p + Epsilon Attack

A more sophisticated bribe attack than the simple bribe attack is the p + Epsilon attack. Imagine we have the following situation. We have a player (actor) who has to vote for option 1 or 2. Along with the player, there is a crowd of other players who have to vote. If our player votes for the same option as the majority of all other players, he receives a reward. The following pay-off matrix shows player A’s pay-offs for every possible strategy combination.

Each player has the incentive to vote for the option everybody else votes for.

An attacker could try to influence the players. The attacker promises to pay P + e to all voters who vote for option 2, if the majority votes for option 2. This changes the pay-off matrix in the following way.

The result is that for player A the dominant strategy is to vote for option 2, no matter what the rest of the crowd does. The interesting thing here is that if every player votes for option 2, the pay-off switches to P (instead of P + e). In this case, the attacker doesn’t need to pay anything. In a blockchain context, this reward is the block reward + transaction fees. The options are branches of a fork. However, the commitment to the conditional bribe is difficult to design because the fork would need to “know” that it is the winning fork (the majority of miners build upon it).