  The success of modern civilization is built upon widespread cooperation in
human society, deciphering the mechanisms behind has being a major goal for
centuries. However, a crucial fact is missing in most prior studies that games
in the real world are typically played simultaneously and interactively rather
than separately as assumed. Here we introduce the idea of interacting games
that different games coevolve and influence each other's decision-making. We
show that as the game-game interaction becomes important, the cooperation phase
transition dramatically improves, a fairly high level of cooperation is reached
for all involved games when interaction goes to be extremely strong. A
mean-field theory indicates that a new mechanism -- the dynamical reciprocity,
as a counterpart to the well-known network reciprocity, is at work to foster
cooperation, which is confirmed by the detailed analysis. This revealed
reciprocity is robust against variations in the game type, the population
structure, and the updating rules etc, and more games generally yield a higher
level of cooperation. Our findings point out the great potential towards high
cooperation for many issues occurred in the real world, and may also provide
clues to resolve those imminent crises like the climate change and trade wars.
