點對點檔案傳輸之賽局分析

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國立臺灣大學電機資訊學院資訊工程學系碩士論文

Department of Computer Science and Information Engineering College of Electrical Engineering and Computer Science

National Taiwan University Master Thesis

點對點檔案傳輸之賽局分析

A GameTheoretic Analysis of P2P FileSharing Systems

蔡瑋倫 WeiLun Tsai

指導教授：陳健輝博士 Advisor: GenHuey Chen, Ph.D.

共同指導教授：陳和麟博士 CoAdvisor: HoLin Chen, Ph.D.

中華民國 108 年 7 月

July, 2019

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摘要

點對點式網路架構常用於使用者之間的檔案傳輸與分享，藉以改善傳統主從式架構伺服器負擔過重以及易受攻擊等問題。然而，實驗結果發現點對點網路架構容易造成搭便車問題，於是我們必須借助賽局理論以設計良好的獎勵機制督促使用者貢獻自己的資源，以維持系統運作 [16]。

我們的研究從 Chiranjeeb Buragohain 等人在 2003 年所提出的模型 [2] 延伸而來。原論文根據每位使用者的貢獻來決定他/她是否能從社群獲得資源的機率函數，貢獻與機率成正相關，而效益函數則是所獲得資源去扣除自己開放頻寬給其他使用者下載的成本，在兩個人的環境下恰有兩個不崩潰的均質納許均衡，促使社群高貢獻的均衡點是穩定的。在我們的論文額外考慮了使用者對其他人所擁有資源的需求有所節制以及多重使用者的情況。在此情形下，我們發現當需求幾乎沒有節制的時候不影響原本的納許均衡；當需求有些節制的時候會壓低原本促使社群高貢獻的均衡點的貢獻量，同時該均衡點轉為不穩定，可能收斂到其他均衡點；當使用者的需求極低 (資源同質性高) 的時候整個系統反而會崩潰 (使用者均不貢獻)。此外，我們也觀察了不同條件之下納許均衡的效率隨著模型參數 (單位資源所產生之效益、需求的節制、社群人數) 的變化。

關鍵字：賽局理論、納許均衡、點對點、檔案分享、獎勵機制

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Abstract

A peertopeer (P2P) network is commonly used for filesharing among different users. This kind of structure can solve some common problems of centralized networks. However, experiments show that freeriding is a major problem for the P2P networks, so we have to design a good incentive mech

anism with the help of game theory in order to encourage users to contribute to the community and maintain the network [16].

We use the model proposed by Buragohain et al. [2] in 2003. In the orig

inal paper, the author determines the probability function, from the contri

bution of each user, which controls the probability that a user can retrieve resources from the community. The probability increases with the contribu

tion. The utility function is determined by the retrieved resources with the contribution cost subtracted. In a twoplayer filesharing game, there are two noncollapsing Nash equilibria, one of which with a greater contribution is stable. In our thesis, we further consider a multiplayer filesharing game where the need for resources of each user is limited. In this game, we’ve dis

covered that when the limitation is not obvious, the original Nash equilibria are not affected. When the limitation is a little influential, the contribution of the Nash equilibrium with a greater contribution will be lowered and it will become unstable. When the limitation is drastic, the system will col

lapse. Besides, we’ve also observed how the efficiency of Nash equilibria changes with system parameters under different conditions. The parameters include the benefit drawn by one unit of resources, the limitation of need for

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resources, and the number of users in the network which will be defined later.

Keywords: Game Theory, Nash Equilibrium, PeertoPeer, FileSharing, In

centive Mechanism

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List of Figures

2.1 A geometric illustration of Lemma 2.1 . . . 7

2.2 A geometric solution of do . . . 8

3.1 A simple diagram of Definition 3.1. The symbols R_2b₁ and R_2b₂ will be defined later. . . 12

3.2 A simple diagram of the remaining regions after the first stage of the elim ination procedure . . . 14

3.3 A simple diagram of the remaining regions after the second stage of the elimination procedure . . . 16

3.4 A geometric perspective of the condition do< K such that u(do) > 0 . . 16

3.5 A geometric solution of d_ℓ and d_h in Definition 3.3 . . . 18

3.6 A simple diagram of N_side1and N_side2in Theorem 3.10 and its corollary . 19 3.7 A simple diagram of N_ℓ and N_h in Theorem 3.12 . . . 20

3.8 A simple diagram of N_o in Theorem 3.13 . . . 21

3.9 A geometric illustration of Theorem 3.14 . . . 22

3.15 A geometric illustration of the case d_ℓ < d_o < d_h . . . 28

3.16 A geometric illustration of the case d_ℓ < d_o = d_h . . . 30

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4.2 A geometric illustration of Definition 4.2 . . . 47

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List of Tables

3.1 The maximum total utility of twoplayer filesharing games . . . 17 3.2 Summary of Nash equilibria of twoplayer filesharing games in ascending

order of their total utility . . . 30 3.3 Summary of the PoS and PoA with K as the only varying parameter. We

assume d_ostarts at d_ℓ and keeps increasing. . . 35 3.4 Summary of the PoS and PoA with b as the only varying parameter. We

assume b starts at its valid minimum value (i.e. bxp^′(x) = 1 has exactly one solution.) and keeps increasing. . . 36

4.1 The maximum total utility of threeplayer games. . . 42 4.2 Condition matching for each Nash equilibrium. . . 44 4.3 Summary of the PoS and PoA with K as the only varying parameter. We

assume d_ostarts at d_ℓℓ and keeps increasing. . . 56 4.4 Summary of the PoS and PoA with b as the only varying parameter. We

assume b starts at its valid minimum value (i.e. bxp^′(x) = 1

2 has exactly one solution.) and keeps increasing. . . 57

5.1 The maximum total utility of multiplayer games. . . 62 5.2 Summary of the PoS and PoA with K as the only varying parameter. We

assume d_ostarts at d_Land keeps increasing. . . 69 5.3 Summary of the PoS and PoA with b as the only varying parameter. We

assume b starts at its valid minimum value (i.e. bxp^′(x) = 1 n− 1 has exactly one solution.) and keeps increasing. . . 70

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5.4 Summary of the PoS and PoA with n as the only varying parameter. We assume n starts at its valid minimum value (i.e. bxp^′(x) = 1

n− 1 has exactly one solution.) and keeps increasing. . . 70

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Chapter 1 Introduction

A peertopeer (P2P) network is a distributed system that consists of many users which are often directly connected to each other, and they can be both providers and consumers of resources at the same time in the network. In constrast to P2P networks, a centralized network also consists of many users, but only servers provide all the resources and are con

nected to clients. The clients can only consume the resources and they are not necessarily connected to each other.

The most significant advantages of P2P networks over centralized networks are scal

ability and robustness. When a new user joins a P2P network, he/she not only increases the network load but also provides some resources to the system (as a small server), so the network load is usually balanced and the P2P network is scalable. When a node is attacked or fails to work for some reason, the other parts of the network can still work as usual because only a very small part of the system is affected. In a centralized network, an attack against one of the main servers can severely reduce the performance since the resources are completely on the servers. Therefore P2P networks are more robust than centralized networks.

However, a major problem for the P2P networks is “freeriding.” Freeriding means that most users only consume the resources but forget to provide enough resources to maintain the network. Since making contribution definitely takes some cost, it is intuitive that freeriding is a dominant strategy. Unfortunately, if everyone chooses this dominant strategy, there will be no resources in the network and therefore the system will collapse.

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Experiments in [9] showed this phenomenon. Hence, some incentive mechanisms are needed to overcome this freeriding problem.

Incentive mechanisms incorporated by the P2P filesharing networks in the past were mainly based on monetary payment schemes or reciprocitybased schemes [5]. In mon

etary payment schemes, users must pay money before consuming resources and can get paid when providing resources to others. Mojonation and Karma [12], and some stud

ies such as [6, 10, 14, 17] used this kind of schemes. The implementation is not easy in practice since it requires infrastructure for accounting and micropayments. Contrary to monetary payment schemes, we can also use reciprocitybased schemes. They include direct reciprocity and indirect reciprocity. In direct reciprocity schemes, the quality of re

sources user A wants to provide to user B is based on the quality of resources A retrieved from B in the past. BitTorrent [3] uses this kind of schemes based on the titfortat strat

egy. In indirect reciprocity schemes, also called reputation based schemes, the quality of resources a user deserves to obtain highly depends on his/her “overall” generosity. The word “overall” here means that as long as user A’s reputation is high, it is not necessary for A to provide good quality resources to user B even if A wants to retrieve good quality re

sources from B. Some studies such as [2, 7] used this kind of schemes. We should note that this is an advantage when a user is not interested in anything the other one can offer. It is the only difference between direct reciprocity and indirect reciprocity. Nowadays, the in

centive mechanisms are further enhanced. For example, Hu et al. [8] combined monetary payment schemes and indirect reciprocity schemes. Zhang et al. [18] used a Blockchain

based mechanism to resolve the difficulty of finding a trusted third party (TTP) in a real P2P system.

[2] is a representative paper about reputation based schemes. In [2] the authors pro

posed a differential servicebased incentive scheme to improve the system’s performance (i.e., reduce freeriding). First, they considered the case of a “homogeneous” system where the value of resources is independent of users who own them and users who retrieve them.

In this case, there exists two noncollapsing Nash equilibria with different contribution levels. Only the one resulting in the better overall performance is stable (i.e., easily real

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ized). Second, they studied the case of a “heterogenous” system through simulation, since no closed form solution is possible. In this case, the numerical experiments showed that the system also converges to the desirable Nash equilibrium if a good initial condition is given, and that the average contribution is almost independent of the number of users. Fi

nally, they gave some suggestions on how to modify current P2P systems to implement the proposed incentive scheme. We need a function of the contribution level of user A to con

trol the probability that A can retrieve resources from another user B. Also, the probability function should be a part of the system’s architecture. It means that the setting should be exactly the same for all users and cannot be modified by them. In order to prevent users from reporting their contribution levels incorrectly, a neighbour audit scheme in which users can verify the information of their neighbors is required. In order to encourage new users to join the system, they can be given a default contribution level at the beginning.

Our research is continued from [2]. In the original paper, the resources a player pos

sesses are not limited. To our best knowledge, there are almost no research papers dis

cussing the case of limited resources, so we will consider this environment in our thesis.

We only study the case of a homogeneous system of two players, three players, and mul

tiple players, but with a fixed maximum benefit of resources from each player, and the probability function satisfying some “good” assumptions that we will introduce in the next chapter. Our main contribution is to find some important Nash equilibria under different parameter settings, analyze their stability and efficiency including the price of anarchy (PoA) and price of stability (PoS), and observe how they vary with related parameters.

We define the PoA to be the ratio of the maximum total utility among all possibilities to that of the “worst” Nash equilibrium, and define the PoS to be the ratio of the maximum total utility among all possibilities to that of the “best” Nash equilibrium.

The rest of the thesis is organized as follows. In Chapter 2, we explain the meaning of our newly proposed model and introduce the related parameters. In Chapter 3, we analyze a homogeneous system of two players. In Chapter 4, we analyze a homogeneous system of three players, but without considering the stability of Nash equilibria. In Chapter 5, we analyze a homogeneous system of multiple players, but only considering symmetric Nash

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equilibria. That is, all players have the same strategy. Finally in Chapter 6, we conclude our analysis, describe additional possibly extended models, and discuss some aspects that can be improved in the future.

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Chapter 2 Model

In this chapter, we’re going to introduce the system parameters inherited from [2] that will be used in this thesis. Assume that there are N players (users) P₁, P₂, ..., P_N in the system.

All parameters, as in the original paper, are dimensionless.

Definition 2.1 (Contribution). Let di be the contribution of P_i which is a nonnegative number. The meaning of the contribution can be very widespread. For example, [2] says we may think of d_i as the disk space contribution integrated over a fixed period of time, or the number of downloads served by this peer to other peers. In this thesis, we usually see d_ias the amount of downloadable resources owned by P_i. Since this parameter is also a strategy one player can decide, the term “strategy” and “contribution” have the same meaning in this thesis.

Definition 2.2 (Benefit). The value of resources owned by a player may vary depending mainly on other users who retrieve them. For example, if Alice has lots of music, whereas Bob has lots of Japanese animation, I may prefer Bob’s resources to Alice’s. Hence we let b denote how much the “unit” contribution made by one player is worth to another player in a homogeneous system. That is, if a player P_i retrieves one unit of contribution from another player Pj, then Pi’s utility will increase by b. Details of the utility function will be introduced later.

Definition 2.3 (Probability as Service Differentiator). In a differential service, the prob

ability that a player P_i can retrieve resources from other players should increase with

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his/her contribution di. This mechanism encourages the players to share their file re

sources. In this thesis, a player P_i can retrieve resources from other players with proba

bility p(d_i), and is rejected with probability 1−p(d_i).

Proposition 2.1. To achieve the goal of a service differentiator, the probability function p(d) must be nondecreasing (i.e., p^′(d)≥ 0 for d ≥ 0). To meet the definition of “prob

ability,” p should satisfy p(0) = 0 and lim

d→∞p(d) = 1. To ensure each player has only one best strategy in each iteration, we assume p^′(d) to be decreasing (i.e., p^′′(d) < 0 when 0 ≤ p(d) < 1). To ensure bdp^′(d) = C has at most two solutions for every constant C > 0, we also assume dp^′(d)

d=0

= lim

d→∞dp^′(d) = 0, and there exists a threshold d₀such that (dp^′(d))^′ > 0 for d < d₀ and (dp^′(d))^′ < 0 for d > d₀. We assume all probability functions p(d) satisfy all our assumptions in this proposition unless otherwise specified.

Definition 2.4 (Utility). Let the total utility u_ithat P_iwill derive in the homogeneous sys

tem be u_i =−di+X

j̸=i

min{K, bdjp(d_i)}. The term −diis the cost of P_ito join the system, which is proportional to his/her contribution. The other termX

j̸=i

min{K, bdjp(d_i)} is the total expected benefit of Pi. It is obvious that min{K, bdjp(di)} for some j is the ex

pected benefit gained from some player P_j. In this term, d_j is the amount of resources P_j can provide, so multiplying it by p(d_i) gives the expected amount of resources P_ican acquire. Multiplying it by b again obtains the expected “benefit.” In that term K denotes the maximum benefit one player can derive from another player. Normally K is greater than 0.

Proposition 2.2. Suppose all d_i’s have the same value of d. If b ≤ 1

n− 1, the utility function uiis therefore not greater than (n− 1) b d p(d) − d = d

(n− 1) b p(d) − 1

≤ d

(n− 1) b − 1

≤ d (1 − 1) = 0. This means that any homogeneous solution is not better than the origin. It may cause the system to collapse. To avoid this problem, we should assume b > 1

n− 1 in this thesis.

After the definitions and propositions, here is one important lemma about Proposition 2.1 that will commonly be referred to when the parameter b varies.

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Lemma 2.1. Assume the two equations b1xp^′(x) = C and b2xp^′(x) = C, where 0 < b1 <

b₂, have solutions. Let the solutions to b₁xp^′(x) = C be d_1ℓ and d_1h, where d_1ℓ ≤ d1h. Let the solutions to b₂xp^′(x) = C be d_2ℓand d_2h, where d_2ℓ ≤ d2h. Then d_2ℓ < d_1ℓand d_2h> d_1h.

Proof. Since xp^′(x)

x=0

= 0 and xp^′(x)

x=d1ℓ

= C/b₁, by the intermediate value theorem there must exist at least one x_ℓ < d_1ℓ such that xp^′(x)

x=xℓ

= C/b₂ (∵ b2 > b₁). Since xp^′(x)

x=d1h

= C/b₁ and lim

x→∞xp^′(x) = 0, by the intermediate value theorem there must exist at least one x_h > d_1hsuch that xp^′(x)

x=x_h = C/b₂ (∵ b2 > b₁). Therefore x_ℓ <

d_1ℓ ≤ d1h < x_h. Since b₂xp^′(x) = C has at most two solutions, we can simply say xℓ= d2ℓand xh = d2h. ∴ d2ℓ< d1ℓand d2h > d1h.

Figure 2.1: A geometric illustration of Lemma 2.1

After introducing the system parameters, we’re going to derive some important lem

mas related to the probability function that will be heavily used in the later chapters.

2.1 Useful Properties

Before the lemmas, we also define two symbols that will be used in the whole thesis.

Definition 2.5. Let uopt be the maximum total utility in an nplayer filesharing game.

That is, u_opt = max

di≥0 for 1≤i≤n

u(d₁, d₂, ..., d_n).

Definition 2.6. Let dobe the unique solution to the equation bxp(x) = K. Since K > 0, d_ocannot be 0 and we can see it as the intersection of p(x) and K

bx.

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Figure 2.2: A geometric solution of d_o

Lemma 2.2. If 0 < d₁ < d₂, then d₁p(d₂) < d₂p(d₁).

Proof. We’re going to prove this lemma with the technique of “change of variables” cov

ered in the calculus course. Write the probability function in an integral form,

p(d₂) = Z d2

0

p^′(x) dx========^x=(d²^/d¹^)u

Z d2·^d1_d2 0·^d1_d2

p^′

d₂ d₁u

d

d₂ d₁u

= d₂ d₁

Z d1

0

p^′

d₂ d₁u

du,

and it can be rearranged intod1

d₂p(d₂) = Z d1

0

p^′

d2

d₁u

du. Compare it with

p(d₁) = Z d1

0

p^′(u) du.

Since d₂ > d₁ (which implies ^d_d²

1u > u for u > 0) and p^′(x) is decreasing if greater than zero, we can always pick some d₀ ∈ (0, d1) such that p^′(u) > 0

i.e., p^′(u) > p^′(^d_d²

1u)

for all u∈ (0, d0) and p^′(u) = 0

i.e., p^′(^d_d²

1u) = 0

for all u∈ (d0, d₁). Hence

d₁

d₂p(d₂) = Z d1

0

p^′

d₂ d₁u

du =

Z d0

0

p^′

d₂ d₁u

du +

Z d1

d0

p^′

d₂ d₁u

du

<

Z d0

0

p^′(u) du + Z d1

d0

p^′

d2

d₁u

du

= Z d0

0

p^′(u) du + Z d1

d0

p^′(u) du

= Z _d₁

0

p^′(u) du = p(d₁).

We can obtain the result by multiplying both sides of the above inequality by d₂.

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Corollary 2.3. If d1p(d2) > d2p(d1) > 0, then d1 > d2 > 0. If d1p(d2) = d2p(d1) > 0, then d₁ = d₂ > 0. If 0 < d₁p(d₂) < d₂p(d₁), then 0 < d₁ < d₂.

Proof. By the law of trichotomy, exactly one of the three conditions d₁ < d₂, d₁ = d₂, or d₁ > d₂ is true. Consider the first statement in our corollary. If 0 < d₁ < d₂, by Lemma 2.2 we can deduce d₁p(d₂) < d₂p(d₁). If 0 < d₁ = d₂, then d₁p(d₂) = d₂p(d₁). Both of the above assumptions violate the statement, so only d₁ > d₂ > 0 can be the conclusion of it. The reader can use the same method to prove the remaining two statements.

Lemma 2.4. If d1p^′(d₂) = d₂ p^′(d₁) > 0, then d₁ = d₂.

Proof. The structure of this proof is very similar to Corollary 2.3. If 0 < d₁ < d₂ and p^′(d₂) > 0, then p^′(d₁) > p^′(d₂) and d₂p^′(d₁) > d₁p^′(d₂) since p^′(x) is decreasing.

Similarly if 0 < d₂ < d₁ and p^′(d₁) > 0, then p^′(d₂) > p^′(d₁) and d₁p^′(d₂) > d₂p^′(d₁).

The above two cases both violate the lemma assumption. From the above, only d₁ = d₂ >

0 can satisfy the assumption, so it is our conclusion.

Lemma 2.5. If p(d1) = p(d2) for some d1 < d2, then p(x) = 1 and p^′(x) = 0 for all x≥ d1.

Proof. Write the probability function p(x) in an integral form.

p(d₂)− p(d1) = Z d2

0

p^′(x) dx− Z d1

0

p^′(x) dx

= Z d2

d1

p^′(x) dx = 0. (2.1)

Suppose for contradiction that p^′(d₁) > 0. Then we can definitely find a d_mid ∈ (d1, d₂) such that p^′(x) > 0 for all x ∈ (d1, d_mid), and

Z d2

d1

p^′(x) dx = Z dmid

d1

p^′(x) dx + Z d2

dmid

p^′(x) dx >

Z d2

dmid

p^′(x) dx ≥ 0,

which violates Equation (2.1). Hence p^′(d₁) = 0 and p^′(x) = 0 for all x≥ d1since p^′(x) is decreasing. In addition p^′(x) = 0 implies p(x) = 1, so p(x) = 1 for all x≥ d1.

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Corollary 2.6. If p(d1) = p(d2), p^′(d1) > 0 and p^′(d2) > 0, then d1 = d2.

Proof. W.L.O.G., assume d₁ ≤ d2. If d₁ < d₂, then by Lemma 2.5 p^′(d₁) = p^′(d₂) = 0 causes a contradiction. Therefore, d₁ = d₂.

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Chapter 3 Nash Equilibrium Analysis for TwoPlayer FileSharing Games

In the previous chapter, we’ve introduced some basic elements of our model. For simplic

ity, we consider a homogeneous system of two players first. It’s easy to see that the model can be simplified to the following.











u₁(d₁) = −d1+ min{K, b d2 p(d₁)} u₂(d₂) = −d2+ min{K, b d1 p(d₂)} u(d₁, d₂) = u₁(d₁) + u₂(d₂).

We also use the notation u(d) = u(d, d) if both d₁ and d₂have the same value of d.

In this chapter, we are going to find all Nash equilibria under different parameter set

tings, analyze their stability and efficiency (PoA and PoS), and observe how they vary with system parameters b and K. Before calculating the PoA and PoS, we should find the points where the maximum total utility occurs.

3.1 Maximum Total Utility

In this section, we hope to find the maximum total utility in different parameter settings.

The method used in this chapter is to calculate the gradient with respect to d₁or d₂at each point in the domain of u(d₁, d₂). Since u is bounded above (u≤ 2K), we can guarantee

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the existence of a maximum, and it cannot occur at the points where (_∂d^∂u

1)⁺= (_∂d^∂u

1)⁻ ̸= 0 or (_∂d^∂u

2)⁺ = (_∂d^∂u

2)⁻ ̸= 0. Based on this observation, we can exclude these points first (called an elimination procedure), then compare the values of the remaining points, and then finally choose the optimal points from them.

Observing the formula in this model, the reader may guess that u_opt occurs when bd₂p(d₁) = K and bd₁p(d₂) = K. In fact this is true under some “good” parameter settings. In this section, we will introduce these “good” conditions, and explain why u_opt occurs at such places.

By symmetry, it suffices to consider only the upper left part of the domain of u(d₁, d₂) in the following analysis. It can be partitioned into three regions with respect to the two equations bp(d₂) = 1 and bd₁p(d₂) = K. These regions can be defined formally.

Figure 3.1: A simple diagram of Definition 3.1.

The symbols R_2b₁ and R_2b₂ will be defined later.

Definition 3.1. Let region 1 be R1 ={ (d1, d2)| 0 ≤ d1 ≤ d2 ∧ bp(d2)≤ 1 }. Let region 2 be R₂ ={ (d1, d₂)| 0 ≤ d1 ≤ d2 ∧ bp(d2) > 1 ∧ bd1p(d₂) ≤ K }. Let region 3 be R₃ ={ (d1, d₂)| 0 ≤ d1 ≤ d2 ∧ bp(d2) > 1 ∧ bd1p(d₂)≥ K }. Let R2bbe the rightmost boundary of R₂. That is, R_2b = R₂∩ { (d1, d₂)| d1 = d₂∨ bd1p(d₂) = K }. Let R3bbe the leftmost boundary of R₃. That is, R_3b= R₃∩ { (d1, d₂)| bd1p(d₂) = K } ⊆ R2b. Let P be the point (d, d) where bdp(d) = K. The letter “b” here means “boundary.”

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The reason why we want to partition the domain is explained as follows. Utilities in R₁are always nonpositive (which will be proven later), which is obviously not better than the origin (0, 0), so this region will be excluded eventually if we hope that the model has a positive u_opt. We also note that the term min{K, bd1p(d₂)} causes the gradient of u(d1, d₂) to be discontinuous at the curve bd₁p(d₂) = K, so the values should be calculated in R₂ and R₃ separately.

In the following several pages we’re going to perform our elimination procedure. The procedure can be divided into three stages. The first stage is to remove the points where uopt cannot occur in R2 and R3. By Lemma 3.1 R2 can be minimized to R2b, and by Lemma 3.2 R₃ can be minimized to R_3b. Since R_2b contains R_3b, we only consider the points where u_opt cannot occur in R_2band remove them in the second stage. By Lemma 3.3 and Lemma 3.4 R_2bcan be minimized to the point P or completely eliminated. Finally, we’ll show that the maximum total utility within R₁is exactly 0 and find out circumstances in which P will be better than R₁.

Now we perform the first stage of the elimination procedure.

Lemma 3.1. After we remove these points where u_opt cannot occur in R₂, the region R₂ should be minimized to R_2b.

Proof. One property of R₂ is the inequality bd₁p(d₂)≤ K. According to this, the utility u =−d1+ min{K, bd1p(d₂)} − d2+ min{K, bd2p(d₁)} can be simplified to u = −d1+ bd₁p(d₂)−d2+min{K, bd2p(d₁)}. Since the “min” term may cause the partial derivatives to be discontinuous, for simplicity we use the notation of partial derivatives as usual to represent the less of the left derivative and right derivative.

∵ ∂

∂d₁K = 0 and ∂

∂d₁bd2p(d1) = bd2p^′(d1)≥ 0. ∴ ∂

∂d₁ min{K, bd2p(d1)} ≥ 0.

∴ ∂u

∂d₁ =−1 + bp(d2) + 0 + ∂

∂d₁ min{K, bd2p(d₁)} ≥ −1 + bp(d2) > 0.

According to this derivative, we can say for each pair of points (ℓ, d₂) and (r, d₂) in R₂, u(ℓ, d₂) < u(r, d₂) if ℓ < r. Hence the result follows.

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Lemma 3.2. After we remove these points where uopt cannot occur in R3, the region R3

should be minimized to R_3b.

Proof. One property of R₃ is the inequality bd₁p(d₂) ≥ K. According to this, we can further deduce bd₂p(d₁) ≥ bd1p(d₂) ≥ K by Lemma 2.2. The utility u = −d1 − d2 + min{K, bd2p(d1)} + min{K, bd1p(d2)} is then simplified to u = −d1 − d2 + 2K, and therefore _∂d^∂u

1 = −1 in this region. According to this derivative, we say for each pair of points (ℓ, d₂) and (r, d₂) in R₃, u(ℓ, d₂) > u(r, d₂) if ℓ < r. Hence the result follows.

Figure 3.2: A simple diagram of the remaining regions after the first stage of the elimination procedure

Now we perform the second stage of the elimination procedure.

Definition 3.2. Let R2b1 = R_2b∩ { (d1, d₂)| d1 = d₂ }, and let R2b2 = R_2b∩ { (d1, d₂)| bd₁p(d₂) = K }. Then R3b = R_2b₂ and R_2b= R_2b₁ ∪ R2b2.

Lemma 3.3. If R_2b₁ exists, it should be minimized to the single point P after we remove these points where uoptcannot occur in R2b1.

Proof. In R_2b₁ there is a condition d₁ = d₂, so bd₂p(d₁) = bd₁p(d₂) ≤ K. If we let d = d₁ = d₂, then the utility can be simplified to u =−d1+ bd₁p(d₂)− d2+ bd₂p(d₁) = 2(bdp(d)− d), and the derivative is

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∂u

∂d = 2 ∂

∂d

bdp(d)− d

= 2

bp(d) + bdp^′(d)− 1

≥ 2

bp(d)− 1

> 0

by the condition bp(d) > 1 stated in R₂. Hence the result follows.

Lemma 3.4. After we remove these points where u_opt cannot occur in R_2b₂, the region R_2b₂ should be minimized to the point P or completely eliminated.

Proof. According to the constraint d₁p(d₂) = K/b stated in R_2b₂, we first differentiate both sides of the equation with respect to d2, in order to obtain ∂d1/∂d2.

∂

∂d₂

d₁p(d₂)

= ∂

∂d₂

K b

=⇒ ∂d₁

∂d₂p(d₂) + d₁p^′(d₂) = 0.

According to the property bd₁p(d₂) = K in R_2b₂, we can further deduce bd₂p(d₁) ≥ bd1p(d2) = K by Lemma 2.2, so the utility is then simplified to u = −d1 − d2 + 2K.

Differentiate u with respect to d₂.

∂u

∂d₂ = ∂

∂d₂

− d1− d2 + 2K

=−∂d₁

∂d₂ − 1 = d₁p^′(d₂)

p(d₂) − 1 = d₁ p^′(d₂)− p(d2) p(d₂)

= d₁ p^′(d₂)−Rd2

0 p^′(t)dt

p(d₂) < d₁ p^′(d₂)− d2p^′(d₂)

p(d₂) = p^′(d₂)

p(d₂) d₁− d2

.

Therefore ∂u

∂d₂ < 0 since p^′(d2)≥ 0, p(d2) > 0, and d1 ≤ d2. According to this derivative, we can increase u only by decreasing d2. As ∂d₁

∂d₂ ≤ 0, only the condition d1 = d2 can stop our traversal. If this condition is reached, then we arrive the point P . If this condition can never be reached (i.e., d₁ = d₂ can only happen when b p(d₂) ≤ 1), then the whole region R_2b₂ should be eliminated. In this case P ∈ R1. Hence the result follows.

Lemma 3.5. Themaximum total utility achieved in R₁ is 0.

Proof. We first observe that bp(d₁) ≤ bp(d2) ≤ 1 because of our assumption d1 ≤ d2. Multiplying d₂ on both sides of bp(d₁)≤ 1 gives bd2p(d₁)≤ d2. Multiplying d₁ on both sides of bp(d₂)≤ 1 gives bd1p(d₂)≤ d1. Then they can be applied to the following.

u₁(d₁) = −d1+ min{K, b d2p(d₁)} ≤ −d1+ b d₂p(d₁)≤ −d1+ d₂. u₂(d₂) = −d2+ min{K, b d1p(d₂)} ≤ −d2+ b d₁p(d₂)≤ −d2+ d₁.

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Adding these two inequalities together, we’ll discover that

u₁(d₁) + u₂(d₂)≤ (−d1+ d₂) + (−d2+ d₁) = 0.

Hence the result follows.

Figure 3.3: A simple diagram of the remaining regions after the second stage of the elimination procedure

The comparison between R₁ and P in the final stage is illustrated in the following theorem.

Theorem 3.6. If d_o ≥ K, then uopt = 0. If d_o < K, then u_opt = 2(K− do) > 0.

Proof. By Lemma 3.5, the maximum achieved in R₁ is 0. If d_o ≥ K, then the total utility at the point P is u(d_o, d_o) = −do + bd_op(d_o)− do + bd_op(d_o) = 2(K− do)≤ 0.

∴ uopt = 0 in this case. If do < K, then u(do, do) = 2(K− do) > 0, which is better than R₁. ∴ uopt = 2(K − do) > 0 in this case.

Figure 3.4: A geometric perspective of the condition do < K such that u(do) > 0

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Corollary 3.7. Let dℓ be the less solution to bxp^′(x) = 1. If do ≥ dℓ, then uopt = u(do).

Proof. ∵ bp(do) ≥ bp(dℓ) > bd_ℓp^′(d_ℓ) = 1 ∴ do < K by Definition 2.6. In this case u_opt > 0 by Theorem 3.6. However the maximum total utility within R₁ is not greater than 0, so u_optcan only occur at the point P = (d_o, d_o).

We close this section with the following conclusive table.

Table 3.1: The maximum total utility of twoplayer filesharing games Condition Utility

d_o ≥ K 0

d_o < K 2(K− do)

3.2 Nash Equilibria

After analyzing the maximum total utility, we still have to find Nash equilibria in order to analyze the PoA and PoS.

Lemma 3.8. The player P_idoes not want to change his/her strategy d_iif and only if one of the following cases occurs.

Case I.

∂u_i

∂di

₋

does not exist (i.e., d_i = 0) and

∂u_i

∂di

+

≤ 0.

Case II.

∂u_i

∂di

₋

≥ 0 and ∂u_i

∂di

+

≤ 0 :









∂ui

∂d_i

₋

=

∂ui

∂d_i

+

= 0 ...(A)

∂u_i

∂d_i

₋

≥ 0 and∂u_i

∂d_i

+

=−1 ...(B) In case I, d_i = 0 and bd_jp^′(0)≤ 1. In case II(A), 0 < bdjp(d_i) < K and bd_jp^′(d_i) = 1.

In case II(B), bdjp(di) = K and bdjp^′(di)≥ 1.

Proof. Recall the utility function u_i = −di + min{K, bdjp(d_i)}. Differentiate it with respect to d_i.

∂ui

∂d_i =









∂

∂di

− di+ bd_jp(d_i)

= bd_jp^′(d_i)− 1 ≥ −1 if bd_jp(d_i)≤ K

∂

∂d_i

− di+ K

=−1 if bd_jp(d_i)≥ K.

Since bd_j is a fixed nonnegative number, and p^′(x) is a nonnegative nonincreasing func

tion, ∂u_i

∂d_i is nonincreasing for all d_i ≥ 0. Hence the result follows.

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Now we are going to discuss the places where these Nash equilibria occur by cases in the following theorems.

Lemma 3.9. In a Nash equilibrium (d1, d₂), if a player P_i’s strategy d_i satisfies case I, then the other player P_j’s strategy d_j must also satisfy case I. That is, d₁ = d₂ = 0.

Proof. If d_i = 0, then u_j = −dj + min{K, b · 0 · p(dj)} = −dj for all d_j ≥ 0, and

∂uj

∂dj =−1. According to Lemma 3.8, the player Pj can make an optimal strategy only by letting d_j = 0. In this case ^∂u_∂dⁱ

i is also−1, so the Nash equilibrium can only be (0, 0).

Definition 3.3. If bxp^′(x) = 1 has two different solutions, let dℓ be the less one, and let d_hbe the greater one. If the equation has only one solution, let d_ℓ and d_h both denote it.

Figure 3.5: A geometric solution of d_ℓ and d_h in Definition 3.3

Theorem 3.10. There exists a Nash equilibrium (d₁, d₂) such that d₁ satisfies case II(A) and d₂ satisfies case II(B), if and only if d₁and d₂both satisfy

(b d₂ p^′(d₁) = 1 and b d₂p(d₁) < K

b d₁ p^′(d₂)≥ 1 and b d1 p(d₂) = K.

In addition, the Nash equilibrium Nside1 = (d1, d2) is unique and exists if and only if dℓ

and d_h both exist and d_ℓ < d_o< d_h.

Proof. First, we prove necessity by contradiction. Assume neither d_ℓ nor d_h exists, or both d_ℓ and d_h exist but d_ℓ = d_h, or both d_ℓ and d_hexist, d_ℓ < d_h but d_o ̸∈ (dℓ, d_h). Each condition implies bd_op^′(d_o) ≤ 1. Now we’re going to explain why bd2p^′(d₁) < 1 in this assumption. By Definition 2.6 the point (d_o, d_o) must lie on the curve bxp(y) = K, and

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by Corollary 2.3 the constraint bd1p(d2) = K > bd2p(d1) tells us d1 > d2. We start from (x, y) = (d_o, d_o) and move along that curve in the correct direction (x ≥ do > y). If p^′(d_o) = 0, then p^′(x) = 0 and byp^′(x) = 0. If p^′(d_o) > 0, then byp^′(x) ≤ byp^′(d_o) <

bd_op^′(d_o)≤ 1. ∴ byp^′(x) < 1. If (x, y) = (d₁, d₂), then bd₂p^′(d₁) < 1 is a contradiction.

Nash equilibria cannot exist in this case.

Second, we prove sufficiency. Assume both d_ℓ and d_h exist, and d_o ∈ (dℓ, d_h). This condition implies bd_op^′(d_o) > 1 instead. By Corollary 2.3, we should move from (d_o, d_o) in the same direction (x ≥ do > y) again, so that byp(x) < K and bxp(y) = K always hold. Besides, we know bxp^′(y)≥ bdop^′(d_o), and byp^′(x) is decreasing, as the point (x, y) goes far away from (d_o, d_o). Since there is a point at infinity lim

xo→∞

yo→0

(x_o, y_o) on the curve such that lim

xo→∞

yo→0

by_op^′(x_o) = 0, by the intermediate value theorem there must exist one point (x, y) in this direction such that byp^′(x) = 1. In this case if (x, y) = (d₁, d₂), the Nash equilibrium N_side1exists. If (x, y) ̸= (d1, d₂), then either (x ≥ d1 and y < d₂) or (x ≤ d1 and y > d₂) happens. If the former happens and p^′(x) > 0, then byp^′(x) ≤ byp^′(d₁) < bd₂p^′(d₁) = 1. If the former happens and p^′(x) = 0, then byp^′(x) = 0. If the latter happens, then byp^′(x) ≥ byp^′(d₁) > bd₂p^′(d₁) = 1. The reader may discover that byp^′(x)̸= bd2p^′(d1) = 1 in both cases, so the point is unique.