Game Theory

where bi is a part of contents of block i, and the one is at position i. And then we can denote by X the r × ( r + d ) matrix whose i^th row is xi. All the xi form a set of r independent vectors which can span a subspace Π_X. Because any linear combination of the vectors {x₁, x₂,…, xr} belongs to Π_X , we know that the Π_X is closed under randomized linear combinations.

In the nullkey, the set of the signature called null key is the set of the null space of Π_X, denotes as Π^⊥_X. According to rank-nullity theorem, the dimension of Π^⊥_X is equal to d. The subspace Π^⊥X is spanned by the vectors {z1, z2,…, zd}, so we denote by Z the d × ( r + d ) matrix whose i^th row is zi. With network coding, all the encoded blocks are randomized linear combination of {x₁, x₂,…, x_r}, and belong to Π_X. Each encoded blocks is orthogonal to randomized linear combination of {z1, z2,…, zd} which belongs to Π^⊥X. The client verifies an encoded block is valid if the encoded block w satisfies the following condition:

(9)

where K_i is the matrix which is formed by the null keys.

2.4 Game Theory

Game Theory is a branch of mathematics which is used in social sciences, economics especially, as well as in biology engineering, political science,

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international relations, computer science, and philosophy. Game theory aspires to mathematically catch behavior in strategic situations, or game, in which an individual’s success depends on the other’s options.

Traditional applications of game theory attempt to find equilibriums in their games. In equilibrium, each player of the game has chosen a strategy, or made a decision. The types of games include cooperative or non-cooperative, symmetric or asymmetric, zero-sum or non-zero-sum, complete information or incomplete information…etc.

A non-cooperative game is a game that each player in the game makes decisions independently. A cooperative game is a game where groups of players enforce cooperative behavior. A symmetric game is a game where the rewards for playing a particular strategy depend only on the other strategies, not on the other’s identity. A zero-sum game means a game has a situation in which a player’s gain or cost is exactly equal to the other’s cost or gain. In non-zero-sum games, a player’s gain does not necessarily correspond with another. The difference between complete information games and incomplete information games is that in complete information game, every player knows the strategies and payoffs of the other player. For instance, Poker is a non-cooperative, asymmetric, incomplete information and zero-sum game, prisoner’s dilemma is a non-cooperative, symmetric, complete information and non-zero-sum game.

In game theory, Nash equilibrium is a solution of a game involving two players or multi player game. In Nash equilibrium situation, each player knows the equilibrium strategies of the other players and for each player, and no other strategy can reward more utility than equilibrium strategy. If each player has chosen a strategy and no player can reward by changing his or her strategy and the other player keep

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their strategy unchanged, then the current set of strategy and the utility constitute Nash equilibrium.

The prisoner's dilemma is a fundamental problem in game theory. This problem illustrates why two people might not cooperate. If the payoff matrix of prisoner’s dilemma is as following:

Prisoner B stays silent Prisoner B betrays Prisoner A stays silent Each serves 6 months Prisoner A: 10 years

Prisoner B: goes free Prisoner A betrays Prisoner A: goes free

Prisoner B: 10 years

Each serves 5 years

Table 2.4-1: The payoff matrix of prisoner’s dilemma

In table 2.4-1, if both prisoner A and prisoner B stay silent, they just only server 6 months, but if one of them betrays, the betrayer can go free and the other must server 10 years. If both prisoner A and prisoner B betray each other, they must serve 5 years. According to above table description, the best strategy in the table 2.4-1 is that both of them stay silent. However, we obtain that either prisoner A or prison B chooses the strategy of betraying is better than staying silent. If they want to choose the best strategy, they must satisfy the cooperative situation. The cooperative situation does not exist in the prisoner dilemma problem because the strategy of staying silent has fewer benefits than the strategy of betraying. In this game, the Nash equilibrium is both prisoner A and prisoner B choose the strategy of betraying. The prisoner dilemma illustrates that the best strategy may not be the Nash equilibrium in the game theory.

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In 2003, Buragohain et al. proposes a game theoretic framework for incentive in the peer-to-peer system [11]. In [11], the author assumes that all players are rational under the game environment given by the author. The players are rational because they wish to maximize their own benefit. There are three key components in this framework: strategy, utility and Nash equilibrium. The strategy for each player is the behavior interacting with other players. The player’s utility is the benefit derived from his interaction with other players. If no player can improve his utility by changing his strategy, the collection of players are said to be at Nash equilibrium. The reaction function is the best reaction for player, given a strategy for other. If the result of reaction function is equal to the result of reaction function at past, then the Nash equilibrium is found.

In 2008, K. J. Ray Liu et al. proposes another game theoretic framework for incentive- based peer-to-peer live streaming social network [12]. In [12], it illustrates two-player peer-to-peer live streaming game with complete information and different optimality criteria such as Pareto-Optimality, proportional fairness and absolute fairness. The author considers the cheating behaviors that the player gives the cheating information to mislead the players into disadvantageous situation. The author proposes the cheating-proof strategy in which the player in game should not send more data than what the other has sent. However, there is a contradiction at the cheating-proof strategy if the two-player game is incomplete information game. The contradiction means that the player in the game will not offer the better strategy because of the restriction of the cheating-proof strategy. In our study, we attempt to find Nash equilibrium under the complete information situation and incomplete information situation with network coding environment. On the other hand, we also consider the content pollution problem situation. However, in [12], it is based on

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peer-to-peer live-streaming social network without content pollution problem and the author only considers the complete information situation.

In 2008, M. K. H. Yeung et al have proposed the packet exchange game for scalable peer-to-peer media streaming system [13]. In the packet exchange game, the author uses the punish-k strategy to achieve the equilibrium strategy. It is different from the above frameworks in which they tend to give some incentive strategy to reward more utility, and the punish-k strategy offers the punishment to prevent the players from changing their strategy. The author mathematically demonstrates that the loss utility of punishment is larger than the reward of leaving the Nash equilibrium.

Recent results in [13], [19], [20], and [21] have focused on using game theory to solve packet forwarding problem in mobile ad hoc networks or peer-to-peer system without network coding technique or with network coding technique. The packet forwarding problem means the procedure to route the packets from the source to the destination. Recent results in [17] have focused on using game theory to solve the resource distribution problem based on network coding technique. Recent results in [22] and [26] have focused on using game theory to solve the joint optimization problem. In [22], the author attempts to increase the capacity of multi-channel mesh network and proposes the joint optimization problem which is concerned with routing, channel assignment, and network coding. In [26], the author attempts to improve the bandwidth efficiency in OFDMA based wireless network and proposes the joint optimization problem is concerned with dynamic subcarrier assignment and network coding. Recent results in [23] and [27] have focused on using game theory to solve the rate allocation and control problem. Recent results in [24] have focused on using game theory to solve the power management problem in ad-hoc opportunistic radio.

Recent results in [25] have focused on using game theory to solve the open spectrum

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sharing problem.In 2008, C. Wu et al have proposed a dynamic auction game for multi-overlay peer-to-peer streaming using network coding [17]. The game attempts to resolve the conflicts among coexisting streaming overlays in their bandwidth competition. The player in the dynamic auction game can minimize their streaming cost and satisfy the streaming rate for each coexisting streaming overlay.

In 2009, X. Zhang et al seek to use a novel concept to describe the coding based peer-to-peer content distribution system as a peer-to-peer market system [18]. The authors have proposed entry price and expected payoff for each coded block, and claimed that this market system can maintain stability if peer follows the operation guidelines for a peer-to-peer market. Finally, the author characterizes the pricing strategies as many subgame perfect Nash equilibrium.

In 2010, T. Chen et al have proposed INPAC for the wireless mesh network using network coding [19]. The authors attempt to solve the incentive compatible packet forwarding problem and incentive compatible routing problem by the analysis of game theory. The author assumes the players in this game are required by the MORE protocol and considers this game as a repeated game. They claim INPAC is the first incentive scheme for packet forwarding in wireless mesh networks using network coding.

However, the dynamic auction game focuses on the optimal distribution of streaming rate based on the minimum of streaming cost, but not considers the malicious or cheating situation on its game. In our study, we pay attention to the maximum of player’s reward based on the Nash Bargaining Solution [15] and also consider the malicious and cheating situation. In [18], the author focuses on a theoretical framework that quantifies the market power of network coding in a non-cooperative P2P content distribution system. In our study, we focus on how to

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distribute the resource to maximize the player’s reward based on the Nash Bargaining Solution. The game in the INPAC is the repeated game to deal with the incentive compatible packet forwarding problem under the wireless Networks using network coding. However, the author does not consider the content pollution problem on their environment. In our study, we have proposed the learning-based game under the coded-based peer-to-peer system to deal with the resources distribution problem with the malicious situation which is the malicious player will randomly modify the contents of encoded block. According to the bargaining procedure, the player in our proposed game will update the coefficient to evaluate a player’s property and share the part of message to other players.

In our study, we attempt to address the problem on the network coding environment. The network coding technique is a branch of the channel coding technique. The channel coding technique is popular and suitable for the content distribution system; especially the transmission type is broadcast as wireless network.

Most of the channel coding techniques can be regarded as the procedure of finding the solution from the set of the linear equations. If the peer can receive enough encoded blocks, the peer can decode the part or all of the original blocks. As above mentioned, we know the proposed rank-based game is also suitable for the environment with channel coding technique.

In this thesis, we consider how to maximize player’s utility through the negotiation of game theory even if there are some of players who maybe perform malicious or cheating behavior. According to the above mentioned, we know our problem is belonging with resource distribution problems and security problems. The resource distribution problems under network coding technique concerned with security is a novel opinion. It is essential and important in the future. If there is no

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effective method to restrict to malicious behaviors, the effect of resource distribution will be reduced or even the whole network based on network coding technique will destroy. To solve this problem, we consider both of malicious behaviors and cheating behaviors with network coding technique and attempt to use the game theory to analyze the player's behavior.

In our study, we consider the content pollution problem with network coding technique and after completing each game, measure the alteration of each player’s contribution and update the player’s information of game. We will start from the analysis of two-player game and then extend the two-player game to the multi-player game. We also consider the impacts of the cheating behaviors and the malicious behaviors and the impacts of above will be restricted in our proposed method. We will propose a novel architecture and rank-based game at Chapter 3.

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