the origin and will always be within [−1, 1]. Under simple majority rule, whichever point, −1 or 1, the vector is closest to wins. Under non-minority rule the vector has to be over half 1 2 way towards the point, see figure 10. The indecisive region in the centre is symmetric. For non-minority rule, any boundary can be chosen, so long as it is symmetric about the origin. It should be clear from figure 10 that a simple majority voting system can determine policy if only one voter is not indifferent to either of the policies, the others abstaining. In this respect, non-minority rule seems to impose a natural quorum, since it requires that at least half of the voters have chosen one or the other policy. In this respect, non-minority rule is less questionable as a decision-making device than simple majority rule. Most electoral systems do in fact operate a non-minority rule system.
The number of electoral ties can be reduced by downgrading Strongly Neutral to Neutral and adding Strongly Decisive. If Neutral is dropped then an electoral preference is given to one policy over the other. The policy that is preferred is usually already in force and is therefore called the incumbent, the other policy is the challenging policy. See figure 10 for the asymmetry of the indecisive region under an absolute majority rule. Definition B.6 (Absolute Special Majority Rule). As for absolute majority rule, but the function is also (Unanimity unambiguous) and (Pro-biased). The collective choice function can be implemented by:

B.3 Non-Egalitarian: Weighted
The effect of the Egalitarian property is that one voter's preference can be exchanged for another within the decision profile and it will have no effect on the evaluation of the collective choice function. An alternative system is to use a weighted system, which is often used by some committees, where the chairman is given both a deliberative and a casting vote. Weighted systems have a number of attractions because they can be designed so that: • No one person can dictate policy to the group.
• Ties can be readily resolved without a further vote.
The simplest way to define a weighted voting function is to use a weighting vector.

Weighted Voting
Definition B.7 (Weighting Vector and Vote). A weighting vector can be defined thus: where ρ i ≥ 0, ρ i ∈ Z + 0 is the weight assigned to each voter Then It does not follow that n( ρ) ≥ n because it is possible to set any number of ρ i equal to zero, but at least one must be non-zero. The conditions can be readily tested by substituting F ( ρ · D) for F ( D). The other systems can be added thus: 1. Non-minority rule Otherwise

Dictators and Vetoers
The final choice may be wholly determined by only one of the voters, in which case that voter is either a dictator or a vetoer. This is not particularly useful, since there may legitimately be a voter whose vote is always in line with the choice of the group as a whole.

Vetoer
Definition B.10 (Vetoer). The first voter, 1, is a Vetoer with regard to a collective choice function F and a weighting vector ρ if: Whether a vetoer can effect those two very specific decision profiles is a condition on the behaviour of the collective choice function and the weighting vector, not on the voters, which is as follows. This, again, is not particularly useful, since there may legitimately be a voter whose always votes against the group.

Sensitivity
Definition B.12 (Essential). A voter i is said to be Essential with regard to a collective choice function, and weighting vector, if at least one of the following conditions holds: For at least one of the vectors D 1 , D 0 , D −1 constructed as follows: This condition requires that a voter can be decisive in at least one decision profile. It prevents a voter from being given so ineffectual a vote that it is never decisive in any election.

B.3.1 Vetoer, Dictator and Essential
By this it is meant safe from dictators and vetoers and sensitive to voters; it is desirable if a collective choice function and weighting vector could be chosen so that for all decision profiles in {1, 0, −1} n there is no voter who is either a Dictator or a Vetoer. It would also be desirable there is at least one voter who is Essential.
The collective choice functions simple majority, non-minority and absolute majority are demonstrably safe from dictators and vetoers when used under an egalitarian regime so only the weighting vector needs to be checked.

Weighted majority and weighted non-minority rule
If the collective choice function is either of the above, for ρ = (ρ 1 , . . . , ρ n ) and the weighting vector has been scaled so that ρ 1 has weight 1. In which case, the worst case for the dictator is that all vote against, so for the dictator to succeed ρ dictator > W ( ρ) / 2. Because both of these collective choice functions are dual; it should be clear ρ dictator = ρ vetoer .
Weighting Vectors that are Essential to Voters Under egalitarian rule, all voting functions are sensitive to all voters, because if any voter is sensitive, a permutation can put another voter in his place, so again the collective choice function will not be at fault should a system of rule prove insensitive, it will be the weighting vector.
There are two possibilities: • The voter has a weighting of zero • The voter has a weighting which can never be decisive Whether a voter has a non-zero vote can only be tested for; probably by using the weighted majority rule function with all other voters not voting.

B.4 Representative Systems
A representative system can be thought of as a heirachy of voting councils in which the outcomes of votes in lower councils become votes in higher councils. A voter in one of the higher councils may be a voter or a voting council. Each voting council can use weighted majority rule between its members. Voters (or councils) can vote more than once in different councils.
Definition B.13 (Representation). Let a heirachy of voting councils be defined as a representation R. Let each level of representation be denoted by a suffix, the lowest level being R 0 . To make the lowest level similar in mathematical structure to higher levels, we shall introduce a selection function S i which selects from a preference profile, D, the preference of voter i. Thereafter, there is a level m ∈ N which is such that: R m = {s(F 1 , . . . , F K )(R m−1 )} This is effectively voting using a tree structure and is probably the most used organizational control system. Unfortunately, it is proving to be very difficult to analyze. Hopefully, more results will arise.