# The Retrievability Consortium - Cryptoecon Analysis

###### tags: `Retrievability Pinning`

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Up to date by August 2022
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*Process Diagram for the Assured Deal Journey on the Retrievability Consortium*

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This section is WIP, don't read :)
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*Diagram of the decisions and processes behind the client-facing DRC protocol*

### Assumptions

- Time value and discounting will not be taken into account.
- The interval duration between decisions and processes will be immediate
- There's no deposit costs for the provider

### Development

#### Decision Structure

The base decision variables (which can assume 0 or 1) are:

- $d_1$: Client decides to create proposal $\zeta$
- $d_2 | \zeta$: Provider decides to accept proposal $\zeta$
- $d_3$: Client decides to appeal
- $d_4$: Referees decide to slash provider

Auxiliary variables

- $\pi_r^C \in \mathbb{R}$: File retrieval utility to the client.
- $r \in \{0, 1\}$: file is retrieved
- $p \in \mathbb{R}_+:$ payment
- $m_c$: committee multiplier
- $m_s$: slashing multiplier
- $N_{a, max}: maximum number of appeals

The probabilities of the decision variables will be taken by assuming an Best Response form, eg:

$P(d_1|\zeta, I) = (\pi_{d_1} > \pi_{\bar{d_1}})$

Given that, we can assume the following payoffs for each base decision:

- $d_1$
- $\pi_{d_1, \zeta}^C = \pi_r^C P(r|d_2) + p\cdot P(d_2|\zeta, d_1, I) \cdot [2 P(d_4|d_2, I) - \langle N_a\rangle m_c - 1)]-\pi_{\bar{d_2}, \zeta}^C P(\bar{d_2}|\zeta, d_1, I)$
- Equivalent form: $\pi_{d_1, \zeta}^C = \pi_r^C P(r|d_2) + p\cdot P(d_2|\zeta, d_1, I) \cdot [P(d_4|d_2, I) - \langle N_a\rangle m_c - P(\bar{d_4}|d_2, I))]-\pi_{\bar{d_2}, \zeta}^C P(\bar{d_2}|\zeta, d_1, I)$
- $\pi_{\bar{d_2}, \zeta}^C$ (The opportunity cost of having the proposal not accepted) will be assumed to be zero.
- $\langle N_a \rangle = \sum_{n_a} P(\prod_{j < n_a} d_{d_3, j} | \prod_{i<j} d_{4, i}, d_2, d_1, I)$
- $\pi_{\bar{d_1}, \zeta}^C = \pi_r^C P(r | \bar{d_2})$
- This inspires an assumption: $P(r|\bar{d_2})<P(r|d_2)$, which leads to the following definition (the marginal increase in retrieval probability): $\delta_r = P(r|d_2)-P(r|\bar{d_2})$
- The above says that the client will create a deal proposal if $\pi_{d_1, \zeta} > \delta_r$
- If $P(d_2)=1$, then the rationality constraint becomes $\pi_r^C \delta_r + 2p P(d_4)> p m_c \langle N_a \rangle + p$
- Alternate form:
- $\pi_r^C = \frac{p}{\alpha} \leftrightarrow \frac{\delta_r/\alpha + 2 P(d_4) - 1}{{\langle N \rangle}} > m_c$
- $\alpha$ can be understood as an "insurance leverage". Values lower than 1.0 means that the client is under insuring in regards to the retrieval utility
- This means that the following things:
- Increasing the number of expected appeals reduces the acceptable value of $m_c$
- Increasing the a priori slash probability increases the acceptable value of $m_c$
- Increasing the marginal retrieval probability increases the acceptable value of $m_c$
- Under-leveraging increases the acceptable value of $m_c$

- $d_2$
- $\pi_{d_2, \zeta}^P = p [P(\bar{d_4}|d_2, I) - m_s P(d_4|d_2, I)]$
- Note: $P(\bar{d_4})=1-P(d_4) \leftrightarrow \pi=p(1-[P(m_s+1)])$
- $\pi_{\bar{d_2}, \zeta}^P = 0$
- The above payoffs implies that **the payment value is unimportant for the provider to accept an deal** and the sole criteria for acceptance is $P(d_4|d_2, I) < \frac{1}{m_s + 1}$
- Equilbrium: $m_s = \frac{1}{P(d_4)}-1$. Interpretation: the maximum value of $m_s$ for the deal to be rational to the provider.
- If $P(d_4)$ for an honest provider is 0.1%, then the equilibrium value of $m_s$ is 999.
- Another expansion is $P(d_4|d_2) = \langle N_a \rangle P(d_4|d_3, d_2)$
- $d_3$
- $\pi_{d_3}^C=\pi_r^C P(r|d_3, I) + p P(d_4 | d_3, I) - p m_c$
- $\pi_{\bar{d_3}}^C=\pi_r^C P(r | \bar{d_3})$
- Note: those are similiar to $d_1$, but with more immediate priors.
- $d_4$
- $\pi_{d_4}^R= \delta_sP(\bar{r}|d_3)$
- $\pi_{\bar{d_4}}^R= \delta_\bar{s} P(r|d_3)$

#### Optimization Objectives

- The system goals are to maximize:
- Adoption of the protocol
- $\pi^g \propto P(d_2|d_1) + P(d_1)$
- Retrievability when taking a deal
- $\pi^g \propto \frac{P(r|d_2)}{P(r|\bar{d_2})}$
- Cost mitigating when data is not retrievable on a deal
- $\pi^g \propto \frac{p P(\bar{r}, d_4|d_2)}{\pi_r^C}$
- Referee fairness
- $\pi_g \propto P(d_4|\bar{r})-P(d_4|r)$
- Taking all together, the global utility is defined as:
- $\pi^g = \beta_0 P(d_2|d_1) + \beta_1 P(d_1) + \beta_2 \frac{P(r|d_2)}{P(r|\bar{d_2})} + \beta_3 \frac{p P(\bar{r}, d_4|d_2)}{\pi_r^C} + \beta_4 P(d_4|\bar{r})- \beta_5 P(d_4|r)$

### Conclusions

- The committee multiplier ($m_c$) is upper bounded by the client belief about the retrieval utility, how much appeals he needs to do and the associated probabilities.
- **Numerical suggestion: $m_c=0.2$ with an acceptable region being between 0.1 and 1.0**
- Rationality constraint: $m_c < \frac{\frac{\delta}{\alpha} + 2P(slash)-1}{\langle N_{appeals} \rangle}$, where $\delta$ is the marginal increase in probability of retrieving the deal and $\alpha$ quantifies how much the `payment` captures the retrieval utility.
- For most cases, it means that $m_c$ can be made relatively high if we assume that the levarage is low (usually on the order of 10 to 1000)
- **$m_c > 1$ will feel weird to the client UX**, as it will cost more to appeal rather than to insure. This has an potentially damaging effect to the protocol adoption. **We recommend therefore setting $m_c$ way lower than the equilibria** condition in order to foster demand.
- Setting up **higher than 1.0 has an potential for misunderstanding and the biggest risk of it being too low is associated to non-honest referees.**
- The above risk is partially mitigated by having an maximum number of appeals
- The `slashing_multiplier` ($m_s$) is upper bounded by the provider belief about his chance of being slashed and there's no trivial lower bound. We suggest setting this parameter as high as possible.
- **Numerical Suggestion: $m_s=100$, with an acceptable region being between 10 and 1000.**
- Rationality constraint: $m_s < \frac{1}{P(slash)}-1$
- For a $P(slash)=0.1\%$, $m_s<999$
- The maximum number of appeals ($N_{a, max}$)
- **We recommend between 3 and 5.**