Mitigating P+e bribery with Collateralized Agreements

Schelling games provide an interesting game theoretical avenue for creating oracle mechanisms (e.g. Augur, Kleros, and TCRs). The mechanism assumes that in the absence of coordination between a group of participants, the natural focal point is for participants to act honestly. However, an attacker can potentially manipulate the focal point by presenting a credible bribe, and if they are clever they potentially can do with a high budget but zero cost. This type of bribe, called “P+epsilon”, was explored by Vitalik in 2015.

The purpose of this post is to discuss one way to make a Schelling point more resistant to bribery attacks using a meta-game that gives honest participants a significant advantage over dishonest participants.

First lets look at payoff in a basic game:

  1. Participants are asked to blindly commit to an option. For simplicity, lets assume it is a binary option A or B.
  2. After all parties have made a commitment, the commitments are revealed.
  3. Participants who vote along with the majority of other Participants are rewarded with P.

Payouts for such a game look like:

Payouts You vote A You vote B
Others Vote A P 0
Others Vote B 0 P

Now lets assume there is a credible P+e style bribe for B:

Payouts You vote A You vote B
Others Vote A P P+e
Others Vote B 0 P

In general there is a social cost to taking bribes, but that cost is spread across the entire group, so even if it will tank the value of the mechanism, an attacker may succeed because users may expect others in the group to defect, and if they do the social cost is out of their control, and therefore defecting may be a rational way to minimize their losses.

In this case everyone would like to be honest, but because they might be worse off if they act honestly they might defect.

Let’s introduce a more complicated Schelling game that designed to be more resistant to bribery using collateralized agreements. This mechanism is both used in and enabled by the Aragon Court, a subjective oracle for dispute resolution, but the approach could be used to support honest participation in other voting mechanism, including TCRs.

The proposed Aragon Court is functionally very similar to the dispute resolution mechanism proposed by Kleros, participants are randomly selected from a subset so that not all participants are involved in every dispute, but any dispute can be escalated through an appeals process until all participants are involved, in which case a bribe would need to be sufficient to sway a majority of all participants. The appeals process does a good job of ensuring that the average cost for settling a dispute remains low, but the cost to effectively bribe generally has to sway at least 51% of all participants. This might be sufficient in practice to deter most attempts at bribery. However, it places the financial burden of repeated appeals on victims, who in some cases may not be able to continue to escalate through appeals.

A simple improvement to this mechanism provides two key benefits:


  1. In order to participate, agents are required to put up a deposit and commit to taking specific actions in the event of a bribe.
  2. If an agent is dishonest or fails to take the prescribed action, another agent can pay a fee to start a dispute against the dishonest agent.
  3. This dispute can lead to additional appeals and disputes if more jurors are found to be dishonest.

Now the cost for escalating is paid by opportunists who are incentivized by the prospect of taking collateral from dishonest jurors, rather than by victims of bribery attacks.

It also enables us to require jurors have a large pool of collateral that is generally not at risk based on the standard subjective Schelling game, but only if they specifically violate their individual commitment.

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