DS Proof Equilibrium
In what follows, we focus on an equilibrium in which the cryptocurrency has a positive value and all
trades are double-spending proof.24 Given R, define the set of optimal money demand as Γ(ε; R).
For a given selection from the solution set Γ(ε; R), the aggregate transfer of balances in the night
market and aggregate money demand are described by
D = BσE(d) = Bσ Z ∞
0
z
∗
(ε; R)dFε(ε) (35)
Z = BE(z) = B
Z ∞
0
z
∗
(ε; R)dFε(ε). (36)
In equilibrium, nominal balances are growing at rate µ and so do prices. Our definition has only
used real balances which stay constant across time.
Definition 6. A DS-proof cryptocurrency equilibrium with (µ, τ ) is given by offers (x(ε), d(ε), N(ε)),
a money demand z(ε) > 0 and a mining choice q such that
- for all ε the money demand and the contract maximizes the buyer’s utility
- the mining choice is a Nash equilibrium of the mining game in every subperiod
- for all ε the contract satisfies condition (20)
- the day market for balances clears.
Note that when the DS constraint (20) is binding, a buyer may consider offering a DS contract in
order to increase the consumption and/or to reduce the confirmation lag. However, the settlement
of a DS contract must be probabilistic (P(d, N) < 1) and, according to the seller’s participation
constraint (33), the buyer has to offer a higher balance d for the seller to accept the offer. Consequently, buyers will offer DS proof contracts when the net gain for carrying extra balances is small.
24Bitcoin – like cryptocurrencies in general – is designed to discourage double spending attacks. Under the current
system where sellers wait for multiple confirmations, double spending activities do not seem to be a significant concern.
However, some successful double-spendings were reported when the confirmation lag was insufficient. According to
Bicoinwiki, “in November 2013 it was discovered that the GHash.io mining pool appeared to be engaging in repeated
payment fraud against BetCoin Dice, a gambling site.