A bisection protocol for political redistricting

The authors conceived of the bisection protocol during a research meeting discussing recent political redistricting literature, in particular, the I-cut-you-freeze protocol preprint. After establishing the theoretical results for the continuous nongeometric setting, they discussed ways to implement both protocols on real-world data, culminating in the Iowa case study and computational experiments with 17 other states. Political redistricting in the United States is the process of drawing congressional and state legislative district boundaries. This work introduces the bisection protocol, a two-player, zero-sum, extensive-form game motivated by political redistricting in which two players alternately divide pieces of a region in half (up to rounding) to obtain a district plan. A subgame perfect Nash equilibrium is presented for the protocol in a relaxed continuous nongeometric setting, and a recurrence is given for the optimal strategies. The bisection protocol is compared with the recently proposed I-cut-you-freeze protocol across a variety of standard fairness metrics. A hardness result is presented for the bisection protocol in the more realistic discrete geometric setting along with exact equilibrium strategies for small grid graphs. Because equilibrium computation is intractable for practical instances, heuristics are developed for both protocols that model players’ drawing strategies as mixed-integer linear programs. When the heuristics are applied to congressional redistricting in Iowa with counties preserved, both protocols produce district plans that are fairer (according to three popular metrics) than Iowa’s 115th congressional districts. Finally, the bisection heuristic is used to generate congressional district plans from census tracts for 18 states, demonstrating its potential for practical use. Funding: This work was supported by National Science Foundation Graduate Research Fellowship Program [Grant DGE-1746047] and the fourth author is supported in part by the Air Force Office of Scientific Research [Grant FA9550-19-1-0106]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2022.0084 .