Agent-Based Models for Higher-Order Theory of Mind - Advances in Social Simulation - page 216

The ToM 1 allocator also considers the offers that he believes the competing
allocator to make. However, the ToM 1 allocator does so without considering the
possibility that the competing allocator is trying to predict the offer he is going to
make himself. Instead, the ToM 1 allocator assumes that the competing allocator
is a ToM 0 allocator. Using the procedure outlined in the previous subsection, the
ToM 1 agent determines which offers the competing allocator is likely to make.
The ToM 1 allocator then chooses to offer the trade that he expects will yield
him the highest score. Example 2 illustrates the behaviour of a ToM 1 allocator.
Example 2. Consider the setup illustrated by Figure 1, and suppose that agent
A 2 is a ToM 1 allocator. Initially, agents A 2 and R can each move three steps
towards their own goal location (see Figure 3a). There is a trade that would
allow agent A 2 to reach his goal (see Figure 3b), but responder R would only be
able to move one step towards her goal in this case. Using his first-order theory
of mind, agent A 2 concludes that responder R would not accept this trade.
It is not possible for agent A 2 to offer a trade that will allow him to reach his
goal, and also increase the responder's score. However, the ToM 1 allocator can
compromise by offering either the trade shown in Figure 3c or the one shown in
Figure 3d. Although the ToM 1 agent is indifferent between these outcomes, he
knows that the responder prefers the outcome of Figure 3d. Moreover, he knows
that if agent A 1 makes an offer that allows the responder to move exactly four
tiles closer to her goal, the offer shown in Figure 3c could be rejected by the
responder, while the offer shown in Figure 3d would still be accepted. The ToM 1
allocator A 2 therefore chooses to make the offer as shown in Figure 3d.
3.5 Higher-Order Theory of Mind Allocator
Similar to the first-order theory of mind allocator discussed above, the second-
order theory of mind ( ToM 2 ) allocator forms beliefs about the trades that other
allocators will offer, as well as the likelihood that a responder will accept a
given offer. Note that since the responder does not make use of theory of mind,
allocators do not benefit from considering the beliefs, goals, and intentions the
responder may be attributing to others. As a result, both the ToM 1 allocator
and the ToM 2 allocator believe that the responder will accept the trade that will
yield her the highest score. The difference in performance of ToM 1 agents and
ToM 2 agents is determined only by their ability to compete with other agents.
While the ToM 1 allocator believes that competing allocators offer a trade
that maximizes their personal score, the ToM 2 allocator believes that competing
allocators also take the point of view of the responder into account. That is,
the ToM 2 allocator believes that competing allocators know that the goal of
a responder is to approach her goal location as closely as possible. The ToM 2
allocator also believes that competing allocators try to predict the trade he is
going to offer himself, and takes this into account when making his offer.
For increasingly higher orders of theory of mind, theory of mind allocators con-
tinue this pattern of forming increasingly deeper nested beliefs, and assuming
that other agents are more sophisticated. In this paper, we restrict our investi-
gation to ToM i agents for i =0 , 1 , 2 , 3 , 4.