Verifier’s Dilemma in Ethereum Blockchain: A Quantitative Analysis

Abstract

A blockchain is an immutable ledger driven by a distributed consensus protocol. In public blockchains such as Bitcoin and Ethereum consensus is established through a computational effort called Proof-of-Work (PoW). Special users called miners contribute to the PoW computational effort in exchange for a fee and also verify the data stored in blocks mined by the other miners. Here is where the Verifier’s Dilemma emerges. To maximise their profits, miners are incentivized to invest their resources in PoW, because they do not receive any incentives for the verification phase. In this paper, we study the Verifier’s Dilemma using a quantitative model based on PEPA. The analysis demonstrates the circumstances under which non-verifying miners gain fees higher than that of verifying miners. Moreover, we consider a mitigation approach consisting of the injection of invalid blocks to disturb the mining process of non-verifying miners. The model allows us to derive the optimal rate at which invalid blocks must be injected, so that skipping the verifying phase becomes economically disadvantageous while the throughput of the blockchain is only minimally reduced. The impact on miners’ rewards and overall performance is also assessed.

Appendices

Appendix

A Tables of Notations

The following tables provide a description of the notations used in the PEPA specification of the BM and IBIM models. In particular they describe the notations used to model the behaviours of single fair and unfair miners and those used for the environments.

Table 7. BM component description.

Table 8. IBIM component description.

B Steady State Probabilities for BM

We report the symbolic expressions of the steady-state probabilities for the BM model. In particular, the steady-state probabilities for the CTMC depicted in Fig. 2 are as follows:

$$\begin{aligned} \pi _1= & {} \frac{(\beta ^2 (2 (\beta + \gamma + \lambda ) - \lambda p))}{K}\\ \pi _2= & {} \frac{(\beta ((\gamma + \lambda ) (2 \gamma + \lambda ) + \beta (3 \gamma + \lambda )))}{K}\\ \pi _3= & {} \frac{(\beta (\beta (\gamma + \lambda ) + (2 \gamma + \lambda ) (\gamma + \lambda - \lambda p)))}{K}\\ \pi _4= & {} \frac{((\beta + \gamma + \lambda ) (\beta (\gamma + \lambda ) + (2\gamma + \lambda ) (\gamma + \lambda - \lambda p)))}{K} \end{aligned}$$

where $\sum _{i=1}^4\pi _i=1$ and K is the normalising constant whose expression is

$$\begin{aligned}&K=(2 \beta ^3 + \beta ^2 (7 \gamma - \lambda (-5 + p)) + \beta (7 \gamma ^2 + \gamma \lambda (11 - 4 p) -\\&\qquad \qquad \qquad \qquad \qquad \quad \;\; - 2 \lambda ^2 (-2 + p)) + (\gamma + \lambda ) (2 \gamma + \lambda )(\gamma + \lambda - \lambda p))\,. \end{aligned}$$

C Steady-State Probabilities for IBIM

We report the symbolic expressions of the steady-state probabilities for the IBIM model. In particular, the steady-state probabilities for the CTMC depicted in Fig. 5 are as follows:

$$\begin{aligned} \pi _1= & {} \frac{\beta ^2 \left( \lambda ^3 r^2 \left( -2 \beta ^2 (r-2)-\beta \varepsilon (r-2)\right) +\lambda ^2 r \left( 3 \beta ^2 \varepsilon (r+2)+2 \beta ^3 (r+2)\right. \right. }{K}\\[2mm]&\qquad \frac{\left. \left. +\beta \varepsilon ^2 (r+2)\right) +\lambda r (\beta +\varepsilon ) \left( 6 \beta ^2 \varepsilon +4 \beta ^3+2 \beta \varepsilon ^2\right) \right) }{K}\\[2mm] \pi _2= & {} \frac{\beta \left( \beta ^2 \lambda r (\varepsilon +\lambda ) \left( 4 \varepsilon ^2+3 \varepsilon \lambda (r+1)+2 \lambda ^2 r\right) +2 \beta ^4 \lambda r (\varepsilon +\lambda )+\right. }{K}\\[2mm]&\qquad \frac{\left. +\beta ^3 \lambda r (\varepsilon +\lambda ) (5 \varepsilon +2 \lambda (r+1))+\beta \varepsilon \lambda r (\varepsilon +\lambda )^2 (\varepsilon +\lambda r)\right) }{K}\\[2mm] \pi _3= & {} \frac{\beta (\beta +\varepsilon +\lambda ) \left( 2 \beta ^3 \lambda r (\varepsilon +\lambda )+\beta ^2 \lambda r (\varepsilon +\lambda ) (5 \varepsilon +2 \lambda )\right. }{K}\\[2mm]&\qquad \frac{\left. +\beta \lambda r (\varepsilon +\lambda ) \left( 4 \varepsilon ^2+3 \varepsilon \lambda -2 \lambda ^2 (r-1) r\right) +\varepsilon \lambda r (\varepsilon +\lambda ) (\varepsilon +\lambda -\lambda r) (\varepsilon +\lambda r)\right) }{K}\\[2mm] \pi _4= & {} \frac{\beta ^2 \left( 2 \beta ^3 \lambda r (\varepsilon +\lambda )+\beta ^2 \lambda r (\varepsilon +\lambda ) (5 \varepsilon +2 \lambda )\right. }{K}\\[2mm]&\qquad \frac{\left. +\beta \lambda r (\varepsilon +\lambda ) \left( 4 \varepsilon ^2+3 \varepsilon \lambda -2 \lambda ^2 (r-1) r\right) +\varepsilon \lambda r (\varepsilon +\lambda ) (\varepsilon +\lambda -\lambda r) (\varepsilon +\lambda r)\right) }{K} \end{aligned}$$

$$\begin{aligned} \pi _5= & {} \frac{\beta ^2 \varepsilon (2 \beta +2 \varepsilon +\lambda r)}{(\beta +\varepsilon ) (2 \beta +\varepsilon ) (\varepsilon +\lambda r) (\beta +\varepsilon +\lambda r)}\\[2mm] \pi _6= & {} \frac{\beta \varepsilon }{(\beta +\varepsilon ) (2 \beta +\varepsilon )}\\[2mm] \pi _7= & {} \frac{\varepsilon }{2 \beta +\varepsilon }\\[2mm] \pi _8= & {} \frac{\beta \varepsilon }{(2 \beta +\varepsilon ) (\beta +\varepsilon +\lambda r)}\\ \end{aligned}$$

where $\sum _{i=1}^8\pi _i=1$ and K is

$$\begin{aligned}&K=(\beta +\varepsilon ) (2 \beta +\varepsilon ) (\varepsilon +\lambda r) (\beta +\varepsilon +\lambda r) \left( 2 \beta ^3+\beta ^2 (5 \varepsilon +5 \lambda -\lambda r)\right. \\&\qquad \qquad \qquad \qquad \qquad \left. + \, 2 \beta (\varepsilon +\lambda ) (2 \varepsilon +2 \lambda -\lambda r)+(\varepsilon +\lambda )^2 (\varepsilon +\lambda -\lambda r)\right) . \end{aligned}$$