Mode Research: An Introduction to Cooperative Game Theory

Mode is building the Onchain Cooperative, an Ethereum L2 designing new economic systems for human coordination. Being a member of the OP Superchain, we are looking to evolve some of the great work done via RetroPGF and define our own flavour of impact. Our mission is to grow Ethereum by focusing on new incentive models for both users and builders.We are tired of seeing purely individualistic and PvP approaches to ecosystem growth. Mode was created under the principle that ‘cooperation leads to greater outcomes for all’ and the field of Cooperative Game Theory was our inspiration for much of what we will be releasing in 2024.

This article is Part One in a series of introductory research pieces that dive into different elements of game theory, its practical implications and how we plan on putting this theory onchain. The culmination of this research will be a sequence of economic mechanisms implemented by Mode Network throughout 2024 and beyond. James Ross, Mode Founder

Blockchain protocols promise many innovations, from decentralized finance to supply chain tracking. However, as public blockchains grow, questions arise about fairly rewarding the many contributors that make these networks valuable. The hundreds of developers, validators, governance participants and users that comprise open protocols, need an alternative to traditional corporate structures to determine rewards and incentives.

Game theory provides conceptual tools for analyzing this challenge. Non-cooperative game theory studies settings where players make decisions independently and cannot negotiate binding agreements. In contrast, cooperative game theory allows groups of players to coordinate strategies and share payoffs as a team.

One key solution concept from cooperative theory is the Shapley value. Initially conceived for n-person cooperative games, Shapley values allocate payouts to each player based on their marginal contribution, averaged across all possible coalition subgroups. This approach satisfies intuitive fairness criteria while incentivizing team collaboration.

As we explore in this article, applying Shapley values to distribute protocol rewards could enable fair, performance-based incentives for open blockchain contributors. However, operationalizing this in a manipulation-resistant way poses research problems at the intersection of mechanism design, cryptography, and economics. If solved carefully, Shapley-style payouts may help growth and sustainability for Web3 protocols looking to compete with traditional technical solutions.

Shapley Values

Shapley values are a solution concept in the field of cooperative game theory that fairly assigns payouts amongst a group of individuals working together. The payouts are based on the marginal contribution of each group member; that is, the payout of the group with the individual included minus the payout of the group without the individual included. Intuitively, Shapley values represent the expected marginal contribution of a player to the game.

Shapley values also adhere to certain invariants. First, each player in the game should earn at least as much as they would from working individually. Players that have identical marginal contributions should receive equal payouts and tangentially, deadweight players with no payout. Finally, the sum of payoffs to each player should be equal to the total payoff generated when all players cooperate.

We can extend the concept of Shapley values to incorporate weights and coalitions. Adding weights to players allows the payoff vector to depend on more than just marginal contribution (e.g. a protocol can favour early adopters or token-holders). Separately, we can assume that players form pre-determined coalitions (smaller teams) and adjust the Shapley value to only account for orderings in which these players are working together.

Interpreting Shapley Values

Because players are guaranteed to earn at least as much as working together versus individually, Shapley value payoffs encourage collaboration. Furthermore, we can view marginal contribution as the “bargaining power” that a player brings to a group. The more the player can increase the group’s current payoff, the more bargaining power (and hence higher payoff) this player should receive proportional to other members in the group.

From the bargaining power perspective, it also follows that players can optimize their payouts by finding groups that can complement their skills. We define the marginal contribution between coalitions in a similar manner. In theory, this creates an economic incentive for players and groups to find synergies and increase the efficiency of their work through cooperation.

Calculating Shapley Values

Let i(v) denote the payoff of the game defined by the function v for the ith player. We can compute this payoff using the formula below:

Intuitively, we’re considering every subset S of the N players and the marginal contribution that i would have on S with the expression on the right. We’re then multiplying this by the number of ways that S can be formed and averaging the marginal contribution over all possible orderings of the players in the fractional term.

One limitation is that the formula cannot be computed efficiently. In current applications, i(v)is estimated with high probability by simulating the formation of different subsets S and averaging the marginal contribution over a large number of trials.

Applications

Shapley values are currently being used to quantify the importance of different inputs to a machine-learning model. In this framework, the players are the set of variables eligible as inputs and the payoff is the accuracy of the model that was run on these inputs.  They’re also used in the finance industry to quantify the risk exposure of firms weighed on variables such as firm size, asset distribution, etc.

The idea of Shapley values in a blockchain protocol is similarly intuitive. For example, we can think of coalitions as different projects or sub-teams (e.g. development, marketing, legal). We can add weights to create or correct imbalances set by protocol policy, such as favouring early adopters, token holders, or certain teams over others when encouraging the protocol to move in a new direction. Users themselves can (naturally) belong to multiple coalitions as well. The Shapley value solution that accurately models this framework will fairly distribute payouts to players proportional to their marginal contribution, while also encouraging collaboration and efficiency.

However, there are a few challenges in their implementation that we pose as open questions below:

Open Questions and Moving Forwards

Defining the Game

A game is defined as a function that maps all subsets of the players to some real number pay-off. While this is intuitive for inherently quantitative applications like ML or Finance, it’s less clear for a blockchain protocol. Do we define payoff as the increase in revenue, TVL, or protocol liveness and how do we accurately keep track of such a metric?

Deterministic vs. Randomized Computation

Following from above, our payoff methodology must continue to encourage collaboration while making bad behaviour economically disadvantageous. If our payoff function is known beforehand and deterministic, can we prove that users cannot exploit by cutting corners or forming collusion groups that would otherwise be financially inexplicable?

We also saw above that the computation of a Shapley value can only be estimated efficiently. Is it acceptable to have payoffs that are randomized between range [a,b] for a player?

Sybil Attacks

Blockchain protocols also differ from real-world applications because a user can have a one-to-many identity mapping. For the protocol to operate safely and efficiently, we want to ensure that any payoff function we have does not reward an individual more if they split up their profile across multiple accounts.

Conclusion

In conclusion, Shapley values allow blockchain protocols to distribute rewards proportional to expected marginal contributions from different users and teams. This helps ensure fairness while incentivizing collaboration between individuals and groups.

However, applying the mathematical framework of Shapley values to real-world blockchain protocols poses challenges. Defining appropriate payoff functions that map user actions to measurable impact is non-trivial, alongside preventing exploitation by bad actors through collusion or Sybil attacks, requires rigorous mechanism design. As we continue researching solutions, the invariants guaranteed by Shapley values provide a useful blueprint for designing fair and collaborative blockchain protocols.

Key open questions remain around quantifying user contributions, balancing efficiency and collaboration incentives, and preventing manipulation by individuals creating multiple identities. Overcoming these challenges can enable Shapley value-based rewards that align user incentives, improve cooperation, and support protocol growth.

This introductory game theory piece by is Mode Research in collaboration with FranklinDAO. In Part Two, we’ll be diving deeper into our implementation strategies for Mode Network. Until next time…