History-dependent equilibrium points in dynamic games

A dynamic system, which is observed (and influenced) by two players simultaneously at discrete points in time, is considered. Both players receive rewards depending on the time path of the system and on the actions taken. The game is played noncooperatively. At an observation point a player can base his decision on the state of the system at that moment (memoryless strategy) or on the entire history of the system (history-dependent strategy). Some examples are given of games in which there exists an equilibrium point in history-dependent strategies, which gives both players a greater reward than any equilibrium point in memoryless strategies does. When the players play according to such a history-dependent equilibrium point, they make implicit agreements and hence cooperate implicitly. These implicit agreements attain stability by the use of threats. Although only some examples are given, it will become clear that the phenomenon is present in a wide class of dynamic games.