Cooperation and coalitional stability in decentralized wireless networks

Abstract

In this paper we consider a wireless contextualization of the local routing protocol on scale-free networks embedded in a plane and analyze on the one hand how cooperation affects network efficiency, and on the other hand the stability of cooperation structures. Cooperation is interpreted on k-cliques as local exchange of topological information between cooperating agents. Cooperative activity of nodes in the proposed model changes the routing strategy at the level of the coalition group and consequently influences the entire routing process on the network. We show that the proposed cooperation model enhances the network performance in the sense of reduced passage time and jamming. Payoff of a certain node is defined based on its energy consumption during the routing process. We show that if the payoff of the nodes is the energy saving compared to the all-singleton case, basically coalitions are not stable, since increased activity within coalition increases costs. We introduce coalitional load balancing and net reward to enhance coalitional stability and thus the more efficient operation of the network. As in the proposed model cooperation strongly affects routing dynamics of the network, externalities will arise and the game is defined in a partition function form.

Appendices

Appendix A

A game in partition function form [51] is a pair (N, V), where $V:\mathcal {E}\rightarrow \mathbb {R}$ is the partition function, which assigns a real payoff to each embedded coalition. Each partition is composed of coalitions, e.g. the partition $\{1,2\}\{3,4,5\}$ of $\{1,2,3,4,5\}$ is composed of coalitions $\{1,2\}$ and $\{3,4,5\}$.

Table 10 Nodal payoffs in various coalitional structures

We define the residual game over the set $R\subsetneq N$ as follows. $\varPi (S)$ denotes the set of partitions of S. Assume $\overline{R}=N\setminus R$ have formed $\overline{{\mathcal {P}_R}}\in \varPi (\overline{R})$. Then the residual game $(R,V_{{\mathcal {P}_{\overline{R}}}})$ is the partition function form game over the player set R with the partition function given by $V_{{\mathcal {P}_{\overline{R}}}}(C,{\mathcal {P}_R})=V(C,{\mathcal {P}_R}\cup {\mathcal {P}_{\overline{R}}})$.

Definition 1

((Pessimistic) Recursive Core [30]) For a single-player game the (pessimistic) recursive core is trivially defined. Now assume that the (pessimistic) recursive core C(N, V) has been defined for all games with $\left| N\right| <k$ players. We call a pair $\omega =(x,\mathcal {P})$ consisting of a payoff vector and a partition $\mathcal {P}\in \varPi (N)$ an outcome. Let us denote the set of outcomes in (N, V) by $\Omega (N,V)$. Then for an $\left| N\right|$-player game an outcome $(x,\mathcal {P})$ is dominated if there exists a coalition Q forming partition $\mathcal {P}_Q$ and a feasible payoff vector $y_Q\in \mathbb {R}^Q$, such that for all $(y_Q,y_{\overline{Q}},\mathcal {P}_Q\cup \mathcal {P}_{\overline{Q}})\in \Omega (N,V)$ we have $y_Q> x_Q$ and if $C(\overline{Q},V_{\mathcal {P}_Q})\ne \varnothing$ then $(y_{\overline{Q}},\mathcal {P}_{\overline{Q}})\in C(\overline{Q},V_{\mathcal {P}_Q})$. The pessimistic recursive core C(N, V) of (N, V) is the set of undominated outcomes.

Appendix B

To give some further impression into coalitional stability of the model we analyze some more cases. Let us consider the coalitions $\{5,7,9\}$, $\{10,13,14\}$ and $\{11, 18, 23\}$ and the values summarized in Table 10.

Considering $\{5,7,9\}$, the stability analysis shows that $\{5,7\},\{9\}$ is the stable partition with $x(5)+x(7)=373$, $x(9)=38$ and $233<x(5)<235$. Considering $\{10,13,14\}$ and $\{11,18,23\}$ the grand coalitions are stable, with payoffs depicted in Fig. 8. Again, it can be seen that stable partitions correspond to the most efficient network operation modes.

Fig. 8

Recursive cores of coalitions $\{10,13,14\}$ and $\{11,18,23\}$ in the payoff space. Equality $x(10)+x(13)+x(14)=328$ holds