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title: Allocation Equilibrium Analysis of The Graph

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# Allocation Equilibrium Analysis of The Graph
**Prysm Group analysis Version: August 9, 2021**


### Questions and Clarification Request
1. **Previous Reports**
    1. Do the notations used in _C01 and C02 Economic Attacks Review_ [EAR] identical to those used in _Allocation Equilibrium Analysis (Updated)_ [AEA]? If not, could you please provide a mapping?
    A: We kept most of the notations consistent, but there are changes in how some variables are denoted (a mapping to follow). 
    2. Please elaborate on the assumptions related to _inflationary rewards_ $\Phi$ (p18)
    A: $\Phi$ was treated as exogenous constant in [EAR], which is quite a restrictive assumption. That is why in the next engagement we conducted an equilibrium analysis in the [AEA] to examine the determinants of $\Phi$.
    3. Are we attributing the same properties to the indexing reward functions $X$ in [EAR] and in [AEA]?
    A: Yes. 
    4. Please elaborate on _expected value_, _expected indexing reward_ and _expected query reward_ which seem to interchangeably be designated as $E[X]$
    5. Is $E[X]$ a true expectation? If so please elaborate on underlying distribution. Is the equation on p18 a result or definition?
    A: There is no parametric assumptions imposed on the distribution of X. In numeric analysis, we used average cost figures Pedro provided as its empirical estimate. 

2. **System Agents**
    1. What are the behavioral decisions (strategy space) facing indexers? 
        1. Are the variables $w_{ist}$ (self stake), $\omega_{it}$ (allocation), _indexingRewardCut_ and _queryFeeCut_, endogenous or exogenous?
        A: All four variables are free choice variables for the indexers. After observing the amount of delegated stake $W_{it}$, indexer $i$ opens an allocation and chooses the amount of self-stake $w_{ist}$, the amount allocated to the allocation $\omega_{it}$, the fraction of indexing rewards they will collect $p_{Iit}$, and the fraction of query rebates they will collect $p_{Qit}$.
        2. Does $s$ in the subscript of $w_{ist}$ represent a variable/index? If so, what does it represent? If not, would that mean $w_{ist}$ represents the aggregate self stake of indexer $i$ across all subgraphs?
        A: We are fixing the subgraph in the analysis. The $s$ in the subscript stands for "self-stake". 
    2. What are the behavioral decisions facing delegators?
        1. Are the variables $w_{ij}$ endogenous? 
        A: A delegator $j$ is free to choose how much to delegate to an indexer $i$. In Lemma 1 of the equilibrium analysis, we used a zero-profit condition to determine the equilibrium level of delegation to indexer $i$.
        2. Is $w_{ij}$ time dependent? Is it the same as $w_{ijt}$ as appears in $W_{it}=\sum_jw_{ijt}$ on middle of p4?
        A: In principle, the amount of delegation $w_{ijt}$ can vary across different allocations. We drop the time subscript when it is not necessary. In particular, a delegator can delegate and undelegate at anytime; however, rewards are only distributed to the delegation pool at an allocation's closing so delegators do not have incentives to exit from an honest allocation. 

2. **Equilibria**
    1. What is the framework and solution concepts used in the model?
    A: We abstract away from the role of information and belief formation. The equilibrium concept is Subgame Perfect Equilibrium. 
    2. Is the equilibria presented in the model unique?
    3. If not, how are multiple equilibria accounted for?
    A: The equilibria are not unique. We chose to rule out equilibria where there is no delegation $(W_{i,t+l}=0)$ because they are undesirable from the perspective of network design. The conditions necessary for an interior solution pins down the equilibrium presented in the analysis (page 8-10).
    4. Is there any way to assess the allocational efficiency of multiple equilibria? 
    A: We have not formally examined the allocational efficiency. Brandon have brought up this metric in our prvious discussion and we are open to incorporate mechanics that allow us to do that. 

3. **Query Fees**
    1. Does Proposition 4 in Appendix D make a claim on equilibrium outcomes?
    A: We are not able to explicitly solve the equilibrium for the general case with query fees. Instead, we present comparative statics that decribe the impact of query fees on the equilibrium. 
    2. Why is the statement _"Holding the sum of all other indexers’ allocations $\Omega_{-it}$ constant"_ necessary?
    A: To highlight the fact that this is a partial equilibrium result, as in a full equilibrium changing total query fees $\Theta$ will change other indexer's equilibrium allocation as well. 
    3. Please add a bit more details on the proof of Proposition 4 in Appendix D. 
    - In progress.
    4. The equilibrium presented in the report assumes indexer and delegator utility is derived only from indexer rewards and not from query fees, how would these results change under the following assumptions
        1. Query fees account for over 50% of indexer income
        2. Query fees account for 100% of indexer income
        3. Please specify any model assumptions made on relation between indexer choice and query fees?
    A: The reason we focused on the no query fee case is because explicit solutions will not be possible for the general case. We can discuss numeric methods in the context of the next section. 

4. **Simulation and Numerical methods**
    1. Please describe in a concise manner the system of equations necessary for numerical solutions and comparative statics suggested on p3
    - Clarify, p3 of which document?
    3. What are the numerical methods recommended for solving this system?
    - Require further research and experimentation. 

4.	**Opportunity Costs**
    1.	How are opportunity costs assessed? 
    2.	Are opportunity costs estimated against money market returns?
    A: Opportunity costs are benchmarked to average annual returns from equity markets, e.g. S&P500.
    3.	Are we implicitly assuming indexers and delegators exit GRT and The Graph ecosystem altogether with unallocated capital?
    A: We currently assume no resource constraints. Indexers and delegators can move capital into GRT up to the point of zero (excess) profit and they can deploy unused capital elsewhere without incurring additional cost. 



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1.	Explicit solution only with no fees, based only on prizes, would this be easier to simulate?
2.	Choose a different and consistent set of variable names
3.	What are the choices made by each type of agent (indexers and delegators)? 
4.	Clarify flow of time, why are delegation and allocation amounts sometimes indexed by time and sometimes not? 
5.	Clarify what is the dynamics between the start and end of allocation? Can and indexer change allocation after start and before end of allocation? How are the rewards distributed in this case? 
6.	How are query fees distributed during an allocation? Is it proportional to the time a subgraph was allocated? 
7.	How are the indexers rewarded for good subgraph allocation choices?
8.	Clearer reference to previous work, like profit function p6
9.	Are the pie cutting parameters $p_Ii$ and $p_Qi$ a choice (strategy) of the indexer/delegator? If yes, could there be multiple equilibria? Do they all have the same efficiency? What can we say about strategical uncertainty?  






Big Questions
1.	What is the objective of delegation
2.	What is the information exchanged and elicited in this mechanism?
3.	What are the rational expectations of the agents? 
4.	The model works in continuous time, what is the impact of discrete block time?

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