Predictive Theory of Mind Models Based on Public Announcement Logic

Abstract

Epistemic logic can be used to reason about statements such as ‘I know that you know that I know that $\varphi$’. In this logic, and its extensions, it is commonly assumed that agents can reason about epistemic statements of arbitrary nesting depth. In contrast, empirical findings on Theory of Mind, the ability to (recursively) reason about mental states of others, show that human recursive reasoning capability has an upper bound.

In the present paper we work towards resolving this disparity by proposing some elements of a logic of bounded Theory of Mind, built on Public Announcement Logic. Using this logic, and a statistical method called Random-Effects Bayesian Model Selection, we estimate the distribution of Theory of Mind levels in the participant population of a previous behavioral experiment. Despite not modeling stochastic behavior, we find that approximately three-quarters of participants’ decisions can be described using Theory of Mind. In contrast to previous empirical research, our models estimate the majority of participants to be second-order Theory of Mind users.

Appendices

Appendix A

This appendix describes how to extend our work beyond Aces and Eights.

In [22], concatenation of sequences is defined: $e \circ e' = (i_1,\dotsc ,i_m,j_1,\dotsc ,j_k)$ for $e = (i_1,\dotsc ,i_m)$, $e' = (j_1,\dotsc ,j_k)$. The empty sequence is $\epsilon$, and $e \circ \epsilon = \epsilon \circ e = e$.

The epistemic depth $\delta (F)$ of a formula F is inductively defined as follows:

D0::

$\delta (p) = \{\epsilon \}$ for any $p \in P$;

D1::

$\delta (\lnot F) = \delta (F)$;

D2::

$\delta (F \rightarrow G) = \delta (F) \cup \delta (G)$;

D3::

$\delta (\wedge \varPhi ) = \delta (\vee \varPhi ) = \cup _{F\in \varPhi }\delta (F)$;

D4::

$\delta (K_i(F)) = \{(i)\circ e : e \in \delta (F)\}$.

D5::

$\delta ([F]G) = \{f \circ e : e \in \delta (F), f \in \delta (G)\}$

We added D5, which is not present in [22]. Moving to novel work, we define the ToM structure $\mathcal {T}_{(p,l)}$, with $p \in A$ and $l \in \mathbb {N}$ inductively as follows:

Base Case::

$e \in \mathcal {T}_{(p,l)}$ for every $e = (i_1, \dotsc , i_m)$ where $0 \le m \le l$, and for every $i_j \in e$ we have that $i_j \in A$ and [if $0 < j < m$, then $i_j \ne i_{j+1}$]. If $m \le 0$ then $e = \epsilon$.

Inductive Step 1::

If $e \in \mathcal {T}_{(p,l)}$ and $l \ge 0$, then $(p) \circ e \in \mathcal {T}_{(p,l)}$

Inductive Step 2::

If, for any $e_1, i, e_2$; $e_1 \circ ((i) \circ e_2) \in \mathcal {T}_{(p,l)}$, then

$(e_1 \circ (i)) \circ ((i) \circ e_2) \in \mathcal {T}_{(p,l)}$

Our base case corresponds to our requirement that the number of ‘perspective switches’ is limited by an agent’s ToM order. Inductive steps 1 and 2 correspond to not switching perspectives, not requiring additional ToM.

For zero or more repetitions of i we write $i^*$. As an example, consider $A = \{0, 1\}$. Then, $\mathcal {T}_{(0,2)} = \{\epsilon , (0^*), (1^*), (0^*,1^*), (1^*,0^*), (0^*,1^*,0^*)\}$.

We then modify our semantic definition of $[\varphi ]\psi$ in Definition 3:

$$\begin{aligned} \begin{array}{ccc}M, (s, (i,l)) \models [\varphi ]\psi & \quad \Leftrightarrow \quad & M, (s, (i,l)) \models \varphi \text { implies }M|\varphi , (s, (i,l)) \models \psi \end{array} \end{aligned}$$

where we define the model restriction $M|\varphi = (S, R, V, T')$ with $(i,l)\in T'(s)\text { iff}$ $(i,l)\in T(s){\ \textit{and}\ }[ M, (s, (i,l))\models \varphi$ or $[\delta (\varphi ) \not \subseteq \mathcal {T}_{(i,l)}] ]$.

Note that $\delta (\varphi ) \not \subseteq \mathcal {T}_{(i,0)}$ is equivalent to “$\varphi$ contains an operator $K_j$ with $i \ne j$”, as $\mathcal {T}_{(i,0)} = \{\epsilon ,(i^*)\}$. With this substitution, our proofs for Theorems 1–3 hold, and our models can be used with any announcements.

Appendix B

Table 1. Tuples at each relevant state during a series of announcements.

There are two games where non-stochastic EL-2 answers correctly whereas our ToM-2 models answer incorrectly.⁶ In both of these, the participant is player 0. The distribution of cards in these games is AA8A88 and 8A8AAA. For the former, we show the removal of tuples after each announcement in Table 1, where each column is a relevant state, and each row corresponds to an announcement. Column ordering corresponds to the order of states in Fig. 1. The rightmost column shows the next announcement, where the index denotes the player, k is ‘I know my cards’, and $k\lnot$ is ‘I do not know my cards’. Tuples that will be removed after the next announcement are . After six announcements, player 0 at ToM-2 will incorrectly answer ‘I know my cards’, whereas at ToM-5 she will answer ‘I do not know my cards’, which is the correct answer. When working through the example, it is recommended to use Fig. 1 as a companion.