The study of a new gerrymandering methodology

The study of a new gerrymandering metho dology ∗ P an Kai, T an Y ue, and Jiang Sheng Departmen t of Modern Physics, Univ ersit y of Science and T ec hnology of C h ina (USTC), Hefei, Anhui 23002 6, P .R.C h ina Abstract This pap er is to obtain a simple dividing-diagram of the congressional districts, where the only limit is that eac h d istrict should cont ain the same p opulation if p ossibly . In order to solv e th is problem, we in trod uce three d if- feren t standards of the ”simple” shap e. The first stand ard is that the fi nal shap e of the congressional districts should b e of a simp lest figure and we apply a mo dified ”sh ortest sp lit line algorithm” where the factor of the same p op- ulation is considered on ly . The second standard is that the gerrym andering should ensure the in tegrit y of th e current administrativ e area as the con v e- nience for managemen t. Th us we com bine the f actor of the administrativ e area with the first standard, and generate an impro v ed mo del resulting in the new diagram in w h ic h the perim eters of the districts are a long the b oundaries of some current counties. Moreo v er, th e gerrymandering sh ould consider the geographic features.The third stand ard is in tro duced to describ e this situa- tion. Finally , it can b e prov ed that the difference b etw een the su pp orting ratio of a certain part y in eac h district and th e av erage su pp orting ratio of that particular part y in the whole sta te ob eys the Chi-square distribution appro ximately . Consequen tly , we can obtain an archet ypal formula to c hec k whether th e gerrymandering w e p rop ose is f air. ∗ Suppo rted b y our deep in terests in ma thematical modeling. 1 Con ten ts 1 In tro duction 3 1.1 Issues in the Mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Previous W orks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Our Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Our Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Basic Ger r ymandering Mo del 5 2.1 Acquiremen t of the P opulation Densit y Matr ix . . . . . . . . . . . . . 5 2.2 A Simple Mo del fo r Redistricting . . . . . . . . . . . . . . . . . . . . 5 2.3 Mo del Mo dification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3.1 A Basic Assumption . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.2 Mo dification Metho d . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.3 Our Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 F urther Optimization of the Mo del 11 3.1 The Mo dification of Avoiding “Gerrymandering” . . . . . . . . . . . . 11 3.1.1 Brief Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.2 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 The Mo dification of G eographic F eatures . . . . . . . . . . . . . . . . 15 3.2.1 Brief Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.2 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Conclusion and Analysis 16 4.1 Comparison of Our Tw o Mo dels . . . . . . . . . . . . . . . . . . . . . 16 4.2 Comparison With Other Mo dels . . . . . . . . . . . . . . . . . . . . . 17 4.3 Comparison With Curren t Districts . . . . . . . . . . . . . . . . . . . 17 5 W eakness and F urther Dev elopmen t 18 5.1 V erification o f the Impro v ed Mo del . . . . . . . . . . . . . . . . . . . 18 5.2 Realization of the Comprehensiv e Considerations . . . . . . . . . . . 19 6 App endix 20 References 22 2 1 In tro duc t ion Figure 1: (a)The ma p o f p opulation densit y [1] . (b)The shap e of districts pro duced b y simple mo del. (c)The shap e of districts pro duced b y improv ed mo del T o ensure the fairness of the election, the foremost task is to obtain a sc ien tific arrangemen t of the b oundaries of the congressional districts. Although the states constitution provide s the n um ber of represen tativ es eac h state may hav e, it articu- lates nothing ab out ho w the district shall b e determined geographically . This o v ersight has now aday s led to an unnatural district shap e whic h includes man y long and narro w areas. T a king the state of New Y or k a s an example, on the map of current congressional districts [2] , we can see that there are some unnatural- shap ed districts, suc h as districts 20 th and 22 nd (sho wn in F igure 12(a)). T o create the “simplest” shap es, where t he only limit is that each district should con tain the same p o pulation if p ossibly , w e in tro duce thr ee differen t standards of the s implicit y: 1. Eac h district is appro ximativ ely a rectangle. 2. Based on the a b o v e standard, the outline s are b est along the b o undar ies of coun ties. 3. The mo difications o f geographical features and fairness are com bined in the final mo del. Therefore, we should take in to accoun ts comprehensiv e factors including the p opulation, t he b o undaries of counties, geographic features and fairness when g er- rymandering. 1.1 Issues in the Mo del • The first ob jectiv e is to divide the state in to some districts follow ing the rule that the p opulation of each district m ust b e same. 3 • As the s econd ob jectiv e, w e wish to mak e the b o undaries of the districts a s simple as p ossible according to the 1 st standard we ha v e pro p osed ab ov e. • The third ob jectiv e is that w e nee d further modifications considering the stan- dard 2 and standard 3. 1.2 Previous W orks There a r e many existing metho ds to r ealize the gerrymandering, with the considera- tion of different asp ects. F or instance , the “shortest split line algorithm” [3] considers the simplicit y of the shap e as k ey fa ctors. Another metho d, whic h uses the statis- tical ph ysics appro a c h, takes ev ery c oun t y as a s ingle elemen t of a matrix, mak es an a na logue to the q-state p ots [7] mo del, and gets the optimal solution kee ping t he in tegralit y of a count y . Other approac hes suc h as the “ fixed district algorit hm” [3] and the “c hanging the v oting system metho d” [3] mainly consider the un biasednes s of the election as the crucial f actor. In our work, the fo remost task is to make sure the simplicit y of the shap es of districts. Therefore, we mo dify the “shortest split line algorithm” to establish o ur initial simple mo del. The main mo dification is that we only use horizontal line and v ertical line instead of diagona ls. In this w a y , w e can get simple rectangles other than irregular p olygonal districts. While in the se ction of Mo del Mo dification w e dev elop a new metho d to ensure the p erimeter of the districts are along the b oundaries of the curren t coun ties. 1.3 Our Ap p roac h • First, w e obtain the p opulation density matrix from t he densit y map. • In order t o obtain a simple shap e of districts, we only tak e in to a ccoun t the rule that ev ery district should ha v e the same p opulatio n to establish a simple mo del. • Then w e optimize our mo del b y adding the mo dification of the presen t shap es of coun ties and g ain a more nature lo oking figure. • Based on the results from the forgoing simulation, w e finally in v estigate in to the factor of geog raph y and fairness in detail when gerrymandering. 4 1.4 Our Result According to the steps men tioned ab ov e, F igure 1 show s our results when applying our mo del to the state of New Y ork. 2 Basic Ge r r ymande ring Mo del In o r der to establish our basic mo del, w e first obtain a p o pulation density mat r ix using the p opulation map. W e then e stablish a simple mo del with the only con- strain t b eing t ha t eac h district ha v e the same p opulation. Finally , w e o ptimize our mo del b y adding additional constrain ts suc h a s prev enting the division of original administrativ e ar eas. 2.1 Acquiremen t of the P opulation Densit y Matrix T o calculate the p opula t io n eac h district con tains, we m ust first extract the p opula- tion density matrix for later computer progr a mming from the existing census data of some big cities and the macroscopic p o pulation densit y map. F or the or iginal colored p opulation map, w e use Matlab to iden tify the color of ev ery pixel. Th us w e can get the populatio n densit y corresp onding to the particular pixel though whic h w e obta in the p opulation densit y matrix. Moreo v er, this matrix can b e sho wn as a grey leve l figure(Figure 2). Figure 2: (a )The colored map of p opulation densit y . (b)The grey fig ure of p opulation densit y 2.2 A Simple Mo del fo r Re districting In order to obtain simple shap es for congressional districts, w e mo dified the origina l “shortest split line algorit hm” and dev eloped a simple algorithm to divide districts. 5 In our mo del: • W e fo cus on k eeping an ev en distribution of p opula t io n in eac h district, ignor- ing other factors. • The algorithm t ak es as input only the p opulation densit y matrix and ignor es other factors suc h as party lo y alties o f the citiz ens, th us gua r an teeing un biased results. • W e use horizontal and v ertical lines to separate the state result in fairly rect- angle districts. The pro cedure of the algorithm can b e sho wn in F igure 3. This district-dividing algorithm has the adv a n tage of simplicit y , clear unbiased - ness, and it pro duces fairly nice-lo oking rectangle districts. The adv an tages of the simple mo del: • Simplicit y • Clear un biasedness • F airly nice-lo oking rectangle districts. The disadv an tages: • F ails to ta k e in to consideration other fa ctors suc h a s geographic features and in tegralit y o f the counties. 2.3 Mo del Mo dific ation Despite o f its adv antage of simplicit y , the original mo del has the disadv antage of ignoring the shap e of administrativ e area and geographic features. As the figures sho w, the b o undaries pro duced b y the original a lg orithm some time s divide a count y whic h is an administrative a r ea into differen t parts. This can pro ve incon v enie n t and unnatural when carrying out the election pro cedures. T o disp el this disadv antage, w e made the f o llo wing improv emen ts to our orig inal mo del: 6 Figure 3 : The pro cedure of the algorithm of the simple mo del. 2.3.1 A Basic Assumption Before taking into cons ideration additional factors in dividing the coun ties, we need to mak e the follo wing assumption: we assume that tw o dis tricts share approximately the same p opulation if the difference in their num b ers of v oters is no more than 5%. In reality , it is imp ossible to divide districts into absolutely equal n um b ers of v oters due to the influence of p opulation flo w and the fluctuation of birth and death rates. And most of t he constitutions also a llow the the standard deviation of n um ber of v oters to fa ll within 10 − 15 % [7] . 7 2.3.2 Mo dification Metho d After making up the designs of landform, w e are faced with a problem - how to adopt the b orderline to a v oid dividing up administrative districts. T o solv e this problem, w e hav e to adopt the b orderlines b etw een the administrative districts. W e follo w the lines in bulge and conca v e in some district. As w e think the densit y in p o pulation almost equals close to b orderline b et w een districts. T aking into account the fa cto r s of populatio n, la ndform a nd managemen t, w e tak e this division as a simple meaning in manag ement. 2.3.3 Our A lgorithm The basic consideration o f this improv ed mo del is the same with the ab o v e simple mo del. How ev er, in that mo del, the dividing lines are all straigh t and th us cannot k eep a single coun t y in tegrate. Now we are trying to o ve rcome this shortcoming and do our b est to mak e the dividing line coincide with the b oundary of each adminis- trativ e area, although coun t y dividing is una v oidable. Step 1: Use the simple mo del t o find the straight dividing line. Figure 4: step 1 8 Step 2: Find all the in tersections b et w een the straight line and the count y b oundaries. Then in the progra m, use “ coun t” the in teger to represen t ho w man y crosso v er p oin ts there are on the straigh t line and use array point[coun t] to r ecord the lo cation o f all these interse ctions. Figure 5: step 2 Step 3: There must b e 2 or 3 direct paths along the straig h t line or the b oundary of the coun t y the straight line go es through connecting the t w o inters ection p oin ts. W e are trying to decide which of the t w o paths a lo ng the b oundary is more “ simple”. The “simple” path should b e the b oundary whic h connects the adjoining p oin ts and more close to the straight line. First w e find the p oint on the straig ht line whose co ordinate p osition is 2 or 3 units a wa y from the in tersection. W e separately searc h the p o ints left and right to it (tak e ve rtical line as a n example , for ho r izon tal lines the 2 directions w ould b e a b o v e and b elow ). The direction w e finally c ho ose would b e the o ne on whic h we arriv e at the b oundary earlier. W e record the direction c hosen in array path[count-1], if the direction is left o r up w ard, pa th[i]=1, if it is right or down w ard, path[i]=3. This metho d ma y not b e v ery strict, but it is easy to realize and tak es v ery little time for the computer to execute. Step 4: Ev ery straigh t line b etw een t w o adjo ining p oints and the b oundary line c hosen in step 3 can confine a small a r ea. W e calculate the p opulation in each small area b y sum up the elemen ts of the p opulation densit y mat r ix including in the area. P a y atten tion that if the area is to the left or b elow the straig h t line, prescrib e that s[i] is negative , otherwise, it is p ositiv e. W e sav e the result in array s[coun t-1]. 9 Figure 6: step 3 Figure 7: step 4 Step 5: In this step w e will finally decide the dividing line - its shap e and lo cation. W e no w kno w all the p oints on the dividing line and w an t to decide the lines jo ining the p oin ts. W e use arra y path[coun t- 1] to describe these lines. path[i] stands for the line connecting the i th p oin t and the ( i+ 1 ) th p oin t. path[i] =1,2 ,3 stands for the 1 st , 2 nd , 3 th path count from left to r igh t or do wn w ards to upw ards. The principles w e use t o decide whic h path to c ho ose is lik e this: First, while w e replace the initial straight line with the b oundary line, the p op- ulation of differen t congressional districts will v ary to eac h other. So the differences m ust b e con trolled within a certain range. Second, w e should use as little as p ossible straight lines, b ecause c ho osing a straigh t path means one count y will b e divided into t w o. The metho d is as fo llo ws: 10 Figure 8: step 5 In step 4 w e get the ar r a y s[coun t-1 ], we first sum up all the elemen ts in s[coun t- 1] and get the result S. Judge whether S < delta , here, delta can b e the theoretical p opulation of a congressional district m ultiple with the allow ed r a nge of error (in our progra m w e c ho ose the range as 5 %), if so, w e consider curren t path is qualified and we get the final arra y path[count-1], exit lo op; if not, we find s[i] whic h is closest to S ( | S − s [ i ] | is the minim um), and let path[i]=2 (c hange t he path to the stra ig h t line)let S = S − s [ i ], return t o judge whether S < delta , ... ,and lo op like this. After exiting the lo op, prin t t he lo cation of t he initial straigh t line and its end p oin ts, p oint[coun t] and pat h[count-1]. The final dividing line w e get with this metho d ma y not b e the optimal line, but it is qualified. The metho d can av oid considering a ll the p ossible combination of paths and thu s can op erate v ery fast. Step 6: D o the recursion a s the simple mo del and get all the dividing line. 3 F urther Optimiz ati o n of the Mo del Except shap e and coun t y integrit y , o ther factors should also b e take n into a ccount in a ctual gerrymandering. F or example, how to a v oid biased “gerrymandering, how to mo dify our mo del in sp ecial landform, as w e are g oing to represen t. 3.1 The Mo dification of A v oiding “Gerrymandering” 3.1.1 Brief In tro duction There are t w o principal strategies b ehind gerrymandering [3 , 8] : maximizing the ef- fectiv e v otes of supp orters, and minimizing the effectiv e v otes of opp onen ts. One 11 form of gerrymandering, pac king is to place as man y v oters of one ty p e into a single district to reduce their influence in other districts. A second form, crac king, in volv es spreading out v oters of a particular type among man y districts in o r der to reduce their represen tation b y den ying them a sufficien tly large v oting blo c k in an y par t ic- ular district. The metho ds are t ypically com bined, creating a few “forfeit” seats for pac k ed v oters of one type in order to secure eve n greater represen ta tion f or v oters of another ty p e. Figure 9: [3] Redra wing the balanced electoral districts in this example creates a guaran teed 3-to- 1 adv an ta ge in represen tation for the blue v oters. Here, 14 red v oters a re pac k ed into the y ello w district and the remaining 18 are crack ed across the 3 blue districts. 3.1.2 Resolution Because the simple algo rithm we based on uses only the shap e o f the state, the n um ber m of districts w an ted, and the p opulation distribution as inputs - and do es not kno w the party loy alties of the v oters in an y giv en region - the result cannot b e biased [3 , 9] . Ho w ev er w e can not completely prev en t t he certain v oter biases from o ccurring due to the random redistricting pro cess - although the probability is small. As a result, we need mak e a mat hematical analysis to determine whether the gerryman- dering our mo del pro p osed is fair. • Model assumption : Due to the the influence of p opulation flo w and the fluctuatio n of birth, death and the supp orting ratio of a p olitical party . It is reasonable to assume tha t the supp orting ratio of a particular part y in eac h district is incorrelate o f one another appro ximately . Next we w ould analyze how to judge the fa ir ness of the gerryman- dering to a specific party (Republican in our example) ; other parties and situations can b e judged via the same metho d. 12 • Mathema tical Analysis : x j i =  1 I f the i th v oter in the j th distr ict su ppor t the Republ ican 0 O ther w ise (1) If p j represen ts the supp orting ratio of the Republican in j th district a nd this district has n voters, then w e would expect to get (3.2) p j = P n i =1 x j i n . (2) Let p b e the last-few-y ear av erage supp orting ratio of the Republican in the whole state. Thus w e can exp ect that p ≈ P m i =1 p j m . (3) Ob viously , x j i is distributed b y B(1,p). X j = r n p (1 − p ) ( p j − p ) = r n p (1 − p ) ( P n i =1 x j i − np n ) = P n i =1 x j i − np p np (1 − p ) ∼ N (0 , 1) (4) According to the statistics t heory , X j ob eys N(0 ,1), if n is large enough. Then it is reasonable to obtain the fo llo wing deduction: Y = m X j =1 X 2 j ∼ χ 2 m , (5) Here m indicates the num b er of districts in a certain state. Th us, w e can use Equ.(3.5) to generate a simple but effectiv e standard to judge the fairness o f the original gerrymandering. 13 According to the statistics theory , if w e get the p opulation densit y of the sup- p orting ra t io of t he Republican, it is easy to gain Y using the data p j and p. After that, the fo remost task is to determine a parameter α allow scien tifically . The parameter α allow functions as a threshold to judge whether the curren t gerry- mandering is bias or not. Then from the data ta ble of distribution χ 2 , we can get the v alue α where P ( X > Y ) = α . Here X stands for the random v ariable distributed b y χ 2 m . Figure 1 0: (a)P ac king strategy (b)Crack ing strategy On one hand, if there happ ens the “pac king” situation, we w o uld exp ect α to b e extremely small. Therefore, if α allow > α , w e can judge this g errymandering as pac king situation leading to unfairness. On the other hand, if “crac king” situation happ ens, w e can exp ect 1 − α t o b e extremely small. W e consider t his gerryman- dering biased as the same (sho wn in Fig ure 10). In conclusion, if the gerrymandering is fair, the v alue of α pro duced b y it w ould comply with following limits. α > α allow N ot “ pack ing ” 1 − α > α allow N ot “ cr ack ing ” . (6) W e can briefly obtain the final limit o n v a lue α (sho wn in Figur e 11), that is α allow < α < 1 − α allow . (7) • Conclusion: 14 Figure 11: Perm itted Y Domain After g enerating the “simp le” congressional districts, w e can use Equ.(3.5) and (3.7) to judge whether it satisfies the requiremen t of f airness, if we can collect the sup- p orting r a tio of the Republican in eac h district, the last-few-y ear av erage supporting ratio in the whole state a nd the parameter α allow . If it is not a fair gerrymandering, w e need to reconsider the pro cedure o f redistricting. 3.2 The Mo dification of G eographic F eatures Up till no w, man y mo dels that can realize gerrymandering ha v e existed. Th us comparison b etw een all kinds of gerrymandering results is necessary . W e are going to further analysis the characteristic of our tw o mo dels while comparing the results w e get with others. 3.2.1 Brief In tro duction Ab o v e-men tio ned metho d has ignored landfo rm factors such as main riv ers, moun- tains, lakes in pro cess of choosing partition, th us it has made district part it io n unnatural. Here w e make some improv emen ts. As to main rivers , mountains and so on, we hold the view that along the riv er, p opulation b enefit in econom y comparative ly resem bles. On this basis, the areas on the rive r should b e one district. 15 3.2.2 Resolution Our o p erations are as follows: 1. A new district with a large p opulation is based on expanding along the riv er and mountains. Considering the adv an t ages of t he long river or deep moun- tains, as w ell as conv enience in managemen t, more districts will b e formed if the p opulat io n increases. 2. T o the large lak e, we ex pand the area in to sev eral districts along the bank, for they are blessed with the same interes t in econom y . 3. T o the area of moun tains and plain, we consider dev eloping them separately as a result of differen t demands in b enefit. 4. The s ame thinking is practical along t he coastline. W e assem ble p eople on the coast in to sev eral districts, to separate the areas of coastline fro m inland to help the dev elopmen t fro m the coastline to the inland. 5. Considering the a ppro ac hing in terest in economy , the area of islands a nd b y- lands is b eing considered alone. A sp ecial zone will b e built. With these fa cto r s of landform b eing conside red alone, priorit y to ev ery electoral district deducted, to the part remaining again, w e will carry out the prev ious simple op eration according to the simple mo del. 4 Conclus ion and Analysis 4.1 Comparison of Our T w o Mo dels W e ha v e fully established tw o mo dels - the simple one and the refined one, b ot h of whic h a re easy to realize. Eac h o f these tw o mo dels has its ow n prominen t adv antages. As to the simple mo del, first of all, the final shap es of the congressional districts are all v ery simple, constructing with horizontal and v ertical lines o nly . Moreov er, the simple mo del guaran tees that eac h district con tains the same p opulation if giv en a precise p opulation division. While to the refined mo del, w e can maintain t he in tegrit y of ev ery single coun t y to a large exten t, whic h ease the managemen t of actual v oting. Whats more, a lthough the p opulatio n of districts v a ries fro m eac h other, the diff erences can b e controlled theoretically to any prop osed precision. 16 4.2 Comparison With Other Mo dels There are many existing metho ds to realize the gerrymandering with the consider- ation of different asp ects. Here w e compares our mo del with the “shortest split line algorithm” and the “q-state p ots mo del”. F rom the theoretically a nalysis and the final result, w e can see that o ur metho d has its merits. First, one imp orta nt task for us is to mak e sure the simplicit y of the shap es of districts. Therefore, we mo dify the “shortest split line algorithm” to establish o ur initial simple mo del. The main mo dification is that w e o nly use horizon tal line a nd v ertical line instead of diagona ls. In this w a y , w e can get simple rectangles other than irregular p olygonal districts pro duced b y original “shortest split line a lg orithm”. Secondly , we think the “q-state p otts mo del” is excellen t. The foremost ch arac- teristic of it is that this mo del k eeps the in tegrit y of counties. How ev er, the difference of dis trict p opulations can just b e controlled within 15%, while this difference in our mo del is no more than 5%. In sum, the adv antages of our mo del is that: 1. Easy to carr y out in the computer. 2. Simple shap es o f the districts. 3. Com bined with the factor of av oiding dividing up the curren t administrativ e ares. 4. The difference of district p opulations can b e con trolled in any precision. 4.3 Comparison With Curren t Districts F rom Figure 12, w e can clearly see the difference b etw een the curren t (the left one) districts a nd the redistricting prop osed b y our mo del. Through careful compar ison, w e can draw the conclusion that each has its adv antages. The c urren t shape of districts o wn its great merit that it tak e in to t he geographi- cal consideration. F or example, the district 20 th is built along the Lak e Ontario and the district 22 nd mainly consists of plain. Both sho w the mo dification of geogra phic features. Although there is some merit in the curren t district, the gerrymandering result pro duced by our mo del still sho w its adv antages. 17 Figure 1 2: Comparison With Curren t Districts 1. The shap e of most districts are r ectangula r satisfying the retiremen ts of the “simplicit y”. 2. Most of the p erimeters of the district a re alo ng the b o undaries o f curren t coun ties resulting in t he con v enience of manag emen t. 3. Last but no t least, the diff erence of district p opulations is no more than 5%, and can b e controlled in an y precision. Finally , we ha v e generated a simple but effectiv e criterion to judge the fairness of the gerrymandering via the reasonably mathematical analysis whic h mak es the mo del as a whole. 5 W eakness and F urt h er Dev elopmen t Although our t w o mo dels a r e reasonable and easy to realize, the final gerrymandering mo del still hav e some weak ness whic h needs f urther improv emen t. 5.1 V erification of the Impro v ed M o del Although we ha v e implemen ted the improv ed mo del to tak e in to consideration the fairness of v oter distribution by pa rt y , w e cannot obtain the data on populat io n den- sit y f av o ring resp ectiv e parities or on supp orting ratio o f a sp ecific party . Therefore w e can not put that part of the mo del in to practice. In a no ther word, if w e w an t to pro v e t he effectiv enes s and fairness of t he mo del w e should tr y our b est to in v estigate suc h data and further correct our simulations a nd mathematical analysis. 18 5.2 Rea lization of the Comprehensiv e Considerations Although w e hav e noticed the imp ortance of geographic features when gerryman- dering, due to the complexit y of our curren t mo del considering another key factor - av oiding separating administrative areas, w e ha v e not achie v ed the goal that pro- grams the a lgorithm considering the landform mo dification. Th erefore, our next mission is to realize the comprehensiv e considerations including geographic fa cto r s. 19 6 App endix Applications to Other States Figure 1 3: (a)W ashington (b)P ennsylv ania Figure 14: (a) W est Virginia (b)New Jersey 20 Figure 1 5: (a)Wisconsin (b)Uta h 21 References [1] P opula t ion densit y maps of all the states, http://www.ans w ers.com , Accessed F eb 10th, 2007 [2] Curren t congressional districts of a ll the states, h ttp://www.cens us.go v , Ac- cessed F eb 10th, 20 07 [3] Gerrymandering - Wikip edia, the free encyclopedia, h ttp://en.wikipedia.or g /wiki/Gerrymandering , Accessed F eb 9th, 2007 [4] Cognitiv e Asp ects of Gerrymandering, h ttp://www.springerlink.c om/index , Accesse d F eb 10th, 2007 [5] P opula t ion list of all the counties in the state of New Y ork, h ttp://www.cit yp opula t io n,de/USA-New Y ork.html, Accessed F eb 9th, 2007 [6] Curren t congressional districts of all the states, h ttp://nationalatlas.gov/prin ta ble/congress.h tml , Accessed F eb 11st, 2007 [7] Hun ting t o n News, h ttp://www.h untingtonnews.ne t/national/06 1006-shns-elections.h tml , Accesse d F eb 11st, 2007 [8] F airV ote - F airness and Indep endence in Redistricting Act (H.R. 2642) , h ttp://www.fairv ote.org, Accesse d F eb 10th, 2007 [9] P ar tisan Gerrymandering, h ttp://www.senate.le g.state.mn.us/departmen ts/scr/redist/red2000/ch5parti.h tm , Accesse d F eb 10th, 2007 [10] Range v oting, http://www.rangev oting.or g, Accessed F eb 10th, 200 7 [11] The theory of statistics, h ttp://www.statistics.go v, Accessed F eb 1 0th, 2007 [12] F air V ote Gerrymander Examples, http://www .w estmiller.com [13] K Sherst yuk. “ How to gerrymander: A formal ana lysis”, Springer , 1998 [14] TW G illigan, JG Matsusak a. “Structural constrain ts on partisan bia s under the efficien t gerrymander”, Springer ,1999 22 [15] JD Cranor, GL Crawle y , RH Sc heele. “The Anatomy of a Gerrymander”, Amer- ic an Journal of Politic al S cienc e , 1989 [16] BE Cain. “Assessing the P artisan Effects of R edistricting”, The Americ an Po- litic al S cienc e R eview , 1985 23 The study o f a new gerrymandering metho d ology ∗ P an Kai, T an Y ue, and Jiang Sheng Departmen t of Mo dern Physic s, Univ ersit y of Science and T ec hnology of C h ina (USTC), Hefei, Anhui 23002 6, P .R.C h ina Abstract This pap er is to obtain a simple dividing-diagram of the congressional districts, w here the only limit is that eac h d istrict should cont ain the same p opulation if p ossibly . In order to solv e th is problem, we in trod uce three d if- feren t standards of the ”simple” shap e. The first stand ard is that the fi nal shap e of the congressional districts should b e of a simp lest figure and we ap p ly a mo dified ”sh ortest sp lit line algorithm” where the factor of the same p op- ulation is considered on ly . The second standard is that the gerrym andering should ensure the in tegrit y of th e current administrativ e area as the con v e- nience for managemen t. Th us we com bine the f actor of the administrativ e area with the first standard, and generate an impro v ed mo del resulting in the new diagram in w h ic h the perim eters of the districts are a long the b oundaries of some current counties. Moreo v er, th e gerrymandering sh ould consider the geographic features.The third stand ard is in tro duced to describ e this situa- tion. Finally , it can b e prov ed that th e difference b etw een th e su pp orting ratio of a certain part y in eac h district and th e av erage su pp orting ratio of that p articular p arty in the w h ole state ob eys the C hi-square distr ib ution appro ximately . Consequ ently , we can obtain an arc het ypal form ula to c hec k whether th e gerrymandering w e p rop ose is f air. ∗ Suppo rted b y our deep in terests in ma thematical modeling. 1 Con ten ts 1 In tro duction 3 1.1 Issues in the Mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Previous W orks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Our Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Our Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Basic Ger r ymandering Mo del 5 2.1 Acquiremen t of the P opulation Densit y Matr ix . . . . . . . . . . . . . 5 2.2 A Simple Mo del fo r Redistricting . . . . . . . . . . . . . . . . . . . . 5 2.3 Mo del Mo dification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3.1 A Basic Assumption . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.2 Mo dification Metho d . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.3 Our Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 F urther Optimization of the Mo del 11 3.1 The Mo dification of Avoiding “Gerrymandering” . . . . . . . . . . . . 11 3.1.1 Brief Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.2 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 The Mo dification of G eographic F eatures . . . . . . . . . . . . . . . . 15 3.2.1 Brief Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.2 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Conclusion and Analysis 16 4.1 Comparison of Our Tw o Mo dels . . . . . . . . . . . . . . . . . . . . . 16 4.2 Comparison With Other Mo dels . . . . . . . . . . . . . . . . . . . . . 17 4.3 Comparison With Curren t Districts . . . . . . . . . . . . . . . . . . . 17 5 W eakness and F urther Dev elopmen t 18 5.1 V erification o f the Impro v ed Mo del . . . . . . . . . . . . . . . . . . . 18 5.2 Realization of the Comprehensiv e Considerations . . . . . . . . . . . 19 6 App endix 20 References 22 2 1 In tro duc t ion Figure 1: (a)The ma p o f p opulation densit y [1] . (b)The shap e of districts pro duced b y simple mo del. (c)The shap e of districts pro duced b y improv ed mo del T o ensure the fairness of the election, the foremost task is to o btain a scien t ific arrangemen t of the b oundaries of the congressional districts. Although the states constitution provide s the n um ber of represen tativ es eac h state may hav e, it articu- lates nothing ab out ho w the district shall b e determined geographically . This o v ersight has now aday s led to an unnatural district shap e whic h includes man y long and narro w areas. T a king the state of New Y or k a s an example, on the map of current congressional districts [2] , we can see that there are some unnatural- shap ed districts, suc h as districts 20 th and 22 nd (sho wn in F igure 12(a)). T o create the “simplest” shap es, where t he only limit is that each district should con tain the same p o pulation if p ossibly , w e in tro duce thr ee differen t standards of the s implicit y: 1. Eac h district is appro ximativ ely a rectangle. 2. Based on the ab ov e standard, t he outlines are b est along t he b oundaries of coun ties. 3. The mo difications o f geographical features and fairness are com bined in the final mo del. Therefore, we should take into accounts comprehensiv e factors including the p opulation, t he b o undaries of counties, geographic features and fairness when g er- rymandering. 1.1 Issues in the Mo del • The first ob jectiv e is to divide the state in to some districts follow ing the rule that the p opulation of each district m ust b e same. 3 • As the second ob jectiv e, we wish to mak e the b oundar ies of the districts as simple as p ossible according to the 1 st standard we ha v e pro p osed ab ov e. • The third ob jectiv e is that w e nee d further modifications considering the stan- dard 2 and standard 3. 1.2 Previous W orks There a r e many existing metho ds to r ealize the gerrymandering, with the considera- tion of different asp ects. F or instance , the “shortest split line algorithm” [3] considers the simplicit y of the shap e as k ey fa ctors. Another metho d, whic h uses the statis- tical ph ysics appro a c h, takes ev ery coun t y as a single elemen t of a matrix, make s an a na logue to the q-state p ots [7] mo del, and gets the optimal solution kee ping t he in tegralit y of a count y . Other approac hes suc h as the “ fixed district algorit hm” [3] and the “c hanging the v oting system metho d” [3] mainly consider the un biasednes s of the election as the crucial f actor. In our work, the fo remost task is to make sure the simplicit y o f t he shap es of districts. Therefore, we mo dify the “shortest split line algorithm” to establish o ur initial simple mo del. The main mo dification is that w e o nly use horizon tal line a nd v ertical line instead of diagona ls. In this w a y , w e can get simple rectangles other than irregular p olygonal districts. While in the section of Mo del Mo dificatio n we dev elop a new metho d to ensure the p erimeter of the districts are along the b oundaries of the curren t coun ties. 1.3 Our Ap p roac h • First, w e obtain the p opulation density matrix from t he densit y map. • In order t o obtain a simple shap e of districts, we only tak e in to a ccoun t the rule that ev ery district should ha v e the same p opulatio n to establish a simple mo del. • Then w e optimize our mo del b y adding the mo dification of the presen t shap es of coun ties and g ain a more nature lo oking figure. • Based on the results from the forgoing simulation, w e finally in v estigate in to the factor of geog raph y and fairness in detail when gerrymandering. 4 1.4 Our Result According to the steps men tioned ab ov e, F igure 1 show s our results when applying our mo del to the state of New Y ork. 2 Basic Ge r r ymande ring Mo del In o r der to establish our basic mo del, w e first o bta in a p opulation densit y matrix using the p opulation map. W e then establish a simple mo del with the only con- strain t b eing t ha t eac h district ha v e the same p opulation. Finally , w e optimize our mo del b y adding additional constrain ts suc h a s prev enting the division of original administrativ e ar eas. 2.1 Acquiremen t of the P opulation Densit y M atrix T o calculate the p opula t io n eac h district con tains, we m ust first extract the p opula- tion density matrix for later computer progr a mming from the existing census data of some big cities and the macroscopic p o pulation densit y map. F or the or iginal colored p opulation map, w e use Matlab to iden tify the color of ev ery pixel. Thus w e can get the p opulation densit y correspo nding to the particular pixel though whic h w e obta in the p opulation densit y matrix. Moreo v er, this matrix can b e sho wn as a grey leve l figure(Figure 2). Figure 2: (a )The color ed map of p opulation densit y . (b)The grey fig ur e o f p opulation densit y 2.2 A Simple Mo del for Redistricting In order to obtain simple shap es for congressional districts, w e mo dified the origina l “shortest split line algorit hm” and dev eloped a simple algorithm to divide districts. 5 In our mo del: • W e fo cus on k eeping an ev en distribution of p opula t io n in eac h district, ignor- ing other factors. • The algorithm t ak es as input only the p opulation densit y matrix and ignor es other factors suc h as party lo y alties o f the citiz ens, th us gua r an teeing un biased results. • W e use horizontal and v ertical lines to separate the state result in fairly rect- angle districts. The pro cedure of the algorithm can b e sho wn in F igure 3. This district-dividing algorithm has the adv a n tage of simplicit y , clear unbiased - ness, and it pro duces fairly nice-lo oking rectangle districts. The adv an tages of the simple mo del: • Simplicit y • Clear un biasedness • F airly nice-lo oking rectangle districts. The disadv an tages: • F ails to ta k e in to consideration other fa ctors suc h a s geographic features and in tegralit y o f the counties. 2.3 Mo del Mo dific ation Despite o f its adv antage of simplicity , the original mo del has the disadv antage of ignoring the shap e of administrativ e area and geographic features. As the figures sho w, the b o undaries pro duced b y the original a lg orithm some time s divide a count y whic h is an administrative a r ea into differen t parts. This can pro ve incon v enie n t and unnatural when carrying out the election pro cedures. T o disp el this disadv antage, w e made the f o llo wing improv emen ts to our orig inal mo del: 6 Figure 3 : The pro cedure of the algorithm of the simple mo del. 2.3.1 A Basic Assumption Before taking into cons ideration additional factors in dividing the coun ties, we need to mak e the follo wing assumption: we assume that tw o dis tricts share approximately the same p opulation if the difference in their num b ers of v oters is no more than 5%. In reality , it is imp ossible to divide districts into absolutely equal n um b ers of v oters due to the influence of p opulation flo w and the fluctuation of birth and death rates. And most of t he constitutions also a llow the the standard deviation of n um ber of v oters to fa ll within 10 − 15 % [7] . 7 2.3.2 Mo dification Metho d After making up the designs of landform, w e are faced with a problem - how to adopt the b orderline to a v oid dividing up administrative districts. T o solv e this problem, w e hav e to adopt the b orderlines b etw een the administrative districts. W e follo w the lines in bulge and conca v e in some district. As w e think the densit y in p o pulation almost equals close to b orderline b et w een districts. T aking into account the fa cto r s of populatio n, la ndform a nd managemen t, w e tak e this division as a simple meaning in manag ement. 2.3.3 Our A lgorithm The basic consideration o f this improv ed mo del is the same with the ab o v e simple mo del. How ev er, in that mo del, the dividing lines are all straigh t and th us cannot k eep a single coun t y in tegrate. Now we are trying to o ve rcome this shortcoming and do our b est to mak e the dividing line coincide with the b oundary of each adminis- trativ e area, although coun t y dividing is una v oidable. Step 1: Use the simple mo del t o find the straight dividing line. Figure 4: step 1 8 Step 2: Find all the in tersections b et w een the straight line and the count y b oundaries. Then in the progra m, use “ coun t” the in teger to represen t ho w man y crosso v er p oin ts there are on the straigh t line and use array point[coun t] to r ecord the lo cation o f all these interse ctions. Figure 5: step 2 Step 3: There must b e 2 or 3 direct paths along the straig h t line or the b oundary of the coun t y the straight line go es through connecting the t w o inters ection p oin ts. W e are trying to decide which of the t w o paths a lo ng the b oundary is more “ simple”. The “simple” path should b e the b oundary whic h connects the adjoining p oin ts and more close to the straight line. First w e find the p oint on the straig ht line whose co ordinate p osition is 2 or 3 units a wa y from the in tersection. W e separately searc h the p o ints left and right to it (tak e ve rtical line as a n example , for ho r izon tal lines the 2 directions w ould b e a b o v e and b elow ). The direction w e finally c ho ose would b e the o ne on whic h we arriv e at the b oundary earlier. W e record the direction c hosen in array path[count-1], if the direction is left o r up w ard, pa th[i]=1, if it is right or down w ard, path[i]=3. This metho d ma y not b e v ery strict, but it is easy to realize and tak es v ery little time for the computer to execute. Step 4: Ev ery straigh t line b etw een t w o adjo ining p oints and the b oundary line c hosen in step 3 can confine a small a r ea. W e calculate the p opulation in each small area b y sum up the elemen ts of the p opulation densit y mat r ix including in the area. P a y atten tion that if the area is to the left or b elow the straig h t line, prescrib e that s[i] is negative , otherwise, it is p ositiv e. W e sav e the result in array s[coun t-1]. 9 Figure 6: step 3 Figure 7: step 4 Step 5: In this step w e will finally decide the dividing line - its shap e and lo cation. W e no w kno w all the p oints on the dividing line and w an t to decide the lines jo ining the p oin ts. W e use arra y path[coun t- 1] to describe these lines. path[i] stands for the line connecting the i th p oin t and the ( i+ 1 ) th p oin t. path[i] =1,2 ,3 stands for the 1 st , 2 nd , 3 th path count from left to r igh t or do wn w ards to upw ards. The principles w e use t o decide whic h path to c ho ose is lik e this: First, while w e replace the initial straight line with the b oundary line, the p op- ulation of differen t congressional districts will v ary to eac h other. So the differences m ust b e con trolled within a certain range. Second, w e should use as little as p ossible straight lines, b ecause c ho osing a straigh t path means one count y will b e divided into t w o. The metho d is as fo llo ws: 10 Figure 8: step 5 In step 4 w e get the ar r a y s[coun t-1 ], we first sum up all the elemen ts in s[coun t- 1] and get the result S. Judge whether S < delta , here, delta can b e the theoretical p opulation of a congressional district m ultiple with the allow ed r a nge of error (in our progra m w e c ho ose the range as 5 %), if so, w e consider curren t path is qualified and we get the final arra y path[count-1], exit lo op; if not, we find s[i] whic h is closest to S ( | S − s [ i ] | is the minim um), and let path[i]=2 (c hange the path t o the straigh t line)let S = S − s [ i ], return t o judge whether S < delta , ... ,and lo op like this. After exiting the lo op, prin t t he lo cation of t he initial straigh t line and its end p oin ts, p oint[coun t] and pat h[count-1]. The final dividing line w e get with this metho d ma y not b e the optimal line, but it is qualified. The metho d can av oid considering a ll the p ossible combination of paths and thu s can op erate v ery fast. Step 6: D o the recursion a s the simple mo del and get all the dividing line. 3 F urther Optimiz ati o n of the Mo del Except shap e and coun t y integrit y , o ther factors should also b e take n into a ccount in a ctual gerrymandering. F or example, how to a v oid biased “gerrymandering, how to mo dify our mo del in sp ecial landform, as w e are g oing to represen t. 3.1 The Mo dification of A v oiding “Gerrymandering” 3.1.1 Brief In tro duction There are t w o principal strategies b ehind gerrymandering [3 , 8] : maximizing the ef- fectiv e v otes of supp orters, and minimizing the effectiv e v otes of opp onen ts. One 11 form of gerrymandering, pac king is to place as man y v oters of one ty p e into a single district to reduce their influence in other districts. A second form, crac king, in volv es spreading out v oters of a particular type among man y districts in o r der to reduce their represen tation b y den ying them a sufficien tly large v oting blo c k in an y par t ic- ular district. The metho ds are t ypically com bined, creating a few “forfeit” seats for pac k ed v oters of one type in order to secure eve n greater represen ta tion f or v oters of another ty p e. Figure 9: [3] Redra wing the balanced electoral districts in this example creates a guaran teed 3-to- 1 adv an ta ge in represen tation for the blue v oters. Here, 14 red v oters a re pac k ed into the y ello w district and the remaining 18 are crack ed across the 3 blue districts. 3.1.2 Resolution Because the simple algo rithm we based on uses only the shap e o f the state, the n um ber m of districts w an ted, and the p opulation distribution as inputs - and do es not kno w the party loy alties of the v oters in an y giv en region - the result cannot b e biased [3 , 9] . Ho w ev er w e can not completely prev en t t he certain v oter biases from o ccurring due to the random redistricting pro cess - although the probability is small. As a result, we need mak e a mat hematical analysis to determine whether the gerryman- dering our mo del pro p osed is fair. • Model assumption : Due to the the influence of p opulation flo w and the fluctuatio n of birth, death and the supp orting ratio of a p olitical party . It is reasonable to assume tha t the supp orting ratio of a particular part y in eac h district is incorrelate o f one another appro ximately . Next we w ould analyze how to judge the fa ir ness of the gerryman- dering to a specific party (Republican in our example) ; other parties and situations can b e judged via the same metho d. 12 • Mathema tical Analysis : x j i =  1 I f the i th v oter in the j th distr ict su ppor t the Republ ican 0 O ther w ise (1) If p j represen ts the supp orting ratio of the Republican in j th district a nd this district has n voters, then w e would expect to get (3.2) p j = P n i =1 x j i n . (2) Let p b e the last-few-y ear av erage supp orting ratio of the Republican in the whole state. Thus w e can exp ect that p ≈ P m i =1 p j m . (3) Ob viously , x j i is distributed b y B(1,p). X j = r n p (1 − p ) ( p j − p ) = r n p (1 − p ) ( P n i =1 x j i − np n ) = P n i =1 x j i − np p np (1 − p ) ∼ N (0 , 1) (4) According to the statistics t heory , X j ob eys N(0 ,1), if n is large enough. Then it is reasonable to obtain the fo llo wing deduction: Y = m X j =1 X 2 j ∼ χ 2 m , (5) Here m indicates the num b er of districts in a certain state. Th us, w e can use Equ.(3.5) to generate a simple but effectiv e standard to judge the fairness o f the original gerrymandering. 13 According to the statistics theory , if w e get the p opulation densit y of the sup- p orting ra t io of t he Republican, it is easy to gain Y using the data p j and p. After that, the fo remost task is to determine a parameter α allow scien tifically . The parameter α allow functions as a threshold to judge whether the curren t gerry- mandering is bias or not. Then from the data ta ble of distribution χ 2 , we can get the v alue α where P ( X > Y ) = α . Here X stands for the random v ariable distributed b y χ 2 m . Figure 1 0: (a)P ac king strategy (b)Crack ing strategy On one hand, if there happ ens the “pac king” situation, we w o uld exp ect α to b e extremely small. Therefore, if α allow > α , w e can judge this g errymandering as pac king situation leading to unfairness. On the other hand, if “crac king” situation happ ens, w e can exp ect 1 − α t o b e extremely small. W e consider t his gerryman- dering biased as the same (sho wn in Fig ure 10). In conclusion, if the gerrymandering is fair, the v alue of α pro duced b y it w ould comply with following limits. α > α allow N ot “ pack ing ” 1 − α > α allow N ot “ cr ack ing ” . (6) W e can briefly obtain the final limit o n v a lue α (sho wn in Figur e 11), that is α allow < α < 1 − α allow . (7) • Conclusion: 14 Figure 11: Perm itted Y Domain After g enerating the “simp le” congressional districts, w e can use Equ.(3.5) and (3.7) to judge whether it satisfies the requiremen t of f airness, if we can collect the sup- p orting r a tio of the Republican in eac h district, the last-few-y ear av erage supporting ratio in the whole state a nd the parameter α allow . If it is not a fair gerrymandering, w e need to reconsider the pro cedure o f redistricting. 3.2 The Mo dification of G eographic F eatures Up till no w, man y mo dels that can realize gerrymandering ha v e existed. Th us comparison b etw een all kinds of gerrymandering results is necessary . W e are going to further analysis the characteristic of our tw o mo dels while comparing the results w e get with others. 3.2.1 Brief In tro duction Ab o v e-men tio ned metho d has ignored landfo rm factors such as main riv ers, moun- tains, lakes in pro cess of choosing partition, th us it has made district part it io n unnatural. Here w e make some improv emen ts. As to main rivers , mountains and so on, we hold the view that along the riv er, p opulation b enefit in econom y comparative ly resem bles. On this basis, the areas on the rive r should b e one district. 15 3.2.2 Resolution Our o p erations are as follows: 1. A new district with a large p opulation is based on expanding along the riv er and mountains. Considering the adv an t ages of t he long river or deep moun- tains, as w ell as conv enience in managemen t, more districts will b e formed if the p opulat io n increases. 2. T o the large lak e, we ex pand the area in to sev eral districts along the bank, for they are blessed with the same interes t in econom y . 3. T o the area of moun tains and plain, we consider dev eloping them separately as a result of differen t demands in b enefit. 4. The s ame thinking is practical along t he coastline. W e assem ble p eople on the coast in to sev eral districts, to separate the areas of coastline fro m inland to help the dev elopmen t fro m the coastline to the inland. 5. Considering the a ppro ac hing in terest in economy , the area of islands a nd b y- lands is b eing considered alone. A sp ecial zone will b e built. With these fa cto r s of landform b eing conside red alone, priorit y to ev ery electoral district deducted, to the part remaining again, w e will carry out the prev ious simple op eration according to the simple mo del. 4 Conclus ion and Analysis 4.1 Comparison of Our T w o Mo dels W e ha v e fully established tw o mo dels - the simple one and the refined one, b ot h of whic h a re easy to realize. Eac h o f these tw o mo dels has its ow n prominen t adv antages. As to the simple mo del, first of all, the final shap es of the congressional districts are all v ery simple, constructing with horizontal and v ertical lines o nly . Moreov er, the simple mo del guaran tees that eac h district con tains the same p opulation if giv en a precise p opulation division. While to the refined mo del, w e can maintain t he in tegrit y of ev ery single coun t y to a large exten t, whic h ease the managemen t of actual v oting. Whats more, a lthough the p opulatio n of districts v a ries fro m eac h other, the diff erences can b e controlled theoretically to any prop osed precision. 16 4.2 Comparison With Other Mo dels There are many existing metho ds to realize the gerrymandering with the consider- ation of different asp ects. Here w e compares our mo del with the “shortest split line algorithm” and the “q-state p ots mo del”. F rom the theoretically a nalysis and the final result, w e can see that o ur metho d has its merits. First, one imp orta nt task for us is to mak e sure the simplicit y of the shap es of districts. Therefore, we mo dify the “shortest split line algorithm” to establish o ur initial simple mo del. The main mo dification is that w e o nly use horizon tal line a nd v ertical line instead of diagona ls. In this w a y , w e can get simple rectangles other than irregular p olygonal districts pro duced b y original “shortest split line a lg orithm”. Secondly , we think the “q-state p otts mo del” is excellen t. The foremost ch arac- teristic of it is that this mo del k eeps the in tegrit y of counties. How ev er, the difference of dis trict p opulations can just b e controlled within 15%, while this difference in our mo del is no more than 5%. In sum, the adv antages of our mo del is that: 1. Easy to carr y out in the computer. 2. Simple shap es o f the districts. 3. Com bined with the factor of av oiding dividing up the curren t administrativ e ares. 4. The difference of district p opulations can b e con trolled in any precision. 4.3 Comparison With Curren t Districts F rom Figure 12, w e can clearly see the difference b etw een the curren t (the left one) districts a nd the redistricting prop osed b y our mo del. Through careful compar ison, w e can draw the conclusion that each has its adv antages. The c urren t shape of districts o wn its great merit that it tak e in to t he geographi- cal consideration. F or example, the district 20 th is built along the Lak e Ontario and the district 22 nd mainly consists of plain. Both sho w the mo dification of geogra phic features. Although there is some merit in the curren t district, the gerrymandering result pro duced by our mo del still sho w its adv antages. 17 Figure 1 2: Comparison With Curren t Districts 1. The shap e of most districts are r ectangula r satisfying the retiremen ts of the “simplicit y”. 2. Most of the p erimeters of the district a re alo ng the b o undaries o f curren t coun ties resulting in t he con v enience of manag emen t. 3. Last but no t least, the diff erence of district p opulations is no more than 5%, and can b e controlled in an y precision. Finally , we ha v e generated a simple but effectiv e criterion to judge the fairness of the gerrymandering via the reasonably mathematical analysis whic h mak es the mo del as a whole. 5 W eakness and F urt h er Dev elopmen t Although our t w o mo dels a r e reasonable and easy to realize, the final gerrymandering mo del still hav e some weak ness whic h needs f urther improv emen t. 5.1 V erification of the Impro v ed M o del Although we ha v e implemen ted the improv ed mo del to tak e in to consideration the fairness of v oter distribution by pa rt y , w e cannot obtain the data on populat io n den- sit y f av o ring resp ectiv e parities or on supp orting ratio o f a sp ecific party . Therefore w e can not put that part of the mo del in to practice. In a no ther word, if w e w an t to pro v e t he effectiv enes s and fairness of t he mo del w e should tr y our b est to in v estigate suc h data and further correct our simulations a nd mathematical analysis. 18 5.2 Rea lization of the Comprehensiv e Considerations Although w e hav e noticed the imp ortance of geographic features when gerryman- dering, due to the complexit y of our curren t mo del considering another key factor - av oiding separating administrative areas, w e ha v e not achie v ed the goal that pro- grams the a lgorithm considering the landform mo dification. Th erefore, our next mission is to realize the comprehensiv e considerations including geographic fa cto r s. 19 6 App endix Applications to Other States Figure 1 3: (a)W ashington (b)P ennsylv ania Figure 14: (a) W est Virginia (b)New Jersey 20 Figure 1 5: (a)Wisconsin (b)Uta h 21 References [1] P opula t ion densit y maps of all the states, http://www.ans w ers.com , Accessed F eb 10th, 2007 [2] Curren t congressional districts of a ll the states, h ttp://www.cens us.go v , Ac- cessed F eb 10th, 20 07 [3] Gerrymandering - Wikip edia, the free encyclopedia, h ttp://en.wikipedia.or g /wiki/Gerrymandering , Accessed F eb 9th, 2007 [4] Cognitiv e Asp ects of Gerrymandering, h ttp://www.springerlink.c om/index , Accesse d F eb 10th, 2007 [5] P opula t ion list of all the counties in the state of New Y ork, h ttp://www.cit yp opula t io n,de/USA-New Y ork.html, Accessed F eb 9th, 2007 [6] Curren t congressional districts of all the states, h ttp://nationalatlas.gov/prin ta ble/congress.h tml , Accessed F eb 11st, 2007 [7] Hun ting t o n News, h ttp://www.h untingtonnews.ne t/national/06 1006-shns-elections.h tml , Accesse d F eb 11st, 2007 [8] F airV ote - F airness and Indep endence in Redistricting Act (H.R. 2642) , h ttp://www.fairv ote.org, Accesse d F eb 10th, 2007 [9] P ar tisan Gerrymandering, h ttp://www.senate.le g.state.mn.us/departmen ts/scr/redist/red2000/ch5parti.h tm , Accesse d F eb 10th, 2007 [10] Range v oting, http://www.rangev oting.or g, Accessed F eb 10th, 200 7 [11] The theory of statistics, h ttp://www.statistics.go v, Accessed F eb 1 0th, 2007 [12] F air V ote Gerrymander Examples, http://www .w estmiller.com [13] K Sherst yuk. “ How to gerrymander: A formal ana lysis”, Springer , 1998 [14] TW G illigan, JG Matsusak a. “Structural constrain ts on partisan bia s under the efficien t gerrymander”, Springer ,1999 22 [15] JD Cranor, GL Crawle y , RH Sc heele. “The Anatomy of a Gerrymander”, Amer- ic an Journal of Politic al S cienc e , 1989 [16] BE Cain. “Assessing the P artisan Effects of R edistricting”, The Americ an Po- litic al S cienc e R eview , 1985 23