# DEA Game Cross-Efficiency Model to Urban Public Infrastructure Investment Comprehensive Efficiency of China

Authors: Yu Sun; Huixia Huang; Chi Zhou
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

In managerial application, data envelopment analysis (DEA) is used by numerous studies to evaluate performances and solve the allocation problem. As the problem of infrastructure investment becomes more and more important in Chinese cities, it is of vital necessity to evaluate the investment efficiency and assign the fund. In practice, there are competitions among cities due to the scarcity of investment funds. However, the traditional DEA model is a pure self-evaluation model without considering the impacts of the other decision-making units (DMUs). Even though using the cross-efficiency model can figure out the best multiplier bundle for the unit and other DMUs, the solution is not unique. Therefore, this paper introduces the game theory into DEA cross-efficiency model to evaluate the infrastructure investment efficiency when cities compete with each other. In this paper, we analyze the case involving 30 provincial capital cities of China. And the result shows that the approach can accomplish a unique and efficient solution for each city (DMU) after the investment fund is allocated as an input variable.


---

## Body

## 1. Introduction

China has experienced rapid urbanization since the reform and opening up process began in 1978 [1]. Along with the developmental speed, the national new urbanization planning (2014–2020) predicted that the urbanization rate of China will exceed 60%, and it will exceed 60%–70% population living in cities by the end of 2020. However, the urbanization causes numerous social and environmental problems and leads to a great pressure on constructing the resource-conserving and environment-friendly society. Therefore, it is of great importance for municipalities to improve the use efficiency of resources with full consideration of environmental condition.Under that case, urban public infrastructure plays a fundamental and decisive role in urban development. In order to speed up urban development, each city has invested lots of fund in related public infrastructure. However, the World Bank estimated that the decision-making errors made by China in infrastructure investment were at about 30%, which caused waste of about 400 billion to 500 billion RMB [2]. Hence, it is of vital significance to improve the investment efficiency.The efficiency of urban public infrastructure investment has been a hot topic both in China and worldwide during the past decades. Morais et al. [3] analyzed the fiscal sustainability (FS) of an infrastructure investment project and believed that the infrastructure investment has a great influence on economic growth. Dahlgren and Leung [4] developed a framework for evaluating repeated infrastructure investments and used it to analyze the critical investment cost. Kemmerling and Stephan [5] studied the political institution in the regional allocation of public infrastructure investments in France, Germany, Italy, and Spain. Zheng et al. [6] examined determinants of regional allocation of infrastructure investment made by the central government and believed that the balance of equity and efficiency is important to the central government’s decision-making. Pradeep and Nair [7] and Mishra et al. [8] studied the transport infrastructure investment. Since the investments are usually costly and irreversible, they believed that it is important to find a reliable and logical appraisal method to assign the investment.Throughout the existing research, there is no doubt that investing in urban public infrastructure scientifically will benefit the comprehensive development. And to improve the urban infrastructure investment efficiency, it is necessary to establish a scheme. As the infrastructure investment efficiency is usually measured by comparing performance indicators where data envelopment analysis (DEA) [9, 10] performs very well, DEA has been widely used in recent literature to evaluate the relative urban infrastructure investment efficiency of different regions. For instance, Z.-F. Li and Y.-L. Li [11] regarded the DEA method as a nonparameter method to evaluate the performance of infrastructure investment in China. Yang and Gao [12] used the DEA model to evaluate the relative efficiency of city infrastructure investment of 10 cities in Shaanxi province from 2008 to 2013. Fancello et al. [13] used DEA to evaluate the urban road system. However, the conventional DEA model is a pure self-evaluation model and does not consider the impacts coming from other DMUs. Thus, the DEA cross-efficiency model was proposed [14, 15]. For example, Sun et al. [16] used DEA cross-efficiency model to evaluate and analyze the urban infrastructure economic efficiency of 35 Chinese large and medium-sized cities in 2012. However, the DEA cross-efficiency model also has limitations: the solution depends on the solving software and it is not unique; the average cross-efficiency is not Pareto solution. Therefore, Liang et al. [17] introduced the game theory into DEA cross-efficiency model. In existing literature, there are some researches discussing about efficiency evaluation with the DEA game cross-efficiency model. Yang et al. [18] proposed a DEA cross-efficiency method with both competition and corporation among DMUs, which considers a specific DMU’s individual preference in differential treatment to other DMUs. Wu et al. [19] used the DEA game cross-efficiency to evaluate the Olympic games and ranked the countries participating in it. Ma et al. [20] used this method to assess supplier performance and then to obtain the Pareto solution. Ma et al. [21] developed a DEA game cross-efficiency with two-stage structures to evaluate the performance of top 30 US commercial banks and got a unique reasonable cross-efficiency for each DMU. In order to study the efficiency of infrastructure investment and the fund allocation of Chinese provincial capital cities, we treat the cities as DMUs. As there are competitions existing among the cities, it is rational to use the DEA game cross-efficiency model to evaluate the comprehensive efficiency of each city in urban public infrastructure and allocate funds.The rest of this paper is organized as follows. Section2 introduces the comprehensive efficiency of infrastructure investment, selects indexes about the comprehensive efficiency of public infrastructure investment, and then describes the DEA game cross-efficiency model. And the model analysis and the validation are presented in Section 3. Section 4 illustrates the results of the model and makes detailed discussion. In the end, Section 5 summarizes the main conclusions of the research.


## 2. Game Cross-Efficiency for Public Infrastructure Investment

### 2.1. Index Selection

Urban public infrastructure, which provides convenience for the city producing and living, includes transportation infrastructure, water infrastructure, energy infrastructure, ecological infrastructure, disaster-preventing infrastructure, and telecommunications infrastructure. In the long-term interactive relationship with economic development, investing in public infrastructure construction could promote the development of economy. Besides, it could improve employment rate, labor productivity, sociocultural integration level, and living environment quality and reduce the urban heat island effect while guaranteeing the residents’ health. Generally, the comprehensive efficiency of urban infrastructure investment can be divided into economic efficiency, society efficiency, and environment efficiency, which influence, interact, and depend on each other. In order to improve the comprehensive efficiency of infrastructure investment, all these three dimensions should be coordinated and cooperated synergistically.There is no doubt that increasing infrastructure investment could benefit the economy development of a city. Meanwhile, it could also provide channels for the improvement of urban society and environment. For the formation and development of society and environment, the economy provides material and basic supports. And the development of the society offers the human resource, technology, and policies to develop economy and improve environment. Mutually, a good urban living environment could speed up economy development and improve the living situations of residents. The synergies of public infrastructure investment benefits are shown in Figure1.Figure 1
Synergies of benefits of public infrastructure investment.The indexes of urban public infrastructure keep connection with each other, while possessing independence, respectively. Considering there are positive (preference value is bigger and better) and negative (preference value is smaller and better) factors that affect the assessment synthetically [22], in order to simplify calculation, we design the positive indexes for the comprehensive benefits. With consideration about most important elements of the investment purposes and data access, we use the following evaluation index system of urban infrastructure investment (seen in Table 1).Table 1
The index system of urban public infrastructure investment comprehensive benefit.
Macroareas
Input
Output
Economy
Completed investment (10,000 RMB)
Revenue of urban maintenance and construction fund (10,000 RMB)
Society
Water system investment (10,000 RMB)Energy system investment (10,000 RMB)Transportation system investment (10,000 RMB)
Water supply (10,000 tons)Gas supply (10,000 cubic meters)Surface area of roads (10,000 m2)Person employed (10,000 persons)Collection of public libraries per 100 persons (copy, piece)
Environment
Environmental system investment (10,000 RMB)
Green coverage area-built district (hectare)


### 2.2. DEA Game Cross-Efficiency Model

For allocating the fund scientifically and reasonably, the DEA game cross-efficiency model is adapted to evaluate the comprehensive efficiency of urban infrastructure investment. Assume that there weren cities with m inputs and s outputs. For the city j, x
i
j
i
=
1,2
,
…
,
m represents i
t
h input and y
r
j
r
=
1,2
,
…
,
s represents r
t
h output. Suppose that the total capital C would be distributed among all cities. The fund c
j
j
=
1,2
,
…
,
n is allocated to the city j and ∑
j
=
1
n
c
j
=
C. In the model, the fund c
j is regarded as a new input variable [23], and then the efficiency score of the whole system for the city d can be calculated with the modified CCR model as follows:(1)
m
a
x
∑
r
=
1
s
u
r
d
y
r
d
∑
i
=
1
m
v
i
d
x
i
d
+
v
m
+
1
d
c
d
d
=
θ
d
s.t.
∑
r
=
1
s
u
r
d
y
r
j
∑
i
=
1
m
v
i
d
x
i
j
+
v
m
+
1
d
c
j
d
≤
1
,
j
=
1,2
,
…
,
n
,
∑
j
=
1
n
c
j
d
=
C
,
c
j
d
≥
0
,
j
=
1,2
,
…
,
n
,
v
i
d
≥
0
,
i
=
1,2
,
…
,
m
,
u
r
d
≥
0
,
r
=
1,2
,
…
,
s
.In order to simplify calculation, we use Charnes-Cooper’s method to transform model (1), and set ω
m
+
1
d
c
j
d as a new variable c
^
j
d, and then get a LP model: (2)
m
a
x
∑
r
=
1
s
μ
r
d
y
r
d
=
θ
d
s.t.
∑
i
=
1
m
ω
i
d
x
i
j
+
c
^
j
d
-
∑
r
=
1
s
μ
r
d
y
i
j
≥
0
,
j
=
1,2
,
…
,
n
,
∑
i
=
1
m
ω
i
d
x
i
d
+
c
^
d
d
=
1
,
∑
j
=
1
n
c
^
j
d
=
ω
m
+
1
d
C
,
c
^
j
d
≥
0
,
j
=
1,2
,
…
,
n
,
ω
i
d
≥
0
,
i
=
1,2
,
…
,
m
+
1
,
μ
r
d
≥
0
,
r
=
1,2
,
…
,
s
.For each cityd under evaluation, we obtain a set of weight c
^
j
d
∗
j
=
1,2
,
…
,
n, ω
i
d
∗
i
=
1,2
,
…
,
m
+
1, μ
r
d
∗
r
=
1,2
,
…
,
s. Using this set of weights, the d-cross-efficiency for any city j can be calculated as(3)
E
d
j
=
∑
r
=
1
s
μ
r
d
∗
y
r
j
∑
i
=
1
m
ω
i
d
∗
x
i
j
+
c
^
j
d
∗
,
d
,
j
=
1,2
,
…
,
n
.For each cityj, the average of all of the d-cross-efficiency is as follows:(4)
E
-
j
=
1
n
∑
d
=
1
n
E
d
j
.However, the cross-efficiency is not unique here, and it yields nonunique allocation plans. According to Liang et al. [17] and Cheng et al. [24], we regard the cities as DMUs and assume that they are players of noncooperative game. Cities compete with each other. Suppose that, in the sense of noncooperative game, a player D
M
U
d gets an efficiency score α
d which cannot be decreased when another player D
M
U
j tries to maximize its own efficiency. Then the game d-cross-efficiency of overall system for D
M
U
j relative to D
M
U
d is defined as (5)
α
d
j
=
∑
r
=
1
s
μ
r
d
y
r
j
∑
i
=
1
m
ω
i
d
x
i
j
+
c
^
j
d
,
d
,
j
=
1,2
,
…
,
n
,where μ
r
d
r
=
1,2
,
…
,
s, ω
i
d
i
=
1,2
,
…
,
m
+
1, and c
^
j
d
j
=
1,2
,
…
,
n are the optimal solutions of the following game d-cross-efficiency model:(6)
m
a
x
∑
r
=
1
s
μ
r
d
y
r
d
=
θ
d
s.t.
∑
i
=
1
m
ω
i
d
x
i
j
+
c
^
j
d
-
∑
r
=
1
s
μ
r
d
y
i
j
≥
0
,
j
=
1,2
,
…
,
n
,
α
d
×
∑
i
=
1
m
ω
i
d
x
i
d
+
c
^
d
d
-
∑
r
=
1
s
μ
r
d
y
i
d
≤
0
,
∑
i
=
1
m
ω
i
d
x
i
d
+
c
^
d
d
=
1
,
∑
j
=
1
n
c
^
j
d
=
ω
m
+
1
d
C
,
c
^
j
d
≥
0
,
j
=
1,2
,
…
,
n
,
ω
i
d
≥
0
,
i
=
1,2
,
…
,
m
+
1
,
μ
r
d
≥
0
,
r
=
1,2
,
…
,
s
,where the parameter α
d
≤
1.According to the above analysis, we propose the following iterative algorithm, which begins with solving model (2) and then obtains an original cross-efficiency score defined in (4). And that process would be repeated for every d, and the optimal value of model (6) comes as a revised α
d. Assuming μ
r
d
∗
α
d was an optimal solution to model (6), for each D
M
U
j, α
j
=
1
/
n
∑
d
=
1
n
∑
r
=
1
s
μ
r
d
∗
α
d
y
r
j is called the average game cross-efficiency. Through circular computations, the algorithm terminates until α
j converges to a stable value.Algorithm 1.
Step 1. Solve the traditional DEA game cross-efficiency model and obtain a set of initial cross-efficiency scores defined in (4). Let t
=
1 and α
d
=
α
d
1
=
E
-
d.
Step 2. Solve model (6) with α
d
=
α
d
1; let α
j
2
=
1
/
n
∑
d
=
1
n
∑
r
=
1
s
μ
r
d
∗
α
d
1
y
r
j
j
=
1,2
,
…
,
n. The general form is calculated as α
j
t
+
1
=
1
/
n
∑
d
=
1
n
∑
r
=
1
s
μ
r
d
∗
α
d
t
y
r
j
j
=
1,2
,
…
,
n.
Step 3. If α
j
t
+
1
-
α
j
t
≥
ε (where ε is a given small positive value) for some j, set α
d
=
α
d
t
+
1 and go back to Step 2; otherwise, stop the iteration and α
j
t
+
1 is the best solution of the system.


## 2.1. Index Selection

Urban public infrastructure, which provides convenience for the city producing and living, includes transportation infrastructure, water infrastructure, energy infrastructure, ecological infrastructure, disaster-preventing infrastructure, and telecommunications infrastructure. In the long-term interactive relationship with economic development, investing in public infrastructure construction could promote the development of economy. Besides, it could improve employment rate, labor productivity, sociocultural integration level, and living environment quality and reduce the urban heat island effect while guaranteeing the residents’ health. Generally, the comprehensive efficiency of urban infrastructure investment can be divided into economic efficiency, society efficiency, and environment efficiency, which influence, interact, and depend on each other. In order to improve the comprehensive efficiency of infrastructure investment, all these three dimensions should be coordinated and cooperated synergistically.There is no doubt that increasing infrastructure investment could benefit the economy development of a city. Meanwhile, it could also provide channels for the improvement of urban society and environment. For the formation and development of society and environment, the economy provides material and basic supports. And the development of the society offers the human resource, technology, and policies to develop economy and improve environment. Mutually, a good urban living environment could speed up economy development and improve the living situations of residents. The synergies of public infrastructure investment benefits are shown in Figure1.Figure 1
Synergies of benefits of public infrastructure investment.The indexes of urban public infrastructure keep connection with each other, while possessing independence, respectively. Considering there are positive (preference value is bigger and better) and negative (preference value is smaller and better) factors that affect the assessment synthetically [22], in order to simplify calculation, we design the positive indexes for the comprehensive benefits. With consideration about most important elements of the investment purposes and data access, we use the following evaluation index system of urban infrastructure investment (seen in Table 1).Table 1
The index system of urban public infrastructure investment comprehensive benefit.
Macroareas
Input
Output
Economy
Completed investment (10,000 RMB)
Revenue of urban maintenance and construction fund (10,000 RMB)
Society
Water system investment (10,000 RMB)Energy system investment (10,000 RMB)Transportation system investment (10,000 RMB)
Water supply (10,000 tons)Gas supply (10,000 cubic meters)Surface area of roads (10,000 m2)Person employed (10,000 persons)Collection of public libraries per 100 persons (copy, piece)
Environment
Environmental system investment (10,000 RMB)
Green coverage area-built district (hectare)


## 2.2. DEA Game Cross-Efficiency Model

For allocating the fund scientifically and reasonably, the DEA game cross-efficiency model is adapted to evaluate the comprehensive efficiency of urban infrastructure investment. Assume that there weren cities with m inputs and s outputs. For the city j, x
i
j
i
=
1,2
,
…
,
m represents i
t
h input and y
r
j
r
=
1,2
,
…
,
s represents r
t
h output. Suppose that the total capital C would be distributed among all cities. The fund c
j
j
=
1,2
,
…
,
n is allocated to the city j and ∑
j
=
1
n
c
j
=
C. In the model, the fund c
j is regarded as a new input variable [23], and then the efficiency score of the whole system for the city d can be calculated with the modified CCR model as follows:(1)
m
a
x
∑
r
=
1
s
u
r
d
y
r
d
∑
i
=
1
m
v
i
d
x
i
d
+
v
m
+
1
d
c
d
d
=
θ
d
s.t.
∑
r
=
1
s
u
r
d
y
r
j
∑
i
=
1
m
v
i
d
x
i
j
+
v
m
+
1
d
c
j
d
≤
1
,
j
=
1,2
,
…
,
n
,
∑
j
=
1
n
c
j
d
=
C
,
c
j
d
≥
0
,
j
=
1,2
,
…
,
n
,
v
i
d
≥
0
,
i
=
1,2
,
…
,
m
,
u
r
d
≥
0
,
r
=
1,2
,
…
,
s
.In order to simplify calculation, we use Charnes-Cooper’s method to transform model (1), and set ω
m
+
1
d
c
j
d as a new variable c
^
j
d, and then get a LP model: (2)
m
a
x
∑
r
=
1
s
μ
r
d
y
r
d
=
θ
d
s.t.
∑
i
=
1
m
ω
i
d
x
i
j
+
c
^
j
d
-
∑
r
=
1
s
μ
r
d
y
i
j
≥
0
,
j
=
1,2
,
…
,
n
,
∑
i
=
1
m
ω
i
d
x
i
d
+
c
^
d
d
=
1
,
∑
j
=
1
n
c
^
j
d
=
ω
m
+
1
d
C
,
c
^
j
d
≥
0
,
j
=
1,2
,
…
,
n
,
ω
i
d
≥
0
,
i
=
1,2
,
…
,
m
+
1
,
μ
r
d
≥
0
,
r
=
1,2
,
…
,
s
.For each cityd under evaluation, we obtain a set of weight c
^
j
d
∗
j
=
1,2
,
…
,
n, ω
i
d
∗
i
=
1,2
,
…
,
m
+
1, μ
r
d
∗
r
=
1,2
,
…
,
s. Using this set of weights, the d-cross-efficiency for any city j can be calculated as(3)
E
d
j
=
∑
r
=
1
s
μ
r
d
∗
y
r
j
∑
i
=
1
m
ω
i
d
∗
x
i
j
+
c
^
j
d
∗
,
d
,
j
=
1,2
,
…
,
n
.For each cityj, the average of all of the d-cross-efficiency is as follows:(4)
E
-
j
=
1
n
∑
d
=
1
n
E
d
j
.However, the cross-efficiency is not unique here, and it yields nonunique allocation plans. According to Liang et al. [17] and Cheng et al. [24], we regard the cities as DMUs and assume that they are players of noncooperative game. Cities compete with each other. Suppose that, in the sense of noncooperative game, a player D
M
U
d gets an efficiency score α
d which cannot be decreased when another player D
M
U
j tries to maximize its own efficiency. Then the game d-cross-efficiency of overall system for D
M
U
j relative to D
M
U
d is defined as (5)
α
d
j
=
∑
r
=
1
s
μ
r
d
y
r
j
∑
i
=
1
m
ω
i
d
x
i
j
+
c
^
j
d
,
d
,
j
=
1,2
,
…
,
n
,where μ
r
d
r
=
1,2
,
…
,
s, ω
i
d
i
=
1,2
,
…
,
m
+
1, and c
^
j
d
j
=
1,2
,
…
,
n are the optimal solutions of the following game d-cross-efficiency model:(6)
m
a
x
∑
r
=
1
s
μ
r
d
y
r
d
=
θ
d
s.t.
∑
i
=
1
m
ω
i
d
x
i
j
+
c
^
j
d
-
∑
r
=
1
s
μ
r
d
y
i
j
≥
0
,
j
=
1,2
,
…
,
n
,
α
d
×
∑
i
=
1
m
ω
i
d
x
i
d
+
c
^
d
d
-
∑
r
=
1
s
μ
r
d
y
i
d
≤
0
,
∑
i
=
1
m
ω
i
d
x
i
d
+
c
^
d
d
=
1
,
∑
j
=
1
n
c
^
j
d
=
ω
m
+
1
d
C
,
c
^
j
d
≥
0
,
j
=
1,2
,
…
,
n
,
ω
i
d
≥
0
,
i
=
1,2
,
…
,
m
+
1
,
μ
r
d
≥
0
,
r
=
1,2
,
…
,
s
,where the parameter α
d
≤
1.According to the above analysis, we propose the following iterative algorithm, which begins with solving model (2) and then obtains an original cross-efficiency score defined in (4). And that process would be repeated for every d, and the optimal value of model (6) comes as a revised α
d. Assuming μ
r
d
∗
α
d was an optimal solution to model (6), for each D
M
U
j, α
j
=
1
/
n
∑
d
=
1
n
∑
r
=
1
s
μ
r
d
∗
α
d
y
r
j is called the average game cross-efficiency. Through circular computations, the algorithm terminates until α
j converges to a stable value.Algorithm 1.
Step 1. Solve the traditional DEA game cross-efficiency model and obtain a set of initial cross-efficiency scores defined in (4). Let t
=
1 and α
d
=
α
d
1
=
E
-
d.
Step 2. Solve model (6) with α
d
=
α
d
1; let α
j
2
=
1
/
n
∑
d
=
1
n
∑
r
=
1
s
μ
r
d
∗
α
d
1
y
r
j
j
=
1,2
,
…
,
n. The general form is calculated as α
j
t
+
1
=
1
/
n
∑
d
=
1
n
∑
r
=
1
s
μ
r
d
∗
α
d
t
y
r
j
j
=
1,2
,
…
,
n.
Step 3. If α
j
t
+
1
-
α
j
t
≥
ε (where ε is a given small positive value) for some j, set α
d
=
α
d
t
+
1 and go back to Step 2; otherwise, stop the iteration and α
j
t
+
1 is the best solution of the system.


## 3. Model Analysis and Validation

### 3.1. Data

In this section, we select 30 Chinese cities for evaluation and ranking, including 1 national capital city (i.e., Beijing), 3 direct-controlled municipalities (i.e., Tianjin, Shanghai, and Chongqing), and 26 provincial capital cities. Without loss of generality, Lhasa, Hong Kong, Macao, and Taiwan are not included in the analysis due to the lack of statistical data. In this paper, the data is derived from theChina City Statistical Yearbook 2013 (which covers the main socioeconomic statistical data for 2012), which is compiled by the Department of Urban Society and Economic Statistics, National Bureau of Statistics of China, and the2012 China Urban Construction Statistical Yearbook (which covers the statistical data on urban construction in 2012), which is compiled by Ministry of Housing and Urban-Rural Development, China.According to the statistical yearbook and the input indexes of Table1, we organize the original data which is shown in Table 2. Table 2 indicates that the water system investment includes the investment of water supply and sewerage, and the energy system investment is made up of gas supply investment and central heating investment. And the transportation system investment is composed of urban rail transit system investment and road and bridge investment. In addition, the environment system investment consists of landscaping investment and environmental sanitation investment. Similarity, depending on the output indexes of Table 1, the original data is shown in Table 3.Table 2
The urban infrastructure construction statistic data of input indexes in 2012.
DMU
Cities
Completedinvestment
Water systeminvestment
Energy system investment
Transportationsystem investment
Environmentalsystem investment
1
Beijing
12162405
600802
944859
7208204
1783322
2
Tianjin
6540958
136451
218055
3965879
667863
3
Shijiazhuang
1030696
115700
86600
655508
743435
4
Taiyuan
1209732
17600
766330
239136
111656
5
Hohhot
923993
83653
77033
536149
227158
6
Shenyang
4966546
493390
286049
3273487
820659
7
Changchun
1917746
75284
89164
1537984
57744
8
Harbin
2058076
1160
108703
1226271
159573
9
Shanghai
3393269
425108
138720
2375115
342831
10
Nanjing
3999657
375792
60437
3270255
270516
11
Hangzhou
1440705
41717
45569
1278534
65404
12
Hefei
810014
43640
25542
591975
127896
13
Fuzhou
1070667
56826
7083
847390
134051
14
Nanchang
2761493
57901
121947
1933313
364760
15
Jinan
1003467
130111
148154
558054
45901
16
Zhengzhou
1079420
51766
51359
865713
110582
17
Wuhan
6645218
786184
—
5140003
524932
18
Changsha
1679690
58059
9200
935186
232585
19
Guangzhou
2022147
124400
29378
1552371
75824
20
Nanning
757615
28779
8305
646653
73860
21
Haikou
474786
111782
—
288898
74106
22
Chongqing
4607586
273448
26200
3565109
680812
23
Chengdu
3934694
66407
5189
3761150
17896
24
Guiyang
789180
20595
12091
755616
878
25
Kunming
538755
71982
—
419849
45291
26
Xi’an
2477694
44173
33591
1391493
172742
27
Lanzhou
1472719
35963
11093
1208824
198611
28
Xining
388893
13800
19116
280303
—
29
Yinchuan
127955
3901
28266
66723
—
30
Urumqi
2269254
41600
614472
479445
—
Note: “—” means the missing data.Table 3
The urban infrastructure construction statistic data of output indexes in 2012.
DMU
Cities
Revenue of urban maintenanceand construction fund
Water supply
Gassupply
Surface areaof roads
Personemployed
Collection of publiclibraries per 100 persons
Green coverage area-built district
1
Beijing
18436572
159646
924763
13509
118.89
445.5
68204
2
Tianjin
1687948
77218
256241
11611
53.23
176.27
25191
3
Shijiazhuang
623340
33531
24613
4285
19.49
180.46
8868
4
Taiyuan
766172
31107
106822
2904
33.19
171.61
12112
5
Hohhot
218142
13718
42050
1949
7.29
174.66
7798
6
Shenyang
1666059
56641
43332
6647
25.34
242.05
19210
7
Changchun
761351
34951
44615
6457
17.84
244.23
15220
8
Harbin
583346
38653
33323
4624
34.45
162.25
14181
9
Shanghai
2372764
309704
721564
9717
98.82
525.75
38242
10
Nanjing
2081571
121401
82413
11424
27.93
274.37
28756
11
Hangzhou
1427371
58182
55623
5284
96.24
351.53
18135
12
Hefei
830742
34573
32924
4854
42.96
288.47
15088
13
Fuzhou
978086
30166
13608
2563
50.55
76.17
9750
14
Nanchang
318475
39397
20213
2391
34.7
230.55
9245
15
Jinan
996423
33250
39300
7251
43.99
296.96
13803
16
Zhengzhou
1189035
35824
75742
3564
35.75
82.56
13456
17
Wuhan
475834
121552
125131
9027
64.9
2187.39
19870
18
Changsha
617456
41997
64298
3958
26.55
319.23
10729
19
Guangzhou
2191312
191432
—
10140
57.89
39.97
40895
20
Nanning
1524615
40215
6554
3334
19.39
168.46
10165
21
Haikou
104762
19066
12143
2439
8.6
42.94
5191
22
Chongqing
3623844
95903
324965
11936
206.39
7.34
45157
23
Chengdu
1632416
76021
210902
7441
64.47
165.05
20301
24
Guiyang
133750
32855
26909
1348
29.05
98.82
22766
25
Kunming
2091018
21984
30866
4056
39.63
35.47
16884
26
Xi’an
2170117
43848
138050
6333
38.72
71.62
15750
27
Lanzhou
369607
25035
88493
2219
14.88
288.95
5963
28
Xining
215088
14633
102749
823
10.33
208.84
2812
29
Yinchuan
146256
11468
147354
1809
7.96
45.6
5632
30
Urumqi
707983
30363
146244
2225
11.8
123.55
13630
Note: “—” means the missing data.Then, we apply the DEA game cross-efficiency model to analyze the performance of urban infrastructure investment and apply a scientific method to allocate the fund of investment reasonably. We use MATLAB 7.0 to calculate the comprehensive efficiencies of infrastructure investment for the chosen cities. In the algorithm, we use the traditional DEA cross-efficiency asα
d
1 and set ε
=
0.0001. Considering the cross-efficiency is not unique, we impose the secondary goals and propose three alternative methods. The first method is the aggressive strategy, which maximizes DEA efficiency for a DMU as the primary goal and minimizes the other DMUs’ cross-efficiencies as a secondary goal [14]. The second method is the benevolent strategy, which maximizes the cross-efficiencies of other DMUs’ as a secondary goal [15]. The third method is the arbitrary strategy, which does not impose the secondary goal. However, no matter which method is chosen, the DEA game cross-efficiency should converge to a solution no matter which method is chosen [19]. Therefore, we adopt the aggressive strategy as the original value, and the corresponding results of DEA game cross-efficiency are shown in Table 4.Table 4
Allocation results and game cross-efficiency for cities in 2012.
DMU
Cities
Assignment weight
Ranking
Game cross-efficiency
Ranking
1
Beijing
0.0794
2
0.9990
23
2
Tianjin
0.0393
8
0.9990
22
3
Shijiazhuang
0.0141
22
0.9991
21
4
Taiyuan
0.0243
17
0.9989
24
5
Hohhot
0.0047
30
0.9985
26
6
Shenyang
0.0118
24
0.9964
30
7
Changchun
0.0135
23
0.9995
19
8
Harbin
0.0273
14
0.9997
17
9
Shanghai
0.0760
4
0.9997
12
10
Nanjing
0.0180
20
0.9995
20
11
Hangzhou
0.0761
3
0.9999
5
12
Hefei
0.0343
11
0.9999
4
13
Fuzhou
0.0388
9
0.9998
10
14
Nanchang
0.0250
16
0.9974
28
15
Jinan
0.0345
10
0.9997
14
16
Zhengzhou
0.0273
15
0.9997
13
17
Wuhan
0.0422
7
0.9985
27
18
Changsha
0.0207
19
0.9998
8
19
Guangzhou
0.0467
6
0.9999
3
20
Nanning
0.0144
21
0.9997
15
21
Haikou
0.0059
29
0.9989
25
22
Chongqing
0.1594
1
0.9998
9
23
Chengdu
0.0488
5
0.9999
6
24
Guiyang
0.0231
18
0.9998
7
25
Kunming
0.0295
13
0.9997
16
26
Xi’an
0.0320
12
0.9999
2
27
Lanzhou
0.0110
25
0.9996
18
28
Xining
0.0081
26
0.9997
11
29
Yinchuan
0.0069
28
0.9999
1
30
Urumqi
0.0071
27
0.9964
29


### 3.2. Model Validation

By solving the DEA game cross-efficiency model, it is obvious that the game cross-efficiency is connected with the input and output indexes of urban public infrastructure. In order to test the credibility of the results, we need to establish the relationship between the results and the indexes. Considering various facts of situations, the multiple linear regression method is used to test the model. We regard the game cross-efficiency as dependent variables and take the input and output indexes of urban public infrastructure as independent variables.The multiple linear regression model is as follows:(7)
y
=
β
0
+
β
1
x
1
+
⋯
+
β
12
x
12
+
ε
,where y is the game cross-efficiency of the public infrastructure investment, which is shown in column 5 of Table 4. And x
i
i
=
1,2
,
…
,
5 is the input index of urban public infrastructure, which is shown in Table 2; x
i
i
=
6,7
,
…
,
12 is the output index of urban public infrastructure, which is shown in Table 3.The multiple linear regression model is employed by using SPSS, and the test results are shown in Tables5 and 6. It can be seen that the results show a high consistency. For example, the R achieves 0.909, the R square achieves 0.826, and the adjusted R square achieves 0.704. In addition, the Analysis of Variance (ANOVA) shows that the P value is 0.000: that is to say, the DEA game cross-efficiency model passes the significance testing. Hence, the results show that the DEA game cross-efficiency model is valid in the problem of evaluating the infrastructure investment efficiency.Table 5
The summary of multiple linear regression model.
Model
R
R square
Adjusted
R square
Std. error of the estimate
1
.909a
.826
.704
.00052
aPredictors: (constant) ×12, ×6, ×7, ×11, ×3, ×5, ×9, ×4, ×2, ×8, ×10, and ×1.Table 6
The ANOVAb of multiple linear regression model.
Model
Sum of squares
df
Mean square
F
Sig.
1
Regression
.000
12
.000
6.741
.000a
Residual
.000
17
.000
Total
.000
29
aPredictors: (constant) ×12, ×6, ×7, ×11, ×3, ×5, ×9, ×4, ×2, ×8, ×10, and ×1.
bDependent variable: y.


## 3.1. Data

In this section, we select 30 Chinese cities for evaluation and ranking, including 1 national capital city (i.e., Beijing), 3 direct-controlled municipalities (i.e., Tianjin, Shanghai, and Chongqing), and 26 provincial capital cities. Without loss of generality, Lhasa, Hong Kong, Macao, and Taiwan are not included in the analysis due to the lack of statistical data. In this paper, the data is derived from theChina City Statistical Yearbook 2013 (which covers the main socioeconomic statistical data for 2012), which is compiled by the Department of Urban Society and Economic Statistics, National Bureau of Statistics of China, and the2012 China Urban Construction Statistical Yearbook (which covers the statistical data on urban construction in 2012), which is compiled by Ministry of Housing and Urban-Rural Development, China.According to the statistical yearbook and the input indexes of Table1, we organize the original data which is shown in Table 2. Table 2 indicates that the water system investment includes the investment of water supply and sewerage, and the energy system investment is made up of gas supply investment and central heating investment. And the transportation system investment is composed of urban rail transit system investment and road and bridge investment. In addition, the environment system investment consists of landscaping investment and environmental sanitation investment. Similarity, depending on the output indexes of Table 1, the original data is shown in Table 3.Table 2
The urban infrastructure construction statistic data of input indexes in 2012.
DMU
Cities
Completedinvestment
Water systeminvestment
Energy system investment
Transportationsystem investment
Environmentalsystem investment
1
Beijing
12162405
600802
944859
7208204
1783322
2
Tianjin
6540958
136451
218055
3965879
667863
3
Shijiazhuang
1030696
115700
86600
655508
743435
4
Taiyuan
1209732
17600
766330
239136
111656
5
Hohhot
923993
83653
77033
536149
227158
6
Shenyang
4966546
493390
286049
3273487
820659
7
Changchun
1917746
75284
89164
1537984
57744
8
Harbin
2058076
1160
108703
1226271
159573
9
Shanghai
3393269
425108
138720
2375115
342831
10
Nanjing
3999657
375792
60437
3270255
270516
11
Hangzhou
1440705
41717
45569
1278534
65404
12
Hefei
810014
43640
25542
591975
127896
13
Fuzhou
1070667
56826
7083
847390
134051
14
Nanchang
2761493
57901
121947
1933313
364760
15
Jinan
1003467
130111
148154
558054
45901
16
Zhengzhou
1079420
51766
51359
865713
110582
17
Wuhan
6645218
786184
—
5140003
524932
18
Changsha
1679690
58059
9200
935186
232585
19
Guangzhou
2022147
124400
29378
1552371
75824
20
Nanning
757615
28779
8305
646653
73860
21
Haikou
474786
111782
—
288898
74106
22
Chongqing
4607586
273448
26200
3565109
680812
23
Chengdu
3934694
66407
5189
3761150
17896
24
Guiyang
789180
20595
12091
755616
878
25
Kunming
538755
71982
—
419849
45291
26
Xi’an
2477694
44173
33591
1391493
172742
27
Lanzhou
1472719
35963
11093
1208824
198611
28
Xining
388893
13800
19116
280303
—
29
Yinchuan
127955
3901
28266
66723
—
30
Urumqi
2269254
41600
614472
479445
—
Note: “—” means the missing data.Table 3
The urban infrastructure construction statistic data of output indexes in 2012.
DMU
Cities
Revenue of urban maintenanceand construction fund
Water supply
Gassupply
Surface areaof roads
Personemployed
Collection of publiclibraries per 100 persons
Green coverage area-built district
1
Beijing
18436572
159646
924763
13509
118.89
445.5
68204
2
Tianjin
1687948
77218
256241
11611
53.23
176.27
25191
3
Shijiazhuang
623340
33531
24613
4285
19.49
180.46
8868
4
Taiyuan
766172
31107
106822
2904
33.19
171.61
12112
5
Hohhot
218142
13718
42050
1949
7.29
174.66
7798
6
Shenyang
1666059
56641
43332
6647
25.34
242.05
19210
7
Changchun
761351
34951
44615
6457
17.84
244.23
15220
8
Harbin
583346
38653
33323
4624
34.45
162.25
14181
9
Shanghai
2372764
309704
721564
9717
98.82
525.75
38242
10
Nanjing
2081571
121401
82413
11424
27.93
274.37
28756
11
Hangzhou
1427371
58182
55623
5284
96.24
351.53
18135
12
Hefei
830742
34573
32924
4854
42.96
288.47
15088
13
Fuzhou
978086
30166
13608
2563
50.55
76.17
9750
14
Nanchang
318475
39397
20213
2391
34.7
230.55
9245
15
Jinan
996423
33250
39300
7251
43.99
296.96
13803
16
Zhengzhou
1189035
35824
75742
3564
35.75
82.56
13456
17
Wuhan
475834
121552
125131
9027
64.9
2187.39
19870
18
Changsha
617456
41997
64298
3958
26.55
319.23
10729
19
Guangzhou
2191312
191432
—
10140
57.89
39.97
40895
20
Nanning
1524615
40215
6554
3334
19.39
168.46
10165
21
Haikou
104762
19066
12143
2439
8.6
42.94
5191
22
Chongqing
3623844
95903
324965
11936
206.39
7.34
45157
23
Chengdu
1632416
76021
210902
7441
64.47
165.05
20301
24
Guiyang
133750
32855
26909
1348
29.05
98.82
22766
25
Kunming
2091018
21984
30866
4056
39.63
35.47
16884
26
Xi’an
2170117
43848
138050
6333
38.72
71.62
15750
27
Lanzhou
369607
25035
88493
2219
14.88
288.95
5963
28
Xining
215088
14633
102749
823
10.33
208.84
2812
29
Yinchuan
146256
11468
147354
1809
7.96
45.6
5632
30
Urumqi
707983
30363
146244
2225
11.8
123.55
13630
Note: “—” means the missing data.Then, we apply the DEA game cross-efficiency model to analyze the performance of urban infrastructure investment and apply a scientific method to allocate the fund of investment reasonably. We use MATLAB 7.0 to calculate the comprehensive efficiencies of infrastructure investment for the chosen cities. In the algorithm, we use the traditional DEA cross-efficiency asα
d
1 and set ε
=
0.0001. Considering the cross-efficiency is not unique, we impose the secondary goals and propose three alternative methods. The first method is the aggressive strategy, which maximizes DEA efficiency for a DMU as the primary goal and minimizes the other DMUs’ cross-efficiencies as a secondary goal [14]. The second method is the benevolent strategy, which maximizes the cross-efficiencies of other DMUs’ as a secondary goal [15]. The third method is the arbitrary strategy, which does not impose the secondary goal. However, no matter which method is chosen, the DEA game cross-efficiency should converge to a solution no matter which method is chosen [19]. Therefore, we adopt the aggressive strategy as the original value, and the corresponding results of DEA game cross-efficiency are shown in Table 4.Table 4
Allocation results and game cross-efficiency for cities in 2012.
DMU
Cities
Assignment weight
Ranking
Game cross-efficiency
Ranking
1
Beijing
0.0794
2
0.9990
23
2
Tianjin
0.0393
8
0.9990
22
3
Shijiazhuang
0.0141
22
0.9991
21
4
Taiyuan
0.0243
17
0.9989
24
5
Hohhot
0.0047
30
0.9985
26
6
Shenyang
0.0118
24
0.9964
30
7
Changchun
0.0135
23
0.9995
19
8
Harbin
0.0273
14
0.9997
17
9
Shanghai
0.0760
4
0.9997
12
10
Nanjing
0.0180
20
0.9995
20
11
Hangzhou
0.0761
3
0.9999
5
12
Hefei
0.0343
11
0.9999
4
13
Fuzhou
0.0388
9
0.9998
10
14
Nanchang
0.0250
16
0.9974
28
15
Jinan
0.0345
10
0.9997
14
16
Zhengzhou
0.0273
15
0.9997
13
17
Wuhan
0.0422
7
0.9985
27
18
Changsha
0.0207
19
0.9998
8
19
Guangzhou
0.0467
6
0.9999
3
20
Nanning
0.0144
21
0.9997
15
21
Haikou
0.0059
29
0.9989
25
22
Chongqing
0.1594
1
0.9998
9
23
Chengdu
0.0488
5
0.9999
6
24
Guiyang
0.0231
18
0.9998
7
25
Kunming
0.0295
13
0.9997
16
26
Xi’an
0.0320
12
0.9999
2
27
Lanzhou
0.0110
25
0.9996
18
28
Xining
0.0081
26
0.9997
11
29
Yinchuan
0.0069
28
0.9999
1
30
Urumqi
0.0071
27
0.9964
29


## 3.2. Model Validation

By solving the DEA game cross-efficiency model, it is obvious that the game cross-efficiency is connected with the input and output indexes of urban public infrastructure. In order to test the credibility of the results, we need to establish the relationship between the results and the indexes. Considering various facts of situations, the multiple linear regression method is used to test the model. We regard the game cross-efficiency as dependent variables and take the input and output indexes of urban public infrastructure as independent variables.The multiple linear regression model is as follows:(7)
y
=
β
0
+
β
1
x
1
+
⋯
+
β
12
x
12
+
ε
,where y is the game cross-efficiency of the public infrastructure investment, which is shown in column 5 of Table 4. And x
i
i
=
1,2
,
…
,
5 is the input index of urban public infrastructure, which is shown in Table 2; x
i
i
=
6,7
,
…
,
12 is the output index of urban public infrastructure, which is shown in Table 3.The multiple linear regression model is employed by using SPSS, and the test results are shown in Tables5 and 6. It can be seen that the results show a high consistency. For example, the R achieves 0.909, the R square achieves 0.826, and the adjusted R square achieves 0.704. In addition, the Analysis of Variance (ANOVA) shows that the P value is 0.000: that is to say, the DEA game cross-efficiency model passes the significance testing. Hence, the results show that the DEA game cross-efficiency model is valid in the problem of evaluating the infrastructure investment efficiency.Table 5
The summary of multiple linear regression model.
Model
R
R square
Adjusted
R square
Std. error of the estimate
1
.909a
.826
.704
.00052
aPredictors: (constant) ×12, ×6, ×7, ×11, ×3, ×5, ×9, ×4, ×2, ×8, ×10, and ×1.Table 6
The ANOVAb of multiple linear regression model.
Model
Sum of squares
df
Mean square
F
Sig.
1
Regression
.000
12
.000
6.741
.000a
Residual
.000
17
.000
Total
.000
29
aPredictors: (constant) ×12, ×6, ×7, ×11, ×3, ×5, ×9, ×4, ×2, ×8, ×10, and ×1.
bDependent variable: y.


## 4. Results and Discussions

In this section, we apply the DEA game cross-efficiency model presented in the previous sections to analyze the efficiency and the assignment weight of the urban public infrastructure investment.The game cross-efficiency and its ranking can be seen in column 5 and column 6 of Table4. It shows that the top ten cities are mainly from west region, and the bottom ten cities are mainly from east region. However, it is the east region that mainly controls the assignment weight. One possible reason for that could be that the cities in east region are economically advanced ones needing urban infrastructure of high level. Nevertheless, these cities have high population density and serious environmental pollution, which causes difficulties in improving the performance of economy, society, and environment simultaneously. Under this case, one of the options to improve the comprehensive efficiency is to increase investment. Comparing with the cities in east region, the cities of the west region have lower population density, and less investment could keep high comprehensive efficiency. As a result, the investment ratio is relatively low while the overall efficiency value of input-output is higher than the east cities.From column 6 in Table4, Yinchuan is more efficient than other cities. However, it should be noted that Yinchuan is in the underdeveloped west area. Although its input value is small, the output value is relatively large, and then the game cross-efficiency scores are better than the other cities. According to the third column of Table 4, the allocation weight of infrastructure investment of Chongqing takes the largest proportion, followed by Beijing, Hangzhou, and Shanghai.According to regional economic situation, the Chinese cities can be divided into four parts. The first part is the east region, including Beijing, Tianjin, Shijiazhuang, Shanghai, Nanjing, Hangzhou, Fuzhou, Jinan, Guangzhou, and Haikou. The second part is the central region, including Taiyuan, Hefei, Nanchang, Zhengzhou, Wuhan, and Changsha. The third part is the west region, including Hohhot, Nanning, Chongqing, Chengdu, Guiyang, Kunming, Xi’an, Lanzhou, Yinchuan, Xining, and Urumqi. The fourth part includes three provinces in the northeast of China, that is, Shenyang, Changchun, and Harbin.This paper analyzes the convergence program of game cross-efficiency. For plotting convenience, we divide the east region cities and the west region cities into two parts and draw iterated graphs, respectively (seen in Figures2–7). With further comparison, it shows that the iterative curves of most cities in west region are better than the rest of the cities. However, the differences between cities are obviously large. Figures 4 and 5 show that the comprehensive efficiency of Urumqi and that of Hohhot are worse than others in every iteration, and the comprehensive efficiency of Xi’an and that of Yinchuan are better than other cities. The iterative processes of east region cities (seen in Figures 2 and 3) are relatively more centralized. Figures 6 and 7 show the iteration of central region cities and the three provinces in the northeast of China, respectively. It shows that even though there is not a big gap, the average iterative process is worse. The iteration diagrams indicate the imbalanced circumstance of urban infrastructure investment in China. In order to construct the urbanization better, the national government should invest reasonably in cities according to the complexity of the cities.Figure 2
Game cross-efficiency calculation for the east region cities.Figure 3
Game cross-efficiency calculation for the east region cities.Figure 4
Game cross-efficiency calculation for the west region cities.Figure 5
Game cross-efficiency calculation for the west region cities.Figure 6
Game cross-efficiency calculation for the central region cities.Figure 7
Game cross-efficiency calculation for Shenyang, Changchun, and Harbin.


## 5. Conclusions

In this paper, we use a DEA game cross-efficiency model to evaluate the urban infrastructure investment efficiency and find a scientific method to distribute the fund. We figure out a Nash equilibrium point and solve the nonunique problem. The final scores from the DEA game cross-efficiency show that the investment from different infrastructure subsystems is related to many factors influencing the economic, society, and environment developments.Though we have concentrated on the DEA game cross-efficiency model widely used in efficiency evaluation and fund allocation, there are limitations of our analysis when the comprehensive efficiency is supposed to include economy, society, and environment. A city has its own complex infrastructure system that is different from others, but this paper only provides a relative evaluation of the 30 cities in terms of their investment amount. Future researches could establish nonquantitative DEA game cross-efficiency model to explain the relationship between the efficiency and the infrastructure subsystems.



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*Source: 9814313-2016-03-08.xml*