# Zero-sum, ve and ve(3,3)

Solidly proposed to combine both design into an AMM. One of the shortfalls of Curve Solidly intended to fix was better aligning emissions of tokens with beneficials actions and solving the problem that most AMM designs temporarily support liquidity provision with token emissions; while the more sustainable incentive-generating mechanism (the fee generation of the AMM) is not incentivized. Hence the AMM fees should be paid to users locking in their assets in the protocol to ensure longer term sustainability. More specifically, lockers receive only trading fees from the liquidity pools they have voted for.

Further, since Solidly tried to achieve an Olympus DAO type of game theory outcome, but did not have a treasury (and therefore a bonding-like action was no option), Solidly intended to compensate with behavioral incentives as follows 1) weekly emission are adjusted as a percentage of circulating supply 2) lockers increase their holdings proportional to emissions and 3) ve-Tokens are NFTs. However, that design failed because 1) drains liquidity from the protocol 2) inflation on inflation is worse, not better 3) transferability of ve-Tokens is antithetical to the idea of aligned interests as it does not disincentives nefarious behaviour. Here we can assume the net present value of money on hand (selling) in this zero-sum game has a higher payoff than staking and that the game is fundamentally non cooperative.

As established by Chitra et. al. (2022), single-ve designs lack unique equilibria. The CRV bribing game is a zero-sum game. In a zero-sum game, the total amount of rewards available to the users is fixed, so any reward gained by one user is necessarily lost by the other user.

In the context of the CRV bribing game, the rewards available to the players are the CRV tokens that are being used to incentivize liquidity providers in the Curve protocol. If one player is able to control the parameters A, B, C, and D in a way that gives them more CRV tokens, then the other player necessarily receives fewer CRV tokens.

A represents the "drift" of the system, which is like the average rate at which the system changes over time.

B represents the "volatility" of the system, which is like how much the system changes randomly over time.

C represents the "discount rate" of the system, which is like how much the system values future rewards compared to immediate rewards.

D represents the "liquidity" of the system, which is like how easy it is to buy or sell digital assets in the system.

By understanding how these parameters work and how they can be controlled, a protocol can develop better strategies for managing liquidity in DeFi protocols.

Such dynamics and models could be studied Hamilton-Jacobi-Isaacs equations. HJI equations are a type of partial differential equation that can be used to analyze the optimal control of non-linear systems in game theory .

In the context of the CRV bribing game, the HJI equation would be used to find the optimal strategies for the two users to control the parameters A, B, C, and D in order to maximize their expected return. The HJI equation would take into account the fact that the game is zero-sum and that the users are competing against each other to control the system and allow would allow to analyze the optimal control of the system in a rigorous and mathematically sound way, which can help to inform the development of better strategies for managing liquidity in DeFi protocols.

## Risks of single-ve designs

It's important to understand that while these "bribing games" can lead to an increase in total liquidity for a DeFi protocol, they can also introduce additional complexities and potential risks into the system.

If a ve-bribing game is a zero-sum game, it means that the gains of one user are exactly offset by the losses of the other user. In other words, the total gains and losses in the game sum to zero. This can create risks and challenges for the users involved:

**Complexity:**Finding solutions to the CRV bribing games may require solving complex equations that may be difficult to analyze tractably. This complexity could lead to unforeseen consequences or vulnerabilities in the design.**Governance Manipulation:**The CRV bribing game allows entities with significant resources to manipulate the governance mechanism of the protocol to their advantage. This could potentially lead to an imbalance in the distribution of rewards, favoring certain pools over others.**Dependence on Third-Party Protocols:**The CRV bribing game relies on third-party protocols (such as Convex Finance) to incentivize users to deposit assets and reallocate them. If these third-party protocols act maliciously or incompetently, it could negatively impact the users and the overall health of the protocol.**Potential for Market Manipulation:**The CRV bribing game allows for the possibility of market manipulation, as entities can influence the allocation of rewards and other parameters within the protocol.**Risk of Centralization:**The game could potentially lead to centralization and cartelization, as entities with more resources could have a disproportionate influence on the protocol's governance.

It's possible to analyse the dymamics of the game with Hamilton-Jacobi-Isaacs Equations. These equations can provide an equilibrium for the game, known as a Nash equilibrium, where no player can unilaterally improve their outcome by changing their strategy, assuming the other player keeps their strategy unchanged. In other words, it's a state where each player's strategy is the best response to the other player's strategy. Here, HJI equations replace the supremum (the greatest value) in the model with a minimax formulation. This minimax formulation minimizes over one users's control parameters and maximizes over the others. This reflects the zero-sum nature of the game, where one user's gain is the other player's loss.

However, finding solutions to these games using HJI equations can be complex and challenging. The equations can be difficult to solve analytically. Therefore, while HJI equations can theoretically be used to find an equilibrium for the CRV bribing game, doing so is nontrivial.

Instead, we suggest another solution by introducing a non-zero sum bribing game: dual-ve.

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