## Secure Implementation in Queueing Problems

### Katsuhiko Nishizaki

Graduate School of Economics, Osaka University, Toyonaka, Japan Email: [email protected]

Received August 3, 2012; revised September 5, 2012; accepted October 7, 2012

## ABSTRACT

This paper studies secure implementability (T. Saijo, T. Sjöström and T. Yamato, “Secure Implementation,” *Theoreti- *

*cal Economics*, Vol. 2, No. 3, 2007, pp. 203-229) in queueing problems. Our main result shows that the social choice

function satisfies strategy-proofness and strong non-bossiness (Z. Ritz, “Restricted Domains, Arrow-Social Welfare Functions and Noncorruptible and Non-Manipulable Social Choice Correspondences: The Case of Private

Alterna-tives,” *Mathematical Social Science*, Vol. 4, No. 2, 1983, pp. 155-179), both of which are necessary for secure imple-

mentation, if and only if it is constant on the domains that satisfy weak indifference introduced in this paper. Weak in- difference is weaker than minimal richness (Y. Fujinaka and T. Wakayama, “Secure Implementation in Economies with

Indivisible Objects and Money,” *Economics Letters*, Vol. 100, No. 1, 2008, pp. 91-95). Our main result illustrates that

secure implementation is too difficult in queueing problems since many reasonable domains satisfy weak indifference, for example, convex domains.

**Keywords:** Secure Implementation; Dominant Strategy Implementation; Nash Implementation; Strategy-Proofness;

Queueing Problems

## 1. Introduction

In this paper, we consider **queueing problems** of allo-

cating positions in a queue to agents, each of whom has a constant unit waiting cost, with monetary transfers. Ex- amples of such problems are the use of large-scaled ex-

perimental installations, event sites, and so forth1_{. }

**Strategy-proofness** is a standard property for non-

manipulability: The truthful revelation is a weakly
domi-nant strategy for each agent. However, the strategy-
proof mechanism might have a Nash equilibrium which
induces a non-optimal outcome. This problem is solved
by **secure implementation** (Saijo, *et al.* [2]), that is,

dou-ble implementation in dominant strategy equilibria and

Nash equilibria2_{. Previous studies illustrate how difficult }

it is to find desirable and securely implementable social

choice functions: Voting environments (Saijo, *et al.* [2];

Berga and Moreno [4]), public good economies (Saijo,

*et al.* [2]; Nishizaki [5]), pure exchange economies (Mi-

zukami and Wakayama [6]; Nishizaki [7]), the prob-lems of providing a divisible and private good with

monetary transfers (Saijo, *et al.* [2]; Kumar [8]), the

problems of allocating indivisible and private goods with monetary transfers (Fujinaka and Wakayama [9]), Shapley-Scarf housing markets (Fujinaka and

Waka-yama [10]), and allotment economies with single-peak- ed preferences (Bochet and Sakai [11]).

This paper is most closely related to the one written by Fujinaka and Wakayama [9]. They show a constancy result on secure implementation when the domain satis- fies minimal richness (Fujinaka and Wakayama [9]). Our model is a special case of their one and have many rea- sonable domains which do not satisfy minimal richness. On the basis of this fact, we study the possibility of se- cure implementation in queueing problems. Unfortunate- ly, our main result shows that only constant social choice functions satisfy strategy-proofness and strong non-bos- siness (Ritz [12]), both of which are necessary for secure

implementation, on the domains satisfy **weak **

**indiffer-ence**, which is weaker than minimal richness, introduced

in this paper.

This paper is organized according to the following sections. In Section 2, we introduce our model, properties of social choice functions, and domain-richness condi- tions. We show our results in Section 3. Section 4 con- cludes this paper.

## 2. Notation and Definitions

Let *I*

1, ,*n*

*n*2

be a set of**agents**. Let

*n*

*i i I* *I*

_{} be a **queue**, where, for each *i**I*, _{i}

is the **position for agent in the queue *** i* and for

each *i j*, *I* with , _{i}_{j}

1_{For queueing problems, see Suijs [1] and so forth. }

2_{This concept is considered to be a benchmark for constructing }

mecha-nisms which work well in laboratories. See Cason, *et al.* [3] for
ex-perimental results.

.

*i* *j* For each *i**I* , let

where is a **monetary transfer for agent ** . Let

be a profile of monetary transfers and be a profile of consumption bundles,

called an **allocation**. Let

*i*
*t*

*i I*

*t* *I*

**i**

*n*

*i*

*t* *t*

,_{ }

*n*

_{}

*n*

,

*Z* *t*

*i*

is a queue and

*n* *n*
*k*
*k I*
*I*

*t*0

be the set of **feasible allocations**.

For each *I* , let *c _{i}*

_{}

*x*

*x*> 0

be a**unit waiting cost for agent** and *Ci* be a set

of unit waiting costs for agent . For each *i*

* i*

*i* *I*, let

*i* be the **utility function for agent **

such that for each and each

:

*u I* *C* **i**

*i*,

*ti*

*I*

*ci*

*Ci*,

*i*, ;

*i*

*i*

*i*1

*i*

*i*.

*u* *t c* *c* *t*

Let *i I* *i* be the **domain** and *i* be

a profile of unit waiting costs. For each

*C*

*C*

_{i}*I*
*c* *c* _{} *C*

*I*

*i* , let

*i* *j i* *j*

*j i* be a profile of unit waiting

costs for agents other than agent . *i*

*j*

:

*c* *c* *C*

*C*

*i*

Let *f C*

*c**C*

*Z*

*c t*,

be a **social choice function**. For each

, let be the allocation associated

with the social choice function

*c* *Z*

*f* at the profile of unit

waiting costs and be the con-

sumption bundle for agent in the allocation

.

*c*

*c*

*c*

*i*

,

*i* *c ti* *I*
*I*

,

*c t*

Saijo *et al.* [2] show that strategy-proofness and strong

non-bossiness are necessary for secure implementation.

**Definition 1 **The social choice function *f* satisfies
**strategy-proofness** if and only if for each *c c*, *C* and

each *i**I*,

, 1 ,

, 1 ,

*i* *i* *i* *i* *i* *i* *i*

*i* *i* *i* *i* *i* *i* *i*

*c c* *c* *t c c*

*c c* *c* *t c c*

**Definition 2 **The social choice function *f* satisfies
**strong non-bossiness** if and only if for each *c c*, *C*

and each *i**I*, if

, 1 ,

, 1 ,

*i* *i* *i* *i* *i* *i* *i*

*i* *i* *i* *i* *i* *i* *i*

*c c* *c* *t c c*

*c c* *c* *t c c*

*c c*

then

*i*,

*i*

,*t c ci*,

*i*

*c ci*,

*i*

,*t c c*

*i*,

*i*

Fujinaka and Wakayama [9] show a constancy result on secure implementation when the domain satisfies minimal richness.

### Definition 3 The domain satisfies minimal rich-

**ness** if and only if for each

*C*

*i**I*, each *c ci* , *i* *Ci*, each

,

*i* *i* *I*

*i*

_{i}**

_{i}

, and each if , then there exists,

*T*

*i*

*i*

*i*

*T* *c* *ci**Ci*

such that

1)

and

*ci*

*T*

*i* *i*

*i*

*i*

2) *ci*

*i*

*i*

*ci* for each

*i*

*I*\

*i* ,

*i*

.The following example shows that many reasonable

**Example 1 ** Let *i* *I*

domains do not

and *Ci*. Moreover, let

3,*ci* 1, *i* 1, *i* 2 and*T* 2

*i*

*c* . In this case, we

have

**

_{i}* *

_{i}

*c*1

_{i}*T*3

* *

_{i}

_{i}

*c*. Let

_{i}*c*

_{i}*C*

_{i}be such that

*i*

*i*

*ci*

*T*, that is,

*ci*2. This

implies that con finition 3 h On the

other hand, if *c _{i}* 2

dition 1) in De olds.

, then

*i*

*i*

*ci* 2 3

*i*

*i*

*ci* for

*i*3 . This

h re im

### k indif- fe

implies that condition 2) in Definition 3 does not old. Our main result implies a constancy result on secu plementation when the domain satisfies weak indif- ference which is weaker than minimal richness.

**Definition 4 **The domain *C* satisfies **wea**

**rence** if and only if for each *i**I*, each ,*c ci* *i* *Ci*,

each *i* , *i* *I*, and each *T*, if

*i*

*i*

*ci*

*T*

*i*

*i*

*ci*, then there exists

*ci*

*Ci*

such that

*i*

*i*

*ci*

*T*.

**Remark 1 **In our model, weak indifference is equi-

va

## 3. Results

lent to convexity3_{. }

For simplicity of notation, let *i* *i*

*c ci*,

*i*

,

,

*i* *i* *c ci* *i*

_{} , *i**i*

*ci*,

*c*

*i*

,

*c ci*,

*i*

*i* *i*

,

,

*i* *i* *ci* *c*

*i* and *ti* *ti*

*c ci*,

*i*

,*ti*

*ti*

*c ci*,

*i*

,

,

*i* *i* *i* *i*

*t**t c c* _{} , *t _{i}*

*t*

_{i}

*c*,

_{i}*c*

_{}

_{i}

,*t*

_{i}*t c*

_{i}

*,*

_{i}*c*

_{}

_{i}

foreach *c c*, *C* and each *i**I*.

### 3.1. Preliminary Results

that the social choice func-

onetary transfer de

*d each*

In this subsection, we assume

tion *f* satisfies strategy-proofness.

Lemma 1 shows that each agent’s m

pends on her position in the queue given unit waiting costs for other agents. Since the proof is similar to Fuji-naka and Wakayama [9], it is omitted.

**Lemma 1 For each** ,*c c**C* *an* *i**I* , *if *

*i* *i*

,* then* *ti* *ti*.

Lemma 2 shows that if su

there exists a unit waiting cost ch that some two different consumption bundles are indifferent in terms of utility level, then the position associated with the unit waiting cost is in between the two positions. In Lemma 2, we use the following notation:

for each *i**I*, each *c _{i}*

*C*, each

_{i}

*,*

_{i}*t*

_{i}

*I*, and

each *i* *I*, let **t****i**

*i*

*i*

*i*

,*ci*

.**Lemma 2** *Fo*

*i* *i* *ci* *t*

* and each i* *I*

; ,

*ach c c*,

*i*
*t*

*r e* *C* ,* if *

*i* *i*

* and there exists * *ci**Ci such that *

*i*1

*ci*

*ti*

*i* 1

*ci*

*ti*

*, then **i* *i**i*.

here

ex

*Proof.* Suppose, by contradiction, that t ist

,

*c c**C* and *i**I* such that * _{i}*

*,*

_{i}

*i*1

*ci*

*ti*

*i* 1

*ci*

*ti*

for some *ci**Ci*, and

3_{For the relationship among weak indifference and certain }

*i* *i*

or *i**i*. We consider the case of *i**i*.

ypothe e have

By the h sis, w

.
*i* *i*
*i*
*i* *i*
*t* *t*
*c*

, we have

(1)

By the definition of **t****i**

; ,*t*,

*c*

**t**

.

*i* *i* *i* *i* *i*

*i* *i*
*t*

the defi of and **strategy-pr**

*c*

(3) and

*i*
*c* **i**

nition

**t**

uations

(2)

**oofness**, we

(3)
By
ha
By Eq
**i*** t*
ve

_{;}

_{,}

_{,}

_{.}4

*i* *i* *i* *i* *i*

*t* * i*

*t*

*i* *i*

, we have *ci**ci*.

Since we consider the case of *i**i*, this implies

; ,*t*,

*c*

,*t*

,*c*

.**t****i***i* *i* *i* *i* * ti*

*i*;

*i*

*i*

*i*(

definition of

4)

, we have

and **strategy-proofnes**

By the **t**_{i}**s**

; ,*t*,

*c*

*t*

;

,*t*

,*c*

. This is a con-we have a con-

tradiction to **strategy-proofness** in the case of *i* *i*

*i* *i* **i***i* *i* *i* *i*

**t****t**

4). Similarly,

*i* *i*
**i**

Equatio*i* n (

tradiction to

. ■

**3.2. Ma**

“o

*i**I*

### in Result

e the “if”

*i* *i*

by cont

**Theorem 1 **Suppose that the domain * satisfies *

part is obvious, we only

*C*
*f*

*weak *

prove the

r each

*indifference*. *The social choice function * * satisfies stra- *
*tegy-proofness and strong non-bossiness if and only if it *
*is constant*5.

*Proof.* Sinc

nly if” part. Let ,*c c**C*. Firstly, we show

*c c*, ,

*t c c*,

*c c*,

,*t c c*,

foxists *j* *I*

*i* *i* *i* *i* *i* *i*

. Suppose, radiction, that there e

, , s impli

such that*c c*

**strong**

*,*

_{j}

_{j}

*t c c*

### bossiness

, , ,*j* *j* *c c**j* *j* *t c c**j*
** non-** **rategy-proofness**, t

. By es

*j*

hi

and **st**

1 1 .

*j* *cj* *tj* *j* *cj* *tj*

(5)

By Lemma 1, this implies *j* *j* or *j* *j* .

,

*j* *j*

, it also Si

**n**

nce

*c c*,

,*t c c*,

,*t c c*

,by **stro**

implies

*j* *j*

**g n** and **st** **roof**

,
*j* *j*
*c c*
**rategy-p**
*j* *j*
**on-bossiness**

**ness**

1 1 .

*j* *cj* *tj* *j* *cj* *tj*

(6)

ns (5) an

Equatio d (6), we have

By

*j*

*j*

*j*

*j*

*j*

*j*

, this implies tha

*j* *j*

*c* *t*

**ence**

*t* *c*

. Si

t there exi

**weak indiffer**

nce *C*

sts

satisfies

*j* *j*

*c* *C*

such that

*j*1

*c*

*j*

*tj*

*j*1

*c*

*j*

*t*

*j*.

(7)

We consider the case of *j* *j*. In th

Le

is case, by

mma 2, we have *j* *j**j* . If *j* *j* or

*j* *j*

, then, by Equatio nd **st** **on- **

**bossiness**, we have

n (7) a **rong n**

*j* *j*

. This is a contradiction.

Therefore, we know

.

*j* *j* *j*

By applying the above argument to the left inequality

repeatedly, we can find *c c** _{j}*,

**

_{j}*C*such that

_{j}**

_{j}

_{j}and

*j*1

*cj*

*tj*

*j*1

*cj*

*tj*

, wher

exist

between

e there

s no position * _{j}* and

_{j}_{ induced by a }

unit waiting cost for agent *i* given *c*_{}* _{j}* In this case, we

have or

.

*j* *j* *j* *j*

_{} _{}_{. By }

**stron** **non-bossiness**,

these imply _{j}_{}_{j}_{. } **g **

tradictio

This is a contradiction. Similarly,

we have a con n in the case of * _{j}*

*.*

_{j}Without loss of generality, let *i*1. Ther ore, we ef

have

*c c*1, 1 ,

*t c c*1, 1

*c c*1, 1

,*t c c*1, 1

. (8)By the same argument stated above, we also have

*c c c*, , ,

*t c c c*, ,

1 2 1,2 1 2 1,2

1, ,2 1,2 , 1, ,2 1,2 ,

*c c c* *t c c c*

(9)where *c*1,2 is a profile of unit waiting costs for agents

an a

other th gents 1 and 2. By Equations (8) and (9), we

have

1 2 1,2 1 2 1,2

1 2 1,2 1 2 1,2

, , , , ,

, , , , ,

*c c c* *t c c c*

*c c c* *t c c c*

.

By sequentially replacing by *cj* *c**j* for each *j*1, 2

in this manner, we finally prove

*c t c*,

*c* ,

*t c*

. ■rem

**Remark 2 **The above theo does not depen

fin

d on the iteness of the number of positions, which is used to prove Claim 3 in Proposition 1 of Fujinaka and Waka- yama [9].

Obviously, constant social choice functions are se- curely implementable. Therefore, by bringing the above theorem together with a characterization of securely

implementable social choice functions by Saijo *et al.* [2],

we have the following constancy result on secure im- plementation.

**Corollary 1 Suppose that the domain satisfies weak **
*indifference. The social choice function is securely im- *
*plementable if and only if it is constant*.

**Remark 3 **In our model, Maskin monotonicity is not

st 6

4_{Note that the equality does not hold. If it holds, then we have }

ronger than strategy-proofness . This relationship im- plies that our main result is established by secure im-

*i* *i*

*c* *c*
by Equations (1) and (2). This implies that f should be a
correspon-dence since we consider the case of *i* *i*.

5_{For the tightness of this theorem, see Examples 2, 3, and 4 in }

plementability but not by Nash implementability.

**Remark 4 **Saijo [14] shows the following constancy

re

ch uch that

sult on “Nash” implementation: The social choice function satisfies Maskin monotonicity and dual do- minance (Saijo [14]) if and only if it satisfies constancy. In line with such domination, Fujinaka and Wakayama [9] show the following constancy result on “secure” imple- mentation: The securely implementable social choice func- tion satisfies non-dominance (Fujinaka and Wakayama [9]) if and only if it satisfies constancy. Note that non-

dominance is weaker than dual dominance7_{. In our model, }

similar to the relationship between minimal richness and weak indifference, we have a constancy result on secure implementation by a weaker condition than non-domi- nance as follows: for each

,*t*

, ,*t*

*f C*

, eaand

,

*c c* *C* s

*c* ,

*t c*

,*t*

,

,*c* *C* such tha

,

*c* *t*

t

*c* *t*

no

and each *i**I*

*i*

, if there exists

, , , ,

*i* *i* *i* *i* *i*

*L* *t* *c* *L* *t* *c* and

*,*

_{i}*,*

_{i}

_{i}

*,*

_{i}

*L* *t* *c* *L* *t _{i}* ,

*c* , then there exists

_{i}*ci*

*Ci*

such that

*i*

*i*

*ci*

*ti*

*ti*, where

, , , 1

1

*i* *i* *i* *i* *i* *i* *i* *i*

*i* *i* *i*

*L* *t* *c* *t* *I* *c* *t*

*c* *t*

for each *i**I*, each *c _{i}*

*C*and each

_{i}

*,*

_{i}*t*

_{i}

*I*.

## 4. Conclusion

secure implementability in queueing

5. Acknowledgements

.A. thesis presented to the

REFERENCES

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This paper is based on my M

Graduate School of Economics, Osaka University. The author would like to thank an anonymous referee, Ta- tsuyoshi Saijo, Masaki Aoyagi, Yuji Fujinaka, Kazuhiko Hashimoto, Shuhei Morimoto, Shinji Ohseto, Shigehiro Serizawa, Takuma Wakayama, and seminar participants at the 2008 Japanese Economic Association Spring Meeting, Tohoku University, for their helpful comments.

The author is especially grateful to Tatsuyoshi Saijo, Yuji Fujinaka, and Takuma Wakayama for their valuable advices. The author acknowledges for the Global COE program of Osaka University for the financial supports. The responsibility for any errors that remain is entirely the author.

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