4.1 Game Theoretic Consensus

DAN's core operating mechanism is based on the SchellingCoin concept proposed by Vitalik Buterin. Through mechanism design, we make "honest voting" the only strong Nash equilibrium in this game.

4.1.1 The Schelling Point Mechanism

Definition:

In game theory, the Schelling point refers to the "default" or "natural" solution that participants tend to choose in the absence of communication.

Applications in DAN:

In a binary arbitration case (e.g., buyer wins vs. seller wins), arbitrators $J_1, J_2, ..., J_n$ must vote without being able to communicate with each other (due to the Commit-Reveal mechanism).

• Incentive objective: The system does not reward the "correct" answer (because there is no God-like correct answer in a decentralized network), but rather the answer that is "coherent" (in line with the majority).

• The Shared Signal: The arbitrator hopes to select the option that the majority would choose. In this game, the only shared signal is the Truth.

  • The truth is obvious (e.g., look at the shipping photos).

  • Lying requires complex collaboration (e.g., which is the lie? Should we choose A or B?). Without channels of communication, coordinating a specific lie is virtually impossible.

  • Therefore, "truth" becomes the Schelling point in the game. A rational arbitrator, in order to maximize their gains, will expect others to choose the truth, and thus choose the truth themselves.

4.1.2 Nash Equilibrium Analysis in Juror Voting

To prove the security of the system, we construct a formalized payoff matrix model.

1. Model Setup

Suppose we have a simple choice between two options:

• Options : the truth(Honest outcome)

• Options : lie (Dishonest outcome)

• Arbitrators

• Each arbitrator The number of staked tokens is 。

• Voting rewards are,Inconsistency penalty is （in ）。

2. Payoff Matrix

Consider arbitrator Alice's strategy, assuming she believes that a majority of the other arbitrators will vote in favor of one party:

Alice's Vote (vi​)	Majority's Vote (Vmaj​)	Alice's Payoff (Pi​)	Result
A (Truth)	A (Truth)	+R	Profit
A (Truth)	B (Lie)	-As	Loss
B (Lie)	A (Truth)	-As	Loss
B (Lie)	B (Lie)	+R	Profit

3. Equilibrium Analysis

This game has two pure strategy Nash equilibria:

1. Honest Equilibrium: Everyone bets on A. If Alice unilaterally deviates and bets on B, she will lose. Therefore, she will not deviate.

2. Conspiratorial Equilibrium: Everyone bets on B. If Alice unilaterally deviates from betting on A, she will lose.Therefore, theoretically this is also an equilibrium.

4. Why Honesty Prevails?

Although two equilibria exist, DAN breaks the tie through the following mechanism, causing the system to converge to an honest equilibrium:

• Coordination Cost: Achieving a "collusive equilibrium" requires enormous coordination costs. All attackers must act simultaneously, targeting the same case, and be certain that everyone else has received instructions to send the signal to B. Under DAN's anonymity and random selection mechanism, such communication is impossible.

• Simple Honest Agents: We assume that there is at least a small percentage (e.g., 5%) of “altruists” or “lazy people” in the network who will vote for the truth anyway.

  • If Alice knows that 5% of people are definitely going to vote for A, then the risk of voting for B will increase.

  • This will trigger a chain reaction: for safety's sake, even rational fence-sitters will side with A.

• P+epsilon attack resistance: Even if the attacker attempts to bribe (promising a reward of to those who vote for B).Due to the transparency of Ethereum smart contracts, the community can initiate a User-Activated Soft Fork (UASF) or trigger a final jury mechanism to directly confiscate the attacker's bribe funds.

5. Conclusion

Mathematical derivation shows that as long as the basic assumption that "honest arbitrators hold a majority of tokens" is met, and the attacker's coordination cost is greater than 0, {All Vote Truth} is the unique and stable Schelling point in this game.

$$P(\text{Honest Outcome}) \to 1 \quad \text{as} \quad N_{jurors} \to \infty$$

(When the number of arbitrators is large enough, the probability of outputting an honest result approaches 1.)

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By introducing Nash equilibrium and payoff matrix, a solid theoretical foundation is provided for the arbitration mechanism of OmniPact, proving that the system maintains justice not only by relying on "good people" but also by relying on "rational self-interested individuals".