  We study a program game version of the Prisoner's Dilemma, i.e., a two-player
game in which each player submits a computer program, the programs are given
read access to each other's source code and then choose whether to cooperate or
defect. Prior work has introduced various programs that form cooperative
equilibria against themselves in this game. For example, the
$\epsilon$-grounded Fair Bot cooperates with probability $\epsilon$ and with
the remaining probability runs its opponent's program and copies its action. If
both players submit this program, then this is a Nash equilibrium in which both
players cooperate. Others have proposed cooperative equilibria based on
proof-based Fair Bots, which cooperate if they can prove that the opponent
cooperates (and defect otherwise). We here show that these different programs
are compatible with each other. For example, if one player submits
$\epsilon$-grounded Fair Bot and the other submits a proof-based Fair Bot, then
this is also a cooperative equilibrium of the program game version of the
Prisoner's Dilemma.
