Agent-Based Models for Higher-Order Theory of Mind - Advances in Social Simulation - page 217

Example 3. Consider the setup illustrated by Figure 1, and suppose that agent
A 1 is a ToM 2 allocator. Following the process described in Example 2, agent A 1
concludes that agent A 2 is going to make the offer depicted in Figure 3d. This
trade would allow the responder to move five tiles towards her goal location.
The ToM 2 allocator A 1 can choose to match this offer by making the offer
shown in Figure 4a. If responder R were to accept this offer, allocator A 1 can
move four tiles closer to his goal location, increasing his score by 40. However,
allocator A 1 believes that allocator A 2 will make an offer that allows responder R
to move five tiles towards her goal as well. In this case, responder R will randomly
select which offer to accept, which means that there is a 50% probability that
the responder will not accept the offer of allocator A 1 .The ToM 2 allocator A 1
therefore assigns an expected gain of 20 to the offer shown in Figure 4a.
Alternatively, the ToM 2 allocator can make a better offer to responser R by
allowing her to reach her goal location (Figure 4b). Allocator A 1 expects that
responder R will accept this offer, allowing him to move three tiles to his goal
location and increase his score by 30. Since this is the higher expected gain,
ToM 2 allocator A 1 decides to make the offer shown in Figure 4b.
4 Simulation
We performed simulations of single-shot Colored Trails games, designed after
the games in [14]. Games were played on a 4 by 4 board of square tiles. Each
tile on the board was randomly colored with one of five possible colors. Players
were allowed to move horizontally and vertically, but diagonal movements were
not allowed. Each game involved two allocators and one responder. To make
individual game settings more comparable, the responder was always initially
located on the top right tile, while her goal was to reach the bottom left tile.
As a result, the responder has 20 different possible paths to reach her goal,
each using six chips. Both allocators were initially placed on the top left tile,
while their goal location was the bottom right tile. In this setup, the goal of the
responder overlaps partially with the goals of the allocators, but not completely.
At the start of the game, each player received an initial set of six randomly
colored chips. Since each player needs at least six chips to reach his or her goal
location, it is sometimes possible that after a trade, both the allocator and the
responder can reach their respective goals. However, this is not always the case.
To ensure that each allocator has an incentive to negotiate to increase his score,
game settings in which some player can reach his or her goal with the initially
assigned set of chips without trading were excluded from analysis.
To determine the effectiveness of theory of mind, we generated 10,000 random
game settings. In each of these settings, we determined the score of a focal ToM i
allocator in the presence of a competing ToM j allocator, for each combination
of i,j =0 , 1 , 2 , 3 , 4. The average score was measured for the same 10,000 game
settings in each condition. This was both done for agents that base beliefs on
iterated best-response, as well as for agents that hold utility-proportional beliefs.