An equity-based incentive mechanism for persistent virtual world content service

Abstract

Virtual world has the potential to become a future global electronic marketplace, integrating many isolated markets in many areas. To achieve this goal, future virtual world is required to be persistent, implying that a virtual world together with its accumulated content shall exist forever regardless of dynamic changes of its users and owners. Unfortunately, existing virtual worlds, owning by some entities, are not immune from death due to business entity failure. To provide a persistent virtual world, a decentralized architecture is explored, which is constructed on user contributed devices. However, there are many challenges to realize a decentralized virtual world. One important issue is user cooperation in reliable content storage. The devices contributed by users may not be reliable for maintaining all user contents, but users do not have the incentive to provide reliable devices for others. This paper addresses the issue by two steps. First, an indicator, called replica group reliability, is provided to users, which is based on the proposed replicability index. Based on the indicator, users can learn the reliability of their content storage. Then, a new user incentive mechanism, called equity-based node allocation strategy, is proposed to promote user cooperation to collectively maintain reliable content storage. A decentralized algorithm implementing the strategy is designed and the evaluation results show its effectiveness and efficiency.

Appendices

Appendix A: Derivation of replicability

Replicability (w) is measured by the maximal amount of data that can be transferred in a replication. In a logical computer, let N₁ be the expected number of live replica when a replication is triggered, t_r be the average time of all N₂ replica fails, and τ_r be the expected residual life of one live replica. During t_r, the average residual life (τ_r) of a replica can be simply estimated by τ_r = t_r/2 (since the residual life of a replica can be any time between 0 and t_r). Given the minimal upload bandwidth assigned for replication (BW_up), the total amount of data that can be uploaded is bounded to:

$$ \begin{aligned} w & = {\text{BW}}_{\text{up}} \cdot \tau_{\text{r}} \cdot N_{1} \\ & = {\text{BW}}_{\text{up}} \cdot \frac{{t_{\text{r}} }}{2} \cdot N_{1} \\ \end{aligned} $$

(A.1)

The average time of all N₁ replica fails can be estimated by t_r= N₁/R_f, where R_f is the replica failure rate (i.e., the average number of replica failure within a time unit). Its reciprocal is the period of replica failure, denoted by T_f, indicating the expected length of two replica failure. For N replicas, suppose their time-to-failure (TTF) is uniformly distributed with the mean value equal to the global node mean time-to-failure (MTTF). Let t₁ be the TTF of replica 1, t₂ be the TTF of 2, …, and t_N be the TTF of replica N. On average, t₁ = T_f, t₂ = 2T_f, …, t_N= N·T_f, and

$$ \begin{aligned} {\text{MTTF}} & = \mathop \sum \limits_{i = 1}^{N} t_{\rm{i}} \cdot \frac{1}{N} \\ & = \frac{{T_{f} + 2T_{f} + \cdots + NT_{f} }}{N} \\ & = \frac{N + 1}{2}T_{f} \\ \end{aligned} $$

Therefore, T_f and R_f can be estimated by

$$ T_{f} = \frac{1}{{R_{f} }} = \frac{{2 \cdot {\text{MTTF}}}}{N + 1} $$

(A.2)

By bringing t_r= N₁/R_f and (7) into (6), w can be converted to:

$$ \begin{aligned} w & = {\text{BW}}_{\text{up}} \cdot \frac{{t_{\rm{r}} }}{2} \cdot N_{1} \\ & = {\text{BW}}_{\text{up}} \cdot \left( {\frac{{N_{1} }}{{2R_{f} }}} \right) \cdot N_{1} \\ & = {\text{BW}}_{\text{up}} \cdot \frac{{N_{1}^{2} \cdot {\text{MTTF}}}}{N + 1} \\ \end{aligned} $$

(A.3)

In the LCR model, two replication thresholds, the minimal number of replica (n) and the maximal number of replica (n + e), are maintained in a logical computer for replication cost minimization [4]. Specifically, a replication procedure will be triggered once (e +1) replicas are failed. Thus, N₁ can be estimated by (n −1) and (9) can be converted to:

$$ w = {\text{BW}}_{\text{up}} \cdot \frac{{\left( {n - 1} \right)^{2} \cdot {\text{MTTF}}}}{n + e + 1} $$

(A.4)

In (10), the number of extra replicas (e) is related to node stability and content size [4], complicating the estimation of w. To simplify the estimation, e can be restricted to its maximum (E). When e reaches its maximum, w will reach its minimum. So logically, as long as the minimum w can meet replication requirement, the actual one from a smaller e is definitely sufficient, and (10) can be converted to

$$ w = {\text{BW}}_{\text{up}} \cdot \frac{{\left( {n - 1} \right)^{2} }}{n + E + 1} \cdot {\text{MTTF}} $$

(A.5)

Since BW_up, n, and E are constants and selected in configuration, w is only related to MTTF, that is,

$$ w = A \cdot {\text{MTTF,}} \;\;{\text{where}}\;\;A = {\text{BW}}_{\text{up}} \cdot \frac{{\left( {n - 1} \right)^{2} }}{n + E + 1} $$

Appendix B: Proof of Theorem 1

Theorem 1

(Weak equity achievement) For any time t_i and t_j, if a device chain is maintained and sorted both at t_i and t_j, then weak equity can be achieved.

Proof

The theorem can be proved by three steps. Step 1 proves that minimizing the difference user return and user contribution is the only way to minimize the difference of return/contribution ratio among all users. Step 2 proves that a sorted device chain can provide the minimal difference between user return and user contribution. Lastly, Step 3 proves that the equity can be maintained along with time.

• Step 1 Let D₁, D₂, …, D_n be n devices belonging to User 1, User 2, …, User n, MTTF_i be the MTTF of D_i, MTTF_i,c (MTTF_i,r) be the MTTF corresponding to User i’s contributed (received) replicability, and MTTF_i,r be the set of MTTFs of the devices providing User i’s received replicability. Moreover, let avg(·) be the function calculating the average value of a given set.

  To minimize $\left| {\frac{{w_{\rm{r}}^{i} }}{{w_{\rm{c}}^{i} }} - \frac{{w_{\rm{r}}^{j} }}{{w_{\rm{c}}^{j} }}} \right|$, |wⁱ_r  − p·wⁱ_c |, |w^j_r − q·w^j_c |, and |p − q| also have to be minimized. In another word, both wⁱ_r and w^j_r are approximately the p times of wⁱ_c and w^j_c . If p = q  > 1, it means all users receive more replicability than their contributed replicability, which is impossible for D_n without out-of-band resource contribution. On the other hand, if p  =  q <1, it means all users receive less replicability than their contributed replicability, which is impossible for D₁ without out-of-band resource depletion. An exception for p  =  q ≠1 without the above restriction is the condition that wⁱ_r  =  w^j_r , wⁱ_c  =  w^j_c , and wⁱ_r ≠ wⁱ_c . A proof by contradiction can falsify this condition. Specifically, wⁱ_r ≠ wⁱ_c means MTTF_r≠MTTF_c. However, as w  =  A·MTTF, if wⁱ_r  =  w^j_r , then MTTF_i = MTTF_j for any device D_i and D_j. As avg(MTTF_i,r) = MTTF_i,_r, then MTTF_i,_r = MTTF_i,_c leading to a contradiction. Thus, the only way to minimize $\left| {\frac{{w_{\rm{r}}^{i} }}{{w_{\rm{c}}^{i} }} - \frac{{w_{\rm{r}}^{j} }}{{w_{\rm{c}}^{j} }}} \right|$, is to minimize |wⁱ_r −·wⁱ_c | and |w^j_r −·w^j_c |, that is, to make user return approach to user contribution.

• Step 2 Let D_i be a device in a device chain L sorted in devices’ observed MTTF, DN be the set of the k neighboring devices of D_i, and D_f be a device in DN such that $\left| {{\text{MTTF}}_{i} {-}{\text{MTTF}}_{f} } \right|{\text{ }} \ge {\text{ }}\left| {{\text{MTTF}}_{i} {-}{\text{MTTF}}_{m} } \right|$ for any device D_m in DN.

  DN provides the least difference of user return and user contribution to D_i, which can be proved by contradiction. If there is another device D_a (D_a ∉ DN) which can provide a smaller difference of user return and user contribution to D_i, then $\left| {{\text{MTTF}}_{i} {-}{\text{MTTF}}_{f} } \right| > \left| {{\text{MTTF}}_{i} {-}{\text{MTTF}}_{a} } \right|$, which means that L is not sorted in devices’ observed MTTF and a contradiction is found. Therefore, a sorted device chain can provide the minimal difference between user return and user contribution.

• Step 3 The periodical execution of the position change routine ensures that the equity can be maintained along with time.