Agent-Based Models for Higher-Order Theory of Mind - Advances in Social Simulation - page 214

Once the responder has made a choice, a Colored Trails game ends. We do
not consider learning across games, which may allow a responder to influence the
behaviour of an allocator in future games. This means that for a responder, there
is no additional benefit of predicting the offers an allocator is likely to make. The
responders described here therefore do not consider the beliefs, desires, and goals
of other agents, and as a result, do not make use of theory of mind.
3.3 Zero-Order Theory of Mind Allocator
A zero-order theory of mind ( ToM 0 ) allocator understands the game, but is un-
able to attribute any mental content such as beliefs, desires, or goals to a respon-
der. That is, although a ToM 0 allocator can determine what chips a responder
would need to reach her goal location, he is unable to consider the possibility
that she wants to reach her goal location. Instead, the zero-order theory of mind
allocator considers the total set of chips that are owned by himself and the re-
sponder. He then determines the subset of chips C that will allow him to move
to a tile as close as possible to his goal location. If there are multiple subsets of
chips that satisfy this condition, the ToM 0 allocator selects one of these subsets
at random. The allocator then offers to trade in such a way that he receives all
the chips in the subset C , while leaving the remaining chips for the responder.
Since the ToM 0 allocator cannot attribute goals or beliefs to other agents,
we assume that he does not make any predictions about the offers made by the
competing allocator. Example 1 shows the behaviour of a ToM 0 allocator in a
game with two allocators and one responder. In this example, agents form beliefs
based on iterated best-response.
Example 1. Consider the setup illustrated by Figure 1. In this situation, the
game is played on a 4 by 4 board with five different colors. There are two alloca-
tors, indicated by A 1 and A 2 to the left of the board, each with their own set of
chips. Allocators are initially placed on the top left tile (tile 1) and aim to get as
close as possible to the bottom right tile (tile 16). There is a single responder R ,
depicted to the right of the board. Unlike the allocators, she is initially placed
at the top right tile (tile 4) and aims to reach the bottom left tile (tile 13).
Suppose that the agent A 1 is a ToM 0 allocator. Allocator A 1 cannot move with
his own chips, but with the combined set of chips of agent A 1 and agent R ,there
are four possible paths for agent A 1 to reach his target, as depicted in Figure
2. The agent randomly selects one of these paths and makes corresponding offer
that would yield him the chips that he needs to reach his goal.
3.4 First-Order Theory of Mind Allocator
The first-order theory of mind ( ToM 1 ) allocator considers that responders and
other allocators have beliefs and goals. While deciding to offer a trade to a
responder, the ToM 1 allocator considers the viewpoint of the responder to de-
termine whether he would accept if he were in her place. Concretely, the ToM 1
allocator does not make any offers that would decrease the score of the responder.