以下のシラバスは最初の計画だった．受講予定者の1名がドロップすると言ってきて受講者が前期に演習に出ていた1名だけになったため，受講者の関心を考慮し内容を大幅に変えることにした．この授業は前期の演習に引き続いて David Austen-Smith and Jeffrey S. Banks. Positive Political Theory I: Collective Preference. 1999. Chapters 5-7 をやることになった．ノートは前期演習のときほどきちんとしたものは作らない．
香川大学経済学研究科 2003 年度
政治理論特殊講義 (Political Theory)
三原麗珠
2単位 第2学期 とりあえず金曜 5, 6 時限目 (1620-1930) 場所 三原オフィス
合理選択政治理論 (実証政治理論・新制度論) はミクロ経済学に見られるような分析的モデルにより政治現象を説明しようとする．説明の際，ルールなどの社会構造の制約のもとで合理的に行動しようとする個人を想定するのが特徴である．この授業では，社会選択またはゲーム理論をもちいた最近の政治理論の研究成果を検討していく．特に日本政治理解のヒントになりそうなペーパーを重点的に取り上げる．
サンプルとして，戦略的投票とDuverger's law にかんするものをいくつかあげる:
参加者は精読したいペーパーを探してくる．その際，導入部とモデルの部分を中心に読めばよい．精読することにしたペーパーは，講師といっしょに苦労しながら解読していくことになる．自分の研究につなげることを目標とする．
指定しない．
ゲーム理論の政治への応用:
手ごろなゲーム理論テキストと，ゲーム理論がコンパクトにまとまった大学院ミクロ経済学テキスト:
新制度論のアプローチを取る政治過程論．受講前に読んでおくこと:
非協力ゲーム理論，公共経済学など．武藤(2001)レベルの非協力ゲーム理論(不完備情報をふくむ)と学部レベルのミクロ経済学の知識を前提とする．
口頭試験・筆記試験・練習問題アサインメント・授業への貢献度などのうちのいくつかの組み合わせにもとづいて認定する．
1630-1930: Text (Austen-Smith and Banks) 5.1. No notes prepared, but it did not cause much problem since the student read the text in advance. He said the latter half of the section had been too difficult for him to understand before the class. I like Lemma 5.1, which gives a necessary and sufficient condition for a nonempty core, better than Theorem 5.1, which gives a sufficient condition using semi-convexity of preferences. Comments: -P. 125, line -3 in the paragraph just after Definition 5.4: We do not need to use reflexivity of R. -P. 125, line -3 in the paragraph just after Definition 5.4: Is it true that if R is semi-convex but not strictly convex, then R is not transitive?
1620-1950: Text (Austen-Smith and Banks) 5.2-5.3. Notes on the proof of Theorem 5.4. The proof of Theorem 5.4 may be hard to grasp at first (especially if you cant illustrate the construction). Other proofs should be straightforward. 1950-2100: Talked about recent topics like network formation and allocation of indivisible objects, where each person can obtain more than one object. Typo: -P. 132, line -9: Lemma 3.4 (3).
1620-2030. Text (Austen-Smith and Banks) 5.4, up to Theorem 5.6, page 142. Notes on the proof of Theorem 5.5. The student did not visualize some key concepts before the class. After I explained them he realized they were not so difficult. Comments: -P. 134, Proof of Lemma 5.5. I used strict convexity of preferences in my proof of the $\subseteq$ part. -P. 136, last line. To show $xRy$ for any $y$, suppose otherwise and substitute $y$ into $w$. -P. 137, line 16. Theorem 5.5. -P. 139, Figure 5.9. Y refers to the line connecting the arrowheads. -P. 139, last line. Since h is a vector, the second expression should be rewritten. -P. 142, just after Theorem 5.6. The necessity half of Theorem 5.6 uses Lemma 5.5 as well as Lemma 5.6. I doubt the last statement there since my proof of Lemma 5.5 used strict convexity of preferences.
No class because of the university festival.
1620-2150. Text (Austen-Smith and Banks) 5.4, from Theorem 5.6, page 142. Notes on the proof of Theorem 5.8; notes on the proof of Corollay 5.2 (after the class). -P. 151. I had not prepared for Corollary 5.2; we spent more than one hour on this. It was resolved only after the class.
1650-1830, 1900-2110. Text (Austen-Smith and Banks) 6.1, up to Lemma 6.2, page 165. This section is concerned about singularity theory.
1640-2100. Text (Austen-Smith and Banks) 6.1 (from page 165)-6.3 (up to Theorem 6.3, page 173). The proof of Theorem 6.1 involves too many citations to results in differencial topology, not giving engough intuition.
1430-1800 Text (Austen-Smith and Banks) Proof of Theorem 6.3. (pages 173-6, up to Claim 6).
The proof of Claim 5 is very difficult. I had to add an assumption.
P. 174, line 7: Better to say "Since $P_L^{-1}(y)$ is open".
P. 174, line 14: by R complete.
P. 176 line 2 (similarly, line 3): $P(z,y; \rho)$, not $P(y,z;\rho)$.
1710-2240. Text (Austen-Smith and Banks) Pages 176 (Claim 7)-184 (end of chapter 6). It took longer than it should since I found out that I had to work on some proofs that I did not do before the class. There were something that I did not understand about the proof of Lemma 6.3. Example 6.7 is accessible to an undergraduate student.
1620-1800. Text (Austen-Smith and Banks) Chapter 7.
1320-1700. -J. Roemer, 2001, Political Competition: Theory and Applications. Mathematical Appendix, pages 309--322. Especially Section A.1 on basics of probability theory. -立石寛「数学付録」(オーマン『ゲーム論の基礎』1991勁草書房 に所収)の III. 測度と確率，160ページまで． We did not cover any probability approaches to political theory in this course. Instead, we tried to pick the minimal knowledge about the Lebesgue integral in this class. We covered Roemer's appendix fairly carefully, mentioning the connection with an ultrafilter, for example. We then looked into the definition of the Lebesgue integral a bit more carefully, using Tateishi. P. 309, Roemer: The range of a measure should include \infty.
1650-2010.
-William Thomson. On the axiomatic method and its recent applications to game theory and resource allocation.
Social Choice and Welfare, Vol. 18, pp. 327-386, 2001.
part I (pages 327-353).
Pages 337-8: When, say, axioms A1, A2, A3, A4 characterizes a solution F*,
Thomson says A1 is _independent_ of A2, A3, and A4 if there is a solution
different from F* that satisfies values -+++
(that is, satisfies -A1 [the negation of A1], +A2 [that is, A2], +A3, and +A4).
Maybe this definition is widely used, where a characterization theorem is given.
But I think it is more appropriate to use the term ``indispensable'' or ``cannot be dispensed with''
than the term ``independent'' in this context, at least when we talk about the direction
``if F satisfies A1, A2, A3, A4 then F=F*.''
I would define that A1 is _independent_ of A2, A3, and A4 if
for each combination of values of A2, A3, and A4 (there are 2^3 such: +++, ++-, ..., --+, ---),
if there is a solution satisfying the values (e.g., +-+), then there are a solution satisfying
A1 and the values (++-+) and -A1 and the values (-+-+).
[That is, for each combination of values of A2, A3, and A4 (e.g., +-+),
there is a solution satisfying A1 and the values (++-+) iff there is a solution satisfying -A1 and the values (-+-+).]
[Identify each axiom with the set of solutions that satisfy the axiom. Then,
A1 is _independent_ of A2, A3, and A4 if for each subset S of {A2, A3, A4}, we have
$A1\cap (\bigcap S)\setminus (\bigcup S^c)$ is nonempty iff
$(A1)^c\cap (\bigcap S)\setminus (\bigcup S^c)$ is nonempty.
Here $A^c$ is the complement of $A$, and for S={A2}, \bigcup S^c=A3\cutp A4.
]
Page 342: Thomson says that evaluating characterizations by the number of axioms on which
they are based is not a good idea. Maybe that's right, but how we present a theorem is
a matter of strategy too. We need to contrast the contribution of the paper with the literature,
while taking into account the limited cognitive ability of humans.
Page 347: Is it true that L4 is stronger than L2?
What if a solution is f(x)=x,
the smaller domain D the set of negative reals, and the larger domain D' the set of reals,
where the axiom asserts existence of positive value?
How about the axiom asserting the existence of a maximal element?
L4 seems to be stronger thatn L2 to the extent that the axioms involve only universal (``forall'') quantifiers.
Pages 352-3: In the parameterization of segment L, \alpha+\beta should be larger than 1.
Otherwise, M and N would be on the segment.
[It would be more suitable to record this as part of graduate seminar
(which dealt with Young's Equity in the second semester).
It is recorded here since it used the time slot for the graduate Positive Political Theory.
Only one and the same student is attending each of these two graduate courses.]
1220-1500. -William Thomson. On the axiomatic method and its recent applications to game theory and resource allocation. Social Choice and Welfare, Vol. 18, pp. 327-386, 2001. part II, pages 353-367.
1500-1640, 1710-1900. -William Thomson. On the axiomatic method and its recent applications to game theory and resource allocation. Social Choice and Welfare, Vol. 18, pp. 327-386, 2001. part II, pages 367-377. The student had had a difficulty understanding Section 11, which effectively says you have to make sure that there is a correponding concrete problem when working in the abstract. If a proof of a theorem resorts to an abstract problem for which there is no corresponding concrete problem, then the theorem itself is not well grounded. Told the graduate student, who will begin to take my graduate seminar this coming spring, to fill in gaps of Young's appendicies A.1, A.2, A.4 during the spring break.