Chapter 2: Problems and Discussion Questions

Analyzing Politics: Answer Key

Iain Osgood July 7, 2010

Chapter 2: Problems and Discussion Questions

• Either player can identify a top choice (or choices in the case of indierence) for any subset of size 2. This

is always the case as long a person's preferences satisfycomparability. There are 5 subsets with 3 or more outcomes: wxy, wxz, wyz, xyz and wxyz. Mr. i's and Ms. j's most-preferred outcomes over these subsets are shown in Table 1. Where either actor cannot state a most-preferred choice or choices, the table contains a `-'.Mr. i Ms. j Subset Top choice Cycle? Top choice Cycle?wxy x No x,y No wxz - Yes x No wyz w No y No xyz - Yes x,y No wxyz - Yes x,y No

Table 1: Mr. i's and Ms. j's preferences for within subsets of the outcomes.

Preference orderings over a subset of outcomes which contain a strict preference cycle throughallof the elements in the subset have no articulable top choice. Preferences orderings over a subset of outcomes which contain a strict preference cycle throughless than allof the outcomes in the subset may or may not have an articulable top choice. By way of comparison, note that i has no top choice among the subset wxyz, but an individual with the following preferences would strictly prefer w despite the presence of a cycle through

x, y and z: wPx, wPy, wPz, xPy, yPz, and zPx.

Thus, transitive preferences are a sucient condition for identifying top choices with respect to any possible set of outcomes. Technically,acyclicpreferences over all triples of alternatives in a subset are necessary and sucient for each subset to possess a maximal element. Acyclicity is dened, for any x, y, z, as: if xPy, yPz then not zPx. Transitivity implies acyclicity and thus is sucient for the existence of a maximal element.Inasmuch as maximizing behavior often relies on specifying clear ordinal rankings among outcomes, transitive preferences are eectively a prerequisite for a rational choice approach to individual decision-making.

• A reasonable assumption is that Senator Clinton's preferences at that point were as follows:P > S > C.

Because she chose the oce of Secretary of State, it would not be reasonable to assume thatP > C > S.The assumption thatP > Sis stronger, but corresponds with reports in the media at the time.1 Solutions Manual for Analyzing Politics Rationality, Behavior, and Instititutions, 2e Kenneth, Shepsle (All Chapters) 1 / 4

The fact that Senator Clinton chose the position within the administration implies the following:S > (1p)C+p(P). The chance to serve as Secretary of State was preferred to a risky `lottery' over remaining in the Senate and eventually becoming President. We can rearrange the parts of the inequality to derive

the following statement:

SC PC > p. Thus, the lowestpwhich would induce Hillary Clinton to stay in the Senate is SC PC =p, and anypgreater than SC PC would induce a strict preference for remaining in the Senate. This relation also suggests three interesting comparative statics when all other variables are held constant: 1. there is a threshold asSincreases at which one cannot resist the oer of the Secretaryship; 2.there is a threshold asPincreases at which one will reject the Secretaryship and hold out for a chance at the Presidency; and, 3. asCincreases the thresholdpat which one will accept the Secretaryship decreases.This latter eect occurs because as serving in Congress becomes more desirable, the lottery overCandP becomes more appealing relative to theS.

• Recall that the theory of expected utility states that given two dierent lotteries, L and L

, over the same outcomes, then LPL

if and only if P xX p(x)u(x)> P xX p

(x)u(x). Using this denition we can get the

following two relations about L1and L2, and L3and L4:

L1PL2impliesu(y)> :10u(x) +:89u(y) +:01u(z) and L4PL3implies:10u(x) +:90u(z)> :11u(y) +:89u(z) The trick here is to manipulate these expressions to show that they imply a contradiction. Consider the following two steps: add:89u(z) to both sides of the rst expression, and then subtract:89u(y) from both

sides of the rst expression. This yields:

:11u(y) +:89u(z)> :10u(x) +:90u(z) which implies L3PL4, which contradicts our second expression.Two steps in this process require further justication. First, is it okay to add and subtract constants to an expected utility expression? This is ne: expected utilities act like numbers and adding the same number to each side of an expression will not change the overall preference relation. If I prefer 2 apples to 1 orange, then I should also prefer 2 apples and 10 units of utility to 1 orange and 10 units of utility. Second, can we take these `manipulated' expressions and treat them as identical to a genuine lottery? If we are smart about our manipulations (i.e. add and subtract things so that we still have a proper probability distribution where all of the probabilities are between 0 and 1, and also sum to 1) then we can treat the new objects as ordinary lotteries. This is what allows us to compare our manipulated version of expression 1 with expression 2.

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Chapter 3: Problems and Discussion Questions

• Using plurality rule and voting over all four alternatives, plan A wins with two votes. In a round-robin

tournament, B is aCondorcet winner: it beats all of the other plans in a head-to-head competition. If plan B is removed from the competition, there is no Condorcet winner. Plan A defeats plan C in a head-to-head vote; plan C defeats plan D in a head-to-head vote; and, plan D defeats plan A in a head-to-head vote.Thus, there is a group preference cycle and the group has intransitive social preferences among the three outcomes, precluding identication of a top choice.

• The following table provides the outcomes of every head-to-head vote overq; r; s; tfor the original and

the revised preferences:

Matchup: qr qs qt sr st rt

Original r q q r t r Revised r q q s t r Under the original preferences, there are no group preference cycles. Under the revised preferences, there are preference cycles among (q; s; r), (r; t; s) and (q; r; s; t). Under both sets of preferences, each individual's preferences are both complete and transitive, satisfying our very basic rationality requirement. However, for the revised preferences, these individually rational preferences give rise to collectively irrational social preferences which feature several preference cycles. This illustrates the fact that rational preferences at the individual level are no guarantee of rational preferences at the collective level.

• i cannot identify a sequential agenda to secure outcomeqbecause outcomeris a Condorcet winner (it

beats any other outcome so it survives every round of a sequential agenda and thus always beatsqat some point in the voting). After k's preferences change,ris no longer a Condorcet winner and the following agendas will lead toqwinning:rsjtjq,rsjqjtandrtjsjq. This notation, for example withrsjtjq, means that in round 1,sandrface o; the winner of that round facestin round 2; and, the winner of round 2 facesq in round 3.Before k changes his mind, j will secure his most-preferred outcome with any agenda, because his top choice, r, is a Condorcet winner. However, similar to player i, k cannot secure his top choice,t, becauserbeats all challengers. After k changes his mind, j can secure his most-preferred outcome with several agendas:sqjrjt, sqjtjr,stjqjr,stjrjqandqtjsjr. Likewise, k can secure his most-preferred outcome withrqjsjt. In answering this question, you may nd it helpful to enumerate all 12 possible agendas and to determine the victor for each one.It is not possible to fashion a sequential agenda which defeats a Condorcet winner, because by denition such an outcome will survive every head-to-head competition it faces in any agenda. Absent a Condorcet winner, all kinds of manipulation are possible although it is of course not possible to create an agenda which leads to the victory of an outcome whichlosesin a head-to-head competition with all other outcomes.

• If player i votes honestly under agendastjrjqthenris the victor. However, if he misrepresents his

preferences by voting strategically for optiontin Round 2, then he secures his most-preferred outcome,q

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in the nal round (assuming the others vote sincerely). Under agendarqjsjt, if j (as well as i and k) vote sincerely, thentwill be the nal outcome. However, if j strategically votes forqin round 1, thenqwill win (assuming the others vote sincerely) providing a superior outcome for j.

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