上海交通大学：《数学的天空》课程教学资源_十八大专题选举制度

之洲润 Chinanews.com Figure1:2013年两会 课程网站 http://math.sjtu.edu.cn/course/skymath/ 第-0∞2讲2013年诺两会选举 2013年3月14日：全国人民代表大会选举习近平同志为中华人民共和国 主席。至此，中国第四、第五代领导人权力交接正式完成！ 普通认为，我国的选举制度是具有中国特色的，不同于西方的民主选 举，但究竞有何差别、哪一种选举体系更为科学合理呢？ 此问题早在上世纪60年代就有定论，这就是1972年诺贝尔经济学奖得 主、现斯坦福大学经济学教授Kenneth Joseph Arrow的Social Choice and Individual Values(社会选择与个人价值，有中译本)的著名结果：Arrow's impossibility theorem(阿罗不可能性定理)。 1

Figure 1: 2013❝ü➡ ➅➜✤Õ http://math.sjtu.edu.cn/course/skymath/ ✶ −∞2 ù 2013 ❝ìü➡➚Þ 2013❝3✛14❋➭✜■❁➡➇▲➀➡➚Þ❙❈➨Ó➇➃➙✉❁➡✁Ú■ ❒❘✧➊❞➜➙■✶♦✦✶✃➇✰✓❁✙å✂✚✔➟✑↕➐ ✃Ï❅➃➜➲■✛➚Þ➏Ý➫ä❦➙■❆Ú✛➜ØÓ✉Ü➄✛➡❒➚ Þ➜✂➘➽❦Û☛❖✦❂➌➠➚Þ◆❳➁➃❽➷Ü♥◗➸ ❞➥❑❅✸þ➢❱ 60 ❝➇Ò❦➼Ø➜ùÒ➫1972❝ì✓✏➨▲➷ø✚ ❒✦②❞✧✹➀➷➨▲➷✓➬Kenneth Joseph Arrow✛Social Choice and Individual Values(✖➡➚❏❺❻❁❞❾➜❦➙➮✢)✛❮➯✭❏➭Arrow’s impossibility theorem(❈ÛØ➀❯✺➼♥)✧ 1

LOOK INSIDE! h 面过 年围出里片量 Figure 2:Social Choice and Individual Values 2

Figure 2: Social Choice and Individual Values 2

Figure3:1962年诺贝尔经济学奖得主阿罗 阿罗1921年8月23日出生于一个罗马尼亚裔的犹太人家庭，1940年在 纽约市立大学获得数学学士学位，1941年在哥伦比亚大学获得硕士学 位。1951年他在哥伦比亚大学获得哲学博士学位。 3

Figure 3: 1962❝ì✓✏➨▲➷ø✚❒❈Û ❈Û1921❝8✛23❋Ñ✮✉➌❻Ûê❩æ➽✛❣✔❁❬❒➜1940❝✸ Ý✕➼á➀➷➻✚ê➷➷➡➷➔➜ 1941❝✸①Ô✬æ➀➷➻✚❛➡➷ ➔✧1951❝➛✸①Ô✬æ➀➷➻✚ó➷➷➡➷➔✧ 3

阿罗不可能定理源自十八世纪法国经济学家孔多塞(Condorcet)的“投 票悖论”(voting paradox): 例假设甲、乙、尼三个选民对三个候选人A、B、C的偏好排序如下 表： 甲 A>B>C 乙 B>C>A 尼 C>A>B 试给出最好的选举方案 显而易见，这种所谓的“社会偏好次序”包含有内在的矛盾，即社会 偏好A胜于C,而又认为A不如C! 4

❈ÛØ➀❯➼♥✌❣➏❧➢❱④■➨▲➷❬➎õ❧(Condorcet)✛✴Ý ➛✣Ø✵(voting paradox)➭ ⑦ ❜✗❵✦➥✦❩♥❻➚➡é♥❻ÿ➚❁A✦B✦C✛➔Ðü❙❳❡ ▲➭ ❵ A > B > C ➥ B > C > A ❩ C > A > B ➪❽Ñ⑩Ð✛➚Þ➄❨. ✇✌➫❸➜ù➠↕➣✛✴✖➡➔Ð❣❙✵➑➵❦❙✸✛❣ñ➜❂✖➡ ➔ÐA➅✉C➜✌q❅➃AØ❳C➐ 4

课堂练习.（相对多数原则-plurality rule)假设100个选民对三个候选 人A、B、C的投票结果如下 A B C 4035 25 则显而易见，如何合理的选举体系应该选举A.然而，这100个选民对三 个候选人A、B、C的排序结果如下 A B 0 B B C A 4035 25 你又作何感想？（比如使用大多数原则(majority rule)) 5

➅✱ö❙. (❷éõê✝❑-plurality rule) ❜✗100❻➚➡é♥❻ÿ➚ ❁A✦B✦C✛Ý➛✭❏❳❡ A B C 40 35 25 ❑✇✌➫❸➜❳ÛÜ♥✛➚Þ◆❳❆❚➚ÞA. ✱✌➜ù100❻➚➡é♥ ❻ÿ➚❁A✦B✦C✛ü❙✭❏❳❡ A B C B C B C A A 40 35 25 ❭q❾Û❛➂➸(✬❳➛❫➀õê✝❑(majority rule).) 5

一个合理的、充分民主的选举体系应该满足下述公理： 公理1：Decisive(确定性公理)-One and only one winner.(U-公理) 公理2：Pareto principle(帕累托公理或一致性公理)-If all voters rank candidate X above candidate Y,then we oughtn't elect Y.(P) 公理3：Nondictatorship(非独裁公理)-No voter should have the power to always get his way.(D-公理) 公理4：Independence of irrelevant candidates(第三者公理或阿罗公 )-Suppose that,given the voting rule and voters'rankings,candidate X ends up the winner of an election.Now look at another situation that is exactly the same except that some other candidate Y-who didn't win-is no longer on the ballot.Well,candidate Y is,in a sense,"irrelevant;"he didn't win the election in the first place,and so leaving him off the ballot shouldn't make any difference.And so,the independence axiom requires that X should still win in this other situation.(I-) 公理4的含义：合理的选举系统中不应出现搅局者(spoiler)。搅局者的 例子是2000年美国大选中的第三者Ralph.Nader: George W.Bush defeated Al Gore by 537 votes.Nader received 97,421 votes,which led to claims that he was responsible for Gore's defeat.Nader, both in his book Crashing the Party and on his website,states:"In the year 2000,exit polls reported that 25%of my voters would have voted for Bush 38%would have voted for Gore and the rest would not have voted at all." (which would net a 13 12,665 votes,advantage for Gore over Bush.) 6

➌❻Ü♥✛✦➾➞➡❒✛➚Þ◆❳❆❚÷✈❡ãú♥➭ ú♥1➭Decisive(✭➼✺ú♥)-One and only one winner. (U-ú♥) ú♥2➭Pareto principle(ø❭÷ú♥➼➌➋✺ú♥)-If all voters rank candidate X above candidate Y, then we oughtn✳t elect Y. (P-ú♥) ú♥3➭Nondictatorship(➎Õàú♥)-No voter should have the power to always get his way. (D-ú♥) ú♥4➭Independence of irrelevant candidates(✶♥öú♥➼❈Ûú ♥)-Suppose that, given the voting rule and voters✳ rankings, candidate X ends up the winner of an election. Now look at another situation that is exactly the same except that some other candidate Y✮who didn✳t win✮is no longer on the ballot. Well, candidate Y is, in a sense, ✴irrelevant;✵ he didn✳t win the election in the first place, and so leaving him off the ballot shouldn✳t make any difference. And so, the independence axiom requires that X should still win in this other situation. (I-ú♥) ú♥4✛➵➶➭Ü♥✛➚Þ❳Ú➙Ø❆Ñ②✟Ûö(spoiler)✧✟Ûö✛ ⑦❢➫2000❝④■➀➚➙✛✶♥öRalph.Nader➭ George W. Bush defeated Al Gore by 537 votes. Nader received 97,421 votes, which led to claims that he was responsible for Gore’s defeat. Nader, both in his book Crashing the Party and on his website, states: ”In the year 2000, exit polls reported that 25% of my voters would have voted for Bush, 38% would have voted for Gore and the rest would not have voted at all.” (which would net a 13 12,665 votes, advantage for Gore over Bush.) 6

阿罗不可能性定理：如果选民人数与候选人个数均不小于3，则不存在 任何遵循原则U,P,I,D的选举系统。 结论（多少有点出乎意料）： 根本不存在一种能保证效率、尊重个人偏好的多数规则的选举系统。 所有一般条件的民主选择要么是强加的，要么就是独裁的结果。 推论（完全有点出乎意料）： 一个社会不可能有完全的每个个人的自由一一否则将导致独裁： 一个社会也不可能实现完全的自由经济一一否则将导致垄断。 7

❈ÛØ➀❯✺➼♥➭❳❏➚➡❁ê❺ÿ➚❁❻êþØ✂✉3➜❑Ø⑧✸ ❄Û❸❒✝❑U➜P➜I➜D✛➚Þ❳Ú✧ ✭Ø(õ✟❦✿Ñ✂➾✍)➭ ❾✢Ø⑧✸➌➠❯✂②✟➬✦❷➢❻❁➔Ð✛õê✺❑✛➚Þ❳Ú✧ ↕❦➌❸❫❻✛➡❒➚❏❻♦➫r❭✛➜❻♦Ò➫Õà✛✭❏✧ íØ(✑✜❦✿Ñ✂➾✍)➭ ➌❻✖➡Ø➀❯❦✑✜✛③❻❻❁✛❣❞✮✮➘❑ò✓➋Õà➯ ➌❻✖➡➃Ø➀❯➣②✑✜✛❣❞➨▲✮✮➘❑ò✓➋➓ä✧ 7