Symmetric Nash equilibrium of political polarization in a two-party system

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Symmetric Nash equilibrium of political polarization in

a two-party system

Jonghoon Kim1

Department of Physics, Pukyong National University, Busan 48513, Korea

Hyeong-Chai Jeong

Department of Physics and Astronomy, Sejong University, Seoul 05006, Korea

Seung Ki Baek

Department of Scientiﬁc Computing, Pukyong National University, Busan 48513, Korea

Abstract

The median-voter hypothesis (MVH) predicts convergence of two party plat-

forms across a one-dimensional political spectrum during majoritarian elec-

tions. From the viewpoint of the MVH, an explanation of polarization is

that each election has a diﬀerent median voter so that a party cannot please

all the median voters at the same time. We consider two parties competing

to win voters along a one-dimensional spectrum and assume that each party

nominates one candidate out of two in the primary election, for which the

electorates represent only one side of the whole population. We argue that

all the four candidates will come to the same distance from the median of

the total population through best-response dynamics.

Keywords: Political polarization, Median-voter hypothesis, Best-response

dynamics

Email address: seungki@pknu.ac.kr (Seung Ki Baek)

1Current address: Department of Energy Engineering, Korea Institute of Energy Tech-

nology, Naju 58330, Korea

Preprint submitted to Elsevier October 5, 2022

arXiv:2210.01336v1 [physics.soc-ph] 4 Oct 2022

1. Introduction

Statistical physics deals with macroscopic patterns emerging from micro-

scopic interactions. Collective decision-making is an example of such emer-

gent phenomena arising from individual-level interactions [1, 2, 3, 4]. Broadly

speaking, the group of activities associated with collective decision-making

in society can be referred to as politics, and it is why statistical physicists

have viewed the interplay between individual choices and political changes

within the framework of complex systems [5, 6, 7, 8]. One of widely accepted

political values is democracy, according to which the people have the right

to rule. A democratic government thus has an excellent incentive to meet

the people’s demands, but the price is political instability: the governmen-

tal policies may suddenly change if the power is transferred to the opposing

party by election, and the change can be especially drastic when political

opinions are polarized to a great degree.

Whether political polarization is an inevitable part of democracy is not

clear. As will be explained below, in a two-party system, the median-voter

hypothesis (MVH) predicts that the two parties along a one-dimensional po-

litical spectrum will converge to the median voter’s position through majori-

tarian elections [9]. The assumptions of the MVH might look too restrictive,

but it has been reported in congressional voting that very often the issue re-

duces to a one-dimensional matter [10]. In such a one-dimensional political

landscape, a two-party system is robust, as assumed by the MVH, because

a centrist party cannot easily ﬁnd a political “niche” between two existing

parties [11]. Thus, the question still remains: Why are they so polarized?

It may be attributed to social homophily or echo chambers [12, 13], and

temporary polarization may be consistent with Bayesian updating [14]. Many

of the existing approaches assume that people update their political positions

through mechanistic interaction with neighbors such as homophily, assimila-

tion, and diﬀerentiation [15, 16, 17, 18]. A recent study answers this question

by pointing out that voters are not as rational as assumed in the MVH [19]:

According to this idea, voters will rather satisﬁce than maximize their util-

ity functions. By adapting to such voting behavior with a simple gradient

ascent method, the parties can develop symmetric polarization, maintaining

an equal distance from the median voter.

In this work, we wish to examine an alternative explanation [20], which ar-

gues that the median voter’s position experienced by a party may be diﬀerent

from election to election: For example, a party has to nominate a candidate

before a presidential election. To become the nominee, candidates in the

primary should take opinions from their own supporters seriously, even if the

supporters’ overall position diﬀers from the general public opinion, because

their votes are needed to win the primary. By considering the nomination

process as a part of election, this approach explains permanent polarization

without resorting to cognitive biases and bounded rationality. In addition,

our work provides a testable prediction that polarization will only increase

further if the payoﬀ for the loser of the ﬁnal election also becomes valuable

enough, which has not been proposed by other models.

In the next section, we will formulate our model as a game of two-round

competition among four players, whose strategies are their political positions.

The Nash equilibria will be identiﬁed. We then propose how to update the

players’ positions to reach one of the Nash equilibria. Such dynamic con-

sideration is important in several aspects: First, the existence of plausible

dynamics ensures the feasibility of the discovered equilibrium. Second, the

time scale to reach the equilibrium can be estimated: If it diverges, devi-

ation from the equilibrium can prove more pervasive than predicted by a

static analysis [21, 22]. Third, when multiple equilibria exist, the distribu-

tion of convergence points will generally require dynamical consideration. As

a model of dynamics, one could consider an evolutionary process which works

in a large population of simple-minded agents, but we believe that a variety

of strategic moves in the course of election can be better described by best-

response dynamics among the candidates, according to which each candidate

deliberately seeks a position to be the ﬁnal winner given that the other com-

petitors’ positions are held ﬁxed. The whole dynamics is thus projected onto

four candidates, and the electorate react to the candidates instantaneously

with ﬁxed political positions, which is an assumption that we have borrowed

from the MVH. The key point is that everyone in our model society seeks

the best response to the given conﬁguration, and this is one of common ap-

proaches to model human behavior in evolutionary game theory [23]. In

other words, each player tries to maximize his or her own objective function,

as is not uncommon in physics, but an important diﬀerence of game theory

from physics is that such individual optimization may drive the total system

away from an optimal point.

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SymmetricNashequilibriumofpoliticalpolarizationinatwo-partysystemJonghoonKim1DepartmentofPhysics,PukyongNationalUniversity,Busan48513,KoreaHyeong-ChaiJeongDepartmentofPhysicsandAstronomy,SejongUniversity,Seoul05006,KoreaSeungKiBaekDepartmentofScienticComputing,PukyongNationalUniversity,Busan48513,K...

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