Example Staking Dynamics
To illustrate the dynamics of this system, consider a toy scenario with three delegators, Alice, Bob, and Charlie, and two validators, Victoria and William. Tendermint consensus requires at least four validators and no one party controlling more than $1/3$ of the stake, but this example uses only a few parties just to illustrate the dynamics.
For simplicity, the the base reward rates and commission rates
are fixed over all epochs at $r=0.0006$ and $c_{v}=0$, $c_{w}=0.1$.
The PEN
and PENb
holdings of participant $a,b,c,…$ are
denoted by $x_{a}$, $y_{a}$, etc., respectively.
Alice starts with $y_{a}=10000$ PENb
bonded to Victoria, Bob starts with $y_{b}=10000$ PENb
bonded to William, and Charlie starts with $x_{c}=20000$ unbonded PEN
.

At genesis, Alice, Bob, and Charlie respectively have fractions $25%$, $25%$, and $50$ of the total stake, and fractions $50%$, $50%$, $0%$ of the total voting power.

At epoch $e=1$, Alice, Bob, and Charlie’s holdings remain unchanged, but their unrealized notional values have changed.
 Victoria charges zero commission, so $ψ_{v}(1)=ψ(1)=1.0006$. Alice’s $y_{a}=10000$
PENb(v)
is now worth $10006$PEN
.  William charges $10%$ commission, so $ψ_{w}(1)=1.00054$. Bob’s $y_{b}=10000$
PENb(w)
is now worth $10005.4$, and William receives $0.6$PEN
.  William can use the commission to cover expenses, or selfdelegate. In this example, we assume that validators selfdelegate their entire commission, to illustrate the staking dynamics.
 William selfdelegates $0.6$
PEN
, to get $0.6/ψ_{w}(2)=0.6/1.00054_{2}=0.59935…$PENb
in the next epoch, epoch $2$.
 Victoria charges zero commission, so $ψ_{v}(1)=ψ(1)=1.0006$. Alice’s $y_{a}=10000$

At epoch $e=90$:
 Alice’s $y_{a}=10000$
PENb(v)
is now worth $10554.67$PEN
.  Bob’s $y_{b}=10000$
PENb(w)
is now worth $10497.86$PEN
.  William’s selfdelegation of accumulated commission has resulted in $y_{w}=53.483$
PENb(w)
.  Victoria’s delegation pool remains at size $10000$
PENb(v)
. William’s delegation pool has increased to $10053.483$PENb(w)
. However, their respective adjustment factors are now $θ_{v}(90)=1$ and $θ_{w}(90)=0.99462$, so the voting powers of their delegation pools are respectively $10000$ and $9999.37$. The slight loss of voting power for William’s delegation pool occurs because William selfdelegates rewards with a one epoch delay, thus missing one epoch of compounding.
 Charlie’s unbonded $x_{c}=20000$
PEN
remains unchanged, but its value relative to Alice and Bob’s stake has declined.  William’s commission transfers stake from Bob, whose voting power has slightly declined relative to Alice’s.
 The distribution of stake between Alice, Bob, Charlie, and William is now $25.67%$, $25.54%$, $48.65%$, $0.14%$ respectively. The distribution of voting power is $50%$, $49.74%$, $0%$, $0.27%$ respectively.
 Charlie decides to bond his stake, split evenly between Victoria and William, to get $10000/ψ_{v}(91)=9485.85$
PENb(v)
and $10000/ψ_{w}(91)=9536$PENb(w)
.
 Alice’s $y_{a}=10000$

At epoch $e=91$:
 Charlie now has $9468.80$
PENb(v)
and $9520.60$PENb(w)
, worth $20000$PEN
.  For the same amount of unbonded stake, Charlie gets more
PENb(w)
thanPENb(v)
, because the exchange rate $ψ_{w}$ prices in the cumulative effect of commission since genesis, but Charlie isn’t charged for commission during the time he didn’t delegate to William.  William’s commission for this epoch is now $1.233$
PEN
, up from $0.633$PEN
in the previous epoch.  The distribution of stake between Alice, Bob, Charlie, and William is now $25.68%$, $25.54%$, $48.64%$, $0.14%$ respectively. Because all stake is now bonded, except William’s commission for this epoch, which is insignificant, the distribution of voting power is identical to the distribution of stake.
 Charlie now has $9468.80$

At epoch $e=180$:
 Alice’s $y_{a}=10000$
PENb(v)
is now worth $11140.12$PEN
.  Bob’s $y_{b}=10000$
PENb(w)
is now worth $11020.52$PEN
.  Charlies’s $y_{c,v}=9468.80$
PENb(v)
is now worth $10548.37$PEN
, and his $y_{c,w}=9520.60$PENb(w)
is now worth $10492.20$PEN
.  William’s selfdelegation of accumulated commission has resulted in $y_{w}=158.77$
PENb(w)
, worth $176.30$PEN
.  The distribution of stake and voting power between Alice, Bob, Charlie, and William is now $25.68%$, $25.41%$, $48.51%$, $0.40%$ respectively.
 Alice’s $y_{a}=10000$
This scenario was generated with a model in this Google Sheet.