  Nash equilibrium (NE) assumes that players always make a best response.
However, this is not always true; sometimes people cooperate even it is not a
best response to do so. For example, in the Prisoner's Dilemma, people often
cooperate. Are there rules underlying cooperative behavior? In an effort to
answer this question, we propose a new equilibrium concept: perfect cooperative
equilibrium (PCE), and two related variants: max-PCE and cooperative
equilibrium. PCE may help explain players' behavior in games where cooperation
is observed in practice. A player's payoff in a PCE is at least as high as in
any NE. However, a PCE does not always exist. We thus consider {\alpha}-PCE,
where {\alpha} takes into account the degree of cooperation; a PCE is a 0-PCE.
Every game has a Pareto-optimal max-PCE (M-PCE); that is, an {\alpha}-PCE for a
maximum {\alpha}. We show that M-PCE does well at predicting behavior in quite
a few games of interest. We also consider cooperative equilibrium (CE), another
generalization of PCE that takes punishment into account. Interestingly, all
Pareto-optimal M-PCE are CE. We prove that, in 2-player games, a PCE (if it
exists), a M-PCE, and a CE can all be found in polynomial time using bilinear
programming. This is a contrast to Nash equilibrium, which is PPAD complete
even in 2-player games [Chen, Deng, and Teng 2009]. We compare M-PCE to the
coco value [Kalai and Kalai 2009], another solution concept that tries to
capture cooperation, both axiomatically and in terms of an algebraic
characterization, and show that the two are closely related, despite their very
different definitions.
