An analysis of peer-to-peer networks with altruistic peers

Abstract

We develop a new model of the interaction of rational peers in a Peer-to-Peer (P2P) network that has at its heart altruism, an intrinsic parameter reflecting peers’ inherent willingness to contribute. Two different approaches for modelling altruistic behavior and its attendant benefit are introduced. With either approach, we use Game Theoretic analysis to calculate Nash equilibria and predict peer behavior in terms of individual contribution. We consider the cases of P2P networks of peers that (i) have homogeneous altruism levels or (ii) have heterogeneous altruism levels, but with known probability distributions. We find that, under the effects of altruism, a substantial fraction of peers will contribute when altruism levels are within certain intervals, even though no incentive mechanism is used. Our results corroborate empirical evidence of large P2P networks surviving or even flourishing without or with barely functioning incentive mechanisms. We also enhance the model with a simple but powerful incentive scheme to limit free-riding and increase contribution to the network, and show that the particular incentive scheme on networks with altruistic peers achieves its goal.

Appendix

Proof that $U_i(\overrightarrow{Q})$ linearly depends on Q _i .

We prove that under some reasonable assumptions $U_i(\overrightarrow{Q})$ linearly depends on Q _i . We set Q _all  = ∑  _i Q _i and $Q_{all}^{-j}=\sum_{i \neq j}{Q_i}$. If N is the set of all peers, let $K_1 \subseteq N$ include those peers who demand at maximum level D and K ₂ include those demanding at $D_i(\overrightarrow{Q})< D$. The last term of Eq. 7 then becomes:

$$F \cdot Q_i \cdot D \cdot \left\{ \sum\limits_{j \in K_1-\{i\} } \frac{1}{Q_{all}^{-j}} + \sum\limits_{j \in K_2-\{i\}} \frac{b}{A \cdot M_i(\overrightarrow{Q}) } \right\} $$

Let:

$$g_i(\overrightarrow{Q})=\left\{ \sum\limits_{j \in K_1-\{i\} } \frac{1}{Q_{all}^{-j}} + \sum\limits_{j \in K_2-\{i\}} \frac{b}{A \cdot M_i(\overrightarrow{Q}) } \right\} $$

Under the realistic approach for altruism Eq. 7 becomes:

$$\begin{array}{lll} U_i(\overrightarrow{Q})&=& D_i(\overrightarrow{Q}) \cdot b \cdot \sum\limits_{k \neq i}{p_{ik}} + L_i \cdot Q_i \cdot D \cdot b \cdot g_i(\overrightarrow{Q}) \\ & & -S \cdot Q_i-F \cdot Q_i \cdot D \cdot g_i(\overrightarrow{Q}) \end{array}$$

and under the selfish approach:

$$\begin{array}{lll} U_i(\overrightarrow{Q}) & = & D_i(\overrightarrow{Q}) \cdot b \cdot \sum\limits_{k \neq i}{p_{ik}} + E_i \cdot b \cdot Q_i \\ & & -S \cdot Q_i-F \cdot Q_i \cdot D \cdot g_i(\overrightarrow{Q}) \end{array}$$

When the population of participating peers is large and there is a non zero level of total contribution in the network, the assumption that the value of Q _i does not affect Q _all or ${Q_{all}^{-j}}$ can be adopted. Consequently, $g_i(\overrightarrow{Q})$ is independent of Q _i . Since $D_i(\overrightarrow{Q})$ and p _ik are also independent of Q _i by definition, under either approach for altruism $U_i(\overrightarrow{Q})$, linearly depends on i’s contribution Q _i .

Derivation of λ ^*.

Let x the fraction of contributing peers.

$$\begin{array}{lll}&&{\kern-6pt} DU_i^{MAX}(\overrightarrow{Q})\\ &&{\kern4pt} =A \Leftrightarrow P(i~\textrm{contributes})\cdot \frac{b \cdot (x\cdot n -1) \cdot Q} {\frac{(M-1) \cdot (x \cdot n -1)}{n-1}+1 }\\ &&{\kern15pt} + P(i~\textrm{doesn't contribute})\cdot \frac{b \cdot x\cdot n \cdot Q} {\frac{(M-1) \cdot (x \cdot n -1)}{n-1}+1}\\&&{\kern4pt} = A \Leftrightarrow x\cdot \frac{b \cdot (x\cdot n -1) \cdot Q} {\frac{(M-1) \cdot (x \cdot n -1)}{n-1}+1 }\\&&{\kern15pt} + (1-x) \cdot \frac{b \cdot x\cdot n \cdot Q} {\frac{(M-1) \cdot (x \cdot n -1)}{n-1}+1}\\ &&{\kern4pt} = A \Leftrightarrow x=\frac{A \cdot (n-M)}{(n-1)^2 \cdot b\cdot Q-A \cdot n \cdot (M-1)} \equiv \lambda^* \end{array}$$