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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