以賽局理論解決IEEE 802.22網路基地台間的共存問題

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୯ҥᆵ᡼εᏢႝᐒၗૻᏢଣႝߞπำᏢࣴز܌

ᅺγፕЎ

Graduate Institute of Communication Engineering College of Electrical Engineering and Computer Science

National Taiwan University Master Thesis

аᖻֽ౛ፕှ، IEEE 802.22 ᆛၡ୷ӦѠ໔ޑӅӸୢᚒ A Game Theoretic Resource Allocation for

Inter-BS Coexistence in IEEE 802.22

࢒։ᑣ Chun-Han Ko

ࡰᏤ௲௤Ǻ᚟ֻӹ റγ Advisor: Hung-Yu Wei, Ph.D.

ύ๮҇୯ 98 ԃ 6 Д June, 2009

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ᄔा

аᖻֽ౛ፕှ، IEEE 802.22 ᆛၡ୷ӦѠ໔ޑӅӸୢᚒ

࢒։ᑣ

ࡰᏤ௲௤Ǻ᚟ֻӹ റγ

IEEE 802.22ࢂಃ΋ঁаགޕคጕႝࣁਥ୷ޑคጕ೯ૻ኱ྗǶӧፕЎ္Ǵךॺஒ٩Ᏽ IEEE 802.22ᆛၡ୷ӦѠӅӸᐒڋ(inter-BS coexistence mechanism)ύޑ୏ᄊၗྍચၛ (dynamic resource renting and offering)ᆶ ᓎ ၰ ᝡ ݾ (adaptive on demand channel contention)Ǵගр΋ঁᓎ᛼Ӆ٦ᄽᆉݤǴځҞޑࣁၲډനԖਏᆶനϦѳޑᓎ᛼ϩଛǶ ᖻֽ౛ፕ(game theory)ஒ೏ᔈҔܭϩ݋ךॺޑ೛ीǶ२ӃǴ೸ၸฝკБݤǴךॺჹٿ

ঁ୷ӦѠޑس಍຾Չϩ݋Ǵ٠வύளډΑ΋٤ᢀჸ่݀ǶᏵԜᢀჸǴךॺёᏤ᛾рn

ঁ୷ӦѠس಍ஒӸӧڼ೚֡ᑽ(Nash equilibrium)ǴЪӧڼ೚֡ᑽΠӚ୷ӦѠޑਏҔ(ջ ᓎ᛼ϩଛ)ஒࢂ୤΋ޑǶԜѦךॺ׳᛾ܴрǴѤঁᆶਏ౗ǵϦѳ࣬ᜢޑᑽໆࡰ኱Ǵхࡴ

നԖਏ౗ޑϩଛǵ࢙܎კന٫ϯǵу៾ε-λϦѳϩଛǵу៾КٯϦѳϩଛ฻Ǵӧڼ೚

֡ᑽΠ೿ԋҥޑǶനࡕǴаගрޑᓎ᛼Ӆ٦ᄽᆉݤࣁਥ୷Ǵךॺёа೛ीрќ΋ঁύ ѧ໣៾Ԅޑᓎ᛼ϩଛᐒڋǴ٬ள؂ঁ୷ӦѠӧаቚуਏҔࣁ߻ගΠǴ೿཮၈ჴӦᇥр Ծρޑሡ؃ǴӵԜനࡕஒၲԋനԖਏǵനϦѳޑᓎ᛼ϩଛǶ

ᜢᗖӷ: IEEE 802.22; inter-BS coexistence; credit token; game theory; Nash equilibrium;

strategy-proofness.

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Abstract

A Game Theoretic Resource Allocation for Inter-BS Coexistence in IEEE 802.22

Chun-Han Ko

Advisor: Hung-Yu Wei, Ph.D.

IEEE 802.22 is the first cognitive-radio-based wireless communication standard. We propose a spectrum transaction scheme for dynamic resource renting and offering (DRRO) and adaptive on demand channel contention (AODCC) in IEEE 802.22 inter-BS coexistence mechanism to achieve efficient and fair spectrum sharing. Game theory is applied to formulate and analyze the proposed spectrum sharing algorithm. We first analyze the simplest two-base- station (BS) game through a graphical method to gain insights for the solution. Then, the Nash Equilibrium of then-BS game is derived and the utility profile at the Nash equilibrium is shown to be unique. We prove several desirable properties, including allocative efficiency, Pareto optimality, weighted max-min fairness, and weighted proportional fairness, are attained at the Nash equilibrium. Lastly, we design a strategy-proof spectrum allocation mechanism based on the proposed spectrum sharing algorithm so that truthful strategies optimize each BS’s performance.

Keywords: IEEE 802.22; inter-BS coexistence; credit token; game theory; Nash equilibrium;

strategy-proofness.

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List of Figures

1.1 IEEE 802.22 Systems . . . 1

1.2 IEEE 802.22 Inter-BS Coexistence Mechanism . . . 2

2.1 System of AgentA and Three BSs . . . 6

2.2 Unit Spectrum Acquisition and Protection Price of BSi . . . 7

4.1 Best Response Functions and Nash Equilibrium in Trafﬁc Case 1 . . . 14

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List of Tables

2.1 Notations . . . 9 3.1 Spectrum Sharing Game Model . . . 12 5.1 Summary of Extension . . . 17

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Chapter 1 Introduction

IEEE 802.22 is the ﬁrst cognitive-radio-based standard operating over 54-862 MHz licensed TV bands [1]. IEEE 802.22 systems, as depicted in Figure 1.1, are composed of base stations (BSs) and consumer premise equipments (CPEs). IEEE 802.22 systems are enabled to opportunistically access the licensed spectrum bands on the premise of not interfering with the licensed users, e.g. TV stations and wireless microphones. Cognitive radio technique [2, 3] is applied to perform spectrum sensing [4]. Through spectrum sensing, vacant channels over which IEEE 802.22 systems can operate are discovered. However, an important issue arises under a common scenario that multiple IEEE 802.22 BSs operate in the same vicinity and cause severe interference. This issue is called inter-BS coexistence (or self-coexistence) in IEEE 802.22 standard. To address this issue, an inter-BS coexistence mechanism is deﬁned.

CPE CPE CPE

802.22 BS1

Wireless MIC

CPE

CPE CPE

CPE

802.22 BS2

TV Broadcaster

Wireless MIC

CPE

aster ca

CPE CPE CPE

CPE

802.22 BS3

Figure 1.1: IEEE 802.22 Systems

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Enough Channels Channel Selection Begins

1. Spectrum Etiquette

2. Interference-free Scheduling

3. Dynamic Resource Renting and Offering (DRRO)

4. Adaptive on Demand

Channel Contention (AODCC) Channel Selection Completes Y

Y N

N

N Enough Channels

Enough Channels

Y

Figure 1.2: IEEE 802.22 Inter-BS Coexistence Mechanism

1.1 IEEE 802.22 Inter-BS Coexistence Mechanism

In IEEE 802.22 standard, the inter-BS coexistence mechanism [1] consists of four stages:

spectrum etiquette, interference-free scheduling, dynamic resource renting and offering (DRRO), and adaptive on demand channel contention (AODCC), as illustrated in Figure 1.2.

Spectrum etiquette is the ﬁrst stage where BSs try to locally ﬁnd channels that their neighbor BSs cannot or do not use. If no spare channel is available under this rule, BSs will conduct interference-free scheduling.

In interference-free scheduling, BSs share the same channel by scheduling their trafﬁc in a non-interfering manner. It, however, can only occur on the premise that the BS who owns the channel agrees to share it with others. In words, if the owner needs to operate exclusively, interference-free scheduling cannot occur and the inter-BS coexistence mechanism must go to the next stage.

IEEE 802.22 uses credit token for DRRO and AODCC operations. The concept of credit token and its utilization for spectrum sharing are ﬁrst introduced in [5]. In IEEE 802.22, credit token is similar to money except that credit token can be frozen but cannot be transferred. Each BS is assumed to have a pre-given credit token budget. In DRRO, two

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entities are deﬁned: offeror is a BS who currently has unused resources; renter is a BS of the counterpart, who currently has an additional resource requirement. An offeror can offer its unused resources by broadcasting the offering information which includes the available resources and the minimum number of credit tokens (MNCT) required. Renters who hear the offering information can send renting requests which include the desired resources and the number of credit tokens (NCT) willing to pay. After receiving and comparing the renting requests, the offeror derives the best (in term of higher credit tokens) renters. These renters are granted to access their requested resources and NCT they are willing to pay is frozen.

AODCC is the ﬁnal stage of the mechanism. AODCC is triggered when BSs do not get enough resources through the previous three stages. AODCC is very similar to DRRO except that a channel owner, also called contention destination, now passively receives contention requests. When a BS, called contention source, selects a contention destination and makes a contention request, the channel contention procedure occurs at the contention destination.

The contention destination compares NCT the contention source is willing to pay with its MNCT required. If the former is larger, the contention destination shall release the requested resources and NCT the contention source is willing to pay is frozen; otherwise, the contention destination replies with rejection.

1.2 Related Work

Recently, game theory has been applied to model IEEE 802.22 operations. S. Sengupta et al. applied minority game theory to investigate the problem that whether a BS should stay at the present channel or switch to another channel [6]. They showed a mixed strategy Nash equilibrium existed and the mixed strategy space performed better than the pure strategy space in achieving optimal solution. D. Gao et al. modeled the DRRO mechanism as a progressive second price auction [7]. The utilization of this auction mechanism had a major beneﬁt that BSs would make their requests truthfully. D. Niyato et al. formulated the transaction of spectrum bands between licensed users and BSs by a sealed-bid double auction [8]. They also introduced a pricing mechanism to model the service between BSs and CPEs.

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Nash equilibrium was found through a numerical method. If interested in minority game theory, progressive second price auction, and double auction, readers can refer to [19, 20, 21].

In this thesis, we aim to ﬁnd a game theoretic solution for IEEE 802.22 inter-BS coexistence. Compared to [6], it concentrated on spectrum etiquette. Compared to [7], it investigated DRRO without taking credit token budgets into consideration. Compared to [8], it focused on service pricing rather than inter-BS coexistence. In contrast, we propose a spectrum sharing algorithm based on the IEEE 802.22 DRRO and ADOCC mechanisms. We formulate the problem with game theory and discover that a Nash equilibrium always exists.

The Nash equilibrium has some desirable properties, including allocative efﬁciency, Pareto optimality, weighted max-min fairness, weighted proportional fairness. Also, by adopting the allocation rule of the spectrum sharing algorithm, we design a strategy-proof mechanism which ensures efﬁciency and fairness at the truth-revealing dominant-strategy equilibrium.

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Chapter 2 Spectrum Sharing Scheme

2.1 System Model

The system we consider consists of an agent,A, and n BSs, BSi fori = 1, 2, ..., n. Agent A, serving like a marketplace, provides the centralized renting-and-offering and contention procedures for all BSs. Besides, AgentA offers spectrum using time O which is the vacant or to-be-utilized spectrum using time of licensed users. IfO is less than zero, “offering O”

means “retrieving −O.” Each BSi has a single orthogonal spectrum band, spectrum using time T , a credit token budget Bi, and a max trafﬁc requirement xi (in time) additional to T . All of these are assumed to be public information. In other parts of this thesis, we will use “spectrum” to denote spectrum using time for short. Figure 2.1(a) is an illustration of a system of AgentA and three BSs. Figure 2.1(b) is the corresponding max additional trafﬁc requirements.

2.2 Spectrum Sharing Algorithm

Founded on DRRO and AODCC in the IEEE 802.22 inter-BS coexistence mechanism, we propose a spectrum sharing algorithm. Initially, Agent A broadcasts that the renting-and- offering procedure starts and it wants to provide spectrum O. After broadcasting this information, Agent A receives the acquisition/offering requests from all BSs. According to the type of the request, each BS is called an acquirer or an offeror. Agent A collects the

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BS₁

BS₂ BS₃

Renting, offering, and contention

Agent A

(a) System Diagram

Spectrum Using Time 0

x1

x2

x3

BS1

BS2

BS3

O Agent A

T

(b) Max Additional Trafﬁc Requirements

Figure 2.1: System of AgentA and Three BSs

offered spectrum from the offerors and then assignsO and the collected offered spectrum to the acquirers, in decreasing order of the unit acquisition price, for the requested amount until exhaustion. (More details about the acquisition/offering requests and the unit acquisition price will be explained later.) When multiple acquirers have the same unit acquisition price and there is no enough spectrum for them, the amount assigned to them is assumed equal.

A BS, on the other hand, is assumed to aim to increase their spectrum. To increase the spectrum, each BS must use its credit tokens not only to acquire others’ spectrum but also to protect its originally owned spectrum from others’ contention. We assume every unit of the spectrum to be acquired and the spectrum to be protected is equally signiﬁcant for each BS. Hence the credit token budget should be fairly allocated. Speciﬁcally, after hearing the renting-and-offering information, each BS_i makes an acquisition/offering request,yi, which is the spectrum it claims to acquire ifyi > 0 or to offer if yi < 0. At the same time, the unit acquisition price for the spectrum to be acquired,[yi]⁺, and the unit protection price for the spectrum to be protectd, T − [−yi]⁺, are both determined to be equal to pi(y_i) = Bi

T + yi

as depicted in Figure 2.2. The function [·]⁺ gives a non-negative value. When Agent A receives the acquisition/offering requests, it assignsO and the offerors’ provided spectrum to the acquirers under the previously described renting-and-offering procedure. After the

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Spectrum

0 T Ty_i

B

i

i i

B Ty

Spectrum to be acquired Spectrum to

be protected

(a)yi> 0

Spectrum

0 Ty_i T

B

i

i i

B Ty

Spectrum to be protected

(b)yi≤ 0

Figure 2.2: Unit Spectrum Acquisition and Protection Price of BSi

renting-and-offering procedure, if the acquirers do not get enough spectrum, they will turn to contention.

While the contention procedure starts, Agent A ﬁrst collects the spectrum, T − [−yi]⁺, each BS_i wants to protect. The collected

T − [−yi]⁺

s are sorted in increasing order of the unit protection price. Afterwards AgentA assigns the sorted

T − [−yi]⁺

s to the acquirers for the inadequate amount in decreasing order of the unit acquisition price. The assignment ends if the unit protection price is greater than or equal to the unit acquisition price. Finally, Agent A returns the unassigned spectrum back to all BSs. When multiple acquirers have the same acquisition price and there is no enough spectrum for them, we assume the amount assigned to them is equal. When multiple BSs have the same protection price and their spectrum is assigned to others, we assume the assigned amount is equally afforded by these BSs. After both renting-and-offering and contention procedures ﬁnish, the credit tokens the acquirers spend for spectrum acquisition are frozen and data transmission begins.

Lastly, we show, in Table 2.1, the mathematical expressions of the spectrum BS_i acquires and offers in the renting-and-offering procedure and the spectrum BS_iacquires or loses in the contention procedure. The former ismin (yi, ri) where ri(y) is the amount BSi can acquire from renting. The latter is min

[yi− ri]⁺, ci

where [yi− ri]⁺ represents the inadequate amount after renting andcirepresents the amount BS_i can acquire (the ﬁrst term) or will lose from (the other two terms) contention. Then the total spectrum BS_i gains or loses in both procedures ismin (yi, ri) + min

[y_i− r_i]⁺, ci

. For simplicity, we can also usemin(yi, ti) to represent the total spectrum BS_i acquires or loses in both procedures whereti is the total spectrum BS_i can acquire (the ﬁrst term) or will lose (the last two terms.) In Lemma 2.1,

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we provemin (yi, ri) + min

[yi− ri]⁺, ci

= min(yi, ti). Besides, the frozen credit tokens of BS_i isPi(y) = p_i(y_i) [min(y_i, ti)]⁺. All other notations are summarized in Table 2.1 as well.

Lemma 2.1. min(yi, ri) + min([yi− ri]⁺, ci) = min(yi, ti) ∀yi ≥ −T and ∀i ∈ {1, ..., n}.

Proof. See Appendix A.

2.3 Problem Description

The problem we want to investigate is as below.

Problem: Given that Agent A provides the spectrum O and that the original spectrum T , the credit token budget Bi, and the max trafﬁc requirement xi of each BS_i are all public information, if the acquisition/offering request yi is constrained by−T and x_i, i.e. −T ≤ yi ≤ x_i, how does each BS_i make the acquisition/offering request in order to increase the spectrum?

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Table 2.1: Notations

AgentA and BSi,i = 1, 2, ..., n.

T Spectrum owned by each BS.

O Spectrum offered by AgentA.

Bi Credit token budget of BS_i.

xi Max trafﬁc requirement additional to T of BS_i.

yi Acquisition/offering request of BS_i. It is the spectrum BS_i claims to acquire or to offer.

pi(yi) Unit acquisition and protection price of BS_i. pi(yi) = _{T +y}^Bⁱ_i

min (yi, ri) Spectrum BSi acquires or offers in the renting-and-offering procedure.

ri(y) Spectrum BS_i can acquire from renting.

ri(y) = [yi]⁺

j;pj=pi

[yj]⁺

O +ⁿ

j=1[−yj]⁺−

j;pj>pi

[yj]⁺

₊

min

[y_i− r_i]⁺, ci

Spectrum BS_i acquires or loses in the contention procedure.

ci(y) Spectrum BS_ican acquire or will lose from contention.

c_i(y) = [y_i− r_i]⁺

j;pj=pi

[yj− rj]⁺

j;pj<pi

T − [−y_j]⁺

−

j;pj>pi

[y_j− r_j]⁺

₊

−

T − [−y_i]⁺

+

⎡

⎢⎢

⎣

T − [−yi]⁺

− T − [−yi]⁺

j;pj=pi

T − [−y_j]⁺

−

j;pj<pi

T − [−yj]⁺

+

j;pj>pi

[yj− rj]⁺

₊⎤

⎥⎥

⎦

+

min (yi, ti) Spectrum BS_iacquires or loses in both procedures.

min (yi, ti) = min (yi, ri) + min

[yi− ri]⁺, ci ti(y) Spectrum BS_ican acquire or will lose in both procedures.

ti(y) = [y_i]⁺

j;pj=pi

[yj]⁺

O +

j;pj=pi

[yj]⁺−

j;pj≥pi

yj+

j;pj<pi

T

₊

− T

+

⎡

⎢⎢

⎣

T − [−yi]⁺

− T − [−yi]⁺

j;pj=pi

T − [−y_j]⁺

−O −

j;pj=pi

[yj]⁺+

j;pj≥pi

yj−

j;pj<pi

T

₊⎤

⎥⎥

⎦

+

Pi(y) Frozen credit tokens of BS_i. Pi(y) = p_i(y_i) [min(y_i, ti)]⁺

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Chapter 3 Game Formulation

Game theory is utilized to deal with the spectrum sharing problem. From the perspective of each BS, spectrum sharing is intrinsically a game that each BS unitarily optimizes its performance by acquiring or offering spectrum according to its credit token budget and max trafﬁc requirement. In the following, we brieﬂy introduce game theory and then construct the spectrum sharing game model.

3.1 Game Theory

Game theory is a set of mathematical tools used to model and analyze interactive decision processes [9, 10]. The core of a game consists of three primary components:

1. A player setN.

2. A strategy space S formed from the Cartesian product of each player’s strategy set, S =

i∈NSi.

3. A set of utility functionsU = {ui(s)} where s ∈ S and ui(s), i ∈ N, represents player i’s utility under the strategy proﬁle s.

In a game, each player is assumed to choose the best available strategy. Each player’s best available strategy is the one maximizing his utility under the belief that other players do in the same way as well. The collection of all players’ best available strategies forms a steady

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state at which no player has a reason to choose any strategy different from his best available one. Such a steady state is called a Nash equilibrium [11].

Definition 3.1. A strategy proﬁles^∗ =

s^∗_i, s^∗_−i

is a Nash equilibrium if

ui(s^∗) ≥ ui

si, s^∗_−i

∀si = s^∗_i and ∀i ∈ N

Though we can find the Nash equilibria of a game where each player has only a few strategies by examining all the possible strategy profiles to see if they satisfy Definition 3.1, it is always full of difficulties in more complicated games. An alternative method is to work with players’ best response functions.

Definition 3.2. BRi(s−i) is the best response function of player i if

BRi(s−i) = {si : ui(si, s−i) ≥ ui(s_i, s−i) , ∀s_i = si}

The best response function of any player depicts his best (in term of highest utility) strategy given all possibles−i from other players. A Nash equilibrium can also be deﬁned by best response functions.

Definition 3.3. A strategy proﬁles^∗ =

s^∗_i, s^∗_−i

is a Nash equilibrium if

s^∗_i = BRi s^∗_−i

∀i ∈ N

3.2 Spectrum Sharing Game

We now apply game theory to construct a model for the spectrum sharing problem. Besides the three main components, the credit token budget and the max trafﬁc requirement of each BS should be taken into account as well:

1. Player setN: Each BSi is the player of the game. N = {1, 2, ..., n}.

2. Strategy spaceY =

i∈NYi: We treat BS_i’s acquisition/offering requestyias its strategy.

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All possible acquisition/offering requests of BS_i compose the strategy set Yi. Mathematically,Yi = {y_i : −T ≤ y_i ≤ x_i} ∀i ∈ N.

3. Set of utility functions U = {ui(y)}: Since the goal of each BS is to increase its spectrum, it is reasonable to set the spectrum as the utility. We ignore the constant termT for the sake of convenience. The utility is therefore the spectrum acquired or lost from renting, offering, and contention. Mathematically, BS_i’s utility function is ui(y) = min (yi, ti).

4. Set of credit token budgets (CTB set)B = {Bi}.

5. Max trafﬁc setX = {xi}: Without losing generality, we assume {pi(xi)} is sorted in decreasing order. This assumption will simplify our analysis.

The game model is summarized in Table 3.1.

Table 3.1: Spectrum Sharing Game Model

G = (N, Y, U, B, X) Player Set Set of the BSs.

N = {1, 2, ..., n}

Strategy Space Cartesian product of each player’s strategy set which is the set of all possible acquisition/offering requests.

Y =

i∈NYiandYi = {yi : −T ≤ yi ≤ xi} ∀i ∈ N Set of Utility

Functions

Set of the spectrum each player acquires or loses.

U = {ui(y)} and u_i(y) = min (y_i, ti) ∀i ∈ N CTB Set B = {Bi}

Max Trafﬁc Set X = {xi} with {pi(xi)} arranged in decreasing order.

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Chapter 4 Graphical Analysis - Two Players with Same Budget

A graphical method to derive the Nash equilibrium in the simplest 2-same-budget-player game is presented to gain the insights for the solution of the general n-player game, i.e.

GameG. It consists of two main procedures. First the utility functions of both players are drawn to derive their best response functions. Then two best response functions are drawn together. The resulting intersection is Nash equilibrium.

Recall that we have assumed p1(x1) ≥ p2(x2). This assumption reduces to x1 ≤ x2

when both players have the same credit token budget. Accordingly, the trafﬁc requirements can be categorized into three cases. The ﬁrst case isx1+x₂ ≤ O which, by applying x₁ ≤ x₂, can be equivalently expressed asx1 ≤ O

2 andx2 ≤ O − x1. The second case isx1 ≤ O 2 andx1 + x₂ > O, equivalently x1 ≤ O

2 andx2 > O − x1. The ﬁnal case is x1 > O 2 and x1+ x2 > O which are equivalent to x1 > O

2 andx2 > O

2. In the following, we discuss case by case.

4.1 Traffic Case 1 - x

1

≤ O

2 and x

2

≤ O − x

1

The best response function of player 1, illustrated in Figure 4.1(a), is uniquelyx1. It means player 1 will always play the unique dominant strategy, y1 = x₁, regardless of player 2’s

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y1

x1

1 2 1

BR (y) x x1

1 2O u1

(a)u₁andBR₁

y2

Ox1

2 1 2

BR ( )y x x2

u2

x2

(b)u₂andBR₂

y2

Ox1

BR ( )y NE = x x1,2

2 1

BR ( )y x2

y

1 2

BR (y)

y1

x1 1 2O

(c) BR₁,BR₂, and NE

Figure 4.1: Best Response Functions and Nash Equilibrium in Trafﬁc Case 1

u1

x1₁

x

y

1 2 1

BR (y) x

y1

x1 1 2O

(a)u₁andBR₁

x2

y2

y1 Oy1

2 1 1 2

BR ( )y O y~x u2

x1

Oy1

(b)u₂andBR₂

y2

y1

1 2O x1

Ox1

1 2

BR (y) x2 ^{BR ( )}²^y¹

1 1 1 2

NE= ,x Ox ~ x x,

(c) BR₁,BR₂, and NE

strategy. We call such a strategy a dominant one since it always dominates (or results in higher utility than) all other strategies. A similar observation is obtained in Figure 4.1(b) that player 2’s best response function isx2 and therefore player 2 plays the unique dominant strategy,y2 = x2. By drawing two best response functions together in Figure 4.1(c), we ﬁnd their intersection, (x1, x2), a unique Nash equilibrium. The corresponding utility proﬁle is (x1, x2) as well.

4.2 Traffic Case 2 - x

1

≤ O

2 and x

2

> O − x

1

We have already derived that player 1 plays the unique dominant strategy, y1 = x1 when x1 ≤ O

2. As depicted in Figure 4.2(b), player 2’s best response function is BR2(y1) = O − y1 x2 which implies that the strategy,y2 = x2, is player 2’s unique dominant strategy.

However, it is not meaningful to discuss the concept of dominant strategy for player 2 while it is like to play a single-player game. We explain why player 2 is like to play a single-player

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game as follows: whenx1 ≤ O

2, player 1 plays the unique dominant strategy,y1 = x1, and acquiresx1 fromO. (When x1 is less than zero, “acquiringx1” means “offering−x1.”) For player 2, it has (O − x1) remained to acquire without any other player. Therefore player 2 is like to play a single-player game and it can always acquire (O − x₁) by playing y₂ such thatO − x1 ≤ y₂ ≤ x₂. The single-player effect obviously results in multiple Nash equilibria. This can also be shown by drawing the two best response functions together.

The resulting intersection is a line segment between (x1, O − x1) and (x1, x2) which means multiple Nash equilibria, (x1, O − x1 x2), exist. Though multiple Nash equilibria exist, the corresponding utility proﬁle is uniquely(x1, O − x1).

4.3 Traffic Case 3 - x

1

> O

2 and x

2

> O 2

We derive the best response function of player 1 from Figure 4.3(a) to Figure 4.3(c),

BR1(y2) =

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

min (O − y2, x1) x1 if y2 ≤ O 2

y₂⁻ if O

2 < y2 ≤ x1

x1 if x1 < y2

and the best response function of player 2 from Figure 4.3(d) and Figure 4.3(e),

BR2(y1) =

⎧⎪

⎪⎨

⎪⎪

⎩

min (O − y1, x2) x2 if y1 ≤ O 2

y⁻₁ if O

2 < y1

We see neither player 1 nor player 2 has dominant strategy. By drawing two best response functions together, we ﬁnd their intersection,

O 2,O

2

a unique equal-strategy Nash equilibrium. The corresponding utility proﬁle is

O 2,O

2

.

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y1

y2 x₁

1 2 2 1 1

BR (y) min(Oy x, ) ~x

2 1

min(Oy x, )

1 2O Oy2

u1

(a) u₁andBR₁fory₂≤1 2O

y1

y2 x1

1 2 2

BR (y) y

1 2O Oy2

y2

1 2O

u1

(b) u₁ andBR₁for 1

2O < y₂ ≤ x₁

y1

y2

x1

1 2 1

BR (y) x x1

u1

(c)u₁andBR₁forx₁< y₂

x2

y2

1 2O y1 ^O^y¹

2 1 1 2 2

BR ( )y min(Oy x, ) ~x u2

1 2

min(Oy x, )

(d)u2andBR2fory1≤ 1 2O

x2

y2

1 2O 1

2O

y1

Oy1

2 1 1

BR ( )y y u2

(e) u2andBR2fory1> 1 2O

y2

y1

1 2O

1 2

BR (y) x2

2 1

BR ( )y

x1

1 1

NE = , 2O2O

§ ·

¨ ¸

© ¹

(f) BR1,BR2, and NE

We summarize the observations as follows. As will be shown, the listed items, playing the essential roles in the two-same-budget-player game, can be extended to the general n- different-budget-player game.

1. Condition for unique dominant strategies: When x1 ≤ O

2, player 1 plays the unique dominant strategy,y1 = x1. Whenx2 ≤ O − x1, player 2 plays the unique dominant strategy,y2 = x₂.

2. Existence of a Nash equilibrium: A Nash equilibrium always exists in all cases.

3. Condition for multiple Nash equilibria: The only case where multiple Nash equilibria exist is x1 ≤ O

2 andx2 > O − x1. We have explained that because player 2 is like to play a single-player game with(O − x1) offered, it can always acquire (O − x1) by playingO − x1 ≤ y2 ≤ x2. Multiple Nash equilibria,(x1, O − x1 x2), hence exist.

4. Unique utility proﬁle at the Nash equilibrium: Even in the multi-Nash-equilibrium case, the corresponding utility proﬁle is unique.

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Chapter 5 Mathematical Analysis - n Players

In this chapter, we ﬁrst extend the two-same-budget-player game to the general n-different- budget-player game, i.e. GameG. The result is summarized in Table 5.1. Afterwards, we do formal derivations for the Nash equilibrium of GameG.

5.1 Extension from Two-Player Game to n-Player Game

Table 5.1: Summary of Extension

2-Same-Budget-Player Game n-Same-Budget-Player Game n-Different-Budget-Player Game

Trafﬁc Threshold

O 2, O − x1

⎧

⎪⎨

⎪⎩

−^j−1 l=0xl n−j+1

⎫⎪

⎬

⎪⎭

ej,−(j−1)

Traffic x1>Ô₂ andx2>Ô₂; xj≤ ⁻

j−1 l=0xl

n−j+1 ∀j ∈ {1, ..., k}, xj≤ ej,−(j−1)∀j ∈ {1, ..., k}, Case x1≤^O₂ andx2> O − x1; xj> ⁻

k l=0x_l

n−k ∀j ∈ {k + 1, ..., n} xj> ej,−k∀j ∈ {k + 1, ..., n}

x1≤^O₂ andx2≤ O − x1 wherek ∈ {0, N} wherek ∈ {0, N}

Nash

O 2,O

2

;

⎛

⎜⎝x1, ..., xk,⁻

k l=0xl n−k , ...,⁻

k l=0xl n−k

⎞

⎟⎠

x1, ..., xk, ek+1, ..., en,−k

Equilibrium (x1, O − x1∼ x2); ifk = n − 1; ifk = n − 1;

(x1, x2)

x1, ..., xn−1, −ⁿ⁻¹

l=0xl∼ xn

x1, ..., xn−1, en,−(n−1)∼ xn

ifk = n − 1 ifk = n − 1

Recall that we have assumed the max trafﬁc requirements are such that {p_i(x_i)} is arranged in decreasing order. In the two-same-budget-player game, we see there are two

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trafﬁc thresholds, O

2 and O = x1. Accordingly, the trafﬁc can be categorized into three cases: x1 > O

2 andx2 > O

2; x1 ≤ O

2 andx2 > O − x1; x1 ≤ O

2 and x2 ≤ O − x1. The corresponding Nash equilibrium is

O 2,O

2

,(x1, O − x1 ∼ x2), and (x1, x2).

Extended from the two-same-budget-player game, it is reasonably to guess the n- same-budget-player game has the set of n trafﬁc thresholds,

⎧⎨

⎩

−^j−1

l=0xl

n−j+1

⎫⎬

⎭, where x0 = −O.

Accordingly, we can categorize the trafﬁc into (n + 1) cases. The (k + 1)-th case, k ∈ {0, N}, is x_j ≤ ⁻

j−1

l=0xl

n−j+1 ∀j ∈ {1, ..., k} and x_j >

−^k

l=0xl

n−k ∀j ∈ {k + 1, ..., n}. The Nash equilibrium is

⎛

⎝x1, ..., xk,

−^k

l=0xl

n−k , ...,

−^k

l=0xl

n−k

⎞

⎠ if k = n−1 and

x1, ..., xn−1, −ⁿ⁻¹

l=0xl ∼ xn

ifk = n−1. By substituting 2 for n, we can check that the n-same-budget-player game really reduces to the two-same-budget-player game.

To further extend to the n-different-budget-player game, we must know what plays the same role as

−^k

l=0xl

n−k in then-same-budget-player game.

Definition 5.1. For GameG, we deﬁne, with O denoted as −x0,

ej,−k ≡ Bj n−k1

n l=k+1Bl

⎛

⎜⎜

⎜⎝

−^k

l=0xl

n − k

⎞

⎟⎟

⎟⎠+

⎛

⎜⎜

⎝ Bj

n−k1

n l=k+1Bl

− 1

⎞

⎟⎟

⎠ T

∀j ∈ {k + 1, ..., n} and ∀k ∈ {0, N}

ej,−kcan be interpreted as weighted and translated

−^k

l=0xl

n−k with the weight ^B^j

n−k1

n l=k+1Bl

. The term−k in the subscript indicates that player i, i ∈ {1, ..., k}, which has already acquired xi

fromO, is excluded. When k = 0, ej,−0 is denoted asej for short. Following the deﬁnition, there is a corollary stating some properties ofej,−k.

Corollary 5.1. For GameG, the following statements about ej,−k are always true:

1. pj(e_j,−k) =

n−k1

n l=k+1Bl

T +

−^k

l=0xl

n−k

∀j ∈ {k + 1, ..., n}.

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2. ^k

j=1xj+ ⁿ

j=k+1ej,−k = min

# O,ⁿ

j=1xj

$

∀k ∈ {0, N}.

3. xk≤ e_k,−(k−1) ⇔ x_j ≤ e_j,−(j−1)∀j ∈ {1, ..., k}.

4. xk+1 > ek+1,−k ⇔ x_j > ej,−k ∀j ∈ {k + 1, ..., n}.

5. xk≤ e_k,−(k−1) ⇒ p_k(e_k,−(k−1)) ≥ p_j(e_j,−k) ∀j ∈ {k + 1, ..., n}.

Proof. See Appendix B.

Corollary 5.1.1 says that the strategies ei,−k and ej,−k ∀i, j ∈ {k + 1, ..., n} are equal- price, i.e. they result in the same price. Besides, player j who plays yj = ej,−k can be viewed similar to the player having the average credit token budget _n−k¹ ⁿ

l=k+1Bl and playing the strategy

−^k

l=0xl

n−k . In Corollary 5.1.2, when k = n, ^k

j=1xj + ⁿ

j=k+1ej,−k = O.

It can be equivalently represented as ⁿ

j=k+1ej,−k = −^k

l=1xl. Therefore the strategy proﬁle, (e_k+1,−k, ..., en,−k), is called the the sum-

−^k

l=1xl

equal-price strategy profile. Especially whenk = 0, the strategy profile, (e1, ..., en), is called the sum-O equal-price strategy profile.

WhenBi = Bj ∀i, j ∈ N, the weights for all ej,−k become 1 andej,−k reduces to

−^k

l=0xl

n−k . It is intuitively to believe that ej,−k play the same roles as

−^k

l=0xl

n−k in the same-budget case.

Hence Game G should have the set of n trafﬁc thresholds,

ej,−(j−1)

. Besides, we can classify the trafﬁc into(n + 1) cases where the (k + 1)-th case, k ∈ {0, N}, is xk ≤ ek,−(k−1)

and xk+1 > ek+1,−k. From Corollary 5.1.3 and 5.1.4, the (k + 1)-th case can equivalently represented asxj ≤ e_j,−(j−1)∀j ∈ {1, ..., k} and x_j > ej,−k ∀j ∈ {k + 1, ..., n}.

Definition 5.2. For GameG and ∀k ∈ {0, N}, we deﬁne

Traﬃc_k ≡ xj ≤ e_j,−(j−1) ∀j ∈ {1, ..., k} and xj > ej,−k ∀j ∈ {k + 1, ..., n}

The Nash equilibrium under Traﬃc_k should be (x1, ..., xk, ek+1, ..., en,−k) if k = n − 1 and

x1, ..., xn−1, en,−(n−1) ∼ xn

ifk = n − 1. Finally, by letting all credit token budgets be the same, we check the n-different-budget-player game becomes the n-same-budget-player game.

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5.2 n-Player Game

The game model forn different-budget players, Game G, is illustrated in Table 3.1. Before starting, we should mention that we will use ui and ti to express ui(y) and ti(y) at any given strategy proﬁle y for short. If we need to compare the results between two different strategy proﬁles, say(yi, y−i) and (y_i, y−i), we will distinguish by using u_i andt_i to express ui(y_i, y−i) and t_i(y_i, y−i).

First, let us examine the increasing property of utility functions with respect to strategies.

Lemma 5.1. For GameG under Traﬃc_k, k ∈ {0, N}, the following statements are always true:

1. ui = yi ∀i ∈ {1, ..., k}.

2. if yi ≤ e_i,−k for some i ∈ {k + 1, ..., n}, ui= y_i.

Proof. From Corollary 5.1.2, we know^k

j=1ej,−(j−1)+ ⁿ

j=k+1ej,−k = min

#

O, −ⁿ

j=1xi

$

≤ O.

The derivation below is suitable for both 1) and 2). For playeri, we have

O + %

j;pj=pi

[yj]⁺− %

j;pj≥pi

yj+ %

j;pj<pi

T

≥ %^k

j=1

ej,−(j−1)+ %ⁿ

j=k+1

ej,−k+ %

j;pj=pi

[y_j]⁺− %

j;pj≥pi

yj + %

j;pj<pi

T

= %

j;pj=pi

[y_j]⁺+ %

j≤k;pj≥pi

ej,−(j−1)− y_j

+ %

j>k;pj≥pi

(e_j,−k− y_j)

+ %

j≤k;pj<pi

ej,−(j−1)+ T

+ %

j>k;pj<pi

(ej,−k+ T ) ≥ %

j;pj=pi

[yj]⁺ ≥ 0 (5.1)

Following Equation (5.1), ifyi> 0, we have

ti = [yi]⁺

j;pj=pi

[yj]⁺

⎡

⎣O + %

j;pj=pi

[yj]⁺− %

j;pj≥pi

yj + %

j;pj<pi

T

⎤

⎦

+

≥ yi j;pj=pi

[y_j]⁺

⎛

⎝ %

j;pj=pi

[y_j]⁺

⎞

⎠ = yi (5.2)

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Ifyi ≤ 0, we have

ti = −T +

⎡

⎣T − [−yi]⁺

− ^{T −[−y}ⁱ^]⁺

j;pj=pi(^{T −[−y}^j^]⁺)

−O −

j;pj=pi

[yj]⁺+

j;pj≥pi

yj −

j;pj<pi

T

₊⎤

⎦

+

= −T + (T + yi) = yi (5.3)

Equation (5.2) and (5.3) reveals thatti ≥ yiand consequentlyui = min(yi, ti) = yi. Lemma 5.1.1 shows thatui,i ∈ {1, ..., k}, is an increasing function of yiunder Traﬃc_k. The condition for unique dominant strategies is then implied.

Theorem 5.1. For Game G under Traﬃc_k, k ∈ {0, N}, player i, i ∈ {1, ..., k}, plays the unique dominant strategy, yi = xi.

Proof. It is shown in Lemma 5.1.1 that under Traﬃc_k, ui = y_i ∀i ∈ {1, ..., k}. Player i’s utility is an increasing function ofyiand uniquely reaches its maximum atyi = xi. Therefore playeri can always play yi = xi to get the highest utility. In words, player i, i ∈ {1, ..., k}, plays the unique dominant strategy,yi = xi.

Recall we have guessed the Nash equilibrium under Traﬃc_kis(x₁, ..., xk, ek+1, ..., en,−k) ifk = n − 1 and

x1, ..., xn−1, en,−(n−1) ∼ xn

ifk = n − 1. To verify our guess is correct, we prove that all other strategy proﬁles cannot be a Nash equilibrium. The proof is taken into two parts. The ﬁrst part is to prove that yi < xi for anyi ∈ {1, ..., k} or yi < ei,−k for any i ∈ {k + 1, ..., n} is not in any Nash equilibrium. The second part is to prove that yi > ei,−k

for anyi ∈ {k + 1, ..., n} is not in any Nash equilibrium.

Lemma 5.2. For Game G under Traﬃc_k, k ∈ {0, N}, yi < xi for any i ∈ {1, ..., k} or yi < ei,−k for any i ∈ {k + 1, ..., n} is not in any Nash equilibrium.

Proof. Since Theorem 5.1 reveals that yi = xi is the unique dominant strategy for playeri

∀i ∈ {1, ..., k}, yi < xi for any i ∈ {1, ..., k} is not in any Nash equilibrium. Also, from Lemma 5.1.2, we know ui = yi if yi ≤ ei,−k for any i ∈ {k + 1, ..., n}. It means when