Chapters 3-4 of Analyzing Politics  

K.A. Shepsle

A Summary 

Getting Started

  • If a group has to decide between a set of feasible actions one way to proceed is by unanimous agreement. However, preferences may be ordered such that a solution is not possible, tastes are too heterogeneous. E.g. A, B and C to decide between x, y and z. A: x>y>z B: y>z>x C: z>y>x. The group does not unanimously share the first preference. Likewise majority rule fails to provide an answer as no two members share the same first preference. A round robin tournament where each outcome plays each outcome and if one is preferred by a majority to all the other it is the group choice is also possible. x vs. y yields y; and x vs. z yields z; and z vs. y yields y. Thus y is the majority preference of the group. But as different majorities prefer y to each other alternative there is incentive not to vote sincerely, that is to be strategic. For example player C could lie in the first round and vote for x. In that way the outcome would be x, z, y, i.e. the tournament would have no winner. So this method only works if people vote sincerely.
  • The preferences may be ordered such that in a round robin tournament there is a cycle of preferences. That is whilst every individual has transitive preferences, when they are aggregated the group preferences are ordered x>y>z>x. This is a preference cycle, and would occur in the above example if player C: z>x>y. To solve this problem an agenda is needed whereby one player decides upon the ordering of the voting in a system where the first pair is voted on, and the winning item then plays the last item, and the winner of that is the group choice. Any of the players can get their most preferred outcome if they believe the others will vote sincerely.

 Condorcet Cycles

  • Even though individuals have consistent preferences the group may not. As the number of alternatives and the number of voters increases the probability of experiencing a majoritarian cycle approaches 1. This probability is radically smaller in small groups/few alternatives situations. In general though it means we cannot rely on majority rule to produce a coherent sense of what the group wants especially if there are no institutional mechanisms for keeping participation low or the number of alternatives small.
  • In reality we do not always observe such cycles. This is because even though probability approaches 1, the cycle is dependent upon each possible ordering of preferences to be equally likely. However, normally what we observe is some interdependence among individuals, whereby people share the same preferences.
  • e.g. if three mayors {A,B,C} have to divide an amount of money, there is no possible distribution of the funds that is not preferred by a majority to every other distribution. If they all start with a distribution {33.3%, 33.3%, 33.3%} a majority of A and B prefer {500, 500, 0}, but then a majority of A and C prefer {600, 0, 400}, and a majority of B and C prefer {0, 500, 500} and so on ad infinitum. The final outcome in the scenario will depend upon institutional features i.e. who has agenda power and how can they use it.
  • Thus the only way to avoid preference cycles is to impose some sort of antimajoritarian restriction.

 Arrow’s Theorem

  • The Condorcet paradox is a problem for any reasonable method of aggregating group preferences.
  • The impossibility result follows: there exists no mechanism for translating individual preferences into coherent group preferences without some kind of dictatorial power. In other words preference aggregation and choice is either incoherent or dictatorial. There is a tradeoff thus between social rationality and the concentration of power. Concentrated power allows for social coherence as the dictator knows his own mind. So systems without such power may seem fairer, but they are also more likely to be inconsistent in ordering alternatives under consideration.
  • This does not mean choices will always be inconsistent, only that it is impossible to rule out such choices. This means it may be possible for clever manipulative people to be strategic in exploiting this fact.
  • The bottom line is that individuals may be rational, but the group need not be. This makes is difficult to speak of legislative intent when thinking about interpreting the law for example.


  • Duncan Black thought that minimal forms of consensus other than unanimity could be sufficient to allow for coherent group choice.
  • In particular if preferences are single peaked then majority rule works perfectly well despite legislators holding wildly divergent views on what the group ought to do.
  • This will be evaluated further in chapter 5.

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