Structures of Possible Worlds in a Game of Players with Internal Models

Takashi Ikegami and Makoto Taiji

We propose a new way of studying social dilemma and epistemic structures of agents. Usually games are conducted by giving fixed strategy from without. Here we study a game between cognitive players, where each player predicts the other players future moves by generating its internal model. Dynamical recognizer(DR) is used here to generate the internal models and we simulate the iterated prisoner's dilemma game for an example.

The DR constructs internal models from the past sequence of moves. An interesting point of using DR is that the internal models are expressed as geometrical shapes in a state space of the DR. If a player decides his moves by a finite automaton, a player A with DR successfully reveals it and the internal model is expressed by a simple clumps of points in its state space. However, when a player A tries to make a model of a player B who also tries to make a model of the player A, the situation becomes much complicated. In this case, each DR express its opponent with strange geometrical patterns, which are temporally changing. Time courses of those geometrical patterns will give a new chaotic dynamics with varying its dimensionality.

In this conference, we particularly pay attention to uncertainty in the DR method. Since the internal model of players are constructed from a finite set of past moves, there generally exits a huge amount of degeneracy(i.e. many models show equal performance in predicting its opponent) in the model space. Hence without having any criterion, we cannot decide which model is most plausible to choose. Instead of choosing one of the model arbitrarily, we let the world line of players bifurcate whenever the future prediction differs due to the model choice. Depending on the game situation (e.g. payoff structures, length of past sequences to concern, uncertainty level in choosing models,..), we report how a bundle of world lines (i.e. possible worlds) change its structures. Also we inversely characterize the expected epistemic structure of players by the structures of possible worlds.