gas price economics.pdf

8/9/2019 Gas Price Economics.pdf

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Gas Price EconomicsMatthew Wampler-Doty


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Two Gas Price Mechanisms

Clients send in a gasPricealong with their transactions

Mechanism 1 (Ethereum Yellow Paper)

Clients charged the gasPriceincluded in their

transaction

Mechanism 2 (Investigated Here)

All transactions in the block have the same gasPrice,

client cant be charged more than the listed price in

their transaction


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Weak Dominance

A strategy !ifor client iis weakly dominatedby

another strategy !"iif and only if !"ipays out at leastas much as !iregardless of the other clients

strategies

A strategy is !"isaid to be weakly dominantif itweakly dominates all other strategies !i


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Nash Equilibria

Mechanism 1: ??????

Mechanism 2: Each client isubmits gasPrice=Ri


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Symmetric Model

Follows Vickreys Counterspeculation, Auctions, and Competitive Sealed

Tenders(1961)

Simplifying Assumptions

Clients are risk neutral

Clients have similar transactions, in terms of gas used

Reserve prices are independently and identically distributed according to

an1probability distribution function F, and this is common knowledge

Fixed number of clientsM, of whichNhave their transactions run

Drunk under a lamppost model !


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Symmetric Model

Idea

Each client isets gasPrice= #(Ri)according to the

same strictly increasing function #

Satisfy the boundary condition #(0) = 0

Use variational calculus to derive #


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Symmetric Model

Define G(x) = FM-N(x)and denote density function

g(x) = G!(x)

Assuming all clientsjare submitting transactions

with gasPrice = #-1(Rj)

Client isubmitting a transaction with gasPrice =x

can expect a payoff of

G(#-1(x))(Ri-x)


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Symmetric Model

Client iis finding the optimal gasPricex, so assume

she is solving the following equation:

0 =

xG(1(x))(Ri x)

= g(1

(x))0(1(x))

(Rix)G(1(x))


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Symmetric Model

Since we are assuming that client iis using the same

strategy as everyone else, thenx = #(Ri), so the

equation she is solving is equivalent to the ODE:

xg(Ri) =G(Ri)0(Ri) + g(Ri)(Ri)

= Ri

(G(Ri)(Ri))


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Symmetric Model

Given the boundary condition #(0)= 0, the solution to

the ODE is:

(Ri) = 1

G(Ri)

Z Ri

0

xg(x)dx

=Ri Z Ri

0

G(x)

G(Ri)dx

=Ri

Z Ri

0

F(x)

F(Ri)

MNdx


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Nash Equilibria (Redux)

Under the assumptions of the Vickrey style

symmetric model:

Mechanism 1: Each client isubmits

Mechanism 2: Each client isubmits gasPrice=Ri

gasPrice= Ri Z Ri

0 F(x)F(Ri)

MN

dx


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Revenue

E[Revenue for Mechanism 2]

=NMM 1N 1

Z 0

xf(x)F(x)N1

(1F(x))

MN

dx

E[Revenue for Mechanism 1]

=NMM

1

N 2

Z

0

Z

x

(y)f(y)dyf(x)F(x)N2(1 F(x))MN1dx


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Interlude


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!"Vs Hobby

Embedding NP-Hard problems in a cryptocurrency


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KnapsackSet of variablesxi"{0,1}, each with value viand weight

wi

The Knapsack Problemis to maximize

Subject to

The Knapsack Problem is NP-Hard

nX

i=1

vixi

nX

i=1

wixi 6 W


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Revenue (Redux)

Theorem: Optimizing revenue for a miner in

Mechanism 1 is NP-Hard

Proof. It is exactly the knapsack problem. Thexi

reflect transactions, the wiare the gas used by

those transactions, the viis the ETH that client iis

willing to pay based on her gasPriceand the gasher transaction consumes.

!


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Revenue (Redux)

Theorem: Optimizing revenue for a miner inMechanism 2 is alsoNP-Hard

See Aggarwal and Hartlines Knapsack Auctions

(2006)


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Engineering RealityMechanism 1 makes more sense for miners

Dynamic programming solutions exist for Knapsack and havebeen well studied, probably could be deployed for miners in

Mechanism 1

Mechanism 2s optimization problem has not been studied at all

Mechanism 2 makes more sense for clients

Weakly Dominant NE still holds when we consider thecomputational complexity of the miners optimization problem

Vickreys symmetric model clearly does not, no real traditionaleconomic theory at all about client behavior


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Setting Transaction CostsA Proposal:

Use Mechanism 2

Why?

Mining is already subsidized

This optimization problem can probably be solved

Greedy solutions may be good enough

Put the gasPricein the header

Have the default for clients be the minimum gasPriceof the last 256

blocks, minus some small constant $


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The Price of Everything and

the Value of Nothing Causes clients gasPriceto be strongly correlated,hopefully drives clients values for Ether to be strongly

correlated as well

Creates market pressure for the gasPriceto decline over

time in the absence of strong demand

Difficult to game, from the perspective of Vlads crypto

economics framework

An oligopoly would have to mine 256 consecutive

blocks in a row in order to control the default price


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Conclusions

The gas pricing mechanism slated to be in

Ethereum 1.0 is challenging to analyze from a

game theoretic point of view

Both gas pricing mechanisms investigated here

impose NP-Hard optimization problems on miners

The contents of this talk is admittedly just the

beginning to analyzing this particular research

topic

gas price economics.pdf

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