**Cooperative Trust Games**

Dariusz G. Mikulski

18 Will-be-set-by-IN-TECH

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Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53928

## **1. Introduction**

In certain multi-agent systems, the interactions between agents result in the formation of relationships, which can be leveraged for cooperative or collaborative activities. These relationships generally constrain individual-agent actions, since relationships imply that at least one contract (or mutual agreement) between the agents must exist. There is always some uncertainty as to whether or not either agent can or will satisfy some contract requirement – especially at the creation of a new contract. But in order to maintain the existence of a contract, each agent must overcome this uncertainty and assume that the other will do the same. The mechanism that facilitates this "act of faith" is generally regarded as "trust." In essence, each agent (whether a person or organization) in a relationship mutually trusts that the loss of some control will result in cooperative gains that neither agent could achieve alone.

In general, trust helps agents deal with uncertainty by reducing the complexity of expectations in arbitrary situations involving risk, vulnerability, or interdependence [1]. This is because agents rely on trust whenever they need to gauge something they cannot ever know precisely with reasonable time or effort. The benefits of trustworthy relationships include lower defensive monitoring of others, improved cooperation, improved information sharing, and lower levels of conflict [2]. But the reliance on trust also exposes people to vulnerabilities associated with betrayal, since the motivation for trust – the need to believe that things will behave consistently – exposes individuals to potentially undesirable outcomes. Thus, trust is a concept that must not only be managed, but also justified [3].

Since agents in an arbitrary system are always assumed to have selfish interests, the goal of each agent is to try to find the most fruitful relationships in a pool of potential agents [4]. That said, we cannot assume that agents do not already have pre-existing relationships with other agents. Furthermore, some agents may actually be within strongly-connected subsystem groups known as *coalitions*, where every agent in the coalition has a relationship with every other agent in the coalition. A coalition may contain a mixture of trustworthy

© 2013 Mikulski, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Mikulski, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and untrustworthy agents – but as a group, achieves cooperative gains that no sub-coalition could match. Thus, agents may be justified in forming relationships with coalition members who are not ideally trustworthy in order to acquire these cooperative gains as well.

Cooperative Trust Games 235

which have limited value for cooperative applications, but are included for completeness. Then, we discuss superadditive and convex trust games, which show conditions for agents to form a grand coalition. In general, grand coalition solution concepts presented here can also be

 �� �� → ℝ associates with each coalition ��� a real-valued payoff �(�) that is distributed between the agents. Singleton coalitions, by definition, are assigned no

The transferable utility assumption means that payoffs in a coalition may be freely distributed among its members. With regards to payoff value of trust between agents, this assumption can be interpreted as a universal means for agents to mutually share the value of their trustworthy relationships. Trust cultivation often requires reciprocity between two agents as a necessary behavior to develop trust, and a transferable utility is a convenient

In defining a transferable payoff value of trust, one aspect to consider are the "goods of trust". These refer to opportunities for cooperative activity, knowledge, and autonomy. In this chapter, we refer to these goods as *trust synergy* �(�), which is a trust-based result that could not be obtained independently by two or more agents. We may also interpret trust synergy as the value obtained by agents in a coalition as a result of being able to work together due to their attitudes of trust for each other. In defining a set function for trust synergy, it is important to explicitly show how each agent's attitude of trustworthiness for every other agent in a coalition affects this synergy. In general, higher levels of trust in a

The payoff value of trust, however, also includes an opposing force in the form of vulnerability exposure, which we refer to as *trust liability* �(�). Trusting involves being optimistic that the trustee will do something for the truster; and this optimism is what causes the vulnerability, since it restricts the inferences a truster makes about the likely actions of the trustee. However, the refusal to be vulnerable tends to undermine trust since it does not allow others to prove their own trustworthiness, stifling growth in trust synergy. Thus, we see that agents in trust-based relationships with other agents must be aware of the balance between the values of the trust synergy and trust liability in addition

Let the characteristic payoff function of a trust game be the difference between the trust

�(�) = �(�) � �(�) (1)

applied to smaller coalitions within a trust game through the use of a trust subgame.

Let Γ = (�� �) be a coalitional trust game with transferable utility where:

**2.1. Preliminaries** 

� is a finite set of agents, indexed by �

way to model the exchange for this notion.

coalition should produce higher levels of synergy.

to their relative magnitudes.

synergy and trust liability of a coalition �.

value; i.e. �(�) = ���� � �.

As a simple example to illustrate this concept, consider two geographically-separated agents (who never physically met) who would like to conduct a financial transaction in exchange for some good. One agent must provide the good (through the mail) and the other must provide the payment (through the mail or electronically). If both agents follow their economic best interest, then neither agent should participate in the transaction since both agents are vulnerable to betrayal. This is because neither agent can truly verify the intent of the other agent before they act. Thus, if a transaction takes place, it can be entirely attributed to trust since both agents needed to overcome the uncertainty associated with the transaction. Let us suppose, however, that the value of the good and the size of the payment are sufficiently high such that no amount of mutual trust allows a direct transaction to take place. To handle this situation, both agents could form a coalition with a mutually trusted third party, such as an escrow agent. The escrow agent would receive the payment from one agent to verify that the good can be shipped, and then later disperse the payment to the other agent (minus the escrow fee) when the good has been verified as received. Here, each agent benefits from the cooperative gains of the transaction. These gains would not be possible if even one agent chose to disband from the coalition.

This chapter intends to show how one could mathematically describe these types of trustbased interactions via the *cooperative trust game* to predict coalition formation and disbanding. It presents a rigorous treatment of coalition formation using cooperative game theory as the underlying mathematical framework. It is important to highlight that cooperative game theory is significantly different than the more widely recognized competitive (non-cooperative) game theory. Cooperative game theory focuses on what groups of self-interested agents can achieve. It is not concerned with how agents make choices or coordinate in coalitions, and does not assume that agents will always agree to follow arbitrary instructions. Rather, cooperative game theory defines games that tell how well a coalition can do for itself. And while the coalition is the basic modeling unit for coalition game, the theory supports modeling individual agent preferences without concern for their possible actions. As such, it is an ideal framework for modeling trust-based coalition formation since it can show how each agent's trust preferences can influence a group's ability to reason about trustworthiness. We refer the reader to [5] for an excellent primer on cooperative game theory.

#### **2. Classes of trust games**

This section characterizes different classes of trust games within the context of cooperative game theory. Our characterizations provide the necessary conditions for a coalition trust game to be classified into a particular class. We start with additive and constant-sum trust games, which have limited value for cooperative applications, but are included for completeness. Then, we discuss superadditive and convex trust games, which show conditions for agents to form a grand coalition. In general, grand coalition solution concepts presented here can also be applied to smaller coalitions within a trust game through the use of a trust subgame.

#### **2.1. Preliminaries**

234 Game Theory Relaunched

the coalition.

primer on cooperative game theory.

**2. Classes of trust games** 

and untrustworthy agents – but as a group, achieves cooperative gains that no sub-coalition could match. Thus, agents may be justified in forming relationships with coalition members

As a simple example to illustrate this concept, consider two geographically-separated agents (who never physically met) who would like to conduct a financial transaction in exchange for some good. One agent must provide the good (through the mail) and the other must provide the payment (through the mail or electronically). If both agents follow their economic best interest, then neither agent should participate in the transaction since both agents are vulnerable to betrayal. This is because neither agent can truly verify the intent of the other agent before they act. Thus, if a transaction takes place, it can be entirely attributed to trust since both agents needed to overcome the uncertainty associated with the transaction. Let us suppose, however, that the value of the good and the size of the payment are sufficiently high such that no amount of mutual trust allows a direct transaction to take place. To handle this situation, both agents could form a coalition with a mutually trusted third party, such as an escrow agent. The escrow agent would receive the payment from one agent to verify that the good can be shipped, and then later disperse the payment to the other agent (minus the escrow fee) when the good has been verified as received. Here, each agent benefits from the cooperative gains of the transaction. These gains would not be possible if even one agent chose to disband from

This chapter intends to show how one could mathematically describe these types of trustbased interactions via the *cooperative trust game* to predict coalition formation and disbanding. It presents a rigorous treatment of coalition formation using cooperative game theory as the underlying mathematical framework. It is important to highlight that cooperative game theory is significantly different than the more widely recognized competitive (non-cooperative) game theory. Cooperative game theory focuses on what groups of self-interested agents can achieve. It is not concerned with how agents make choices or coordinate in coalitions, and does not assume that agents will always agree to follow arbitrary instructions. Rather, cooperative game theory defines games that tell how well a coalition can do for itself. And while the coalition is the basic modeling unit for coalition game, the theory supports modeling individual agent preferences without concern for their possible actions. As such, it is an ideal framework for modeling trust-based coalition formation since it can show how each agent's trust preferences can influence a group's ability to reason about trustworthiness. We refer the reader to [5] for an excellent

This section characterizes different classes of trust games within the context of cooperative game theory. Our characterizations provide the necessary conditions for a coalition trust game to be classified into a particular class. We start with additive and constant-sum trust games,

who are not ideally trustworthy in order to acquire these cooperative gains as well.

Let Γ = (�� �) be a coalitional trust game with transferable utility where:


The transferable utility assumption means that payoffs in a coalition may be freely distributed among its members. With regards to payoff value of trust between agents, this assumption can be interpreted as a universal means for agents to mutually share the value of their trustworthy relationships. Trust cultivation often requires reciprocity between two agents as a necessary behavior to develop trust, and a transferable utility is a convenient way to model the exchange for this notion.

In defining a transferable payoff value of trust, one aspect to consider are the "goods of trust". These refer to opportunities for cooperative activity, knowledge, and autonomy. In this chapter, we refer to these goods as *trust synergy* �(�), which is a trust-based result that could not be obtained independently by two or more agents. We may also interpret trust synergy as the value obtained by agents in a coalition as a result of being able to work together due to their attitudes of trust for each other. In defining a set function for trust synergy, it is important to explicitly show how each agent's attitude of trustworthiness for every other agent in a coalition affects this synergy. In general, higher levels of trust in a coalition should produce higher levels of synergy.

The payoff value of trust, however, also includes an opposing force in the form of vulnerability exposure, which we refer to as *trust liability* �(�). Trusting involves being optimistic that the trustee will do something for the truster; and this optimism is what causes the vulnerability, since it restricts the inferences a truster makes about the likely actions of the trustee. However, the refusal to be vulnerable tends to undermine trust since it does not allow others to prove their own trustworthiness, stifling growth in trust synergy. Thus, we see that agents in trust-based relationships with other agents must be aware of the balance between the values of the trust synergy and trust liability in addition to their relative magnitudes.

Let the characteristic payoff function of a trust game be the difference between the trust synergy and trust liability of a coalition �.

$$\upsilon(\mathcal{S}) = \mathfrak{s}(\mathcal{S}) - l(\mathcal{S}) \tag{1}$$

$$\upsilon(\mathcal{S}\_1 \cup \mathcal{S}\_2) = \upsilon(\mathcal{S}\_1) + \upsilon(\mathcal{S}\_2) \,\,\forall \mathcal{S}\_1, \mathcal{S}\_2 \subset \mathcal{N} \tag{2}$$

$$\begin{array}{c} \mathbf{s}(\mathcal{S}\_1 \cup \mathcal{S}\_2) - l(\mathcal{S}\_1 \cup \mathcal{S}\_2) = \mathbf{s}(\mathcal{S}\_1) - l(\mathcal{S}\_1) + \mathbf{s}(\mathcal{S}\_2) - l(\mathcal{S}\_2) \\ \{\forall \mathcal{S}\_1, \mathcal{S}\_2 \subset N \colon \mathcal{S}\_1 \cap \mathcal{S}\_2 = \emptyset\} \end{array}$$

$$\begin{array}{c} \mathbf{s}(\mathcal{S}\_1 \cup \mathcal{S}\_2) - \mathbf{s}(\mathcal{S}\_1) - \mathbf{s}(\mathcal{S}\_2) = l(\mathcal{S}\_1 \cup \mathcal{S}\_2) - l(\mathcal{S}\_1) - l(\mathcal{S}\_2) \\ \{\forall \mathcal{S}\_1, \mathcal{S}\_2 \subset N \colon \mathcal{S}\_1 \cap \mathcal{S}\_2 = \emptyset\} \end{array} \tag{3}$$

$$\upsilon(N) = \upsilon(\mathcal{S}) + \upsilon(N|\mathcal{S}) = k \,\,\forall \mathcal{S} \subset N \tag{4}$$

$$\begin{aligned} \text{s(N)} - l(N) &= \text{s(S)} - l(\text{S}) + \text{s(N/S)} - l(N/\text{S}) \,\forall \mathcal{S} \subset N\\ \text{s(N)} - \text{s(S)} - \text{s(N/S)} &= l(N) - l(\mathcal{S}) - l(N/\mathcal{S}) \,\forall \mathcal{S} \subset N \end{aligned} \tag{5}$$

$$\mathbf{s}(N) - \mathbf{s}(l) - \mathbf{s}(N \backslash l) = l(N) - l(l) - l(N \backslash l) \,\,\forall l \in N \tag{6}$$

$$\begin{array}{c} \mathsf{s}(N) - l(N) = \mathsf{s}(l) - l(l) + \mathsf{s}(N\backslash i) - l(N\backslash i) \,\,\forall i \in N\\ \upsilon(N) = \upsilon(i) + \upsilon(N\backslash i) \,\,\forall i \in N \end{array}$$

$$\upsilon(N) = \upsilon(N \backslash \mathfrak{l}) \,\,\forall \mathfrak{l} \in N \tag{7}$$

$$
\upsilon(\mathcal{S}\_1 \cup \mathcal{S}\_2) \ge \upsilon(\mathcal{S}\_1) + \upsilon(\mathcal{S}\_2) \,\,\forall \mathcal{S}\_1, \mathcal{S}\_2 \subset N \tag{8}
$$

$$S \subseteq A \subseteq N \to \mathcal{v}(S) \le \mathcal{v}(A) \le \mathcal{v}(N) \tag{9}$$

$$\begin{aligned} \mathrm{s}(\mathcal{S}\_1 \cup \mathcal{S}\_2) - l(\mathcal{S}\_1 \cup \mathcal{S}\_2) &\cong \mathrm{s}(\mathcal{S}\_1) - l(\mathcal{S}\_1) + \mathrm{s}(\mathcal{S}\_2) - l(\mathcal{S}\_2) \\ &\{\forall \mathcal{S}\_1, \mathcal{S}\_2 \subset N \colon \mathcal{S}\_1 \cap \mathcal{S}\_2 = \emptyset\} \end{aligned}$$

$$\begin{aligned} \mathfrak{s}\left(\mathbb{S}\_1 \cup \mathbb{S}\_2\right) - \mathfrak{s}\left(\mathbb{S}\_1\right) - \mathfrak{s}\left(\mathbb{S}\_2\right) &\geq l\left(\mathbb{S}\_1 \cup \mathbb{S}\_2\right) - l\left(\mathbb{S}\_1\right) - l\left(\mathbb{S}\_2\right) \\ &\{\forall \mathcal{S}\_1, \mathcal{S}\_2 \subset N \colon \mathbb{S}\_1 \cap \mathbb{S}\_2 = \emptyset\} \end{aligned} \tag{10}$$

$$
\upsilon(\mathcal{S}\_1 \cup \mathcal{S}\_2) + \upsilon(\mathcal{S}\_1 \cap \mathcal{S}\_2) \ge \upsilon(\mathcal{S}\_1) + \upsilon(\mathcal{S}\_2) \,\,\forall \mathcal{S}\_1, \mathcal{S}\_2 \subset N \tag{11}
$$

$$\upsilon(\mathbb{S}\cup i) - \upsilon(\mathbb{S}) \le \upsilon(A\cup i) - \upsilon(A) \text{ whenever } \mathbb{S} \subset A \subset N\backslash i \tag{12}$$

$$\begin{aligned} \upsilon\_R(\mathbb{S}\_1 \cup \mathbb{S}\_2) + \upsilon\_R(\mathbb{S}\_1 \cap \mathbb{S}\_2) &= \upsilon(R \cup \mathbb{S}\_1 \cup \mathbb{S}\_2) + \upsilon(R \cup (\mathbb{S}\_1 \cap \mathbb{S}\_2)) - 2\upsilon(R) \\ &= \upsilon\left((R \cup \mathbb{S}\_1) \cup (R \cup \mathbb{S}\_2)\right) + \upsilon\left((R \cup \mathbb{S}\_1) \cap (R \cup \mathbb{S}\_2)\right) - 2\upsilon(R) \\ &\ge \upsilon(R \cup \mathbb{S}\_1) + \upsilon(R \cup \mathbb{S}\_2) - 2\upsilon(R) \\ &= \left\{\upsilon(R \cup \mathbb{S}\_1) - \upsilon(R)\right\} + \left\{\upsilon(R \cup \mathbb{S}\_2) - \upsilon(R)\right\} \\ &= \upsilon\_R(\mathbb{S}\_1) + \upsilon\_R(\mathbb{S}\_2) \end{aligned} \tag{13}$$

$$\begin{aligned} \upsilon\_R(\langle \mathcal{S}\_1 \cup \mathcal{S}\_2 \rangle \backslash R) &\geq \upsilon\_R(\mathcal{S}\_1 \backslash R) + \upsilon\_R(\mathcal{S}\_2 \backslash R) \\ \upsilon(\mathcal{S}\_1 \cup \mathcal{S}\_2) - \upsilon(R) &\geq \upsilon(\mathcal{S}\_1) - \upsilon(R) + \upsilon(\mathcal{S}\_2) - \upsilon(R) \end{aligned}$$

$$\begin{aligned} \upsilon(\mathbb{S}\_1 \cup \mathbb{S}\_2) + \upsilon(R) &\geq \upsilon(\mathbb{S}\_1) + \upsilon(\mathbb{S}\_2) \\ \upsilon(\mathbb{S}\_1 \cup \mathbb{S}\_2) + \upsilon(\mathbb{S}\_1 \cap \mathbb{S}\_2) &\geq \upsilon(\mathbb{S}\_1) + \upsilon(\mathbb{S}\_2) \end{aligned} \tag{14}$$

$$s\_R(\{\mathcal{S}\_1 \cup \mathcal{S}\_2\} \backslash R) - l\_R(\{\mathcal{S}\_1 \cup \mathcal{S}\_2\} \backslash R) \ge s\_R(\mathcal{S}\_1 \backslash R) - l\_R(\mathcal{S}\_1 \backslash R) + s\_R(\mathcal{S}\_2 \backslash R) - l\_R(\mathcal{S}\_2 \backslash R)$$

$$\{\forall \mathcal{S}\_1, \mathcal{S}\_2 \subset N \colon \mathcal{S}\_1 \cap \mathcal{S}\_2 = R\}$$

$$l\_R(\{\mathcal{S}\_1 \cup \mathcal{S}\_2\} \backslash R) - s\_R(\mathcal{S}\_1 \backslash R) - s\_R(\mathcal{S}\_2 \backslash R) \ge l\_R(\{\mathcal{S}\_1 \cup \mathcal{S}\_2\} \backslash R) - l\_R(\mathcal{S}\_1 \backslash R) - l\_R(\mathcal{S}\_2 \backslash R) \tag{15}$$

$$T = \begin{bmatrix} t\_{i,j} \end{bmatrix}\_{\|N\| \times \|N\|} = \begin{cases} t\_{i,j} = 1, & i = j \\ t\_{i,j} \in [0, 1], l \neq j \end{cases} \tag{16}$$

$$\Sigma = \left[ \sigma\_{l,j} \right]\_{|N| \times |N|} = \begin{cases} \sigma\_{l,j} = 0, & i = j \\ \sigma\_{l,j} = \sigma\_{j,l} \ge 0, i \ne j \end{cases} \tag{17}$$

$$
\Lambda = \begin{bmatrix} \lambda\_{i,j} \end{bmatrix}\_{\|N\| \times \|N\|} = \begin{cases} \lambda\_{i,j} = 0, i = j \\ \lambda\_{i,j} \ge 0, i \ne j \end{cases} \tag{18}
$$

$$\text{cs(S)} = \Sigma\_{i,j \in \mathcal{S}} \sigma\_{i,j} \ t\_{i,j} t\_{j,i} \,\forall i > j \tag{19}$$

$$l(\mathbb{S}) = \Sigma\_{i,j \in \mathbb{S}} \lambda\_{i,j} t\_{i,j} \; \forall i \neq j \tag{20}$$

$$\begin{aligned} \boldsymbol{\nu}(\boldsymbol{\Sigma}) &= \sum\_{\substack{\boldsymbol{i},\boldsymbol{j}\in S \\ \forall i>j}} \sigma\_{\boldsymbol{i},\boldsymbol{j}} \, t\_{\boldsymbol{i},\boldsymbol{j}} t\_{\boldsymbol{j},\boldsymbol{i}} - \sum\_{\substack{\boldsymbol{i},\boldsymbol{j}\in S \\ \forall i\neq j}} \lambda\_{\boldsymbol{i},\boldsymbol{j}} t\_{\boldsymbol{i},\boldsymbol{j}} \\ \boldsymbol{\nu}(\boldsymbol{\Sigma}) &= \sum\_{\substack{\boldsymbol{i},\boldsymbol{j}\in S \\ \forall i>j}} t\_{\boldsymbol{i},\boldsymbol{j}} t\_{\boldsymbol{j},\boldsymbol{i}} \left( \sigma\_{\boldsymbol{i},\boldsymbol{j}} - \frac{\lambda\_{\boldsymbol{i},\boldsymbol{j}}}{\mathbf{t}\_{\boldsymbol{j},\boldsymbol{i}}} - \frac{\lambda\_{\boldsymbol{j},\boldsymbol{i}}}{\mathbf{t}\_{\boldsymbol{i},\boldsymbol{j}}} \right) \end{aligned} \tag{21}$$

$$\sum\_{\substack{l,j \in S \\ \forall l > j}} \sigma\_{l,j} < \sum\_{\substack{l,j \in S \\ \forall l \neq j}} \frac{\lambda\_{l,j}}{t\_{j,l}} \tag{22}$$
 
$$\left\{ \forall i, j \in S \colon t\_{i,l} t\_{j,l} = k \right\}$$

$$\upsilon(\mathcal{S}) = \sum\_{\forall i \geqslant j} \sigma\_{i,j} \, k - \sum\_{\forall i \geqslant \mathcal{S}} \frac{\lambda\_{\ell,j}}{\mathbf{t}\_{f,\ell}} k \, \tag{23}$$
 
$$\upsilon(\mathcal{S}) = k \begin{pmatrix} \sum\_{\substack{i,j \in \mathcal{S} \\ \forall i > j}} \sigma\_{i,j} - \sum\_{\substack{i,j \in \mathcal{S} \\ \forall i \not\simeq j}} \frac{\lambda\_{\ell,j}}{\mathbf{t}\_{f,i}} \end{pmatrix}$$


$$
u\_A(B) \le 
u\_A(B \cup l) \qquad A \subseteq B \subseteq N \backslash l \tag{24}$$

$$
\mu\_A(\mathcal{C}) \le \mu\_A(\mathcal{C} \cup \mathcal{t}) \qquad A \subseteq \mathcal{C} \subseteq N/\mathcal{t} \tag{25}
$$

$$
u\_A(B \cup l) \le 
u\_A(C \cup l) \qquad A \subseteq B \subseteq C \subseteq N \backslash l \tag{26}$$

$$
u\_A(B \cup l) - 
u\_A(B) \le 
u\_A(C \cup l) - 
u\_A(C) \, A \subseteq B \subseteq C \subseteq N/l \tag{27}$$

$$
u\_A(\mathbb{C}) \le 
u\_{A \cup i}(\mathbb{C}) \qquad A \subseteq \mathcal{C} \subseteq N \backslash i \tag{28}$$

$$
u\_B(\mathcal{C}) \le 
u\_{B \cup l}(\mathcal{C}) \qquad B \subseteq \mathcal{C} \subseteq N/l \tag{29}$$

$$
u\_{A \cup l}(\mathcal{C}) \le \mathcal{u}\_{B \cup l}(\mathcal{C}) \quad A \subseteq B \subseteq \mathcal{C} \subseteq N \backslash l \tag{30}$$

$$
u\_{A \cup i}(\mathcal{C}) - \mathcal{u}\_A(\mathcal{C}) \le \mathcal{u}\_{B \cup i}(\mathcal{C}) - \mathcal{u}\_B(\mathcal{C}) \; A \subseteq B \subseteq \mathcal{C} \subseteq N \backslash i \tag{31}$$

$$\iota\_R(\mathbb{S}) = \upsilon(R) + \sum\_{l \in R. j \in \mathbb{S} \backslash R} \upsilon(\{l, j\}) \text{ } R \subseteq \mathbb{S} \subseteq N \tag{32}$$

$$a\_R(\mathcal{S}) = \Sigma\_{l \in R, j \in \mathcal{S} \backslash R} \,\,\nu(\{l, j\}) \,\, R \subseteq \mathcal{S} \subseteq N \tag{33}$$

$$\mathcal{L}\_R(\mathcal{S}) = \mathcal{\upsilon}(\mathcal{S}) - \mathcal{\upsilon}(\mathcal{S}\backslash R) - \sum\_{l \in R, j \in S} \mathcal{\upsilon}(\{l, j\}) \text{ } R \subseteq \mathcal{S} \subseteq N \tag{34}$$

$$m\_R(\mathbb{S}) = a\_R(\mathbb{S}) + c\_R(\mathbb{S}) = \upsilon(\mathbb{S}) - \upsilon(\mathbb{S} \backslash R) \, R \subseteq \mathbb{S} \subseteq N \tag{35}$$

$$
\Sigma = \begin{bmatrix} 0 & 2 & 3 & 4 \\ 2 & 0 & 3 & 4 \\ 3 & 3 & 0 & 4 \\ 4 & 4 & 4 & 0 \end{bmatrix} \\ \Lambda = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 \\ 2 & 2 & 0 & 2 \\ 3 & 3 & 3 & 0 \end{bmatrix}
$$

$$\upsilon(\{1,2\}) = 1; \; \upsilon(\{1,3\}) = 1; \; \upsilon(\{1,4\}) = 1; \; \upsilon(\{2,3\}) = 0; \; \upsilon(\{2,4\}) = 0;$$

$$\upsilon(\{3,4\}) = -1; \; \upsilon(\{1,2,3\}) = 2; \; \upsilon(\{1,2,4\}) = 2; \; \upsilon(\{1,3,4\}) = 1;$$

$$\upsilon(\{2,3,4\}) = -1; \; \upsilon(\{1,2,3,4\}) = 2;$$

$$T\_1 = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{bmatrix} \\ T\_2 = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 \end{bmatrix} \\ T\_3 = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{bmatrix} \\ T\_4 = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 \end{bmatrix}$$

$$v(\{1,2\}) = 1; \; v(\{1,3\}) = 1; \; v(\{1,4\}) = 1; \; v(\{2,3\}) = 0; \; v(\{2,4\}) = 0;$$

$$v(\{3,4\}) = 0; \; v(\{1,2,3\}) = 2; \; v(\{1,2,4\}) = 2; \; v(\{1,3,4\}) = 2;$$

$$v(\{2,3,4\}) = 0; \; v(\{1,2,3,4\}) = 3;$$

$$\Sigma = \begin{bmatrix} \sigma\_{l,j} \end{bmatrix}\_{\left| N \right| \times \left| N \right|} = \begin{cases} \sigma\_{l,j} = 0, & l = j \\ \sigma\_{l,j} = \max(\{l, j\}), & l \neq j \end{cases} \tag{36}$$

$$
\Lambda = \begin{bmatrix} \lambda\_{i,j} \end{bmatrix}\_{\vert N\vert \times \vert N\vert} = \begin{cases} \lambda\_{i,j} = 0, & i = j \\ \lambda\_{i,j} = i - 1, i \neq j \end{cases} \tag{37}
$$

$$T = \begin{bmatrix} t\_{i,j} \end{bmatrix}\_{|N| \times |N|} = \begin{cases} \begin{array}{c} t\_{i,j} = 1, & i = j \\ t\_{i,j} = 1, & i \neq j, \min\{\{i, j\}\} = 1 \end{array} \\\ t\_{i,j} = t\_{j,i} \in \{0, 1\}, & i \neq j, \min\{\{i, j\}\} = 2 \\\ t\_{i,j} = 0, & i \neq j, \min\{\{i, j\}\} > 2 \end{cases} \tag{38}$$

$$
\max(\{i, j\}) < (i - 1) + (j - 1)
$$

$$
\max(\{i, j\}) < i + j - 2
\tag{39}
$$

$$\mathcal{V}(\mathcal{S}) = \sum\_{\substack{l,j \in \mathcal{S} \\ \forall l \succ j}} t\_{l,j} t\_{j,l} \left( \max\{ \{ l, j \} \} - \frac{l-1}{t\_{j,l}} - \frac{j-1}{t\_{l,j}} \right) \tag{40}$$

$$\upsilon(\{i, j\}) = \mathfrak{t}\_{i, j} \mathfrak{t}\_{j, l} \left( \max(\{i, j\}) - \frac{i - 1}{\mathfrak{t}\_{j, l}} - \frac{j - 1}{\mathfrak{t}\_{l, j}} \right) \tag{41}$$

$$
\upsilon(\{1, j\}) = t\_{1,j} t\_{j,1} \left( j - \frac{j - 1}{t\_{1,j}} \right)
$$

$$
\upsilon(\{1, j\}) = j t\_{1,j} t\_{j,1} - j t\_{j,1} + t\_{j,1}
$$

$$
\upsilon(\{1, j\}) = t\_{j,1} \{j t\_{1,j} - j + 1\} \tag{42}
$$

$$\upsilon(\{i,1\}) = t\_{i,1}t\_{1,i}\left(i - \frac{i-1}{t\_{1,i}}\right)$$

$$\upsilon(\{i,1\}) = it\_{i,1}t\_{1,i} - it\_{i,1} + t\_{i,1}$$

$$\upsilon(\{i,1\}) = t\_{i,1}(it\_{1,i} - i + 1) \tag{43}$$

$$
\psi(\{2, j\}) = t\_{2,j} t\_{j,2} \left( j - \frac{1}{t\_{j,2}} - \frac{j - 1}{t\_{2,j}} \right)
$$

$$
\psi(\{2, j\}) = t\_{2,j} t\_{j,2} j - t\_{2,j} - j t\_{j,2} + t\_{j,2}
$$

$$
\psi(\{2, j\}) = t\_{j,2} \{j t\_{2,j} - j + 1\} - t\_{2,j} \tag{44}
$$

$$
\upsilon(\{i, 2\}) = t\_{i,2}t\_{2,l} \left( i - \frac{i - 1}{t\_{2,l}} - \frac{1}{t\_{l,2}} \right)
$$

$$
\upsilon(\{i, 2\}) = it\_{l,2}t\_{2,l} - it\_{l,2} + t\_{l,2} - t\_{2,l}
$$

$$
\upsilon(\{i, 2\}) = t\_{l,2}\{it\_{2,l} - i + 1\} - t\_{2,l} \tag{45}
$$


[9] A. Dixit and B. Nalebuff, "Prisoners' Dilemmas and How to Resolve Them," in *The Art of Strategy*. New York: W.W. Norton and Company, 2008, pp. 64-101.

**Chapter 0**

**Chapter 12**

**A Graphical Game for Cooperative**

Antoniou Josephina, Lesta Papadopoulou Vicky,

Additional information is available at the end of the chapter

overlapping channels are selected by neighbouring APs.

tool to model such decentralized, topology-dependent schemes.

cited.

Libman Lavy and Pitsillides Andreas

http://dx.doi.org/10.5772/54394

**1. Introduction**

**Neighbourhoods of Selfishly Oriented Entities**

The chapter presents a graphical game of selfishly oriented players, inspired by considering a dense urban residential area where each home unit has its own IEEE 802.11 based wireless access point (AP), deployed without any coordination with other such units. The motivation for this game is to provide a framework where coordination between the game players is desired even though in the real world environment the APs lack any management regarding the efficient utilization of the communication channels and furthermore, it is quite common for a terminal served by one of the APs to be within the signal range of multiple alternative APs. This may not be desirable since APs can be in competition for the same communication resource (radio channel), and since the current standards dictate that at any given time every terminal must be rigidly associated with one particular AP, this situation results in increased interference and consequently a low utilization efficiency of the radio resource, when same or

The graphical game aims to motivates cooperation of the players, i.e. the APs, to overcome the resulting interference because of the unmanaged dense deployment of the APs. In fact, it would be much better for individual APs that are in physical proximity to each other to form groups, where one member of the group would serve the terminals of all group members in addition to its own terminals, so that the other access points of the group can be silent or even turned off, thereby reducing interference and increasing overall *Quality of Experience* (QoE). In this chapter, these groups include only members whose signal strength is sufficient to serve all group members, so that the access point that would be responsible for serving the terminals of a particular group or *neighbourhood* could change on a rotating basis, to allow all group members to equally serve and be served. Since there is no centralized entity that can control the APs and force them to form cooperative groups, the creation of such groups must be able to arise from a distributed process where each AP makes its own decisions independently and rationally for the benefit of itself and its terminals. *Graphical game theory* [2] is an appropriate

> ©2013 Antoniou et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

©2013 Antoniou et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
