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Solutions Manual for Analyzing Politics Rationality,
Behavior, and Instititutions, 2nd Edition by Kenneth, Shepsle
(All Chapters) A+
Analyzing Politics: Answer Key
Chapter 2: Problems and Discussion Questions
4. Either player can identify a top choice (or choices in the case of indifference) for any
subset of size 2. This is always the case as long a person’s preferences satisfy comparability.
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 l. 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 l: 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 through
all of the elements in the subset have no articulable top choice. Preferences orderings over a
subset of outcomes which contain a strict preference cycle through less than all of 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
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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 sufficient condition for identifying top choices with respect to
any possible set of outcomes. Technically, acyclic preferences over all triples of alternatives in a
subset are necessary and sufficient for each subset to possess a maximal element. Acyclicity is
defined, for any x, y, z, as: if xPy, yPz then not zPx. Transitivity implies acyclicity and thus is
sufficient for the existence of a maximal element. Inasmuch as maximizing behavior often relies
on specifying clear ordinal rankings among outcomes, transitive preferences are effectively a
prerequisite for a rational choice approach to individual decision-making.
5. A reasonable assumption is that Senator Clinton’s preferences at that point were as
follows: P > S > C. Because she chose the office of Secretary of State, it would not be reasonable
to assume that P > C > S. The assumption that P > S is stronger, but corresponds with reports in
the media at the time.
The fact that Senator Clinton chose the position within the administration implies the following:
S > (l — p)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: S—C
> p. Thus, the lowest p which would induce Hillary Clinton to stay in the
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Senate is S—C
= p, and any p greater than S—C
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: l. there is a threshold as S increases at which one cannot resist the offer of the
Secretaryship; 2. there is a threshold as P increases at which one will reject the Secretaryship and
hold out for a chance at the Presidency; and, 3. as C increases the threshold p at which one will
accept the Secretaryship decreases. This latter effect occurs because as serving in Congress
becomes more desirable, the lottery over C and P becomes more appealing relative to the S.
6. Recall that the theory of expected utility states that given two different lotteries, L and L′,
over the same outcomes, then LPL′ if and only if ΣxєX p(x)u(x) > ΣxєX p′(x)u(x). Using this
definition we can get the
following two relations about Ll and L2, and L3 and L4:
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 first expression, and then
subtract .89u(y) from both sides of the first expression. This yields:
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which contradicts our second expression.
Two steps in this process require further justification. First, is it okay to add and subtract
constants to an expected utility expression? This is fine: 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 l orange, then I should also prefer 2 apples and lO units of utility
to l orange and lO 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 O and l, and also sum to l) then we can treat the new objects as ordinary lotteries.
This is what allows us to compare our manipulated version of expression l with expression 2.
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