Bipartisan Gerrymandering

Bipartisan Gerrymandering Hideo Konishi y Chen-Yu Pan z February 15, 2016 Abstract In this paper we propose a tractable model of partisan gerrymandering followed by electoral competitions in policy positions and transfer promises in multiple districts. With such por-barrel considerations, we investigate the optimal policies under two situations: partisan and bipartisan gerrymanderings. With complete freedom in gerrymandering as in Friedman and Holden (2008), partisan gerrymandering policy generates the most one-sidedly biased district pro le, while bipartisan gerrymandering generates the most polarized district pro- le. In contrast, with limited freedom in gerrymandering as in Owen and Grofman (1988), partisan and bipartisan gerrymandering tend to prescribe the same policy: slicing districts from left to right. Friedman and Holden (2009) nd empirically that there is no signi cant di erence between bipartisan and partisan gerrymanderings in explaining incumbent reelection rates. Our result suggests that the gerrymanderers may not be as free in redistricting as popularly thought. Keywords: electoral competition, partisan gerrymandering, bipartisan gerrymandering, policy convergence/divergence, por-barrel politics JEL Classi cation Numbers: C72, D72 Still preliminary. Not for circulations. We than Jim Anderson, Mehmet Emeci, and Ron Segel for their comments. y Hideo Konishi: Department of Economics, Boston College, USA. Email: hideo.onishi@bc.edu z Chen-Yu Pan: School of Economics and Management, Wuhan University, PRC. E-mail panwhu@126.com

1 Introduction It is widely agreed that the US Congress has polarized quite a bit in the last half century. The distribution of the House representatives political positions were more concentrated at the center of political spectrum with considerable overlaps between Republican and Democratic representatives positions in 1960s, while it became sharply twin-peaed without overlaps in 2000s (see Figure 1: by Fiorina, Abrams, and Pope, 2011). 1 Simultaneously, Fiorina, Abrams, and Pope (2011) argue that the US voters have not polarized so much during the same time period. These con icting observations generate an obvious puzzle: How could the Congress polarize if voters didn t? [Figure 1 here] One possible explanation is that voters sorted out into Republican and Democratic parties by their political positions during the period, and that the parties political positions were polarized in party members preference aggregation. Levendusy (2009) suggests that party elites polarization led voter sorting, although it is controversial how much mass polarization actually occurred by voter sorting. 2 A more direct explanation for that is gerrymandering would be the US politics. Since the one-person, one-vote decisions by the US Supreme Court in 1960s, the courts closely examine population equality in districts. In practice, however, redistricting is not simply about equaling the population in each district. It is clear that there is a strong incentive for the party in charge of redistricting to change the political identity of each district to favor the partisan interests. 3 Although Ferejohn (1977) nds little support for gerrymandering being the cause of declines in competitiveness of congressional districts from mid 1960s to 1980s, Fiorina et al. (2011) state Many (not all) observers believe that the redistricting that occurred in 2001-2002 had a good bit to do with this more recent decline in competitive seats the party behaved conservatively, concentrating on protecting their seats rather 1 It is now standard to use a one-dimensional scaling score (DW-Nominate procedure on economic liberal-conservative, Poole and Rosenthal, 1997) to measure representatives political positions. 2 Levendusy (2009) asserts that party elites polarization clearly caused limited scale of mass polarization. Fiorina et al. (2011) say that there is little increase in mass polarization, although they admit that party activists political positions have polarized. 3 One recent example is the 2003 Republican redistricting in Texas that has been considered contributing to Democrat s defeat in the subsequent election. 2

than attempting to capture those of the opposition." (see Fiorina et al. pp. 214-215). They further argue that this decrease in competitiveness from gerrymandering is a driving force behind the recent political polarization in Congress (see also Gilroux, 2001). Recent empirical studies are not quite supportive to these ideas. McCarty et al. (2006, 2009) document that the political polarization of the House of Representatives has increased in recent decades, using data on roll call votes, but they nd only minor relation between polarization and gerrymandering. 4 Regarding recent decline in competitiveness of districts, Friedman and Holden (2009) investigate whether or not gerrymandering caused the rising incumbent reelection rate by using data up to 2004, nding evidence of the opposite e ect all else equal. 5 Friedman and Holden (2009) also investigate whether or not the incumbent reelection rate depends on gerrymandering being partisan or bipartisan. 6 In partisan gerrymandering case, the majority party may try to oust the opposing party s incumbents, and this may be reducing the incumbent reelection rate. Interestingly, they did not nd signi cant di erence between bipartisan and partisan gerrymandering on the e ect on the incumbent reelection rate. As they say, this result suggests that partisan gerrymandering may not be as e ective as popularly thought. In partisan gerrymandering, two tactics are often discussed. The rst one is to concentrate or pac those who are supporting opponent party in losing districts. The second is to evenly distribute or crac supporters in winning districts. Introducing uncertainty in each district s median voter s position, Owen and Grofman (1988) consider the situation where a partisan gerrymanderer redesigns districts in order to maximize the expected number of seats. They assume that the uncertainty in median voter s political position is local and is independent across districts when the objective is expected number of seats. Assuming that the average of the positions of district median voters must stay the same after redistricting (their feasibility constraint), they show that the optimal strategy is pacing the opponents in losing districts, and cracing the rest of voters evenly across the winning districts with substan- 4 Krasa and Polborn (2015) argue that their answer may be incomplete if the political positions of district candidates are mutually interdependent. 5 After 2008, the incumbent reelection rate went down signi cantly. 6 Redistricting in the US is usually conducted by state legislatures (partisan gerrymandering), but in Arizona, Hawaii, Idaho, Montana, New Jersey, and Washington, it is conducted by bipartisan redistricting commissions. In California and Iowa, redistricting lines are drawn by nonpartisan redistricting committees. 3

tial margins so that the party can win districts even in the cases of negative shocs under both cases. 78 Friedman and Holden (2008), on the other hand, assume that a partisan gerrymanderer has full freedom in allocating population over a nite number of districts, and that she maximize the expected number of seats when there are only valence uncertainty in median voters utilities (thus, there is no uncertainty in median voter s political position). In this idealized situation, they nd that the optimal strategy is slice-andmix : mixing a slightly larger number of the most extreme supporters and a slightly smaller number of the most extreme opponents in the rst district, then mixing a slightly larger number of the second most extreme supporters and a slightly smaller number of opponents in the second district, and so on and so forth. Thus, theoretically, the levels of freedom in gerrymandering can a ect the optimal policy to tae. In bipartisan gerrymandering, Gul and Pesendorfer (2010) extend Owen and Grofman (1988) by introducing a continuum of districts, and voters party a liations. Here, bipartisan gerrymandering means that the two parties own their territories, gerrymandering exclusively within each territory. They assume that each party leader maximizes the probability of winning the majority of seats under more sophisticated feasibility constraints 9 by assuming a continuum of districts and generate a pac-and-crac. However, these papers in the literature do not compare the optimal partisan and bipartisan gerrymandering policies. They also do not model spatial competition in policy positions, and the elected representatives positions are implicitly assumed to be the district median voters positions (Downsian competition). In this paper, we assume that policy-motivated party leaders compete with their political positions and por barrel promises, and we nd that policy divergence actually occurs. 10 We investigate the optimal policies in 7 They also consider the case where the majority party maximizes the probability to win a woring majority of seats for her party by assuming that the uncertainty is global. They again get pac-and-crac policy as the optimal policy. 8 The original cracing tactics create the maximum number of winning districts with the smallest margins. In the traditional literature, some argue that gerrymandering will increase political competition by this reason. In this paper, we use cracing tactics in the sense of Owen and Grofman (1988). 9 They consider two feasibility constraints. The rst is the constant mean of median voters positions which is the same as the one in Owen and Grofman (1988). The second one is that the status quo needs to be a mean-preserved spread of a feasible redistricting plan. 10 In two-party electoral competition model a la Wittman (1983), the equilibrium out- 4

both partisan and bipartisan gerrymandering cases within the same political competition model. With por-barrel politics, the party leader is aware of the cost of por-barrel policies, and therefore she has strong partisan and bipartisan gerrymandering incentives to collect their supporters in the winning districts in order to avoid large por-barrel promises. Our simple model is exible enough to allow us to analyze the optimal policies under various feasibility constraints on gerrymandering and partisan and bipartisan gerrymanderings. It turns out that under complete freedom in redistricting as in Friedman and Holden (2008), the predicted policies are very di erent: we also obtain the slice-and-mix policy in partisan gerrymandering, which creates not only extreme districts for the gerrymandering party candidates, but also some relatively competitive districts. In contrast, under bipartisan gerrymandering, the two party enclose their supporters rst, then they each applies a slice-and-mix policy. As a consequence, the outcome in bipartisan gerrymandering would exhibit the most polarization of candidates possible under complete freedom in redistricting. This is consistent with an example in Fiorina et al. (2011 pp. 216-217). However, if the gerrymanderer does not have such high level of freedom as in Owen and Grofman (1988), it is not clear whether or not partisan and bipartisan gerrymanderings mae a big di erence, since Owen and Grofman s result for partisan gerrymandering is "pac and crac," and pacing the opponent party supporters will occur in either type of gerrymandering. In our model with policy preferences, we obtain a very di erent result from the complete freedom case, if we assume that the average of district medians needs to be ept constant (Owen and Grofman 1988). With worable functional speci cation, we can show that pacing occurs anyway, and the partisan or bipartisan gerrymandering does not a ect the results much although "pacand-crac" or "pac-and-slice" can occur depending on parameter values. The rest of the paper is organized as follows. Section 2 discusses some related literature. In Section 3, we start with analyzing political-position and por-barrel competition and characterizing the party leader s payo from each winning district by the district median voter s position (Proposition 1). In Section 4, we investigate the optimal gerrymandering strategy when the party leader has complete freedom as in Freedman and Holden (2008), and come is the Downsian unless there is uncertainty in median voter s position (see Roemer 2001). In our model, if we place the upperbound on the level of por-barrel promise, then the political competition equilibrium converges to the Downsian as the upperbound goes to zero. 5

show that their slice-and-mix is also an optimal strategy in partisan gerrymandering case (Proposition 2). In contrast, in bipartisan gerrymandering case, we obtain a rule that rst partition voters into two consecutive sets in their political positions, and both parties apply slice-and-mix to their groups. This policy generates the most polarized allocation (Proposition 3). In Section 5, we proceed to the cases where gerrymanderer s freedom is limited by indivisibility of localities. For tractability of our analysis, we assume voters and leaders cost function in political distance have common constant elasticity > 1. We also assume that each district has normally distributed voters to justify the feasibility constraint imposed by Owen and Grofman (1988). We show that the optimal gerrymandering policy are liely to be pac-and-slice, in which the gerrymanderer pac the opponent supporters and slice them from moderate to the strongest, and slice her own supporters from the strongest to moderate in order (Proposition 4). This is the same thing as slicing entire localities in order. With bipartisan gerrymandering, the result is the same, since both parties want to slice their supporters and their opponents (Proposition 5). The two parties preferences totally coincide with each other. Thus, if the freedom in gerrymandering is limited in Owen and Grofman s sense, then there is no di erence between partisan and bipartisan gerrymandering, although under complete freedom, the di erence is huge. In order to explain Friedman and Holden s (2009) empirical nding that the di erence between bipartisan and partisan gerrymandering, we may say that gerrymanderers are not facing complete freedom in redistricting. Section 6 concludes. All proofs are collected in Appendix A. 2 Related Literature Our paper is related to two branches of literature. The rst one is the porbarrel literature. In this branch, our model is most related to Lindbec and Weibull (1987) and Dixit and Londregan (1996). The former introduces a two-party competition model in which (extreme) parties use por-barrel policies to attract agents with heterogeneous policy preferences. The latter generalizes Lindbec and Weibull (1987) to allow parties having di erent abilities in practicing por-barrel policies, and this di erence determines the por-barrel policy s target being swing voters or loyal supporters. Our model is di erent from theirs in that we introduce parties platform decisions besides por-barrel politics, and party leaders choose these two policies simul- 6

taneously. 11 Since we try to explain the House representatives polarization, having platform decisions is essential in our analysis. Moreover, the political competition result is deterministic in our model which is di erent from the setup with uncertainty in the literature. Second, other than the optimal gerrymandering literature we discussed in the previous section, there is gerrymandering literature from normative point of view as well. This literature focuses on the how the gerrymandering a ects the relation between seats and the vote shares won by a party, socalled seat-vote curve. Coate and Knight (2007) identify the social welfare optimal seat-vote curve and then the conditions under which the optimal curve can be implemented by a districting plan. With xed and extreme parties policy positions, they nd that the optimal seat-vote is biased toward the party with larger partisan population. However, Bracco (2013) shows that, when parties strategically choose their policy position, the direction of seat-vote curve bias should be the opposite. Besley and Preston (2007) construct a model similar to Coate and Knight (2007) and show the relation between the bias of seat-vote curve and parties policy choices. They further empirically test the theory and the result show that reducing the electoral bias can mae parties strategy more moderate. 3 The Model We consider a two-party (L and R) multi-district model with one party being entitled to redistrict a state. There are many localities in the state, each of which is considered the minimal unit in redistricting (a locality cannot be divided into smaller groups in redistricting). We assume that there are L discrete localities each of which has population 1. The state must have K equally populated districts, and L is a multiple of K. That is, the party in power needs to create K districts by combining L = n localities in each district. Locality ` = 1; :::; L has a voter distribution function K F` : ( 1; 1)! [0; 1], where ( 1; 1) is a space of ideologies in the society and F`() is non-decreasing with F`( 1) = 0 and F`(1) = 1. Ideology 11 Dixit and Londregan (1998) propose a por-barrel model with strategic ideological policy decision based on their previous wor. However, the ideology policy in their paper is the equality-e ciency concern engendered by parties por-barrel strategies. Therefore, the ideology decision in their wor is a consequence of por-barrel politics instead of an independent policy dimension. 7

< 0 is regarded left, and > 0 is right. With a slight abuse of notations, we denote the set of localities also by L f1; :::; Lg. A redistricting plan = fd 1 ; :::; D K g with jd j = n for all = 1; :::; K, is a partition of L. 12 The gerrymandering party s leader chooses the optimal district partition from the set of all possible partitions. 13 In each district, the probabilistic distribution function P F is an average of distribution functions of n localities: F () = 1 n `2D F`(). District s median voter is denoted by x = x (F ) 2 ( 1; 1) with F (x ) = 1. We assume the uniqueness of x 2 in each districting plan. We will consider two cases later: one case is with complete freedom in redistricting as in Friedman and Holden (2008), and the other is with limited ability is redistricting in the line of Owen and Grofman (1988). In either case localties will have naturally ordered by its political positions. We also introduce uncertainty in the position of median voter in actual election (due to turnout or some valence term of two candidates in each district : y is a realization of uncertain term: the median voter in actual election in district is denoted by ^x = x + y. We assume that y is i.i.d. across districts, following probabilistic distribution function G : [ y; y]! R, where y > 0 is the largest value of relative valence term and G(0) = 1. We 2 assume that gerrymandering needs to tae place before y s realize, but that electoral competition occurs after y realizes: the resulting median voter s position after the shoc realization is ^x x + y. We model por-barrel elections in a similar manner with Dixit and Londregan (1996). A type voter in district evaluates party L or R according to the utility function with two arguments: one is the policy position of the candidate representing the corresponding party, 2 R, and the other is the party s por-barrel transfer t 2 R +. We interpret this por-barrel transfer as a promise of local public good provision (measured by the amount of monetary spending) in the case that the party s candidate is elected. Formally, a voter in district s utility when party L is the winner is U (L) = t L c(j Lj) (1) 12 A partition of L is a collection of subsets of L; fd 1 ; :::; D K g, such that [ K =1 D = L and D \ D 0 = ; for any distinct pair and 0 : 13 In reality, there are many restrictions on what can be done in a redistricting plan. For example, a district is required to be connected geographically. Despite the complication involved, our analysis can still be extended to the case with geographic restrictions by introducing the set of admissible partitions A (see Puppe and Tasnadi, 2009) 8

where L is party L s district candidate s policy position, t L 0 is party L s district--speci c por-barrel transfer, and c(d) 0 is the ideology cost function which is increasing in the distance between a candidate s position and her own position. We assume that c() is continuously di erentiable, and satis es c(0) = 0, c 0 (0) = 0, and c 0 (d) > 0 and c 00 (d) > 0 for all d > 0 (strictly increasing and strictly convex). Similarly, we have U (R) = t R c(j Rj) where R is party R s district candidate s policy position and t R R s district--speci c transfer. Therefore, voter votes for party L if and only if is party U (L) U (R) = [c(j Rj) c(j Lj)] + t L t R > 0 (2) Since the (after shoc) median voter s type in district is ^x = x + y, given L, R, t L and t R, L wins in district if and only if U^x (L) U^x (R) = [c(j^x Rj) c(j^x Lj)] + t L t R > 0 (3) Each party leader in the state (of these K districts) cares about (i) the in uence or status within her party based on the number of winning districts in his/her state, (ii) the candidate s policy position in each district, and (iii) the district-speci c por-barrel spending. The party leader We assume that the party leader prefers to win a district with a candidate s position closer to her own ideal ideological position and a less por-barrel promise. The former is regarded as the policy-motivation in the literature. By formulating the latter, we consider a situation where the leader bears some costs when implementing the promised local public goods provision. For example, the bargaining e orts needed to push for federal funding. To simplify the analysis, we assume that the negative utility by por-barrel is measured by the amount of money promised. We denote the ideal political positions of the leaders of party L and R by L and R, respectively, with L < R. Without loss of generality, we will set L = 1 and R = 1 in the end, but we will stic to notations L and R until the gerrymandering analysis starts to help the readers comprehend the model more easily. Formally, by winning in district, party L s leader gets utility V L = Q t L C( L L ); 9

where Q > 0 is the xed payo that party leader obtains from each winning district, and C(d) is a party leader s ideology cost function with C(0) = 0, C 0 (0) = 0, C 0 (d) > 0 and C 00 (d) > 0 (strictly increasing and strictly convex). If she loses in district, she gets zero utility from the district. The national party elites are ultimately interested in the number of seats their party gets, so the number of seats a state party leader wins is important in recognize her contribution to the national party. Since we are considering a state s gerrymandering problem, it is reasonable to assume that the bene t from winning a district does not depend on which district to win. Thus, party L s state leader s utility from K districts in the state is V L = KX I L ()VL = =1 KX I L ()[Q t L C( L L)]; (4) =1 where I L () = 1 if L wins, I L () = 0 if L loses, and I L () = 1 2 R in district. Similarly, the utility for R is de ned as if L ties with V R = KX I R ()VR = =1 KX I R ()[Q t R C( R R)]: (5) =1 We introduce a tie-breaing rule in each district based on the relative levels of the state party leaders utilities VL and V R. We assume that if two parties o ers are tied for the median voter ^x (U^x (L) = U^x (R)) while one party s leader gets a higher (indirect) utility than the other s, the median voter will vote for the party. That is, Assumption 1. (Tie-Breaing) Given two parties o ers are such that U^x (L) = U^x (R), L (R) wins if VL > V R (V L < V R ). This assumption is justi ed by the fact that the higher utility is equivalent to the higher ability to provide a better o er to the median voter. Especially, consider the case in which two parties are tied and, say, VL > V R = 0, party L has the ability to provide > 0 more por-barrel promise. Therefore, we brea the tie by assuming the median voter prefers L, which is the standard assumption. Our second assumption is a simple su cient condition that assures interior solutions for both parties. Assumption 2. (Relatively Strong O ce Motivation) For all feasible ^x, Q min fc(j j j) + c(j ^x j)g holds for j = L; R. 10

Notice that if the party leader gets 0 utility, he must o er por-barrel promise equal to Q C(j j j). Therefore, the median voter get utility U^x = Q C(j j j) c(j ^x j). Then, this assumption means that the payo from winning a district, Q, is large enough so that for any ^x, both party can o er the median voter positive utility, which is a su cient condition for the candidate selection problem has interior solution. Also notice that since the model only allows a nite median voters positions, there must exist a Q to satisfy this assumption. Moreover, the implication of this assumption is that it guarantees that in equilibrium both parties promise positive porbarrel. We will see this more clearly in the next section. 14 The timing of the game is as follows: 15 1. One party, say L, chooses a redistricting plan = fd 1 ; :::; D K g of L, thus a median voter vector (x 1 ; :::; x ; :::; x K ). 2. In each district, y 2 [ y; y] realizes. 3. Given the districting plan in stage 1 and the realized median voter ^x = x + y in stage 2, party leaders L and R simultaneously choose local policy positions and por-barrel promises ( L ; t L )K =1 and ( R ; t R )K =1, respectively. 4. All voters vote sincerely (with our tie-breaing rule). The winning party is committed to its policy position and its por-barrel promise in each district = 1; :::; K. All payo s are realized. We will employ wealy undominated subgame perfect Nash equilibrium as the solution concept. We require that in stage 3, party leaders play wealy undominated strategies so that the losing party leader does not 14 This assumption can be weaened signi cantly. In the Appendix B, we present a weaer condition (Assumption 2 ) for interior solutions, and discuss the case of corner solutions. However, our argument still extend to the cases with corner solutions to some extent. 15 We can separate stage 2 into two: policy position choices followed by por-barrel promises. If we do so, the loser of a district will get zero payo in every subgame, so it becomes indi erent among policy positions. Thus, we need equilibrium re nement to predict the same allocation. By assuming that the loser party chooses the policy position that minimizes the opponent party leader s payo, we can obtain the exactly the same allocation in SPNE. 11

mae cheap promises to the district median voters. 16 We will call a wealy undominated subgame perfect Nash equilibrium simply an equilibrium. 3.1 Stage 3: Electoral Competition with Por-Barrel Politics We solve the equilibria of the game by bacward induction. We start with stage 3, nowing that voters vote sincerely in stage 3. Notice that the ey player is the median voter in the voting stage. Thus, when the leader of party L maes her policy decisions in district, she at least needs to match R s o er in terms of median voter s utility in order to win. First, we consider the case that party L wins (the party R s leader wins only by providing a strictly better o er to the median voter). In this case, the leader of party L tries to o er the same utility to the median voter x and wins in the district with the tie-breaing rule. Formally, the party leader s problem is described by max L ;t L fq t L C( L L )g subject to t L c(j^x Lj) U R, t L 0; and (6) Q t L C( L L ) 0; where U R is the median voter s utility level from R s o er. Notice that t L 0 and Q t L C( L L ) 0 may or may not be binding while t L c(j^x L j) U R must be binding. The solution for this maximization problem is straightforward. De ne ^(^x ; ) by the following equation c 0 (j^x ^(^x ; )j) = C 0 ( ^(^x ; ) ): (7) Notice that (7) is simply the rst order condition of optimization problem (6) after substituting t L = c(j^x L j) + U R into the objective function. The 16 This game is the rst price auction under complete information. There is a continuum of pure strategy equilibria, since the losing party does not su er from cheap promise, since she gets zero utility in losing districts anyway. The winning party needs to match the o er as long as she can get a positive payo by doing so. Demanding players to play wealy undominated strategies, we can eliminate these unreasonable equilibria. Another justi cation for this is to require mixed strategy equilibrium. There is a unique mixed strategy equilibrium in which the winning party plays a pure strategy while losing party plays a mixed strategy equilibrium. The outcome of this mixed strategy equilibrium coincides with the wealy undominated Nash equilibrium in pure strategies. 12

optimal policy L = ^(^x ; L ) when c(j^x ^(^x ; L )j) U R. That is, it is not enough for the winning party to win just by only using the policy platform. Therefore, in this case, it is clear that the optimal por barrel promise is t L = U R + c(j^x ^(^x ; L )j) Although it seems not clear that t L de ned above is positive or not, it turns out that t L is always positive, since the similar optimization problem applies for the losing party and Assumption 2. Now, it is clear that the winning party s decision is related to what the losing part proposes in equilibrium. The following lemma shows that the losing party cannot lose with positive surplus. Lemma 1. Suppose R is the losing party in district. In equilibrium, R proposes the policy pair (R ; t R ) which is the solution of the following problem max U x (R) = t R R c(j^x Rj) ;t R subject to t R 0 and Q t R C( R R ) 0 That is, the losing party leader o ers a policy position and por-barrel promise that leave herself zero surplus in equilibrium. R = ^(^x ; R ) R = Q C( R ^(^x ; R ) ) t Moreover, this policy pair is the best she can o er for the realized median voter x. The intuition of this lemma is straightforward. If the losing party does not o er the median voter the best one, then since the winning party will provide the median voter the same utility level, the losing can always o er the median voter something better than her original o er and win the district. This cannot happen in equilibrium. Therefore, for the losing party R, the equilibrium strategy is R = ^(^x ; R ) and t R = Q C( R R ). The policy pair provides the median voter with the utility U R = Q C( R R ) c(j^x R j). Using this U R, one can solve the winning party s equilibrium por-barrel promise t L = Q C( R R ) c(j^x R j) + c(j^x L j). 13

One thing left to decide is which party should be the winning party. Notice that, by Lemma 1, the losing party always propose the best o er by depleting all her surplus. Therefore, the party can potentially provide the median voter with a higher utility level is the winner. Notice that j party s por-barrel promise is bounded above by j party leader s payo evaluated at j (otherwise, the leader gets a negative utility): Q C( R R ): Substituting this into median voter s utility, we obtain WR = Q C( R R ) c(j^x R j); similarly, for party L, W L = Q C( L L ) c(j^x L j); where WR and W L are the (potential) maximum utilities that the median voter gets from the corresponding party s o er. Therefore, party L wins in the third stage if and only if c(j^x R j) + C( R R ) > c(j^x L j) + C( L L ); (8) which simply means that L wins if and only if L ^x < R ^x : (9) Proposition 1. Suppose that Assumptions 1 and 2 are satis ed. De ne ^(^x ; ) by (7). We have 1. For the losing party j, the optimal choice j the interval (^x ; j ) (or ( j ; ^x )). = ^(^x ; j ) which lies in 2. For the winning party i, i c(j^x j j) + c(j^x i j). = ^(^x ; i ), and t i = Q C(j j j j) 3. Party i wins in the th district if and only if Party s winning probability I i (^x ) is described by 8 < 1 if i ^x < j ^x I i (^x 1 ) = if : 2 i ^x = j ^x 0 if i ^x > j ^x : i ^x < j ^x. 14

4. The payo for the party leader i from district is 8 < ~V i (^x ; i ; j ) = : 1 2 C( j ^x ) C( i ^x ) if i ^x < j ^x C( j ^x ) C( i ^x ) if i ^x = j ^x 0 if i ^x > j ^x : where C( i ^x i ) C( ^(^x ; i ) ) + c(j^x ^(^x ; j )j): 5. The winning payo for party i, ~ V i, increases as ^x moves away from j. If ^x < L < R, then ~ V L (^x ; L ; R ) is convex in x, and if L < R < ^x, then ~ V R (^x ; L ; R ) is strictly convex in x. The last result (Proposition 1-5) holds from the fact that C 0 () > 0 and C 00 () > 0. Note that party leaders policy position choices do not strategically depend on the position decisions of the other party. This property of our model simpli es the analysis. Since there is no strategic issue, the winner of the district is determined by which party leader s position is closer to the realized median voter s position. Moreover, party leader s winning payo from the district is simply the di erence in the two parties sums of ideology cost functions of the realized median voter and the party leader. Now, consider two partitions and 0 such that D 0 () = D 0 ( 0 ) for all 0 6= f; g. ~ Thus, D ( 0 ) and D ~ ( 0 ) are created by swapping some localities between D () and D ~ (): i.e., S = D ()nd ( 0 ) = D ~ ( 0 )nd ~ () and T = D ~ ()nd ~ ( 0 ) = D ( 0 )nd () are swapped. Is such swapping bene cial to the gerrymandering party leader? By swapping localities S and T, x and x ~ must move in the opposite directions. We call a swap cracing if the median voters of two districts involved moves closer to each other after swapping and a swap anti-cracing if the opposite happens. 3.2 Partisan Gerrymandering Problem Now, we can formalize the partisan gerrymandering party leader s optimization problem. Proposition 1 shows that x = x () is the su cient statistics to determine the outcome of the th district. Without loss of generality, assume that party L is in charge of redistricting. Notice that the indirect utility of L, V ~ L (x ; L ; R ), is relevant only when party L wins in district. 15

The choice of = D 1 ; :::; D K a ects the party leader L s payo through x 1 (D 1 ); :::; x K (D K ) via winning chance I(x (D )) in each district and its indirect utility V ~ L (x (D ); L ; R ): the party L leader s utility function is written as E V ~ KX Z y n L (; L ; R ) I L (x (D ) + y ) V ~ o L (x (D ) + y ; L ; R ) dy =1 y From now on, we suppress L and R in indirect utility ~ V L and E ~ V L. Although x is solely determined by D, we can write x = x (D ()) = x () for all = 1; :::; K, since uniquely determines D 1 ; :::; D K. Thus, we can rewrite the party leader L s gerrymandering choice to be the result of the following maximization problem 2 arg max 2 E ~ V L () The SPNE of this game is ( ; ( L )K =1 ; ( R )K =1 ; (t L )K =1 ; (t R )K =1 ).17 3.3 Bipartisan Gerrymandering Problem Since bipartisan gerrymandering requires negotiation between the two parties, there can be many possible formulations. One way is to assume that the number of seats each party currently has, but if the original district structure is already distorted by past gerrymandering, it is hard for the two parties to agree on redistricting. So, here, we will simply assume that based on the global median voter x, parties L and R can create K F (x)c and K (1 F (x))c districts, respectively, where c denotes the largest integer not more than. The two parties negotiate which localties to use in redistricting. In the following, we consider the situations where localties are ordered by their political positions. In that case, given Proposition 1-5, each party collects localties that are the most further away from the other party s leader s position. Thus, there is no con icting incentive between them. 4 Gerrymandering with Complete Freedom As a limit case, let us consider the ideal situation for the gerrymanderer (Friedman and Holden, 2008): there is a large number of in nitesimal locali-. 17 The existence of SPNE is guaranteed by the convexity of function c and C and a nite 16

ties with politically homogeneous population: for all position x 2 ( 1; 1), there are localities `s with F`(x ) = 0 and F`(x + ) = 1 for a small > 0. That is, the gerrymanderer can create any ind of population distributions for K districts freely as long as they sum up to the total population distribution. We as what strategy the gerrymanderer should tae. First, by Proposition 1-5, she is better o by maing the (ex ante) median voter s location as far from the other party s leader s position as possible. Second, the same strategy increases the probability of winning the district the highest. Thus, the gerrymanderer tries to create the furthest district structure from the opponent party leader s position. If the gerrymander loses some districts for sure, she does not care how she loses those districts. Therefore, there may be multiple optimal redistricting plans. 4.1 Partisan Gerrymandering under Complete Freedom In partisan gerrymandering case, the party leader in charge of gerrymandering will try to mae district medians as further away as possible from the other party leader s position. 18 Without loss of generality, we assume that party L is in charge of gerrymandering. In order to create the most extreme district, x 1 should satisfy F (x 1 ) = 1 2K (x1 is the median voter of the district: the most extreme district achievable with population 1 ). Although the remaining population to the right of x 1 can be anything in district 1, wasting K the other party s strong supporters by combining them is a good idea, since it would mae the remaining population more leaning towards her position. Thus, she will create district 1 by combining sets L 1 : F (1 L ) = 1 + 2K K and R 1 : 1 F (1 R ) = 1 where > 0 is arbitrarily small. In district 1, the (ex ante) median voter would be x 1 L de ned by F (x1 L ) = 1. 2K 2K K Similarly, she can create districts 2; :::; K sequentially. Let L be such that F (L ) = + for all = 1; :::; K, and let 2K K R 2 [ 1; 1] be such that 1 F (R ) =. For > 0 small enough, we have 2K K 1 = 0 L < 1 L < ::: < K L = K R < ::: < 1 R < 0 R = 1: 18 As long as there are positive winning probabilities in all districts (if y is large enough), this is true. If not, party L s leader may need to create unwinnable districts, but she would be indi erent how to draw lines for these districts. But the slice-and-mix below is one of the optimal strategies even in that case. 17

We call this redistricting plan a party-l-slice-and-mix policy, which is proposed in Friedman and Holden (2008). Under the slice-and-mix policy, the resulting district median voter allocation is x L (x 1 L ; :::; xk L ) with x L is such that F (x L ) = for each = 1; :::; K, with close to zero. We will 2K show that this is the optimal policy for party L leader. Symmetrically, we can de ne a party-r-slice-and-mix policy with the resulting district median voter allocation is x R (x 1 R ; :::; xk R ) with x R is such that 1 F (x R ) = K +1 2K for each = 1; :::; K, with close to zero. Clearly, these district median voter allocations x L and x R are the most biased district median voter allocations towards left and right, respectively. Proposition 2. Suppose that gerrymanderer can create districts with complete freedom and that party L ( R) is in charge of gerrymandering. Then, the party-l ( R)-slice-and-mix policy is an optimal gerrymandering policy. The resulting district median voter allocation is approximately x L ( x R ). Another interesting observation from this proposition is that if party L is the majority party in terms of the state population (x m < 0), then it can win all seats with probability 50% or higher (x K < 0). However, one party s monopolizing all districts is rare in the US politics partly because of the presence of majority-minority district requirement (see Shotts (2001)). 19 The majority-minority requirement forces the gerrymanderer seeing for secondbest districting plan as a result. It is worthwhile to note that the sliceand-mix strategy is identical to the optimal policy analyzed in Friedman and Holden (2008). Friedman and Holden (2008) and our paper share the features that (i) the party leader prefers a more extreme median voter s position than a moderate one, and (ii) complete freedom in gerrymandering unlie in Owen and Grofman (1988) and in the basic model of Gul and Pesendorfer (2010). 20 However, there are big di erences between our paper and Friedman and Holden (2008). Our model is based on competitions with political positions as well as transfer promises, while Friedman and Holden (2008) have neither element in their model. Nonetheless, we can say that the above two common conditions are the eys for getting the same results. 19 In fact, even though either one of the two parties must be the majority in a state, the majority party usually does not win all districts. This can be attributed to Section 2 of the Voting Act Rights (accompanied with other United States Supreme Court cases) which essentially avoid the minority votes being diluted in the voting process similar to our slice-and-mix strategy. 20 Gul and Pesendorfer (2010) also include aggregate uncertainty, generalizing Owen and Grofman (1988). 18

4.2 Bipartisan Gerrymandering under Complete Freedom With the global median voter x m 2 [ 1; 1] with F (x m ) = 1, ex ante Pareto 2 e cient allocation for the two party leaders can be described as follows. First divide localties at some border x: i.e., party L taes localties with population in ( 1; x) and party R taes localties with population in (x; 1). Then, parties L and R can create F (x)c and (1 F (x))c districts with their own interest, respectively. The leftover district is given to one of the parties depending on F (x) F (x)c R (1 F (x)) (1 F (x))c. Since each party leader wants to mae their districts as extreme as possible, swapping localties across x does not improve their expected payo s. Among all these Pareto e cient allocations, a natural way to set x may be x = x m. Once it is agreed, each party leader applies the slice-and-mix strategy on her localties. Let K(x) = F (x)c. Supposing that F (x) F (x)c < (1 F (x)) (1 F (x))c, parties L and R get K(x) and K K(x) districts. By applying the same method as in the previous section, let L be such that F ( L ) = +, and 2K K let R be such that F (x) F ( R ) = for = 1; :::; K(x). Similarly, let 2K K L be such that F ( K +1 (K +1) L ) = F (x) +, and let 2K K R be such that 1 F (R ) = K +1 (K +1) + for = K(x)+1; :::; K. We call this bipartisan 2K K policy x-bipartisan-slice-and-mix policy, and the resulting median voter pro le is (x 1 L ; :::; x K(x) K(x)+1 L ; x R ; :::; x K R ), since (x K(x)+1 R ; :::; x K R ) are the K K(x) most extreme right median voter pro le, and (x 1 L ; :::; x K(x) L ) are K(x) most extreme left median voter pro le, with small enough. Thus, this is one of the most polarized district median voter allocation, which is very di erent from partisan gerrymandering median voter allocation which has some more competitive districts. If uncertainty y is small, then there may not be any uncertainty in district elections under bipartisan gerrymandering. Proposition 3. Suppose that gerrymanderer can create districts with complete freedom and that bipartisan gerrymandering taes place with party line x. Then, the x-bipartisan-slice-and-mix policy is an optimal gerrymandering policy. The resulting district median voter allocation is approximately x L ( x R ). 19

5 Gerrymandering with Indivisible Localities (needs to be updated totally) In this section, we will explore how the slice-and-mix" result would be modi ed if we drop "complete freedom in gerrymandering" (and "no uncertainty in district median voter s position" in some cases). We will consider indivisible locality case to accommodate actual geographic elements in redistricting, following the spirits of Owen and Grofman (1988) and Gul and Pesendorfer (2010). In this section, we will largely wor on normal distribution case that justi es the paper in Owen and Grofman (1988). Owen and Grofman (1988) analyzed the optimal partisan gerrymandering policy by imposing a constraint such that the sum of x s, P K =1 x (), must stay constant for all. They obtained the famous pac-and-crac result when the party leader maximizes the number of seats under this constraint. In our indivisible locality case, if each locality has normally distributed voters, the feasibility constraint becomes exactly this constraint (the proof is obvious by noting that the median is equivalent to the mean under normality). Lemma 2. Suppose that the voter distribution in each locality is normal distributed, i.e., F` N(`; `) for each ` 2L. Then, the median of district is x () = 1 X `: n `2D () This lemma automatically implies that P K =1 x () must stay constant, since P K P =1 x () = 1 nk `2L `: Therefore, under the normal distribution assumption, swapping the sets of localities S and T between districts and ~, we have = x ( 0 ) x () = P `2T ` while x ( 0 ) + x ~ ( 0 ) = x () + x ~ (). 5.1 Tractable Cost Function n P `2S ` = x ~ () x ~ ( 0 ); We introduce a convenient special ideology cost function such that both voters and party leaders cost functions have common constant elasticity. 20

Let C(d) = a C d and c(d) = a c d, where > 1, a C > 0, and a c > 0 are party leaders and voters cost parameters, respectively. In this case both party leaders and voters have the same elasticity that is constant. In this case, we have the following convenient formula. Claim. Denote A = A(a C ; a c ) = a C 1 a C 1. Then, we have a c ^(x ; i ) = 1+ 1 + i + 1 1 + x + a c 1 1+ > 0 where = 1 Q Suppose that max A x L ; x R holds to assure Assumption 2. Normalizing L = 1 and R = 1, the winning party leader s utility from district, V ~ i (x ; i ; j ), can be described by ^V ( x A x + 1 1 x if x 1 ) = A x + 1 x 1 if x ; 1 where (i) ^V 0 > 0 and (ii) ^V 00 R 0, R 2. When = 2, the formula becomes ^V ( x ) = 4A x Since A is a constant, a parameter is the only relevant one to decide the curvature of payo function. This convenient formula provides a clear insight in the gerrymandering stage. Also, one can see the importance of political competition on por-barrel dimension. Suppose por-barrel promise is forbidden in stage 2, one can see that the only equilibrium is Downsian equilibrium. Therefore, the winning payo is Q C(j i x j) which is always a concave function in x. However, by considering the por-barrel promise, the party has extra spending saving bene t when x move closer to extreme which maes our payo function di erent from the previous one in the literature. Also we use party leaders and voters have cost functions with constraint elasticity denoted by, although we did not need to assume any additional condition on cost functions of party leaders and voters in the complete freedom case. This result (ii) may require some explanation. The readers may be puzzled by having concave ^V function under 1 < < 2, since cost functions C( j ; x ) and C( i ; x ) are always strictly convex in x. The reason for (ii) to happen is that ^V subtracts a strictly convex function from another strictly 21

convex function, and such a function can be anything in general. If the cost function is a constant elasticity type, however, we happen to be able to say that ^V is strictly concave or strictly convex depending on R 2. Let the party leaders and voters cost functions be C(d) = a C d and c(d) = a c d, where > 1, a C > 0, and a c > 0, and let us normalize L and R at 1 and 1, respectively. We assume Q 1 max A x L ; x R to assure interior solutions. We also assume that random variable y follows uniform distribution: i.e., g(y) = 1 for all y 2 [ y; y]. This uniform distribution buys even more 2y mileage. Suppose that x is the ex ante median voter s position. Then, party R s expected payo is Z y E V ~ R = 1 x + y + 1 1 x y dy 2y x 1 h = x + y + 1 +1 + 1 x y i +1 y 2y ( + 1) x Thus, depending on the value of x, functional form of V ~ R changes: 8 h E V ~ < 1 x + y + 1 +1 R 2y(+1) + 1 x y i +1 2 = h : x + y + 1 +1 x y + 1 +1 + 1 x y +1 1 2y(+1) if x 1 x + y +1 i if x Note that E ~ V R is convex if x < y irrespective of the value of, though it can be concave or convex depending on? 2, if x > y. [Figure 2 here] Since V ~ R is convex if x > 1 = R, as long as y is large enough, E V ~ R is convex. Thus, cracing is not a good strategy unlie in Owen and Grofman (1988). The di erence between the current paper and theirs is that our party leaders are also policy-motivated. In order to characterize the optimal partisan gerrymandering policy under our normality assumption, it is useful to order localities by their means. P `2D ` Let ` < `0 mean ` 0`. Then, since x = = 1 hold, if n party L cannot win all P districts, it is bene cial to pac party R supporting localities. Let x = 1 n n `=n( 1)+1 ` < 0 for all = 1; 2; :::; K. Let us call the allocation (x ) K =1 a pac-and-slice gerrymandering policy. This policy pac all opponents in ex ante unwinable districts, and slices supporting voters 22

from the most extreme to moderate in order. The opponent party s ex ante winning localties are also sliced as long as y is large enough so that the party has some chance to win. This involves districts with voter group who are even more extreme than the party leader. This is because with convex ^V, the opponent party s cost to attract the median voter of such a district is so high, and the winning party s leader does not need to pay much in the porbarrel politics. Thus, we have the following characterization of the optimal partisan gerrymandering policy. Proposition 4. Suppose that party leaders and voters have constant elasticity cost functions, and that the voter distribution is normal in each locality. In addition, suppose that y > 1 or > 2 holds. Then, the optimal partisan gerrymandering policy is pac-and-slice with (x ) K =1. How about bipartisan gerrymandering? both parties want slicing anyway. The result is the same, since Proposition 5. Suppose that party leaders and voters have constant elasticity cost functions, and that the voter distribution is normal in each locality. In addition, suppose that y > 1 or > 2 holds. Then, the optimal bipartisan gerrymandering policy is pac-and-slice with (x ) K =1. How about the case with y < 1 and < 2? By the same logic above, it is easy to see that the optimal policy involves some pac-and-crac, but it is hard to characterize, since (i) extreme districts face pac-and-slice anyway, and (ii) the optimal number of winning districts is hard to characterize. For the second point, see the following example. Example 1. Suppose < 2. There are 20 SL-type localities in which the voter distribution is normal and the mean voter is 1 (Strong Left supporter), 5 W R-type localities with mean voter 0:1 (Wea Right supporter), and 5 SR type localities with mean voter 1 (Strong Right supporter). Each district must include 5 localities and there are 6 districts. If the gerrymandering party is L, the optimal districting plan must group the SR type localities together. The gerrymanderer must decide between grouping 4 W R localities with 1 SL in ve winning districts or grouping 5 SL localities in four winning districts. Then the decision is to win 4 districts with x = 1 or to win 5 districts with x = 0:78. The payo di erence between these two plan is D = A[4 2 5 ((1:78) (:22) )] 23