# Investigate the Optimality Gap as an Indicator of Collusion on Gitcoin Grants

###### tags: `Research`

For more background information on this collaboration, read [this Notion TL;DR](https://www.notion.so/blockscience/Gitcoin-Modelling-Co-Lab-TL-DR-4e115525dbc0430289bb78d60ad79a04).

## Intro

**One of the primary considerations of using Quadratic Funding as a governance policy over fund allocation is the threat of collusion**. But what is collusion in graph-theoretical terms? Describing private intent through available public structures is a common hurdle when designing equitable incentive structures, and a comprehensive definition is often out of reach.

However, it is possible to infer what happens when looking into the actual consequences of successful collusion. We can imagine that colluders will optimize their tactics and strategies for maximizing their objective of maximizing funding outcomes, while organic actors will have a more random structure due to the diversity and heterogenity of tactics and choice functions.

For the Gitcoin Grants case, this means that the neighboring graph structures for colluding grants might be highly optimized for funding, while the ones for organic grants would not present a highly optimized collaboration structure.

In this research plan, we explore the above idea by using Neighbor Subgraphs as a proxy for those network structures. We then examine the gap between the transactions in these subgraphs as they occurred in reality, compared with an idealized optimal transaction order that maximizes funding - **we call this measure the 'optimality gap'.**

As a working question, **we hypothesize that this optimality has a bi-modal distribution:**
- **'Cluster of communities with large Optimality Gap'** representing low optimization of funding on behalf of organic grants
- **'Cluster of communities with small Optimality Gap'**, which could indicate grants which are adopting colluding strategies.

*An example of a bi-modal distribution. Source: https://medium.com/precarious-physicist/teaching-a-class-with-a-bimodal-distribution-if-you-have-one-c9629ac15469*:

## Definitions

D0: The full contribution graph $G$, for each vertex is either a grant (whose nodes constitute the set $\mathcal{G}$) or a contributor (constituing the set $\mathcal{C}$). All edges go from a vertex of type contributor to a vertex of type grant.

D1: $NeighborsSubgraph(g)= \{n \in \mathcal{G} \;: d(g, n) \leq 3 \}$
- $n$: node (grant or contributor)
- $g$: grant
- $d(u, v)$: Degree distance between nodes $u$ and $v$

*The Neighbors Subgraph for a given grant is somewhat similiar to the 'expansion 3' subgraph. Source: https://stackoverflow.com/questions/63534977/how-to-get-neighbor-nodes*

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D2: The Quadratic Match of a subgraph is the sum of quadratic match of all the grants contained inside it.

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D3: The metric of interest is the Quadratic Match for a Neighbors Subgraph associated with a grant $g$.
- Alternatively, we can understand the quadratic match of the subgraph as being the utility function.

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D4: The *real subgraph of a grant* is the NeighborsSubgraph that is induced from original graph.

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D5: The *optimal subgraph of a grant* is the *real subgraph* where the edges are rewired such that metric of interest in regards to the induced subgraph is maximized.

*Example of graphs with the same amount of nodes and edges. Source: https://www.researchgate.net/publication/332979502_Global_Robustness_vs_Local_Vulnerabilities_in_Complex_Synchronous_Networks*

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D6: Optimality gap is the one minus the fraction between the metric of the neighbors real subgraph and the optimal subgraph:
$$OptimalityGap=\left(1-\frac{M_r}{M_o}\right)$$

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### On optimality

suppose there is an optimization problem
$\max_x f(x)$

then the optimal arguement is $x^*$ s.t $M_0=f(x^*)$, for any observed value $x$ (in this case the real graph), then we have $M_r = f(x)$

-->

## Hypothesis

H1: The optimality gap per grant distribution on the Gitcoin Grants network is bimodal, with a larger density on the high side.
- This could be indirectly interpreted(*) that the mode associated with small gap is a 'colluders' mode, and the one with large gap would be the 'common' mode.

## Methodology

The execution of this research plan will be done partly through public working sessions, which will have live streams and live coding sessions with community participation, while BSci researchers will implement the required definitions and mechanisms as well as integrating with the available data.

We will use the Rounds 8 dataset and matching algorithm.

Through those, we'll perform a Exploratory Data Analysis over the optimality gap metric, seeking to characterize the associated distribution as well as exploring emergent properties that may arise.

## Applications

If hypothesis is confirmed:

- Small optimality gap implies high optimization for funding,
- Actionability of policies on the optimality gap: set thresholds for human evaluation of grants with small gap.

Else:

- Change D1 so that it uses subgraphs from community detection algorithms
- Formulate an alternative for H1
- Formulate an alternative for D3

Note that these applications are not exhaustive.

## Next Steps

- Testing H1 for the dynamical network instead of static network
- Evaluating the Optimality Gap by using live data on next Grants round
- Expanding D1 or D5 to include temporal neighbors or windows
- use community detection subgraphs and solve optimal rewiring to $m$ grants rather than just $1$.
- **There's a world of science out there to explore - let's get started!**

*Gitcoin Grants Round 7 on Kumu. Red nodes are donors, green nodes are grants. Source: https://gitcoin.co/blog/towards-computer-aided-governance-of-gitcoin-grants/*