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# Cooperative Vaults

telluscoop.com/model.html
We have created the concept of Cooperative Vaults, which simulate a multi-signature account with distributed weights among its participants.
To evaluate the feasibility of this model, we have defined different assumptions and participants.
Participants are signers on a Cooperative Vault
Each participant will have the possibility to vote with the following options (for the time being)
• Vote Cash Out
• Vote for Lock
• Vote for Invest
Outcomes: If more than 50% of participants vote for an action, this will be performed. If a quorum is not reached, the default outcome will be in place (Lock).
Payoffs: The payoff for each participant is the amount of funds they own at the end of the simulation.
Decision tree of participants in a Cooperative Vault
Explanation:
The participant has three options: Cash out, Lock, or Invest
If the participant chooses Cash Out, his payoff will be 100 (Initial amount deposited)
If the participant chooses the option Lock, his expected payoff will be:
$p*a*100=(1-p)*0$
Where:
p = probability of success
⍺ > 1 interest rate
(1-p) = probability of failure
0 for P close to 1
If the participant chooses to Invest, his expected payoff will be
$q*β*100$
For our model, we establish that:
$p>1$
$⍺ < β$
So a high probability of success (p) is balanced by a low ⍺
Now we wonder. How will a rational participant deal with such a decision tree?
He should compare payoffs for different actions:
If 100> max {p100, qβ100} it is best to cash out otherwise if p100 > qβ100 or p/β > q/⍺
then lock is the best otherwise invest is the best.
If we have N identical rational players dynamic of model will be straightforward: They all choose best available option.
Participant's risk assesment
2. Risk attitude and heteregenous players.
To make the model more realistic we need to introduce the concept of risk attitude.
Risk attitude is a function that defines players perception of a situation when he can lose a lot.
For example facing a decision to look or to invest we have two choices:
$a: 0.5(300)+0.5*(0)=150$
$b: 0.75*(150)+0.25*(0)= \frac{3}{4} * 150 = \frac{450}{4}=115$
Choice a is risky, but it gives better reward. To differentiate players willing to take risks for higher profit we introduce
then the player is risk neutral
$|f\Longrightarrow f(x)=x$
Example: Consider
$f(x) = x^2$
​this is risk averse person
Now recalculate option a and b from above
$a:f(0.5)300=0.25300=75$
$b:f(0.75)150=0.5625150=84.375$
So now option b is more profitable because risk is lower.
Now we considere more complicated model when every person can be risk neutral, risk lower, and risk averse. For simplicity we consider X^1/2 and X^2 functions
If we have random pool of players and each plauyer have own function now the question is: Is there a majority?
If the answer is yes → choice of the pool will be he choice of this group. If no → no choice will be made and pool will be locked.
This is not the dynamic we want to model, so we consider extension.
3. Model with mixed strategies
Suppose we have a player that values actions like this:

#### invest>cash out> lock

Suposse that there are no majority for any action, so lock is employed
This is the worst option for the player and he can consider deviation towards out. This is called strategic voting or manipulative voting.
To model this we can consider n rounds of voting, when every player chooses his two best options with certain probablity
$P_i,1-P_i$
So strategy for each player will be
$P_i\varepsilon (0.1)$
for best option,
$1-P_i$
for second best option, and 0 for worst option
Another variant of the model can be such as this:
Player chooses best option on first round with probability 1, if best option is not happened → employ mixed strategy mentioned above.
This approach uses non-cooperative games formalism, which assumes that players cannot make collaborations.