Ambiguity Preference and Context Learning in Uncertain Signaling

Abstract

Lexical ambiguity is present in many natural languages, but ambiguous words and phrases do not seem to be advantageous. Therefore, the presence of ambiguous words in natural language warrants explanation. We justify the existence of ambiguity from the perspective of the context dependence. The main contribution of the paper is that we constructed a context learning process such that the interlocutors can infer opponent’s private belief from the conversation. A sufficient condition is proved to show if the learning can be successful. Furthermore, we investigate when the learning fails, how the interlocutors choose among degrees of ambiguous expressions through an adaptive learning.

Appendix

Theorem 1

Given the SPB game, if any two of receiver’s possible private belief partitions from the sender’s point of view are distinguishable, then the receiver’s private belief is learnable.

Proof:

This theorem can be proved from the players’ reasoning process on inferring opponent’s private belief by playing the SPB game repeatedly. The algorithm of this learning can be described as follows. For convenience, we eliminate the subscribe indicating the players in B in the proof.

• Step 1 Since T is finite, we can list all the possible private belief partitions as a sequence $B: B^1, \dots , B^m$;

• Step 2 Calculate all the expected payoffs yielded by each $B^j, j \in \{1,2, \dots , m\}$ for each state $i, i\in \{1,2, \dots , n\}$;

• Step 3 Pick the first two partitions in the sequence B, $B^1$ and $B^2$, since any belief partition are distinguishable, then there exists a state k such that $u(k, A_k | s_k \cap B^{1k}) \ne u(k, A_k | s_k\cap B^{2k})$. Therefore, once state k happens (the occurrence of state k can be guaranteed because players are playing this game repeatedly and every state is possible to occur.), by comparing the true payoff with the payoffs given by $B^1$ and $B^2$, There are two situations:

  • One of the beliefs yields the true payoff, then the sender just return the correct partition back to the sequence B.

  • Neither belief yields the true payoff, then both beliefs should be eliminated.

• Step 4 Update the sequence B, then repeat from step 1.

Since for any two private belief partitions, they are distinguishable, and at least one of them is wrong. Hence, the list B can be eliminated to only one element in finite steps. The remaining belief partition is receiver’s true private belief. Therefore, receiver’s private belief is learnable.    $\square$