# How Cooperation and Competition Arise in Regional Climate Policies: RICE as
a Dynamic Game

Yijun Chen  Australian Center for Field Robotics, School of Aerospace,
Mechanical and Mechatronic Engineering, The University of Sydney, NSW,
Australia, email: {yijun.chen<EMAIL_ADDRESS>Guodong Shi
Australian Center for Field Robotics, School of Aerospace, Mechanical and
Mechatronic Engineering, The University of Sydney, NSW, Australia, email:
{yijun.chen<EMAIL_ADDRESS>
###### Abstract

One of the most widely used models for studying the geographical economics of
climate change is the Regional Integrated model of Climate and the Economy
(RICE). In this paper, we investigate how cooperation and competition arise in
regional climate policies under the RICE framework from the standpoints of
game theory and optimal control. First, we show that the RICE model is
inherently a dynamic game. Second, we study both cooperative and non-
cooperative solutions to this RICE dynamic game. In cooperative settings, we
investigate the global social welfare equilibrium that maximizes the weighted
and cumulative social welfare across regions. We next divide the regions into
two clusters: developed and developing, and look at the social welfare
frontier under the notion of Pareto optimality. We also present a receding
horizon approach to approximate the global social welfare equilibrium for
robustness and computational efficiency. For non-cooperative settings, we
study best-response dynamics and open-loop Nash equilibrium of the RICE game.
A Recursive Best-response Algorithm for Dynamic Games (RBA-DG) is proposed to
describe the sequences of best-response decisions for dynamic games, which
indicates convergence to open-loop Nash equilibrium when applied to the RICE
game by numerical studies. We also study online receding horizon feedback
decisions of the RICE game. A Receding Horizon Feedback Algorithm for Dynamic
Games (RHFA-DG) is proposed. All these proposed solution concepts are
implemented and open sourced using the latest updated parameters and data. The
results reveal how game theory may be used to facilitate international
negotiations towards consensus on regional climate-change mitigation policies,
as well as how cooperative and competitive regional relations shape climate
change for our future.

##  1 Introduction

The issue of global warming has emerged as a central international
environmental question over decades. As a consequence of industrialization and
economic development, human-caused emissions of greenhouse gases, most notably
carbon dioxide ($\rm CO_{2}$), contribute to a significant increase in the
mean atmospheric temperature by $1.15^{\circ}$C relative to the pre-industrial
age [35]. This temperature deviation yields significant changes in the global
climate and ecosystem, including increasing larger wildfires [3], sea level
rise [15], melting of ice lands [40], and so on. To assess the damages of
anthropogenic greenhouse gases, especially $\rm CO_{2}$, on society,
scientists in this field employ the Integrated Assessment Models (IAMs) [23,
2, 32]. IAMs simulate the dynamics of the economy-climate interactions by
incorporating mathematical models from both economics and geophysical science.
The Dynamic Integrated model of Climate and Economy (DICE) [32, 30], developed
by William Nordhaus, who received the $2018$ Nobel Memorial Prize in Economic
Sciences in large part for this body of work, is one of the most well-known
IAMs. The DICE model is a globally aggregated model and treats global warming
as a single-agent problem. Considering the crucial aspect of regional socio-
economic heterogeneity, Nordhaus also proposed the Regional Integrated model
of Climate and the Economy (RICE) [33, 28], which is a decentralized version
of the DICE model. By dividing the world into several regions, the RICE model
takes the vantage point in determining how multiple regions may jointly design
climate policies and cope with the global warming issue.

Global society has been making efforts in developing sensible strategies to
achieve international consensus on climate-change mitigation over the last
several decades. There are several important international agreements on
climate change. In $1992$, the United Nations Framework Convention on Climate
Change (UNFCC) was the first formal global treaty to be signed by $154$ (now
$198$) parties to explicitly address climate change, which established an
annual forum for international discussions aimed at stabilizing greenhouse gas
concentrations in the atmosphere [5]. These meetings produced the well-known
Kyoto Protocol [6], Copenhagen Accord [38], and Paris Agreement [41]. The
Kyoto Protocol was a binding agreement that was signed in $1997$ and expired
in $2012$. The Kyoto Protocol called for developed countries to reduce
emissions by an average of $5$ percent below pre-industrial levels, and
established a system to monitor countries’ progress [6]. The meeting in
Copenhagen in $2009$ was called to establish a replacement for the Kyoto
Protocol. Although it failed to establish binding emission limits after
$2012$, countries recognized “the scientiﬁc view that the increase in global
temperature should be below $2^{\circ}$C” [38]. The Paris Agreement was signed
in $2016$ and called for all countries to set emissions-reduction
pledges/targets, with the goals of preventing the mean atmospheric temperature
from rising $2^{\circ}$C above pre-industrial levels and pursuing efforts to
keep it below $1.5^{\circ}$C [41]. Despite intensified diplomacy in these
meetings, most existing climate-change treaties are neither sufficient nor
mandatory, some of which even stalled due to a lack of political will [34,
37]. Due to economic competition and political divide, an international
enforceable agreement on specific emission-reduction control has not yet been
reached.

There are two coupled but conflicting sides in regional emission reduction
policies: Regions are affected by the same global climate system; they also
decide their climate strategies to benefit their individual economic benefits
and political self-interests. Therefore, it is actually a decision-making
process where competition occurs. Game theory has been a fundamental tool in
explaining decision-making of independent and self-interested players in a
strategic and competitive setting [45]. In a game, each player takes its
action to maximize its payoff function. The payoff function of each player
relies on not only its own action, but also other players’ actions, resulting
in inherent competitions. A common solution concept in game theory is Nash
equilibrium under which no one will benefit by changing its action when others
remain unchanged [22]. In the RICE model, the regional climate strategies are
associated with a common global climate dynamic, and as a result, the climate
policy decisions fall into the concept of dynamic games established from the
interface between game theory and optimal control theory [4]. In a dynamic
game, the strategic interaction among players recurs over time. The group of
players is associated with a dynamical game state that depends on all players’
actions. The goal is for each player to take an action to maximize each
player’s cumulative payoff function over time that depends on the game state,
its own action, and other players’ actions.

In this paper, we investigate how cooperation and competition in climate
policy across the various regions in the globe under the RICE framework affect
the formation of international climate treaties, the implementation of
regional climate-change mitigation measures, and the resulting implications of
competitive/non-cooperative decisions for climate change. In control
community, there are a few efforts on studying the climate-change mitigation
measures. The work of [25] provided a tutorial introduction to the DICE model
and proposed a receding horizon approach to DICE. A bi-objective optimal
control problem (OCP) on DICE was studied, objectives of which are maximizing
social welfare and minimizing atmospheric temperature deviation [20]. The work
of [8] studied a multi-objective stochastic OCP on DICE, which accounts for
stochastic disturbances and aligns with physical targets posed by
international agreements on climate change mitigation.

The contributions of this work are summarized as follows. We show that the
RICE model is inherently a dynamic game, termed the RICE game. Both
cooperative and non-cooperative solutions to this RICE game are considered:

  * •

For cooperative solutions, we study global social welfare maximization problem
where all regions take actions that maximize the weighted and cumulative
social welfare across all regions, which serves as a benchmark. Next, we
classify regions into two clusters of developed regions and developing
regions, and consider the concept of Pareto equilibrium, describing that any
attempt to benefit one cluster by deviating to some outcome will necessarily
result in a loss in satisfaction of the other cluster. Our simulation results
show that under different Pareto equilibria, the social welfare of developed
regions and developing regions would not drastically change, and the
atmospheric temperature deviation is quite robust. Finally, we apply a
receding horizon approach to approximate the solution to the global social
welfare maximization problem. The receding horizon control presents a
favorable approximation property.

  * •

For non-cooperative solutions, we study best-response dynamics and open-loop
Nash equilibrium of the RICE game. Multiple plays/episodes of dynamic games
are considered, where each player chooses the sequence of control for the next
episode that maximizes its social welfare given other players’ sequences of
control in the current episode.A Recursive Best-response Algorithm for Dynamic
Games (RBA-DG) is proposed. By applying it to the RICE game, simulation
results show that the obtained sequence of actions converges to a steady
point, indicating that RBA-DG is useful for computing the open-loop Nash
equilibrium of the RICE game. We also study online receding horizon feedback
decisions of the RICE game. A single play of dynamic games is considered,
where players observe the current game state and other players’ actions, and
then apply a receding-horizon feedback decision-making approach to predict the
action in the future. A Receding Horizon Feedback Algorithm for Dynamic Games
(RHFA-DG) is proposed. By implementing it over the RICE game, simulation
results show that with the nature of competition, receding horizon, and myopic
assumption about other players’ actions, the emission-reduction rates become
lower in most time steps.

The implementations of our studies are based on the RICE-2010 model with a few
parameters updated from the latest updated data. There have been a few
implementations in the literature on DICE and RICE [31, 18, 19, 24, 1]. As for
the DICE model, Nordhaus provided a GAMS implementation of the DICE-$2013R$
model with the objective of solving a social welfare maximization problem [32,
31]. Faulwasser, Kellett and Weller published a Matlab implementation of the
DICE-$2013R$ model, which replicated the basic functionality of GAMS
implementation [24]. Further, Faulwasser, Kellett and Weller released a Matlab
implementation of the MPC-DICE model which uses model predictive control to
approximately solve the DICE OCP. As for the RICE model, Nordhaus provided an
Excel implementation (without optimization module), and a GAMS implementation
(which is currently inaccessible) [31]. Anthoff and Errickson created an
implementation in Julia programming language [1], which re-coded the Excel
version of RICE without optimization module [1]. Our implementations of
various solution concepts for RICE under the proposed dynamic-game perspective
are developed relying on these previous efforts. We have open-sourced our
implementation as a RICE-GAME framework, with a Matlab and Casadi-based
implementation of RICE dynamic game, Preprint at arXiv.org, code for download
at [12].

The paper is organized as follows. Section 2 provides preliminaries of the
DICE/RICE model and dynamic games. Section 3 represents the RICE model as a
dynamic game. Section 4 considers cooperative solutions for the RICE game
under three cooperative settings: global social welfare maximization, RICE
Pareto frontier, and receding-horizon global social welfare maximization. Two
non-cooperative settings are then considered. Best-response dynamics and open-
loop Nash equilibrium for the RICE game are studied in Section 5. A receding
horizon feedback planning approach for RICE game is proposed in Section 6. The
paper ends with concluding remarks in Section 7.

##  2 Preliminaries

In this section, we introduce some preliminary knowledge on the DICE/RICE
model and dynamic games.

###  2.1 The DICE Model

The DICE model [32] is one of integrated assessment models (IAMs) that
simulate the interplay between economy and climate, and quantify the social
cost of $\rm CO_{2}$ emissions. The DICE model operates in periods of $5$
years and its latest version starts from the year $2015$ as the initial year.
The DICE model is composed of two sectors (see Fig. 1): a geophysical sector
(blue dotted block) that accounts for the global interaction between carbon
and temperature, and an economic sector that is globally aggregated for the
world total (red dotted block).

Figure 1: A block diagram for the DICE model.

Geophysical and Economic Sectors. In the geophysical sector, the DICE model
considers ${\rm CO}_{2}$ emissions as the major contributor to climate change.
The geophysical sector is constructed as follows.

  * •

There are two main sources of $\rm CO_{2}$ emissions: industrial $\rm CO_{2}$
emissions related to the carbon intensity (denoted by $\sigma$) of global
economic activities and natural $\rm CO_{2}$ emissions due to land use
changes, $E^{\rm land}$. The global $\rm CO_{2}$ emissions as the sum of
industrial and natural emissions drive the carbon cycle of the Earth.

  * •

The Carbon dynamics are described by a three-reservoir model [36] on the
carbon flows among three reservoirs: the atmosphere, the upper oceans and
biosphere, and the deep oceans. The average carbon masses in those reservoirs
are represented by $M^{\rm AT}$, $M^{\rm UP}$, and $M^{\rm LO}$, respectively.

  * •

Accumulations of $\rm CO_{2}$ emissions and other greenhouse gases warm the
Earth’s surface through enhanced radiative forcing. Radiative forcing resulted
by $\rm CO_{2}$ emissions has a logarithmic dependence on the atmospheric
carbon mass; greenhouse gases other than $\rm CO_{2}$ emissions contribute to
exogenous radiative forcing $F^{\rm EX}$.

  * •

The rise in temperature at the Earth’s surface is driven by radiative forcing.
Temperature dynamics are captured by a two-layer model [42]. Given the
temperature in year $1750$ as zero reference, $T^{\rm AT}$ and $T^{\rm LO}$
represent the temperature deviation in the atmosphere and in the lower ocean
from those of the reference year, respectively.

The economic sector of DICE is based on the Cobb-Douglas production function
[14], where gross economic output is determined by total factor productivity
$A$, labor $L$, and capital $K$. The total factor productivity and labor
evolve exogenously; the capital dynamics follows the Solow-Swan model [44]
where capital depreciates over time and is replenished by investment.

Climate-Economy Feedback. The DICE model establishes two feedback loops
between the geophysical sector and the economic sector: (i) the industrial
${\rm CO}_{2}$ emissions are a by-product of economic activities; (ii) the
atmospheric temperature rise has an negative impact on economic production.

Control Inputs. The DICE model assumes two control decisions: the saving rate
$s$ and the emission-reduction rate $\mu$. The saving rate $s$ represents the
ratio of investment to the economic output; the emission-reduction rate $\mu$
represents the rate at which industrial $\rm CO_{2}$ emissions are reduced. By
adjusting the saving rate, it is possible to balance consumption today and
consumption in the future. By increasing the emission-reduction rate to slow
down ${\rm CO}_{2}$ emissions as a “climate investment”, the currently
available amount of consumption and investment will be reduced [32]. This
climate investment will lower climate damage and therefore potentially
increase consumption in the future.

System Outputs. The economic output is counted as output net of emission
abatement cost and climate damage. Social welfare is calculated as the
discounted sum of the population-weighted utility of per capita consumption.

DICE Variables. All variables described above are time-dependent, although not
explicitly written. The variables $T^{\rm AT},T^{\rm LO},M^{\rm AT},M^{\rm
UP},M^{\rm LO}$ and $F^{\rm EX}$ belong to the geophysical sector, whereas the
variables $K,A,L,\sigma,$ and $E^{\rm land}$ belong to the economic sector.
Some variables evolve independently, whereas others evolve in an
interdependent manner. The variables that evolve independently as exogenous
signals are $F^{\rm EX},A,L,\sigma,$ and $E^{\rm land}$; the variables
evolving in an interdependent manner are $T^{\rm AT},T^{\rm LO},M^{\rm
AT},M^{\rm UP},M^{\rm LO},$ and $K.$

###  2.2 The RICE Model

The RICE model is a variant of the DICE model that accounts for regional
climate damages and control decisions [28, 29]. Since being proposed in 1990s,
calibration of the RICE model has been updated several times, and the latest
version of the RICE model is the RICE-2011 model on which our study is based.
The RICE-2011 model uses a time-step of $10$ years, starting from the year
2005 as the initial year.

The RICE-2011 model integrates a global geophysical sector with regional
economic sectors (see Fig. 2):

  * •

The global geophysical sector of the RICE-2011 model contains the same carbon
dynamics and temperature dynamics as the DICE model;

  * •

The regional economic sectors of the RICE-2011 model disaggregates the world
into $12$ regions (US, EU, Japan, Russia, Non-Russian Eurasia, China, India,
Middle East, Africa, Latin America, other high income countries, and other
Asian countries), each of which is equipped with region-specific climate
damage level, economic factors, and saving rate and emission-reduction rate as
local control inputs.

Figure 2: The workflow of the RICE-2011 model.

RICE Variables. The variables $T^{\rm AT},T^{\rm LO},M^{\rm AT},M^{\rm UP},$
$M^{\rm LO},$ and $F^{\rm EX}$ in the global geophysical sector of the
RICE-2011 model are inherited from the DICE model, while the variables
$K_{i},A_{i},L_{i},\sigma_{i},$ and $E^{\rm land}_{i},i\in\\{1,2,\dots,12\\},$
in the regional economic sectors of the RICE-2011 model correspondingly depend
on specific regions.

###  2.3 Dynamic Games

The theory of dynamic games lies in the interface between game theory and
optimal control, which involves a dynamic decision process for multiple
players [4]. An $n$-player discrete-time dynamic game over a finite horizon is
defined as follows.

Dynamic Game. The $n$ players are indexed in
${\mathcal{V}}:=\\{1,2,\dots,n\\}$; time is discrete with the steps indexed in
${\mathcal{T}}:=\\{0,1,\dots,T\\}$. Each player can manipulate the game
through its control decisions, and the control decision space of player
$i\in{\mathcal{V}}$ is denoted by ${\mathcal{U}}_{i}\subseteq\mathbb{R}^{d}$.
At each time step $t=0,\dots,T$, the decision executed by player $i$ is
denoted by ${\mathbf{u}}_{i}(t)\in{\mathcal{U}}_{i}$. We also use
${\mathbf{u}}(t)=[{\mathbf{u}}_{1}^{\top}(t);\dots;{\mathbf{u}}_{n}^{\top}(t)]$,
${\mathbf{U}}_{i}=[{\mathbf{u}}_{i}^{\top}(0);\dots;{\mathbf{u}}_{i}^{\top}(T)]$
and ${\mathbf{U}}=[{\mathbf{U}}_{1};\dots;{\mathbf{U}}_{n}]$ to represent the
all-player decision profile at time $t$, the player-$i$ decision throughout
the time horizon, and the decision profile for all players and for all time
stpes. The control decisions of all players excluding player $i$ at time step
$t$ is denoted by ${\mathbf{u}}_{-i}(t)$, and the control decisions of all
players excluding player $i$ over the entire horizon is represented by
${\mathbf{U}}_{-i}$.

For each $t=0,1,\dots,T$, the group of players are associated with a dynamical
state ${\mathbf{x}}(t)\subseteq\mathbb{R}^{m}$ that evolves according to

${\mathbf{x}}(t+1)={\mathbf{f}}(t,{\mathbf{x}}(t),{\mathbf{u}}(t)),\quad{\mathbf{x}}(0)={\mathbf{x}}_{0},\quad
t=0,\dots,T,$ (1)

with ${\mathbf{x}}_{0}$ being the initial state. At each time $t=0,\dots,T$,
upon playing ${\mathbf{u}}_{i}(t)$, the agent $i$ receives a payoff
$g_{i}({\mathbf{x}}(t),{\mathbf{u}}_{i}(t),{\mathbf{u}}_{-i}(t))\in\mathbb{R}$
given other players’ actions ${\mathbf{u}}_{-i}(t)$ and the current state
${\mathbf{x}}(t)$, where
$g_{i}({\mathbf{x}}(t),{\mathbf{u}}_{i}(t),{\mathbf{u}}_{-i}(t))$ is a
continuous function with respect to ${\mathbf{x}}(t)$, ${\mathbf{u}}_{i}(t)$,
and ${\mathbf{u}}_{-i}(t)$. The cumulative payoff of agent $i$ throughout the
time horizon is therefore

$\mathsf{J}_{i}({\mathbf{X}},{\mathbf{U}}_{i},{\mathbf{U}}_{-i})=\sum_{t=0}^{T}g_{i}({\mathbf{x}}(t),{\mathbf{u}}_{i}(t),{\mathbf{u}}_{-i}(t))$
(2)

where
${\mathbf{X}}=({\mathbf{x}}^{\top}(0),\dots,{\mathbf{x}}^{\top}(T))^{\top}$.
Each player’s goal is to make decisions for maximizing its cumulative payoff
function; the system dynamics produces a terminal state ${\mathbf{x}}(T+1)$
towards the end of the time horizon as a result of those decisions.

In what follows, we present cooperative and non-cooperative settings as well
as three solution concepts for discrete-time dynamic game: Pareto solution,
open loop Nash equilibrium (NE) and feedback Nash equilibrium.

####  2.3.1 Cooperative Decisions: Pareto Solutions

In the cooperative setting, players are able to communicate and cooperate with
each other to achieve their objectives, and all players know the system
dynamics and payoff functions of other players. Pareto optimality is an
efficiency concept [26] under which any attempt to benefit one player by
deviating to some other outcome will necessarily result in a loss in
satisfaction of another player.

######  Definition 1 (Pareto Efficiency)

A decision profile ${\mathbf{U}}^{p}$ is Pareto efficient for the dynamic game
if there does not exist another decision profile ${\mathbf{U}}$ such that

  * (i)

there holds for all $i\in\mathcal{V}$
$\mathsf{J}_{i}({\mathbf{X}}^{p},{\mathbf{U}}_{i}^{p},{\mathbf{U}}_{-i}^{p})\leq\mathsf{J}_{i}({\mathbf{X}},{\mathbf{U}}_{i},{\mathbf{U}}_{-i})$;

  * (ii)

there exists at least one $k\in\mathcal{V}$ such that
$\mathsf{J}_{k}({\mathbf{X}}^{p},{\mathbf{U}}_{k}^{p},{\mathbf{U}}_{-k}^{p})<\mathsf{J}_{k}({\mathbf{X}},{\mathbf{U}}_{k},{\mathbf{U}}_{-k})$

Here ${\mathbf{X}}$ and ${\mathbf{X}}^{p}$ are the states evolved under
${\mathbf{U}}$ and ${\mathbf{U}}^{p}$, respectively. The set of all Pareto
solution is called the Pareto frontier.

The following Lemma provides a convenient way of computing Pareto solutions
[17, Lemma 6.1].

###### Lemma 1

Consider a set of parameters $\mathcal{C}:=\\{\mathbf{c}=(c_{1},\dots,c_{n}):\
c_{i}\geq 0,\text{\rm and }\sum_{i=1}^{n}c_{i}=1\\}.$ If a decision profile
${\mathbf{U}}^{p}$ is such that

${\mathbf{U}}^{p}\in\arg\max_{{\mathbf{U}}}\Big{\\{}\sum_{i=1}^{n}c_{i}\mathsf{J}_{i}\Big{\\}},$
(3)

for some $\mathbf{c}\in\mathcal{C}$, then ${\mathbf{U}}^{p}$ is Pareto
efficient.

####  2.3.2 Non-Cooperative Setting: Nash Equilibriums

In the non-cooperative setting of games, Nash equilibriums (NE) mark one of
the most important solution concepts. For a dynamic game, the information
structure in terms of what players know before a decision is made at a
particular time becomes critical in properly defining NE. There are two basic
types of information structure: the open loop information structure and the
feedback information structure.

In the open loop information structure, each player knows the initial state
${\mathbf{x}}_{0}$, and then plans at $t=0$ all the control decisions
${\mathbf{u}}_{i}(t)$ for $t\in{\mathcal{T}}$. Consequently, the open loop
control decision of ${\mathbf{u}}_{i}(t)$ can be written as

${\mathbf{u}}_{i}(t)={\mathbf{u}}_{i}(t,{\mathbf{x}}_{0}).$ (4)

Denote an open loop decision profile by ${\mathbf{U}}^{\ast}$ where
${\mathbf{U}}_{i}^{\ast}:=[{\mathbf{u}}^{\ast}_{i}(0,{\mathbf{x}}_{0});\dots;{\mathbf{u}}^{\ast}_{i}(T,{\mathbf{x}}_{0})],i\in{\mathcal{V}}$.
We introduce the following definition. Here with slight abuse of notation we
also write $\mathsf{J}_{i}({\mathbf{X}},{\mathbf{U}}_{i},{\mathbf{U}}_{-i})$
as $\mathsf{J}_{i}({\mathbf{x}}_{0},{\mathbf{U}}_{i},{\mathbf{U}}_{-i})$
noting the fact that ${\mathbf{X}}$ is uniquely determined by
${\mathbf{x}}_{0}$ and $({\mathbf{U}}_{i},{\mathbf{U}}_{-i})$.

######  Definition 2 (Open Loop NE)

Given the initial state ${\mathbf{x}}_{0}$, a control decision profile
${\mathbf{U}}^{\ast}$ is said to be an open loop Nash equilibrium control
decision profile if there holds for all $i\in{\mathcal{V}}$ and all
${\mathbf{U}}_{i}$ that

$\mathsf{J}_{i}({\mathbf{x}}_{0},{\mathbf{U}}_{i}^{\ast},{\mathbf{U}}_{-i}^{\ast})\geq\mathsf{J}_{i}({\mathbf{x}}_{0},{\mathbf{U}}_{i},{\mathbf{U}}_{-i}^{\ast}).$
(5)

In the feedback information structure, at time step $t\in{\mathcal{T}}$, each
player knows the current system state ${\mathbf{x}}(t)$. As a consequence,
each player $i\in{\mathcal{V}}$ may employ a feedback law
${\bm{\pi}}_{i}:{\mathcal{T}}\times{\mathcal{X}}\to{\mathcal{U}}_{i}$ to
determine its decisions by

${\mathbf{u}}_{i}(t)={\bm{\pi}}_{i}(t,{\mathbf{x}}(t)).$ (6)

In this case, we denote the overall feedback law as
${\bm{\pi}}=[{\bm{\pi}}_{1};\dots;{\bm{\pi}}_{n}]$, and then write
$\mathsf{J}_{i}({\mathbf{X}},{\mathbf{U}}_{i},{\mathbf{U}}_{-i})$ as
$\mathsf{J}_{i}({\mathbf{x}}_{0},{\bm{\pi}}_{i},{\bm{\pi}}_{-i})$ noting
$({\mathbf{X}},{\mathbf{U}}_{i},{\mathbf{U}}_{-i})$ is uniquely determined by
${\mathbf{x}}_{0}$ and ${\bm{\pi}}$. We introduce the following definition.

######  Definition 3 (Feedback NE)

For any initial state ${\mathbf{x}}_{0}$, a feedback profile
${\bm{\pi}}^{\ast}$ is said to be a feedback Nash equilibrium for the dynamic
game if there holds for all $i\in\mathcal{V}$ and all ${\bm{\pi}}_{i}$ that

$\mathsf{J}_{i}({\mathbf{x}}_{0},{\bm{\pi}}^{\ast}_{i},{\bm{\pi}}^{\ast}_{-i})\geq\mathsf{J}_{i}({\mathbf{x}}_{0},{\bm{\pi}}_{i},{\bm{\pi}}^{\ast}_{-i}).$
(7)

##  3 RICE as a Dynamic Game

In this section, we show that the RICE model is inherently a dynamic game
where regional saving rates and emission-reduction rates regulate global
temperature, and then the global temperature has impact on regional social
welfares through climate damage. Our presentation is based on the RICE-2011
model with slight modifications, but the nature of being a dynamic game is
embedded in all RICE models.

There are $12$ regions in the RICE-2011 model. Each region is considered a
player and the regions are indexed in ${\mathcal{V}}=\\{1,2,\dots,n\\}$ with
$n=12$. We operate the RICE dynamic game in periods of $5$ years, starting
from the year 2020 as the initial year111The RICE-2011 model operates in
periods of $10$ years starting from 2005.. Taking the discrete time step index
${\mathcal{T}}=\\{0,1,\dots,T\\}$, the relation between an actual calendar
year and the corresponding discrete time step is determined by

$year(t)=year(0)+5t,\quad year(0)=2020.$ (8)

Note that although most variables in the RICE-2011 model are defined as flows
per year and only some variables are in flows per decade [29, Supplementary
Material], all variables in this present of the RICE dynamic game are defined
as flows per year.

###  3.1 System Dynamics

We define the dynamical state of the RICE model at time step
$t\in{\mathcal{T}}$ as

$\displaystyle{\mathbf{x}}(t)=[T^{\rm AT}(t);T^{\rm LO}(t);M^{\rm
AT}(t);M^{\rm UP}(t);M^{\rm
LO}(t);K_{1}(t);\dots;K_{n}(t)]\in\mathbb{R}^{n+5}.$ (9)

Let the control decision of region $i\in{\mathcal{V}}$ at time step
$t\in{\mathcal{T}}$ be

${\mathbf{u}}_{i}(t)=[s_{i}(t);\mu_{i}(t)]^{\top}:=[{\mathbf{u}}_{i[1]}(t);{\mathbf{u}}_{i[2]}(t)]^{\top}\in[0,1]^{2}.$
(10)

Consequently, the control decisions of the RICE dynamic game at time step
$t\in{\mathcal{T}}$ of all players are

${\mathbf{u}}(t)=[s_{1}(t);\mu_{1}(t);\dots;s_{n}(t);\mu_{n}(t)]\in[0,1]^{24}.$
(11)

According to the RICE model, the dynamics of ${\mathbf{x}}(t)$ can be written
as

${\mathbf{x}}(t+1)={\mathbf{f}}(t,{\mathbf{x}}(t),{\mathbf{u}}(t)),\quad{\mathbf{x}}(0)={\mathbf{x}}_{0},\quad
t\in{\mathcal{T}},$ (12)

where ${\mathbf{f}}:=[f_{1};f_{2};\dots;f_{n+5}]^{\top}$ follows from the
interdependency among the geophysical signals and the economic signals, and
the feedback between the two sectors. In what follows, we briefly describe the
form of entries in the dynamics ${\mathbf{f}}$. For a more detailed
description, please refer to [29].

Carbon dynamics. There are three carbon reservoirs: the atmosphere, the upper
oceans and the biosphere, and the deep oceans. The atmospheric carbon
reservoir has an additional input, the global ${\rm CO}_{2}$ emissions $E(t)$
that is related to economic activities and land use at time $t$. The carbon
dynamics for carbon transition among the three reservoirs are described by

$\begin{bmatrix}M^{\rm AT}(t+1)\\\ M^{\rm UP}(t+1)\\\ M^{\rm
LO}(t+1)\end{bmatrix}=\begin{bmatrix}\zeta_{11}&\zeta_{12}&0\\\
\zeta_{21}&\zeta_{22}&\zeta_{23}\\\ 0&\zeta_{32}&\zeta_{33}\\\
\end{bmatrix}\begin{bmatrix}M^{\rm AT}(t)\\\ M^{\rm UP}(t)\\\ M^{\rm
LO}(t)\end{bmatrix}+\begin{bmatrix}\xi_{1}\\\ 0\\\ 0\end{bmatrix}E(t),$ (13)

where
$\zeta_{11},\zeta_{12},\zeta_{21},\zeta_{22},\zeta_{23},\zeta_{32},\zeta_{33}$
and $\xi_{1}$ are constant parameters.

Temperature dynamics. The evolution of the atmospheric and ocean temperature
is governed by the following equations:

$\begin{bmatrix}T^{\rm AT}(t+1)\\\ T^{\rm
LO}(t+1)\end{bmatrix}=\begin{bmatrix}\phi_{11}&\phi_{12}\\\
\phi_{21}&\phi_{22}\end{bmatrix}\begin{bmatrix}T^{\rm AT}(t)\\\ T^{\rm
LO}(t)\end{bmatrix}+\begin{bmatrix}\xi_{2}\\\ 0\end{bmatrix}F(t),$ (14)

where $\phi_{11},\phi_{12},\phi_{21},\phi_{22}$ and $\xi_{2}$ are constant
parameters, and the radiative forcing at time step $t$, $F(t)$, is computed as

$F(t)=\eta\log_{2}(\frac{M^{\rm AT}(t)}{M^{\rm AT,1750}})+F^{\rm EX}(t).$ (15)

Here $\eta$ and $M^{\rm AT,1750}$ are constants, and $F^{\rm EX}(t)$
represents radiative forcing due to other greenhouse gases at time step $t$.

Economic dynamics. The economy of each region $i\in{\mathcal{V}}$ at time step
$t\in{\mathcal{T}}$ follows from the Cobb-Douglas production function [14]:

$Y_{i}(t)=A_{i}(t)K_{i}(t)^{\gamma_{i}}(t)L_{i}(t)^{1-\gamma_{i}},$ (16)

where $Y_{i}(t),A_{i}(t),K_{i}(t)$ and $L_{i}(t)$ represent region $i$’s gross
economic output, total factor productivity, capital stock and labor at time
step $t$, respectively. Here, $\gamma_{i},i\in{\mathcal{V}},$ are constant
parameters.

Economy-climate feedback. Global ${\rm CO}_{2}$ emissions at time step $t$ are
the sum of natural emissions because of each region’s land use at time step
$t$, $E^{\rm land}_{i}(t)$, and industrial emissions resulted from each
region’s economic activities at time step $t$. Each region $i$’s industrial
emissions depend on each region $i$’s carbon intensity at time step $t$,
$\sigma_{i}(t)$, which is an exogenous variable. Consequently, global ${\rm
CO}_{2}$ emissions at time step $t$ is described by

$E(t)=\sum_{i=1}^{n}\Big{(}\sigma_{i}(t)(1-\mu_{i}(t))Y_{i}(t)+E^{\rm
land}_{i}(t)\Big{)},$ (17)

where $\mu_{i}(t),i\in{\mathcal{V}},t\in{\mathcal{T}},$ are control decisions
representing the emission-reduction rate.

The emission abatement cost fraction as the percentage of gross economic
output spent on emission-reduction effort at time step $t$ is given by

$\Lambda_{i}(t)=1-\theta_{i}^{[1]}(t)\mu_{i}(t)^{\theta_{i}^{[2]}},$ (18)

where $\theta_{i}^{[2]},$ and
$\theta_{i}^{[1]}(t),i\in{\mathcal{V}},t\in{\mathcal{T}},$ are parameters. The
parameters $\theta_{i}^{[1]}(t),i\in{\mathcal{V}},t\in{\mathcal{T}},$ are
calculated by

$\theta_{i}^{[1]}(t)=\frac{pb_{i}}{1000\cdot\theta_{i}^{[2]}}(1-\delta^{pb}_{i})^{t-1}\cdot\sigma_{i}(t),$
(19)

where $pb_{i},i\in{\mathcal{V}},$ represent the price of backstop technology
at time step $t=0$ for region $i$ to replace all carbon fuels, and
$\delta^{pb}_{i},i\in{\mathcal{V}},$ are constant parameters. The damage
function $\Omega_{i}(t)$ is the percentage of gross economic output damaged by
temperature rising. As a result, the net economic output $Q_{i}(t)$ (economic
output after the emission-reduction spending and climate damage), is given by

$Q_{i}(t)=\Omega_{i}(t)\Lambda_{i}(t)Y_{i}(t).$ (20)

The Solow-Swan model [44] gives a description of capital accumulation of each
region $i\in{\mathcal{V}}$:

$K_{i}(t+1)=(1-\delta^{\rm K}_{i})^{5}K_{i}(t)+5s_{i}(t)Q_{i}(t),$ (21)

where $\delta^{\rm K}_{i},i\in{\mathcal{V}},$ are constant parameters, and
$s_{i}(t),i\in{\mathcal{V}},t\in{\mathcal{T}},$ are control decisions
representing the saving rate, i.e., the percentage of net economic output
invested in capital.

###  3.2 Damage Functions

The rising atmospheric temperature has a negative impact on economic
production. Although there are various specifications offering various
estimates of the damage function, there are no substantial discrepancies among
them [7]. In the RICE-2011 model, damages to economic gross output caused by
rising temperatures are considered to be region-specific and dependent on
factors such as atmospheric temperature deviation, sea level rises, and
atmospheric carbon mass. However, the form of the damage function of the
RICE-2011 model is not given explicitly. In the current study, a simplified
form of the damage function is employed, which only depends on the rising
atmospheric temperature deviation:

$\Omega_{i}(t)=1-a_{i}^{[1]}T^{\rm AT}(t)-a_{i}^{[2]}T^{\rm
AT}(t)^{a_{i}^{[3]}},$ (22)

where $a_{i}^{[1]},a_{i}^{[2]},$ and $a_{i}^{[3]},i\in{\mathcal{V}},$ are
parameters calibrated to yield a certain amount of damage loss to regional
economic gross output. Each region’s damage loss to its regional economic
gross output at $2^{\circ}C$ is presented in Table 1. For example, India would
suffer a damage loss of $1.55\%$ to its economic gross output when the
atmospheric temperature deviation reaches $2^{\circ}C$.

| US | EU | Japan | Russia | Eurasia | China
---|---|---|---|---|---|---
Loss | 0.56% | 0.64% | 0.65% | 0.46% | 0.52% | 0.66%
| India | MidEast | Africa | LatAm | OHI | OthAsia
Loss | 1.55% | 1.19% | 1.47% | 0.66% | 0.62% | 1.04%
Table 1: Each region’s damage loss with respect to its gross economic output
when atmospheric deviation is at $2^{\circ}C.$

###  3.3 Payoff Functions

Note that from the net economic output $Q_{i}(t)$, a total amount of
$s_{i}(t)Q_{i}(t)$ has been made as investment. The remaining part
$C_{i}(t)=(1-s_{i}(t))Q_{i}(t)$ can then be used for consumption. In RICE
models, for region $i$, the social welfare of the population $L_{i}(t)$
consuming $C_{i}(t)$ of economic output at time $t$ is defined by the
population-weighted utility of per capita consumption

$g_{i}(C_{i}(t),L_{i}(t))=L_{i}(t)\cdot\frac{(\frac{C_{i}(t)}{L_{i}(t)})^{1-\alpha_{i}}-1}{1-\alpha_{i}},$
(23)

where $\alpha_{i}$ is a constant. The cumulative social welfare of region $i$
across the time horizon is then given by

$\displaystyle\mathsf{J}_{i}$
$\displaystyle=\sum_{t=0}^{T}\frac{g_{i}(C_{i}(t),L_{i}(t))}{(1+\rho_{i})^{5t}}$
$\displaystyle=\sum_{t=0}^{T}\Bigg{(}\Big{[}\frac{A_{i}(t)L_{i}(t)^{1+\alpha_{i}-\gamma_{i}}}{(1-\alpha_{i})(1+\rho_{i})^{5t}}\Big{(}1-{\mathbf{u}}_{i[2]}(t)\Big{)}\Big{(}1-a_{i}^{[1]}{\mathbf{x}}_{1}(t)-a_{i}^{[2]}{\mathbf{x}}_{1}(t)^{a_{i}^{[3]}}\Big{)}\Big{(}1-\theta_{i}^{[1]}(t){\mathbf{u}}_{i[1]}(t)^{\theta_{i}^{[2]}}\Big{)}{\mathbf{x}}_{5+i}(t)^{\gamma_{i}}\Big{]}$
$\displaystyle\ \ \ -\frac{L_{i}(t)}{(1-\alpha_{i})(1+\rho_{i})^{5t}}\Bigg{)}$
$\displaystyle:=\mathsf{J}_{i}({\mathbf{X}},{\mathbf{U}}_{i},{\mathbf{U}}_{-i})$
(24)

where $\rho_{i}$ is a constant discounting factor. For each region
$i\in{\mathcal{V}}$, naturally it will attempt to maximize its cumulative
social welfare.

We have now formally represented the RICE-2011 model as a dynamic game, termed
the RICE game where the regions as players seek to plan their control
decisions in emission-reduction rate and saving rate for the entire time
horizon (600 years)
${\mathbf{U}}_{i}=[s_{i}(0);\mu_{i}(0);\dots;s_{i}(T);\mu_{i}(T)]$ so as to
maximize their payoff functions (3.3) subject to the underlying dynamical
system (12), represented in (14)-(21). In what follows when we implement the
proposed RICE game, the values of the initial state ${\mathbf{x}}(0)$ and the
parameters are updated in the following way: the initial state are calibrated
to match the data in the year $2020$; the parameters in the geophysical sector
use the latest updated values in the DICE-2016 model [30], while the
parameters in the regional economic sector remain unchanged.

###  3.4 The Social Cost of $\rm CO_{2}$

The social cost of $\rm CO_{2}$ (SCC) is a central concept for understanding
and implementing climate change policies. In a market setting like a cap-and-
trade regime, the SCC would serve as the trading price of carbon emission
permits. In a carbon-tax regime, the SCC would be the carbon tax for the
regions who want to emit carbon emissions. The SCC in a particular year is
defined as the decrease in aggregate consumption in that year that would
change the current expected value of social welfare by the same amount as a
one unit increase in carbon emissions in that year [27]. The regional SCC is
then given by

$\displaystyle{\rm SCC}_{i}(t)$
$\displaystyle=-1000\cdot\frac{\partial\mathsf{J}_{i}}{\partial
E_{i}(t)}\Big{/}\frac{\partial\mathsf{J}_{i}}{\partial C_{i}(t)}$
$\displaystyle=-1000\cdot\frac{\partial C_{i}(t)}{\partial E_{i}(t)}.$ (25)

##  4 The RICE Game: Cooperative Solutions

In this section, we study the solutions to the RICE game under cooperative
settings. First of all, we revisit the classical RICE solution concept defined
by a system-level social welfare maximization, and present numerical solutions
under the current calibration of climate damage function and the updated
geophysical and economic parameters and conditions. Next, we move to Pareto
solutions to the proposed RICE game, and present the Pareto frontier of the
RICE game between developing and developed regions. Finally, we introduce a
receding horizon solution to the classical RICE solution as the counter-part
for MPC-DICE developed in [18, 19].

###  4.1 RICE Social Welfare Maximization

The RICE-2011 model focused on the sum of the weighed regional social welfare
across all regions:

$\mathsf{W}_{\mathbf{c}}=\sum_{i=1}^{n}c_{i}\mathsf{J}_{i},$ (26)

where $0<c_{i}<1$ is know as the Negishi weight for region $i$ with
$\sum_{i=1}^{n}=1$. The values of the $c_{i}$ were calibrated in the work of
[29] to be the inverse of the marginal utilities of consumption when
maximizing (26) subject to RICE dynamics (13)-(21) under the assumption that
there is no abatement of ${\rm CO_{2}}$ emissions [33, 32, 43].

####  4.1.1 Solution Concept

One benchmark cooperative solution to the RICE game is for a centralized
climate policy planner to compute the
${\mathbf{U}}_{1},\dots,{\mathbf{U}}_{n}$ for all regions that achieve the
maximal value of $\mathsf{W}_{\mathbf{c}}$, for a given initial condition
$\mathbf{x}_{0}$.

###### Definition 4

A decision profile ${\mathbf{U}}^{\rm w}$ is a global social welfare
equilibrium if it is a solution to the following optimization problem

$\displaystyle\max_{{\mathbf{U}}_{1},\cdots,{\mathbf{U}}_{n}}$
$\displaystyle\mathsf{W}_{\mathbf{c}}({\mathbf{X}},{\mathbf{U}})$ (27)
$\displaystyle{\rm subject\ to}$
$\displaystyle{\mathbf{x}}(t+1)={\mathbf{f}}(t,{\mathbf{x}}(t),{\mathbf{u}}(t)),\
\ {\mathbf{x}}(0)={\mathbf{x}}_{0},t\in{\mathcal{T}}$
$\displaystyle{\mathbf{u}}(t)\in[0,1]^{24},\ \ t\in{\mathcal{T}}.$

####  4.1.2 Results

We run a total of $120$ $5-$year periods, and therefore we set $T=120$. We
obtain the global social welfare equilibrium ${\mathbf{U}}^{\rm w}$ for the
RICE game under Definition 4 by solving the corresponding OCP (27).

First, each region’s optimal control decisions in emission-reduction rate and
saving rate under ${\mathbf{U}}^{\rm w}$ are plotted in Fig 3. Clearly,
regions including China, Africa, India, Eurasia, OthAsia, and Russia have
relatively high emission-reduction rates who should reach carbon neutral
status (i.e., the emission-reduction rate $\mu_{i}(t)$ becoming exactly $1$)
relatively sooner. The reason for that could be twofolds. First, regions such
as Africa and India bear the greatest damage loss caused by rising atmospheric
temperature as presented in Table 1. Second, it takes regions such as China,
Eurasia and Russia comparatively lower cost to reduce carbon emissions (see
the estimated price for backstop technology replacing all carbon fuels in the
year $2020$ in Table 2).

(a) Optimal emission-reduction rate.

(b) Optimal saving rate.

Figure 3: Each region’s optimal emission-reduction rate and saving rate under ${\mathbf{U}}^{\rm w}$. | US | EU | Japan | Russia | Eurasia | China
---|---|---|---|---|---|---
Price | 1051 | 1635 | 1635 | 701 | 701 | 817
| India | MidEast | Africa | LatAm | OHI | OthAsia
Price | 1284 | 1167 | 1284 | 1518 | 1284 | 1401
Table 2: The estimated price for backstop technology in each region that can
replace all carbon fuels in the year $2020.$

Then, in Fig. 4, we examine the trajectory of the atmospheric temperature
deviation when the emission-reduction rates and saving rates under
${\mathbf{U}}^{\rm w}$ are taken. The work of [29] also solved the OCP (27)
with the same horizon length ($600$ years) starting from the year of $2005$.
Although we are inaccessible to the exact and complete data of results in [29,
Fig. 3], the atmospheric temperature deviation trajectory in Fig. 4 appears to
be similar to that in [29]. To be specific, the atmospheric temperature
deviation in [29, Fig. 3] is around $2.8^{\circ}C$, whereas ours is
approximately $3^{\circ}C$.

Finally, in Fig. 5, the social cost of ${\rm CO}_{2}$ in each region under the
global social welfare equilibrium is presented. It shows that the social cost
of ${\rm CO}_{2}$ under ${\mathbf{U}}^{\rm w}$ in India, Africa and OthAsia is
significantly higher than those in other regions. The reason for that could be
that both prices for backstop technology and optimal emission-reduction rates
for these regions are higher than those for other regions.

Figure 4: The atmospheric temperature deviation trajectory under
${\mathbf{U}}^{\rm w}$. Figure 5: The social cost of ${\rm CO}_{2}$ in each
region under ${\mathbf{U}}^{\rm w}$.

###  4.2 RICE Pareto Frontier

Noting the global social welfare equilibrium is a solution concept where the
Negishi weights $c_{i},i=1,\dots,n$ are calibrated in a centralized manner.
Since the weight $c_{i}$ may greatly impact the optimal emission rates and
saving rates for region $i$ and then its own social cost of carbon, the
regions have intrinsic incentives in negotiating on the distribution of the
$c_{i}$’s. In recent years’ international climate policy forums, the divide
between developed and developing regions has been one of the main barriers
from a global consensus on carbon emission rates [38, 16, 11]. In this
subsection, we focus on the social welfare Pareto frontier between developed
and developing regions involved in the RICE game.

####  4.2.1 Solution Concept: Pareto Frontier

We classify the regions in the RICE-2011 model into two clusters of regions:
developed regions (US, EU, Japan and other high income countries) and
developing regions (Russia, Non-Russian Eurasia, China, India, Middle East,
Africa, Latin America and other Asian countries). We denote
${\mathcal{V}}_{\rm developed}=\\{1,2,3,11\\}$ and ${\mathcal{V}}_{\rm
developing}=\\{4,5,6,7,8,9,10,12\\}$. Correspondingly, the social welfare of
the two clusters of regions is defined as, respectively,

$\mathsf{W}_{\rm developed}=\sum_{i\in{\mathcal{V}}_{\rm
developed}}\mathsf{J}_{i},\ \ \mathsf{W}_{\rm
developing}=\sum_{i\in{\mathcal{V}}_{\rm developing}}\mathsf{J}_{i}.$

For the considered RICE game over these two clusters of regions, we consider
the Pareto optimality.

###### Definition 5

For the RICE game under developed and developing clusters of regions, a
decision profile ${\mathbf{U}}^{\rm p}$ is a Pareto social welfare equilibrium
between the developed and developing clusters if there does not exist another
decision profile ${\mathbf{U}}$ such that

  * (i)

there hold

$\displaystyle\mathsf{W}_{\rm developed}({\mathbf{X}}^{\rm
p},{\mathbf{U}}_{i}^{\rm p},{\mathbf{U}}_{-i}^{\rm p})$
$\displaystyle\leq\mathsf{W}_{\rm
developed}({\mathbf{X}},{\mathbf{U}}_{i},{\mathbf{U}}_{-i}),$
$\displaystyle\mathsf{W}_{\rm developing}({\mathbf{X}}^{\rm
p},{\mathbf{U}}_{i}^{\rm p},{\mathbf{U}}_{-i}^{\rm p})$
$\displaystyle\leq\mathsf{W}_{\rm
developing}({\mathbf{X}},{\mathbf{U}}_{i},{\mathbf{U}}_{-i});$

  * (ii)

either one of the above two inequalities holds strictly.

Here ${\mathbf{X}}$ and ${\mathbf{X}}^{\rm p}$ are the states evolved under
${\mathbf{U}}$ and ${\mathbf{U}}^{p}$, respectively.

Based on Lemma 1, we can calculate such Pareto social welfare frontier between
the developed and developing clusters by solving the family of optimization
problems for a given initial condition $\mathbf{x}_{0}$:

$\displaystyle\max_{{\mathbf{U}}_{1},\cdots,{\mathbf{U}}_{n}}$ $\displaystyle
p\cdot\mathsf{W}_{developed}+(1-p)\cdot\mathsf{W}_{developing}$ (28)
$\displaystyle{\rm subject\ to}$
$\displaystyle{\mathbf{x}}(t+1)={\mathbf{f}}(t,{\mathbf{x}}(t),{\mathbf{u}}(t)),\
\ {\mathbf{x}}(0)={\mathbf{x}}_{0},t\in{\mathcal{T}}$
$\displaystyle{\mathbf{u}}(t)\in[0,1]^{24},\ \ t\in{\mathcal{T}}$

where $p$ is selected in the interval $[0,1]$. For any fixed $p\in[0,1]$, we
obtain a Pareto social welfare equilibrium, and their collection forms the
Pareto frontier between the developed and developing clusters. The Pareto
formulation might have the potential to serve as a benchmark for the
interchange of positions between developed regions and developing regions on
climate policies. The parameter $p$ serves as a quantitative characterization
to the allocation of responsibility for climate change mitigation between
developed regions and developing regions: If $p$ is close to $0$, developed
regions will take higher responsibility, as the developed regions will if $p$
is close to $1$.

####  4.2.2 Results

We set a total of $120$ $5$-year periods. We take $999$ linearly spaced values
between $0.001$ and $0.999$ as the values of $p$. For each $p$, we obtain the
Pareto solution ${\mathbf{U}}^{p}$ by solving the respective OCP (28). We plot
the social welfare Pareto frontier between developed and developing regions in
Fig. 6. We also plot the atmospheric temperature deviation at the final time
step, $T^{\rm AT}(120)$, versus the parameter $p$ in Fig. 7.

Figure 6: Social welfare Pareto frontier between developed regions and
developing regions ($\times 10^{4}$ trillion USD). Figure 7: The atmospheric
temperature deviation in the year $2620$, $T^{\rm AT}(120)$, versus the
parameter $p$.

These results reveal a few notable effects.

  * •

From Fig. 6, it can be seen that the values of $p$ can not drastically change
the social welfare received for both the developing and developed clusters of
regions. The maximal ($p=0.999$) and minimal ($p=0.001$) values for
$\mathsf{W}_{\rm developed}$ differs only by $0.35\%$; the maximal ($p=0.001$)
and minimal ($p=0.999$) values for $\mathsf{W}_{\rm developing}$ differs only
by $0.56\%$.

  * •

Fig 7 shows that the atmospheric temperature deviation at the final time step
is quite robust with respect to $p$ under the Pareto equilibrium. In fact, for
all values of $p$ between $0.001$ and $0.999$, $T^{\rm AT}(120)$, the
atmospheric temperature deviation in the year of $2620$, always falls between
$2.8^{\circ}C$ and $3.3^{\circ}C$. It is also clear that a higher or a lower
$p$ will both result in a lower atmospheric temperature deviation in the final
year, while $p=0.862$ leads to highest terminal atmospheric temperature
deviation, i.e., global warming in the worst case.

###  4.3 Receding Horizon RICE

The (27) is an OCP for a nonlinear dynamical system with non-convex cost
function. When the horizon length gets larger, it becomes more difficult to
solve the problem numerically. The work of [25] established a novel receding
horizon solution to DICE, which provides robustness and computational
efficiency compared to solving DICE in a long time horizon directly. In what
follows, we extend the idea of [25] to RICE.

####  4.3.1 Solution Concept: MPC-RICE

For the receding horizon approach, we denote the prediction horizon by
$T_{rh}$ and the simulation horizon by $T_{sim}$. We introduce

$\displaystyle
l(t,{\mathbf{x}}(t),{\mathbf{u}}(t)):=\sum_{i=1}^{n}c_{i}\cdot\frac{g_{i}(C_{i}(t),L_{i}(t))}{(1+\rho_{i})^{5t}},$

and assume a full measurement or estimate of the state ${\mathbf{x}}(t)$ is
available at each time step
$t\in{\mathcal{T}}_{sim}:=\\{0,1,\dots,T_{sim}\\}$. The receding horizon
process to approximate the global social welfare equilibrium
${\mathbf{U}}^{\rm w}$ is proposed in Algorithm 1. A decision profile
${\mathbf{U}}^{\rm rhw}=[{\mathbf{u}}^{\rm
rhw}(0)^{\top},\dots,{\mathbf{u}}^{\rm rhw}(T_{sim})^{\top}]^{\top}$ as output
of Algorithm 1 is said to be a receding horizon global social welfare
equilibrium.

Algorithm 1 MPC-RICE

Input: simulation horizon $T_{sim}$; prediction horizon $T_{rh}$.

1:$t\leftarrow 0$

2:while $t\leq T_{sim}$ do

3: observe ${\mathbf{x}}(t)$

4: compute the optimal solution
${{\mathbf{u}}^{\ast}(s),s\in\mathcal{S}:=\\{t,t+1,\dots,t+T_{rh}\\}},$ to the
following optimization problem over the receding horizon $\mathcal{S}$
$\displaystyle\max_{\mathbf{u}(s),\forall s\in\mathcal{S}}\quad$
$\displaystyle\sum_{s=t}^{t+T_{rh}}l(s,{\mathbf{x}}(s),{\mathbf{u}}(s))$ (29)
$\displaystyle{\rm subject\ to}$
$\displaystyle{\mathbf{x}}(s+1)={\mathbf{f}}(s,{\mathbf{x}}(s),{\mathbf{u}}(s)),\
\ s\in\mathcal{S},$ $\displaystyle{\mathbf{u}}(s)\in[0,1]^{24},\ \
s\in\mathcal{S}.$

5: apply ${\mathbf{u}}^{\rm rhw}(t):={\mathbf{u}}^{\ast}(t)$ to RICE game

6:end while

7:return ${\mathbf{U}}^{\rm rhw}$

####  4.3.2 Results

We set the simulation horizon to be $T_{sim}=120$ (600 years), and implement
MPC-RICE under prediction horizons $T_{rh}\in\\{10,20,60\\}$. We also
reproduce MPC-DICE with a simulation horizon of $600$ years under prediction
horizons $T_{rh}\in\\{10,20,60\\}$.

In Fig. 8, we plot trajectories of the optimal emission-reduction rates under
global social welfare equilibrium ${\mathbf{U}}^{\rm w}$ and the optimal
control decision ${\mathbf{U}}^{\rm rhw}$ obtained by MPC-RICE with
$T_{rh}\in\\{10,20,60\\}$. When the prediction horizon gets larger, the
optimal emission-reduction rates under ${\mathbf{U}}^{\rm w}$ converge towards
those under ${\mathbf{U}}^{\rm w}$ for most time steps. Moreover, the optimal
emission-reduction rates under ${\mathbf{U}}^{\rm rhw}$ are more steady in the
sense that they do not drop back to lower levels towards the end of the
simulation horizon, which is the case under ${\mathbf{U}}^{\rm w}$. Compared
with [18], MPC-RICE and MPC-DICE present similar approximation feature under
large prediction horizons.

In Fig. 9, we plot trajectories of atmospheric temperature deviation for the
entire simulation horizon under the optimal solutions from DICE OCP, MPC-DICE
with $T_{rh}=60$, ${\mathbf{U}}^{\rm w}$, and ${\mathbf{U}}^{\rm rhw}$ with
$T_{rh}=60$. The trajectories of atmospheric temperature deviation under DICE
OCP and MPC-DICE with $T_{rh}=60$ are higher than under ${\mathbf{U}}^{\rm w}$
and ${\mathbf{U}}^{\rm rhw}$ with $T_{rh}=60$ in most time steps except at the
beginning. To this extent, the RICE model and MPC-RICE raise more concerns
about global warming.

Figure 8: The comparison of each region’s optimal emission-reduction rates
under ${\mathbf{U}}^{\rm w}$ and ${\mathbf{U}}^{\rm rhw}$ with different
prediction horizons $T_{rh}\in\\{10,20,60\\}$. Figure 9: The comparison of
atmospheric temperature deviation for the entire simulation horizon under the
optimal solutions from DICE OCP, MPC-DICE with $T_{rh}=60$, ${\mathbf{U}}^{\rm
w}$, and ${\mathbf{U}}^{\rm rhw}$ with $T_{rh}=60$.

##  5 RICE Game: Best-response Dynamics and Open-Loop NE

In this section, we study the best-response dynamics and open-loop Nash
equilibrium of the RICE game.

###  5.1 Best-response Recursions for Dynamic Games

In game theory, best-response dynamics is a classical model that describes how
players strategically behave in repeated plays [39]. In standard best-response
episodes, each player takes a best-response action for the next round that
maximizes its payoff function given other players’ actions in the present
round of play. For certain types of games such as two-player zero-sum games
and potential games, the player actions under best-response dynamics may
converge to a Nash equilibrium [21, 13].

In this subsection, we establish the best-response recursions for the dynamic
game introduced in Section 2.3 with $n$ players over a finite horizon $T$. We
assume the dynamic game is repeatedly and recursively played for $N$ episodes,
where each episode consists of $T$ time steps. We thereby define the
aggregated control decisions of all players in episode $k=1,\cdots,N$ by
${\mathbf{U}}^{(k)}$, and the decisions of player $i$ in episode $k$ by
${\mathbf{U}}_{i}^{(k)}$. Similarly, the decisions of players excluding player
$i$ in episode $k$ is denoted by ${\mathbf{U}}_{-i}^{(k)}$. In view of the
best-response for static games [39], the best-response recursion of the agents
in the dynamic game over the $N$ episodes are described in the following
Recursive Best-response Algorithm for Dynamic Games (RBA-DG) as in Algorithm
2. The RBA-DG produces an output ${\mathbf{U}}^{\rm
NE}_{\ast}={\mathbf{U}}^{(N)}$ after $N$ episodes of updates.

Algorithm 2 Recursive Best-response Algorithm for Dynamic Games (RBA-DG)

Input: Episodes $N$; $c_{i},i\in{\mathcal{V}}$ .

1:compute an optimal cooperative solution ${\mathbf{U}}^{\rm c}$ by the
following problem
$\displaystyle\max_{{\mathbf{U}}_{1},\cdots,{\mathbf{U}}_{n}}$
$\displaystyle\sum_{i\in{\mathcal{V}}}c_{i}\cdot
J_{i}({\mathbf{X}},{\mathbf{U}}_{i},{\mathbf{U}}_{-i})$ $\displaystyle{\rm
subject\ to}$
$\displaystyle{\mathbf{x}}(t+1)={\mathbf{f}}(t,{\mathbf{x}}(t),{\mathbf{u}}(t)),\
\ {\mathbf{x}}(0)={\mathbf{x}}_{0},t\in{\mathcal{T}}$
$\displaystyle{\mathbf{u}}(t)\in[0,1]^{24},\ \ t\in{\mathcal{T}}.$

2:let ${\mathbf{U}}^{(0)}_{i}={\mathbf{U}}^{\rm c}_{i},\forall
i\in{\mathcal{V}}$

3:$k\leftarrow 0$

4:while k < N do

5: for each player $i\in{\mathcal{V}}$ do

6: observe ${\mathbf{U}}^{(k)}_{-i}$

7: compute ${\mathbf{U}}^{(k+1)}_{i}$ by solving the problem
$\displaystyle\max_{{\mathbf{U}}_{i}}$
$\displaystyle\quad\mathsf{J}_{i}({\mathbf{X}},{\mathbf{U}}_{i},{\mathbf{U}}_{-i}),$
(30a) $\displaystyle s.t.$
$\displaystyle\quad{\mathbf{x}}(t+1)={\mathbf{f}}(t,{\mathbf{x}}(t),{\mathbf{u}}(t)),$
(30b) $\displaystyle\quad{\mathbf{U}}_{-i}={\mathbf{U}}^{(k)}_{-i},$ (30c)
$\displaystyle\quad{\mathbf{x}}(0)={\mathbf{x}}_{0}.$ (30d)

8: end for

9:
${\mathbf{U}}^{(k+1)}=[{\mathbf{U}}^{(k+1)}_{1};\cdots;{\mathbf{U}}^{(k+1)}_{n}]$

10: $k\leftarrow k+1$

11:end while

12:return ${\mathbf{U}}^{\rm NE}_{\ast}={\mathbf{U}}^{(N)}$

The implementation of RBA-DG requires two conditions. First, the initial value
of the system ${\mathbf{x}}(0)={\mathbf{x}}_{0}$ needs to be known by all
players at the beginning of the process. Second, at the end of each episode
$k=1,\cdots,N$, every player should be able to observe or know all other
players’ decision sequences over the episode ${\mathbf{U}}_{-i}^{(k)}$. Then
the update of the player decisions for the next episode follows directly from
the best-response dynamics. We present the following result, which holds true
immediately from definition of open-loop Nash equilibrium.

###### Proposition 1

Consider the repeatedly played $n$-player dynamic game. Let $N=\infty$ in the
RBA-DG. Suppose there exists ${\mathbf{U}}^{\ast}$ such that there holds
$\lim_{k\to\infty}{\mathbf{U}}^{(k)}={\mathbf{U}}^{\ast}.$ Then
${\mathbf{U}}^{\ast}$ is an open-loop Nash equilibrium for the dynamic game.

###  5.2 Recursive Best-response for RICE

The RICE model describes the interplay of the various regions in the world in
terms of their climate-change mitigation policy over a few centuries (for
example, RICE-2011 assumes a total time horizon of $600$ years [29,
Supplementary Material]). As a result, despite the fact that the RBA-DG may be
conceptually applied to RICE since RICE has been identified as a dynamic game,
it is important to clarify the real-world implication of recursive best
response for RICE.

####  5.2.1 Solution Concept: Regional Climate Policy Negotiations

We propose to adopt RBA-DG over RICE as a mechanism for regional climate
policy negotiations. The overall negotiations take a prescribed $N$ episodes.
In each round of the negotiations, regions accept RICE as the standing model
for climate-economy integration, and decide their emission reduction rates and
saving rates for a fixed time horizon (e.g., 200 years). At the end of each
round, all regions reveal their current planning of the emission reduction
rates and saving rates for the entire time horizon to other regions. Then,
during the next round of negotiations, regions get to revise their planned
decisions and adopt RBA-DG as their principle of updating such planned
decisions. After the $N$ episodes of negotiations, if all regions realize none
of them can unilaterally change their climate actions and gain significant
increase in social welfare, such a mechanism will produce an approximate open-
loop Nash equilibrium for the RICE game in view of Proposition 1. Such a Nash
equilibrium holds higher promise of being accepted by all regions since no
region is able to benefit from a revised decision when all other regions take
the actions from the equilibrium.

####  5.2.2 Results

Now we implement the RBA-DG over the RICE game. We set the number of episodes
to be $N=21$.

Convergence. In Fig. 10, we plot the trajectory of
$\|{\mathbf{U}}^{(k+1)}-{\mathbf{U}}^{(k)}\|$ versus episode $k$. The result
shows that when applying RBA-DG over the RICE game, the obtained sequence of
${\mathbf{U}}^{(k)}$ converges to a steady point, and after $5$ episodes, the
$\|{\mathbf{U}}^{(k+1)}-{\mathbf{U}}^{(k)}\|$ has become very close to zero.
From Proposition 1, this implies that the RBA-DG may also serve as an
efficient algorithm for computing the open-loop Nash equilibrium of the RICE
game.

Cooperation vs Competition. In Fig. 11, we plot the comparison of each
region’s optimal emission-reduction rates under global social welfare
equilibrium ${\mathbf{U}}^{\rm w}$ (cooperative solution) and the optimal
control decision ${\mathbf{U}}^{\rm NE}_{\ast}$ (competitive solution)
obtained by RBA-DG (Algorithm 2). From Fig. 11, it is not surprising that each
region’s optimal emission-reduction rates under ${\mathbf{U}}^{\rm NE}_{\ast}$
are significantly lower than those under ${\mathbf{U}}^{\rm w}$. This is a
direct reflection that under Nash equilibrium of the RICE game, the regions
are working towards maximizing their own social welfare instead of a
collective social welfare of all regions. As a result, each region has
incentives to reduce its emission-reduction rate and its emission abatement
cost, and thus improve its social welfare. These results partially show the
rationale behind regions leaving signed climate treaties, e.g., Canada opt-out
from Kyoto Protocol [34] in $2012$, and the United States formally quit Paris
Agreement [37] in $2020$ because a less aggressive regional policy leads to
higher economic benefit even facing climate change damages.

In Fig. 12, we plot the comparison of the atmospheric temperature deviation
trajectories under ${\mathbf{U}}^{\rm w}$ and ${\mathbf{U}}^{(\rm
NE)}_{\ast}$. From the results, with competition, substantially higher
atmospheric temperature deviation occurs under ${\mathbf{U}}^{(\rm
NE)}_{\ast}$, as a consequence of lower emission-reduction rates. For example,
in $2620$, the atmospheric temperature deviation resulted from RBA-DG is
around $6^{\circ}$C while the atmospheric temperature deviation under global
social welfare equilibrium is about $3^{\circ}$C.

Figure 10: Convergence of $\|{\mathbf{U}}^{(k+1)}-{\mathbf{U}}^{(k)}\|$ versus
episodes for RBA-DG.

.

Figure 11: The comparison of each region’s optimal emission-reduction rates
under ${\mathbf{U}}^{\rm w}$ and ${\mathbf{U}}^{\rm NE}_{\ast}$. Figure 12:
The comparison of the atmospheric temperature deviation trajectories under
${\mathbf{U}}^{\rm w}$ and ${\mathbf{U}}^{\rm NE}_{\ast}$.

##  6 RICE Game: Receding Horizon Feedback Decisions

In this section, we present a framework for dynamic games where in a single
play over the time horizon, players observe the underlying dynamic process and
other players’ actions, and then apply a receding-horizon feedback decision
making for a prediction horizon. Then we apply this receding horizon feedback
process to RICE game in order to capture the competitive nature of regional
climate policies, and show how such competitions have impact on the global
climate dynamics.

###  6.1 Receding Horizon Feedback Decisions for Dynamic Games

Consider the dynamic game introduced in Section 2.3 with $n$ players over a
finite horizon $T$. The game is played only once, and the players take the
following feedback decision process in the receding horizon sense.

The Receding Horizon Feedback Decision Process. The players apply a receding
horizon approach and compute their feedback decisions $\mathbf{u}_{i}(t)$. At
each time $t=0,\dots,T-1$, each player $i$ observes other players’ played
action $\mathbf{u}_{-i}(t)$ and the system state $\mathbf{x}(t)$. Then, every
player $i$ assumes that $\mathbf{u}_{-i}(t)$ will continue to be played over
$[t+1,t+T_{rh}]$, and therefore decides its best feedback decision plan
$\mathbf{u}_{i|t+1\to t+T_{rh}}^{\rm RHP}(\mathbf{x}(t),\mathbf{u}_{-i}(t))$,
where $\mathbf{u}_{i|t+1\to t+T_{rh}}^{\rm RHP}$ maximizes the cumulative
payoff of player $i$ over the time horizon $[t+1,t+T_{rh}]$ conditioned on
that $\mathbf{u}_{-i}(t)$ be played over $[t+1,t+T_{rh}]$. Finally, each
player $i$ plays the first planned decision $\mathbf{u}_{i|t+1}^{\rm RHP}$ in
$\mathbf{u}_{i|t+1\to t+T_{rh}}^{\rm RHP}$ for the step $t+1$, and the process
moves forward recursively. Denoting $\mathbf{u}_{i}^{\rm RHF}(t)$ as the
actions generated by the receding horizon feedback decision process, clearly
there is an underlying feedback law $\pi_{i}$ such that

$\mathbf{u}_{i}^{\rm RHF}(t)=\pi_{i}(t,\mathbf{x}(t),\mathbf{u}_{-i}^{\rm
RHF}(t)).$

Note that the decisions ${\mathbf{u}}_{i}^{\rm RHF}(t)$ are actually played by
the players at each time $t$. The resulting collective player decisions over
the entire time horizon is written as $\mathbf{U}^{\rm RHF}$. The
computational process of this receding horizon feedback decision framework is
presented in the following Receding Horizon Feedback Algorithm for Dynamic
Games (RHFA-DG).

Algorithm 3 Receding Horizon Feedback Algorithm for Dynamic Games (RHFA-DG)

Input: simulation horizon $T_{sim}$; prediction horizon $T_{rh}$;
$c_{i},i\in{\mathcal{V}}$.

1:compute an optimal cooperative solution ${\mathbf{U}}^{\rm c}$ by the
following problem
$\displaystyle\max_{{\mathbf{U}}_{1},\cdots,{\mathbf{U}}_{n}}$
$\displaystyle\sum_{i\in{\mathcal{V}}}c_{i}\cdot
J_{i}({\mathbf{X}},{\mathbf{U}}_{i},{\mathbf{U}}_{-i})$ $\displaystyle{\rm
subject\ to}$
$\displaystyle{\mathbf{x}}(t+1)={\mathbf{f}}(t,{\mathbf{x}}(t),{\mathbf{u}}(t)),\
\ {\mathbf{x}}(0)={\mathbf{x}}_{0},t\in{\mathcal{T}}$
$\displaystyle{\mathbf{u}}(t)\in[0,1]^{24},\ \ t\in{\mathcal{T}}.$

2:let ${\mathbf{U}}^{\rm RHF}_{i}(0)={\mathbf{U}}^{\rm c}_{i}(0),\forall
i\in{\mathcal{V}}$

3:$t\leftarrow 0$

4:while $t\leq T_{sim}$ do

5: for each player $i\in{\mathcal{V}}$ do

6: Take action ${\mathbf{U}}^{\rm RHF}_{i}(t)$

7: end for

8: for each player $i\in{\mathcal{V}}$ do

9: observe ${\mathbf{x}}(t)$ and ${\mathbf{U}}^{\rm RHF}(t)$

10: compute ${\mathbf{x}}(t+1)$ according to (1)

11: assume all players $j\in{\mathcal{V}}/\\{i\\}$ will continue to play
${\mathbf{U}}^{\rm RHF}_{-i}(t)$ over $[t+1,t+T_{rh}]$

12: compute its optimal solution ${\mathbf{u}}_{i|t+1\to t+T_{rh}}^{\rm RHP}$
to the following receding horizon optimization problem
$\displaystyle\max_{{\mathbf{u}}_{i|t+1\to t+T_{rh}}}\quad$
$\displaystyle\sum_{s=t+1}^{t+T_{rh}}g_{i}(s,{\mathbf{x}}(s),{\mathbf{u}}_{i}(s),{\mathbf{u}}_{-i}(s))$
(31a) $\displaystyle s.t.\quad$
$\displaystyle{\mathbf{x}}(s+1)={\mathbf{f}}(s,{\mathbf{x}}(s),{\mathbf{u}}(s)),$
(31b) $\displaystyle{\mathbf{u}}_{-i}(s)={\mathbf{U}}^{\rm RHF}_{-i}(t),\ \
s\in[t+1,t+T_{rh}].$ (31c)

13: plan ${\mathbf{U}}^{\rm RHF}_{i}(t+1)={\mathbf{u}}_{i|t+1}^{\rm RHP}$

14: end for

15:end while

###  6.2 Receding Horizon Feedback for RICE

In the current climate-change mitigation measures, there has been no
international consensus on the emission reduction rates and the saving rates
for different regions or for the globe collectively, despite the successful
adoption of global or regional climate-change agreements such as the Paris
Agreement. In fact, most climate treaties are neither substantial nor
mandatory [9]. The objective of such treaties has not been the actual
emission-reduction rate, but rather to establish environmental norms at the
international level. The hope is that such international environmental norms
may then be translated into the domestic climate policies according to each
region’s political processes [10].

In the real world, regions revise their climate-change policies from time to
time and attempt to compete with each other while acknowledging the importance
of climate-change mitigation. Indeed, such revisions of regional climate-
change policies may depend on many factors such as the public opinion shifts,
government changes, and new international environmental norms. In the end, the
nature of regional competitions persists as aggressive climate-change
mitigation policy for a region may benefit other regions economically in the
short run, although collectively aggressive climate-change mitigation policy
benefits every region as shown from the comparison between cooperative social
welfare equilibrium and the approximate open-loop Nash equilibrium. Here, we
propose to apply RHFA-DG on RICE as an attempt to model the real-world
regional climate-policy evolution given the economically competitive nature of
such policies.

###  6.3 Results

We implement the RHFA-DG over the RICE game. We set the simulation horizon and
prediction horizon to be $T_{sim}=120$ and $T_{rh}=\\{5,10,20\\}$.

In Fig. 13, we plot the comparison of each region’s optimal emission-reduction
rates under global social welfare equilibrium ${\mathbf{U}}^{\rm w}$, the
approximated open-loop Nash equilibrium ${\mathbf{U}}^{\rm NE}_{\ast}$ solved
from RBA-DG (Algorithm 2), and the receding horizon feedback decisions
${\mathbf{U}}^{\rm RHF}$ solved from RHFA-DG (Algorithm 3) with
$T_{rh}=\\{5,10,20\\}$. First, with competition, the receding horizon feedback
decisions of emission-reduction rates solved from RHFA-DG are significantly
lower than those under global social welfare equilibrium. Second, due to
receding horizon and myopic assumption of other regions’ future decisions, the
emission-reduction rates from RHFA-DG are lower than those obtained from RBA-
DG in the early time steps. However, interestingly, the former continues to
climb in the final time steps, while the latter drops back to lower levels in
the end of simulation horizons.

In Fig. 14, we plot the comparison of the atmospheric temperature deviation
trajectories under ${\mathbf{U}}^{\rm w}$, ${\mathbf{U}}^{\rm NE}_{\ast}$, and
${\mathbf{U}}^{\rm RHF}$ with $T_{rh}=\\{5,10,20\\}$. There are several
observations on the results. First, the atmospheric temperature deviation
under ${\mathbf{U}}^{\rm RHF}$ is substantially higher than that under
${\mathbf{U}}^{\rm w}$, and slightly higher than that under ${\mathbf{U}}^{\rm
NE}_{\ast}$. Second, for ${\mathbf{U}}^{\rm RHF}$ with $T_{rh}=\\{5,10,20\\}$,
the smaller the prediction horizon is, the higher the atmospheric temperature
deviation is. This implies that longer prediction horizon forces the regions
to take the long-term climate damages more into their receding horizon
decisions, and therefore leads to better climate-change mitigation.

Figure 13: The comparison of each region’s optimal emission-reduction rates
under ${\mathbf{U}}^{\rm w}$, ${\mathbf{U}}^{\rm NE}_{\ast}$, and
${\mathbf{U}}^{\rm RHF}$ with $T_{rh}=\\{5,10,20\\}$. Figure 14: The
comparison of the atmospheric temperature deviation trajectories under
${\mathbf{U}}^{\rm w}$, ${\mathbf{U}}^{\rm NE}_{\ast}$, and ${\mathbf{U}}^{\rm
RHF}$ with $T_{rh}=\\{5,10,20\\}$.

##  7 Conclusions

In this paper, we investigated how cooperation and competition arise in
regional climate policies under the RICE framework from the perspective of
game theory and optimal control. The results revealed how game theory may be
used to facilitate international negotiations towards consensus on regional
climate-change mitigation policies, as well as how cooperative and competitive
regional relations shape climate change for our future.

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