Abstract—Most of the literatures on rent-seeking contest focus on the simultaneous contest, i.e. effort levels of contestants are chosen simultaneously. But in reality, effort levels are chosen sequentially, especially when lobbying actions are done publicly. This paper discusses the dynamic consistency of rent-seeking contest. The result of the paper shows that if there are only two contestants, the rent-seeking contest is dynamic consistent, and if there are more than two contestants, it is not. So if there are only two contestants, we do not need to pay our attention on the action sequence of the contest. But if there are more than two contestants, action sequence of the contest must be carefully dealt with.

I. INTRODUCTION

OLLOWING the seminal contributions by Tullock [1]�[2], Krueger [3] and Posner [4], the literature on

rent-seeking analyzes contests in which a number of contestants expend resources to win a “prize” such as a government contract, monopoly license, public good provision or regulatory decision. Much recent work has focused on aspects of multiple winner contest [5], contest with endogenous coalition formation process [6], two-sided contest [7], group competition contest [8]�[9], simultaneous versus sequential contests and the dissipation of rent-seeking [10]�[11].

All the literatures on rent-seeking focus on simultaneous contests, or contestants move simultaneously in stage contest. Obviously simultaneous contests are the most popular phenomena in rent-seeking because the effort of rent-seeking contest, such as lobbying, is usually not public to the society. But in reality, contestants may take action sequentially and use spies to detect their rivals’ lobbying efforts. In this case sequential contest is more suitable to describe the contest process. So why we also assume contestants move simultaneously in these contests?

The motivation of this paper is, therefore, to discuss the dynamic consistency of contest. If contestants take the same action between simultaneous contest and sequential contest, we say the rent-seeking contest is dynamic consistent, otherwise it is not. For our purpose, we model a contest

Changlin Wu is with the Institution of Systems Engineering, Huazhong

University of Science and Technology, Wuhan, China. (e-mail: wuchanglin77@sina.com)

Changchen Liu is with the School of Economics, South–Central University for Nationalities, Wuhan, China. (corresponding author: tel: 8613487072449; e-mail: ccliufree@163.com)

Yunfeng Luo is with the Institution of Systems Engineering, Huazhong University of Science and Technology, Wuhan, China. (e-mail: yfluo@hust.edu.cn)

Yan Li is with the Institution of Systems Engineering, Huazhong University of Science and Technology, Wuhan, China. (e-mail: 54liyaneco@hust.edu.cn)

consisting n contestants who can choose to move simultaneously or sequentially. The result shows that when there are only two contestants to engage in contest, the contest is dynamic consistent, otherwise it is not.

The remainder of the paper is organized as follows. Section 2 lays out the model of the paper. Section 3 discusses the dynamic consistency of rent-seeking contest with only two contestants. Section 4 discusses the dynamic consistency of rent-seeking contest with three contestants. Section 5 is the conclusion of the paper.

II. THE MODEL

A. Simultaneous contest

Consider a rent-seeking contest as a game between n homogeneous contestants who must exert efforts to win a prize that has a value normalize to 1 to all. Each contestant chooses a level of effort 0ie ≥ for 1, 2,...,i n= . The effort

level of all contestants is denoted 1 2( , ,..., )ne e e e= . The effort

level chosen by the contestants determine a portion ( )i eρ for

each contestant i . We interpret a “portion” in one of two ways–either as a fraction of a divisible good or as the probability of receiving a non-divisible good. By the expected theorem, these interpretations are mathematically equivalent if all contestants are risk neutral. We assume that the benefit a contestant receives is linear in the contestant’s portion of the prize and the cost a contestant pays is linear in effort. So we can formulize the contestants’ expected utility as following

( ) ( )i i iu e e eρ= − (1)

The portion of ( )i eρ is given by

0( )

1/ otherwise

ij

ji

eif e

ee

n

ρ

�>�

= ���

��

Then the expected utility of contestant i is

0( )

1/ otherwise

ii j

ji

ee if e

eu e

n

�− >�

= ���

�� (2)

As contestants can always choose nothing to do, so 1ie ≤

for 1, 2,...,i n= . If all contestants move simultaneously, then the contest has

a symmetric equilibrium 1 2 2

1... n

ne e e

n

−= = = .

B. Sequential contest

Suppose that the time of exert effort is chosen by

Dynamic Consistency of Rent-seeking Contest

Changlin Wu, Changchen Liu, Yunfeng Luo, and Yan Li

F

Fourth International Workshop on Advanced Computational Intelligence Wuhan, Hubei, China; October 19-21, 2011

978-1-61284-375-9/11/$26.00 @2011 IEEE 697

contestants, i.e. all contestants not only choose their effort levels but also choose the time to exert effort. There are two cases need to be considered. For case 1, all contestants exert efforts simultaneously and the result is the same as that of chapter II.A. For case 2, some contestants choose their effort levels after observing some other contestants’ effort level. We will pay our attentions on the case in which contestants exert efforts sequentially.

III. SEQUENTIAL CONTEST WITH TWO CONTESTANTS

Suppose that in this contest, contestant 1 moves first and contestant 2 moves next. In his decision point, contestant 2 can observe contestant 1’s effort level 1e . He chooses an

effort level 2e to maximize his expected utility. So contestant

2’s strategy is a function from [0,1] to [0,1] , i.e.

2 : [0,1] [0,1]e → .

We can use adverse selection to solve it. Given 1e ,

contestant 2 chooses 2 1( )e e such that

2

22 1 2 1 2 2

0 1 2

( ) arg max ( , ) [ ]e

ee e u e e e

e e≥

∈ = −+

(3)

If 1 0e = , then contestant 2 can choose 2 1( )e e as an

arbitrary small number to win the prize with probability 1. If

1 0e > , we must solve 2 1( )e e through the first order

condition. According to the first order condition, the following

equation holds

2 1 1 1( )e e e e= − (4)

(4) is contestant 2’s reaction function. The reaction function tells us the effort level contestant 2 will choose when he observes the effort level of contestant 1.

Now we come to the problem faced by contestant 1. Contestant 1 knows that if he chooses effort level 1e ,

contestant 2 will choose effort level 2 1( )e e . So when

contestant 1 chooses effort level 1e , his expected utility is

1 1 2 1 1 1( , ( ))u e e e e e= − (5)

When 1

1

4e = , contestant 1’s expected utility is

1

4, which

is maximum to him. According to (4), when 1

1

4e = , 2

1

4e = ,

equivalent to the effort level chosen by contestant 2 when he moves simultaneously with contestant 1. So when there are only two contestants, rent-seeking contest is dynamic consistent.

IV. SEQUENTIAL CONTEST WITH THREE CONTESTANTS

We discuss the case when 1 and contestant 2 move first and move simultaneously, contestant 3 moves next.

We first solve the problem of contestant 3. Given 1 2,e e ,

contestant 3 chooses 3 1 2( , )e e e such that

3

33 1 2 3 1 2 32

03

0

( , ) arg max ( , ) [ ]e

jj

ee e e u e e e

e e≥

=

∈ = −

+� (6)

If 2

0

0jj

e=

=� , then contestant 3 can choose a 3 1 2( , )e e e

which is arbitrary small to win the prize with probability 1. If 2

0

0jj

e=

=� , we must solve 3 1 2( , )e e e through the first order

condition. According to the first order condition, the following holds

2 2

3 1 20 0

( , ) j jj j

e e e e e= =

= −� � (7)

(7) is contestant 3’s reaction function. The reaction function tells us the effort level contestant 3 will choose when he observes the effort level of contestant 1 and 2. Now we come to the problem faced by contestants 1 and 2. Given contestant 3’s reaction function, contestant 1 and 2 knows that if they choose effort level 1e and 2e , contestant 3 will

chooses effort level 3 1 2( , )e e e . So when contestant 1 and 2

choose effort level 1e and 2e , their expected utilities are

11 1 2 3 1 2 1

1 2 1 2 1 2

11

1 2

( , , ( , ))( )

eu e e e e e e

e e e e e e

ee

e e

= −+ + + − +

= −+

(8)

22 1 2 3 1 2 2

1 2

( , , ( , ))e

u e e e e e ee e

= −+

(9)

Equation (8) and (9) has a unique solution 1 2

9

32e e= = .

According to Equation (7), 3

3

16e = . In this case, the total

effort is 3

4ie =� .When contestants move simultaneously,

1 2 3

2

9e e e= = = . the total effort is

2

3ie =� , which is

smaller than the case when contestants move sequentially. So when two contestants move first, the third contestant moves next, the result is not the same as the result when contestants move simultaneously, and rent dissipation is larger than the case when contestants move simultaneously.

When 1 contestant moves first, the other two contestants

move next, the equilibrium result is 1

3

8e = , 2 3

3

16e e= = . In

this case, the total effort is 3

4ie =� . The result is not the

same as the result when contestants move simultaneously. So when there are 3 contestants, rent-seeking contests is not dynamic consistent.

698

V. CONCLUSION

Dynamic consistency of rent-seeking contest cares about the relation between the contest result and the action sequence of rent-seeking. In traditional literatures of rent-seeking contest, authors suppose that contestants exert effort simultaneously. But rent-seeking is a long lasting process, it is hard to let contestants take action simultaneously, especially when the lobbying action is taken publicly or when contestants can use spies to detect their rivals action. If contestants take action sequentially, will the result of the contest change? If the result of the contest does not change when contestants change their action sequence, the contest structure is immune to the action sequence of the contestants. So it is easy to implement the contest. The result of the paper shows that if there are only two contestants, the rent-seeking contest is dynamic consistent, but if there are three contestants, the rent-seeking contest is not dynamic consistent. Furthermore, it is easy to extend the dynamic inconsistent result to more than 3 contestants. The result means that if there are only two contestants, the contest designer does not need to pay his attention on the action sequence of the contest. But if there are more than two contestants, action sequence of the contest must be carefully dealt with.

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