Arrow s Impossibility Theorem

Transcription

1 Rep Fun Gme Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1

2 Rep Fun Gme Properties Arrow s Theorem Leture Overview 1 Rep 2 Fun Gme 3 Properties 4 Arrow s Theorem Arrow s Impossiility Theorem Leture 12, Slide 2

3 Rep Fun Gme Properties Arrow s Theorem Ex-post expeted utility Definition (Ex-post expeted utility) Agent i s ex-post expeted utility in Byesin gme (N, A, Θ, p, u), where the gents strtegies re given y s nd the gent types re given y θ, is defined s EU i (s, θ) = A j N s j ( j θ j ) u i (, θ). The only unertinty here onerns the other gents mixed strtegies, sine i knows everyone s type. Arrow s Impossiility Theorem Leture 12, Slide 3

4 Rep Fun Gme Properties Arrow s Theorem Ex-interim expeted utility Definition (Ex-interim expeted utility) Agent i s ex-interim expeted utility in Byesin gme (N, A, Θ, p, u), where i s type is θ i nd where the gents strtegies re given y the mixed strtegy profile s, is defined s EU i (s θ i ) = p(θ i θ i ) s j ( j θ j ) u i (, θ i, θ i ). θ i Θ i A j N i must onsider every θ i nd every in order to evlute u i (, θ i, θ i ). i must weight this utility vlue y: the proility tht would e relized given ll plyers mixed strtegies nd types; the proility tht the other plyers types would e θ i given tht his own type is θ i. Arrow s Impossiility Theorem Leture 12, Slide 4

5 Rep Fun Gme Properties Arrow s Theorem Ex-nte expeted utility Definition (Ex-nte expeted utility) Agent i s ex-nte expeted utility in Byesin gme (N, A, Θ, p, u), where the gents strtegies re given y the mixed strtegy profile s, is defined s EU i (s) = θ i Θ i p(θ i )EU i (s θ i ) or equivlently s EU i (s) = p(θ) s j ( j θ j ) u i (, θ). θ Θ A j N Arrow s Impossiility Theorem Leture 12, Slide 5

6 Rep Fun Gme Properties Arrow s Theorem Nsh equilirium Definition (Byes-Nsh equilirium) A Byes-Nsh equilirium is mixed strtegy profile s tht stisfies i s i BR i (s i ). Definition (ex-post equilirium) A ex-post equilirium is mixed strtegy profile s tht stisfies θ, i, s i rg mx s i S i EU i (s i, s i, θ). Arrow s Impossiility Theorem Leture 12, Slide 6

7 Rep Fun Gme Properties Arrow s Theorem Soil Choie Definition (Soil hoie funtion) Assume set of gents N = {1, 2,..., n}, nd set of outomes (or lterntives, or ndidtes) O. Let L - e the set of non-strit totl orders on O. A soil hoie funtion (over N nd O) is funtion C : L - n O. Definition (Soil welfre funtion) Let N, O, L - e s ove. A soil welfre funtion (over N nd O) is funtion W : L - n L -. Arrow s Impossiility Theorem Leture 12, Slide 7

8 Rep Fun Gme Properties Arrow s Theorem Plurlity pik the outome whih is preferred y the most people Plurlity with elimintion ( instnt runoff ) everyone selets their fvorite outome the outome with the fewest votes is eliminted repet until one outome remins Bord ssign eh outome numer. The most preferred outome gets sore of n 1, the next most preferred gets n 2, down to the n th outome whih gets 0. Then sum the numers for eh outome, nd hoose the one tht hs the highest sore Pirwise elimintion in dvne, deide shedule for the order in whih pirs will e ompred. given two outomes, hve everyone determine the one tht they prefer Arrow s Impossiility Theorem eliminte the outome tht ws not preferred, nd ontinue Leture 12, Slide 8 Some Voting Shemes

9 Rep Fun Gme Properties Arrow s Theorem Condoret Condition If there is ndidte who is preferred to every other ndidte in pirwise runoffs, tht ndidte should e the winner While the Condoret ondition is onsidered n importnt property for voting system to stisfy, there is not lwys Condoret winner sometimes, there s yle where A defets B, B defets C, nd C defets A in their pirwise runoffs Arrow s Impossiility Theorem Leture 12, Slide 9

10 Rep Fun Gme Properties Arrow s Theorem Leture Overview 1 Rep 2 Fun Gme 3 Properties 4 Arrow s Theorem Arrow s Impossiility Theorem Leture 12, Slide 10

11 Rep Fun Gme Properties Arrow s Theorem Fun Gme Imgine tht there ws n opportunity to tke one-week lss trip t the end of term, to one of the following destintions: (O) Orlndo, FL (P) Pris, Frne (T) Tehrn, Irn (B) Beijing, Chin Construt your preferene ordering Arrow s Impossiility Theorem Leture 12, Slide 11

12 Rep Fun Gme Properties Arrow s Theorem Fun Gme Imgine tht there ws n opportunity to tke one-week lss trip t the end of term, to one of the following destintions: (O) Orlndo, FL (P) Pris, Frne (T) Tehrn, Irn (B) Beijing, Chin Construt your preferene ordering Vote (truthfully) using eh of the following shemes: plurlity (rise hnds) Arrow s Impossiility Theorem Leture 12, Slide 11

13 Rep Fun Gme Properties Arrow s Theorem Fun Gme Imgine tht there ws n opportunity to tke one-week lss trip t the end of term, to one of the following destintions: (O) Orlndo, FL (P) Pris, Frne (T) Tehrn, Irn (B) Beijing, Chin Construt your preferene ordering Vote (truthfully) using eh of the following shemes: plurlity (rise hnds) plurlity with elimintion (rise hnds) Arrow s Impossiility Theorem Leture 12, Slide 11

14 Rep Fun Gme Properties Arrow s Theorem Fun Gme Imgine tht there ws n opportunity to tke one-week lss trip t the end of term, to one of the following destintions: (O) Orlndo, FL (P) Pris, Frne (T) Tehrn, Irn (B) Beijing, Chin Construt your preferene ordering Vote (truthfully) using eh of the following shemes: plurlity (rise hnds) plurlity with elimintion (rise hnds) Bord (volunteer to tulte) Arrow s Impossiility Theorem Leture 12, Slide 11

15 Rep Fun Gme Properties Arrow s Theorem Fun Gme Imgine tht there ws n opportunity to tke one-week lss trip t the end of term, to one of the following destintions: (O) Orlndo, FL (P) Pris, Frne (T) Tehrn, Irn (B) Beijing, Chin Construt your preferene ordering Vote (truthfully) using eh of the following shemes: plurlity (rise hnds) plurlity with elimintion (rise hnds) Bord (volunteer to tulte) pirwise elimintion (rise hnds, I ll pik shedule) Arrow s Impossiility Theorem Leture 12, Slide 11

16 Rep Fun Gme Properties Arrow s Theorem Leture Overview 1 Rep 2 Fun Gme 3 Properties 4 Arrow s Theorem Arrow s Impossiility Theorem Leture 12, Slide 12

17 Rep Fun Gme Properties Arrow s Theorem Nottion N is the set of gents O is finite set of outomes with O 3 L is the set of ll possile strit preferene orderings over O. for ese of exposition we swith to strit orderings we will end up showing tht desirle SWFs nnot e found even if preferenes re restrited to strit orderings [ ] is n element of the set L n ( preferene ordering for every gent; the input to our soil welfre funtion) W is the preferene ordering seleted y the soil welfre funtion W. When the input to W is miguous we write it in the susript; thus, the soil order seleted y W given the input [ ] is denoted s W ([ ]). Arrow s Impossiility Theorem Leture 12, Slide 13

18 Rep Fun Gme Properties Arrow s Theorem Preto Effiieny Definition (Preto Effiieny (PE)) W is Preto effiient if for ny o 1, o 2 O, i o 1 i o 2 implies tht o 1 W o 2. when ll gents gree on the ordering of two outomes, the soil welfre funtion must selet tht ordering. Arrow s Impossiility Theorem Leture 12, Slide 14

19 Rep Fun Gme Properties Arrow s Theorem Independene of Irrelevnt Alterntives Definition (Independene of Irrelevnt Alterntives (IIA)) W is independent of irrelevnt lterntives if, for ny o 1, o 2 O nd ny two preferene profiles [ ], [ ] L n, i (o 1 i o 2 if nd only if o 1 i o 2) implies tht (o 1 W ([ ]) o 2 if nd only if o 1 W ([ ]) o 2 ). the seleted ordering etween two outomes should depend only on the reltive orderings they re given y the gents. Arrow s Impossiility Theorem Leture 12, Slide 15

20 Rep Fun Gme Properties Arrow s Theorem Nondittorship Definition (Non-dittorship) W does not hve dittor if i o 1, o 2 (o 1 i o 2 o 1 W o 2 ). there does not exist single gent whose preferenes lwys determine the soil ordering. We sy tht W is dittoril if it fils to stisfy this property. Arrow s Impossiility Theorem Leture 12, Slide 16

21 Rep Fun Gme Properties Arrow s Theorem Leture Overview 1 Rep 2 Fun Gme 3 Properties 4 Arrow s Theorem Arrow s Impossiility Theorem Leture 12, Slide 17

22 Rep Fun Gme Properties Arrow s Theorem Arrow s Theorem Theorem (Arrow, 1951) Any soil welfre funtion W tht is Preto effiient nd independent of irrelevnt lterntives is dittoril. We will ssume tht W is oth PE nd IIA, nd show tht W must e dittoril. Our ssumption tht O 3 is neessry for this proof. The rgument proeeds in four steps. Arrow s Impossiility Theorem Leture 12, Slide 18

23 Rep Fun Gme Properties Arrow s Theorem Arrow s Theorem, Step 1 Step 1: If every voter puts n outome t either the very top or the very ottom of his preferene list, must e t either the very top or very ottom of W s well. Consider n ritrry preferene profile [ ] in whih every voter rnks some O t either the very ottom or very top, nd ssume for ontrdition tht the ove lim is not true. Then, there must exist some pir of distint outomes, O for whih W nd W. Arrow s Impossiility Theorem Leture 12, Slide 19

24 Rep Fun Gme Properties Arrow s Theorem Arrow s Theorem, Step 1 Step 1: If every voter puts n outome t either the very top or the very ottom of his preferene list, must e t either the very top or very ottom of W s well. Now let s modify [ ] so tht every voter moves just ove in his preferene rnking, nd otherwise leves the rnking unhnged; let s ll this new preferene profile [ ]. We know from IIA tht for W or W to hnge, the pirwise reltionship etween nd nd/or the pirwise reltionship etween nd would hve to hnge. However, sine oupies n extreml position for ll voters, n e moved ove without hnging either of these pirwise reltionships. Thus in profile [ ] it is lso the se tht W nd W. From this ft nd from trnsitivity, we hve tht W. However, in [ ] every voter rnks ove nd so PE requires tht W. We hve ontrdition. Arrow s Impossiility Theorem Leture 12, Slide 19

25 Rep Fun Gme Properties Arrow s Theorem Arrow s Theorem, Step 2 Step 2: There is some voter n who is extremely pivotl in the sense tht y hnging his vote t some profile, he n move given outome from the ottom of the soil rnking to the top. Consider preferene profile [ ] in whih every voter rnks lst, nd in whih preferenes re otherwise ritrry. By PE, W must lso rnk lst. Now let voters from 1 to n suessively modify [ ] y moving from the ottom of their rnkings to the top, preserving ll other reltive rnkings. Denote s n the first voter whose hnge uses the soil rnking of to hnge. There lerly must e some suh voter: when the voter n moves to the top of his rnking, PE will require tht e rnked t the top of the soil rnking. Arrow s Impossiility Theorem Leture 12, Slide 20

26 Rep Fun Gme Properties Arrow s Theorem Arrow s Theorem, Step 2 Step 2: There is some voter n who is extremely pivotl in the sense tht y hnging his vote t some profile, he n move given outome from the ottom of the soil rnking to the top. Denote y [ 1 ] the preferene profile just efore n moves, nd denote y [ 2 ] the preferene profile just fter n hs moved to the top of his rnking. In [ 1 ], is t the ottom in W. In [ 2 ], hs hnged its position in W, nd every voter rnks t either the top or the ottom. By the rgument from Step 1, in [ 2 ] must e rnked t the top of W. Profile [ 1 ] : Profile [ 2 ] : Arrow s Impossiility Theorem Leture 12, Slide 20

27 Rep Fun Gme Properties Arrow s Theorem Arrow s Theorem, Step 3 Step 3: n (the gent who is extremely pivotl on outome ) is dittor over ny pir not involving. We egin y hoosing one element from the pir ; without loss of generlity, let s hoose. We ll onstrut new preferene profile [ 3 ] from [ 2 ] y mking two hnges. First, we move to the top of n s preferene ordering, leving it otherwise unhnged; thus n n. Seond, we ritrrily rerrnge the reltive rnkings of nd for ll voters other thn n, while leving in its extreml position. Profile [ 1 ] : Profile [ 2 ] : Profile [ 3 ] : Arrow s Impossiility Theorem Leture 12, Slide 21

28 Rep Fun Gme Properties Arrow s Theorem Arrow s Theorem, Step 3 Step 3: n (the gent who is extremely pivotl on outome ) is dittor over ny pir not involving. In [ 1 ] we hd W, s ws t the very ottom of W. When we ompre [ 1 ] to [ 3 ], reltive rnkings etween nd re the sme for ll voters. Thus, y IIA, we must hve W in [ 3 ] s well. In [ 2 ] we hd W, s ws t the very top of W. Reltive rnkings etween nd re the sme in [ 2 ] nd [ 3 ]. Thus in [ 3 ], W. Using the two ove fts out [ 3 ] nd trnsitivity, we n onlude tht W in [ 3 ]. Profile [ 1 ] : Profile [ 2 ] : Profile [ 3 ] : Arrow s Impossiility Theorem Leture 12, Slide 21

29 Rep Fun Gme Properties Arrow s Theorem Arrow s Theorem, Step 3 Step 3: n (the gent who is extremely pivotl on outome ) is dittor over ny pir not involving. Now onstrut one more preferene profile, [ 4 ], y hnging [ 3 ] in two wys. First, ritrrily hnge the position of in eh voter s ordering while keeping ll other reltive preferenes the sme. Seond, move to n ritrry position in n s preferene ordering, with the onstrint tht remins rnked higher thn. Oserve tht ll voters other thn n hve entirely ritrry preferenes in [ 4 ], while n s preferenes re ritrry exept tht n. Profile [ 1 ] : Profile [ 2 ] : Profile [ 3 ] : Profile [ 4 ] : Arrow s Impossiility Theorem Leture 12, Slide 21

30 Rep Fun Gme Properties Arrow s Theorem Arrow s Theorem, Step 3 Step 3: n (the gent who is extremely pivotl on outome ) is dittor over ny pir not involving. In [ 3 ] nd [ 4 ] ll gents hve the sme reltive preferenes etween nd ; thus, sine W in [ 3 ] nd y IIA, W in [ 4 ]. Thus we hve determined the soil preferene etween nd without ssuming nything exept tht n. Profile [ 1 ] : Profile [ 2 ] : Profile [ 3 ] : Profile [ 4 ] : Arrow s Impossiility Theorem Leture 12, Slide 21

31 Rep Fun Gme Properties Arrow s Theorem Arrow s Theorem, Step 4 Step 4: n is dittor over ll pirs. Consider some third outome. By the rgument in Step 2, there is voter n who is extremely pivotl for. By the rgument in Step 3, n is dittor over ny pir αβ not involving. Of ourse, is suh pir αβ. We hve lredy oserved tht n is le to ffet W s rnking for exmple, when n ws le to hnge W in profile [ 1 ] into W in profile [ 2 ]. Hene, n nd n must e the sme gent. Arrow s Impossiility Theorem Leture 12, Slide 22