Algorithmic Cheap Talk

TL;DR

This paper initiates the algorithmic study of cheap talk in a finite environment by showing that it is NP-hard to approximate the sender's expected utility value at the sender-optimal cheap talk equilibrium up to a certain multiplicative or additive constant, and deducing that approximating the welfare-maximizing cheap talk equilibrium up to a certain additive constant or multiplicative factor is also NP-hard.

Abstract

The literature on strategic communication originated with the influential cheap talk model, which precedes the Bayesian persuasion model by three decades. This model describes an interaction between two agents: sender and receiver. The sender knows some state of the world which the receiver does not know, and tries to influence the receiver's action by communicating a cheap talk message to the receiver. This paper initiates the systematic algorithmic study of cheap talk in a finite environment (i.e., a finite number of states and receiver's possible actions). We first prove that approximating the sender-optimal or the welfare-maximizing cheap talk equilibrium up to a certain additive constant or multiplicative factor is NP-hard. We further prove that deciding whether there exists an equilibrium in which the receiver gets utility higher than the trivial utility he can guarantee is NP-hard. Fortunately, we identify two naturally-restricted cases that admit efficient algorithms for finding a sender-optimal equilibrium - a constant number of states or a receiver having only two actions.

Figures (5)

• Figure 1: Figure (a) demonstrates the receiver's utility as a function of the posterior probability $p_{\sigma}(*){\omega_1}$ (the horizontal axis). Dashed lines capture the utilities of all actions; solid lines capture the utility of the receiver's best action. Figure (b) depicts the sender's indirect utility (i.e., her utility as a function of the receiver's posterior $p_{\sigma}(*){\omega_1}$). Again, dashed lines capture the utilities of all actions, while solid lines capture the sender's utility under a receiver's best response action.
• Figure 2: The states, variable pools and clause pools of the $3$CNF formula with the variables $x_1,x_2,x_3,x_4$ and the clauses $x_1 \lor x_2 \lor \neg x_3$ and $\neg x_1 \lor x_2 \lor x_4$. The states are depicted as black rectangles; the variable pools are blue rectangles; the clause pools are lines with dots, with a dot indicating that the state belongs to the pool.
• Figure 3: Geometric Illustration for the Proof of Proposition \ref{['pro:rec-ut']}.
• Figure 4: Illustration of a possible re-pooling in Lemma \ref{['lem:vcont']} proof for $n=4$, $m=2$, with the clauses being $c_1 = x_1 \lor x_2 \lor \neg x_3$ and $c_2 = \neg x_1 \lor x_2 \lor x_4$. The black rectangles represent states; the blue rectangles -- attractive variable pools (note that $P_v(1)$ and $P_v(\neg 1)$ are contradicting, and so are $P_v(2)$ and $P_v(\neg 2)$); the black lines -- clause pools (with dots marking the states in these pools); and the red rectangles -- singleton pools. The modification breaks all contradicting attractive variable pools and creates new $2$ clause and $4$ singleton pools. The clause pools correspond to the assignment $(*){x_1,x_2,x_3,x_4}=(\texttt{True},\texttt{False},\texttt{True},\texttt{True})$ that satisfies both clauses. The signaling policy in states $\neg x_{3,1}$ and $\neg x_{4,2}$ remains unchanged during the modification.
• Figure 5: Illustration of re-pooling in Lemma \ref{['lem:ccont']} proof for $n=4$, $m=2$, demonstrated on the clause $x_1 \lor x_2 \lor x_3$ with the index $j=1$. The black rectangles represent states, the blue rectangles -- attractive variable pools, the black lines -- clause pools (with dots in the states belonging to them) and the red rectangle -- a singleton pool. Under the transformation, the attractive variable pool $P_v(*){1}$ is removed, and a new clause pool $\{*\}{c_1 , x_{1,1}, \neg x_{2,1}, \neg x_{3,1}}$ is created, together with a new singleton pool $\{*\}{x_{1,2}}$. The signaling policy in states $\neg x_{1,1}, \neg x_{1,2}, \neg x_{2,2}, x_{4,2},\neg x_{4,2}$ and $c_2$ remains unchanged during the modification.

Theorems & Definitions (24)

• Remark 2.1
• Definition 2.2: Cheap talk equilibrium
• Proposition 2.4
• proof
• Corollary 2.5: NP Membership of Cheap Talk
• Theorem 3.1
• Corollary 3.2
• Proposition 3.3
• proof : Theorem \ref{['thm:additive']} proof overview
• Definition 3.4