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 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 dPEN holdings of participant $a,b,c,\dots$ are denoted by $x_a$, $y_a$, etc., respectively.

Alice starts with $y_a=10000$ dPEN of Victoria’s delegation pool, Bob starts with $y_b=10000$ dPEN of William’s delegation pool, 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 ${\psi }_v(1)=\psi (1)=1.0006$. Alice’s $y_a=10000$ dPEN(v) is now worth $10006$ PEN.
  • William charges $10\%$ commission, so ${\psi }_w(1)=1.00054$. Bob’s $y_b=10000$ dPEN(w) is now worth $10005.4$, and William receives $0.6$ PEN.
  • William can use the commission to cover expenses, or self-delegate. In this example, we assume that validators self-delegate their entire commission, to illustrate the staking dynamics.
  • William self-delegates $0.6$ PEN, to get $0.6/{\psi }_w(2)=0.6/1.00054^2=0.59935\dots$ dPEN in the next epoch, epoch $2$.

• At epoch $e=90$:

  • Alice’s $y_a=10000$ dPEN(v) is now worth $10554.67$ PEN.
  • Bob’s $y_b=10000$ dPEN(w) is now worth $10497.86$ PEN.
  • William’s self-delegation of accumulated commission has resulted in $y_w=53.483$ dPEN(w).
  • Victoria’s delegation pool remains at size $10000$ dPEN(v). William’s delegation pool has increased to $10053.483$ dPEN(w). However, their respective adjustment factors are now ${\theta }_v(90)=1$ and ${\theta }_w(90)=0.99462$, 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 self-delegates 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 bonded 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 $10000/{\psi }_v(91)=9485.85$ dPEN(v) and $10000/{\psi }_w(91)=9536$ dPEN(w).

• At epoch $e=91$:

  • Charlie now has $9468.80$ dPEN(v) and $9520.60$ dPEN(w), worth $20000$ PEN.
  • For the same amount of unbonded stake, Charlie gets more dPEN(w) than dPEN(v), because the exchange rate ${\psi }_w$ 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.

• At epoch $e=180$:

  • Alice’s $y_a=10000$ dPEN(v) is now worth $11140.12$ PEN.
  • Bob’s $y_b=10000$ dPEN(w) is now worth $11020.52$ PEN.
  • Charlies’s $y_{c,v}=9468.80$ dPEN(v) is now worth $10548.37$ PEN, and his $y_{c,w}=9520.60$ dPEN(w) is now worth $10492.20$ PEN.
  • William’s self-delegation of accumulated commission has resulted in $y_w=158.77$ dPEN(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.

This scenario was generated with a model in this Google Sheet.