An Incentive-Compatible and Computationally Efficient Fog Bargaining Mechanism

Abstract

This work contributes an (approximately) incentive-compatible and computationally efficient bargaining mechanism for pricing fog computing resources. In network settings (e.g., fog computing), it is plausible to think that self-interested and incompletely informed players (represented by software agents) will attempt to maximize their own benefits at the expense of others. Hence, it is crucial that fog bargaining mechanisms give incentives to agents for behaving in a manner consistent with the desired outcome where every agent’s benefit is maximized. Equilibrium analyses validate that the fog bargaining mechanism in this work is approximately Bayesian incentive compatible because every agent can approximately maximize its expected utility by adhering to the strategy recommended by the bargaining mechanism given that all other agents also adhere to their equilibrium strategies. That is, if every agent in the market adheres to the strategy recommended by the bargaining mechanism, then the strategy profile of the agents forms an approximate Bayesian Nash equilibrium. Given that a fog resource market has a large number of buyers and a large number of sellers, computational efficiency is also imperative since every agent needs to process a huge number of trading alternatives. Computational complexity analyses validate that 1) the procedure for carrying out the bargaining strategy has a linear time complexity, and with every passing round, the search space dwindles but the solutions become progressively better, 2) the number of rounds for each agent to complete bargaining is logarithmic in the number of its opponents, and 3) each agent has a linear message complexity.

Appendix: Numerical Examples

Mathematical proofs in Sects. 3 and 4 have already fulfilled the objectives of this paper, which are validating both the properties of (approximate) Bayesian incentive compatibility and computational efficiency of the AFB mechanism. This appendix is added at the request of an anonymous reviewer to aid non-specialist readers in understanding the AFB mechanism. Readers with knowledge in (algorithmic) mechanism design who are only interested in the formal proofs can skip the Appendix entirely.

Since the AFB mechanism is designed for bargaining between many buyers and many sellers, bargaining situations where 1) there are more sellers than buyers, 2) there is an equal number of buyers and sellers, and 3) there are more buyers than sellers were studied with the following parameters.

Agent type: Each agent-based fog resource market is comprised of a large number of buyer agents (BAs) representing owners of IoTs and a large number of seller agents (SAs) representing owners of fog nodes.

Market type: Buyer-to-seller ratios of {2:3, 3:4, 4:5}, {1:1}, and {5:4, 4:3, 3:2} were used to model “buyer-favorable”, “balanced”, and “seller-favorable” fog resource markets, respectively. For example, a seller-favorable fog resource market with a buyer-to-seller ratio of 4:3 can be modeled by having ${10}^{6}$ BAs and $7.5\times {10}^{5}$ SAs.

Market closing time: The market closing time $\tau$ is determined based on Theorem 6. For example, in a fog resource market with ${10}^{6}$ BAs and $7.5\times {10}^{5}$ SAs, i.e., $\left|{\mathbb{B}}\right|={10}^{6}$ and $\left|{\mathbb{S}}\right|=7.5\times {10}^{5}$, it follows from Theorem 6 that $\tau =\frac{1}{2}\lceil{\mathrm{log}}_{2}\mathit{max}(\left|{\mathbb{B}}\right|,\left|{\mathbb{S}}\right|)\rceil=10$.

Price range: To study bargaining situations where there is a feasible chance for BAs and SAs to reach agreements on the subscription price for a fog computing service, it is assumed without loss of generality that there exists an intersection between the range of agreeable prices of BAs and the range of agreeable prices of SAs. Whereas the IP and RP of BAs were uniformly generated from the closed intervals $[\mathrm{10,60}]$ and $[\mathrm{70,120}]$, respectively, the IP and RP of SAs were also uniformly generated from $[\mathrm{90,140}]$ and $[\mathrm{30,80}]$, respectively.

Protocol: All BAs and SAs followed the STDMC protocol (Sect. 2.1). Hence, all agents made their price proposals simultaneously at every round.

Strategy: All BAs and SAs adopted the VO strategy (Sect. 2.2). The proofs of Theorems 1, 2, and 3 validate that there is generally no incentive for agents to diverge from the equilibrium strategy, i.e., the VO strategy. For ease of exposition on how agents execute the VO strategy, consider the bargaining between ${\textit{BA}}_{539611}$ with $\left[\textit{IP}_{B539611},{\textit{RP}}_{B539611}\right]=[\mathrm{59.520,9}0.011]$ and ${\textit{SA}}_{1682}$ with $\left[{\textit{RP}}_{S1682},\textit{IP}_{S1682}\right]=[\mathrm{47.112,93.870}]$, which was only one example among many concurrent bargaining of a large number of BA-SA pairs in the seller-favorable fog resource market described above. At $t=0$, while ${\textit{BA}}_{539611}$ proposed $59.520$, ${\textit{SA}}_{1682}$ simultaneously proposed $93.870$. However, it is reminded that the bargaining between ${\textit{BA}}_{539611}$ and ${\textit{SA}}_{1682}$ is NOT a bilateral bargaining, e.g., at $t=0$, ${\textit{BA}}_{539611}$ also received proposals from many other SAs with the highest price proposal of $132.430$ from ${\textit{SA}}_{67845}$ such that the maximum price difference between ${\textit{BA}}_{539611}$ and its opponent was ${\pi }_{t=0}^{B539611\leftrightarrow max}=132.430-59.520=72.910$ while ${\textit{SA}}_{1682}$ also received proposals from many other BAs with the lowest price proposal of $10.110$ from ${\textit{BA}}_{791336}$ such that ${\pi }_{t=0}^{S1682\leftrightarrow max}=93.870-10.110=83.760$. While the price difference between ${\textit{BA}}_{539611}$ and ${\textit{SA}}_{1682}$ was $93.870-59.520=34.350$, from (4) (Sect. 2.2), the normalized price difference between ${\textit{BA}}_{539611}$ and ${\textit{SA}}_{1682}$ from the perspectives of ${\textit{BA}}_{539611}$ and ${\textit{SA}}_{1682}$ were calculated as $34.350/72.910 =0.471$ and $34.350/83.760 =0.410$, respectively. The average normalized price difference (ANPD) between ${\textit{BA}}_{539611}$ (respectively, ${\textit{SA}}_{1682}$) and all its many opponents was $0.609$ (respectively, $0.504$). Since the normalized price difference between ${\textit{BA}}_{539611}$ and ${\textit{SA}}_{1682}$ from the perspective of ${\textit{BA}}_{539611}$ (respectively, ${\textit{SA}}_{1682}$) was smaller than the ANPD between ${\textit{BA}}_{539611}$ (respectively, ${\textit{SA}}_{1682}$) and all its many opponents, from (5) (respectively, (8)) (Sect. 2.2), in the next round $t=1$, ${\textit{SA}}_{1682}$ (respectively, ${\textit{BA}}_{539611}$) was selected as a viable option of ${\textit{BA}}_{539611}$ (respectively, ${\textit{SA}}_{1682}$). At $t=1$, by the conflict rule of the STDMC protocol, the bargaining between ${\textit{BA}}_{539611}$ (respectively, ${\textit{SA}}_{1682}$) and all its non-viable options ended in a conflict and ${\textit{BA}}_{539611}$ (respectively, ${\textit{SA}}_{1682}$) ignored all non-viable options when calculating the ANPD. At $t=1$, the ANPD between ${\textit{BA}}_{539611}$ (respectively, ${\textit{SA}}_{1682}$) and all its viable options was $0.532$ (respectively, $0.413$). From (1) and (2) (respectively, (7)) (Sect. 2.2), the price proposal of ${\textit{BA}}_{539611}$ (respectively, ${\textit{SA}}_{1682}$) at $t=1$ was calculated as $59.520+\left(0.532\times 34.350 \right)=77.794$ (respectively, $93.870-\left(0.413\times 34.350 \right)=79.683$). A series of simultaneous proposals (SPs) by ${\textit{BA}}_{539611}$ and ${\textit{SA}}_{1682}$ is shown in Table

Table 1 Simultaneous-move bargaining of a buyer-seller pair in a seller-favorable market

1.

Performance measure: The average utility of agents that reached agreements was used as the performance measure. The utility of each BA was calculated using the utility function ${U}^{x}$ defined in (9) and (10) in Sect. 2.2. The utility of each SA was calculated in a similar way. Although the generalized equations in (9) and (10) enable agents to use different scales and origins, for the purpose of the study, the level of satisfaction of each agent for the bargaining outcome was measured on the scale of 1 (least satisfied) to 10 (most satisfied). For BAs, this was done by setting ${\alpha }_{x}=9$ and ${\beta }_{x}=1$ in (9) such that ${U}^{x}\left({P}_{ag}\right)=9[\left( \textit {RP}_{x}-{P}_{ag}\right)/\left(\textit{RP}_{x}-\textit{IP}_{x}\vphantom{{P}_{ag}}\right)]+1$. For example, since $\textit{{BA}}_{539611}$ with $\left[\textit{IP}_{B539611}, \textit{RP}_{B539611}\right]=[\mathrm{59.520,9}0.011]$ reached an agreement with $\textit{SA}_{1682}$ at the price of $\textit{P}_{ag}=78.858$, the utility of $\textit{BA}_{539611}$ was $9\left[\left(90.011-78.858)/(90.011-59.520\right)\right]+1=4.292$. The utility of $\textit{SA}_{1682}$ with $\left[\textit{RP}_{S1682},\textit{IP}_{S1682}\right]=[\mathrm{47.112,93.870}]$ was $9\left[\left(78.858-47.112)/(93.870-47.112\right)\right]+1=7.110$.

Numerical results: Numerical results for seller-favorable, balanced, and buyer-favorable fog resource markets are summarized in Fig. 

Fig. 1

Agents’ average utilities

1. In seller-favorable fog resource markets, since there were more BAs than SAs, there was a higher probability that each SA received price proposals that were closer to its own price proposal and hence, SAs had a higher chance to achieve higher utilities than BAs if agreements were reached. For example, it can be observed in Table 1 that at each round, the ANPD between $\textit{SA}_{1682}$ and all its viable options was smaller than that between $\textit{BA}_{539611}$ and all its viable options. Consequently, $\textit{SA}_{1682}$ made smaller concessions than $\textit{BA}_{539611}$ and obtained a higher utility than $\textit{BA}_{539611}$ when an agreement was reached. On the contrary, in buyer-favorable fog resource markets, BAs made smaller concessions and achieved higher average utilities than SAs. The numerical results are consistent with the specification of the VO strategy in Sect. 2.2 that a smaller (respectively, larger) ANPD between an agent and all its viable options confers more (respectively, less) bargaining power. In a balanced fog resource market, since there was an equal number of BAs and SAs, both BAs and SAs generally achieved moderate utilities.