Cheap talk is a special kind of signaling game, i.e., costless signaling.

<aside> <img src="/icons/anchor_blue.svg" alt="/icons/anchor_blue.svg" width="40px" /> Definition of Cheap Talk Games [1]

Sequence of events

1. Nature chooses the sender’s type.
2. The sender (he) learns his type and chooses a message.
3. The receiver (she) observes the sender’s message, modifies her beliefs about the sender’s type, and chooses an action (response).

Sender’s utility function $u_S(a_R, \theta)$

• The payoff depends on his type $\theta$, and on the action (response) of the receiver $a_R$.
• Cheap Talk: The payoff does not depend on his message $m$.

Receiver’s utility function $u_R(a_R, \theta)$

• The payoff depends on the action (response) of the receiver $a_R$, and on the sender’s type $\theta$.
• The payoff does not depend on his message $m$. The message only affects his beliefs </aside>

Example of Defensive Medicine [1]

<aside> <img src="/icons/apple_red.svg" alt="/icons/apple_red.svg" width="40px" /> Sequence of events

1. Nature moves first determining the value of a test to a patient:
   • with prob $1/3$ the test is beneficial,
   • with prob $2/3$ the test is useless.
   • This information is only known by the doctor.
2. The doctor (he) decides to recommend / not recommend the test. His payoff is
   • The patient takes the test that are beneficial: a+5;
   • The patient takes the test that are useless: a-5;
   • The patient does not take the test: 0.
3. The patient (she) chooses whether to undertake the test or not. Her payoff is
   • Take the test that are beneficial: 5;
   • Take the test that are useless: -5;
   • Doesn’t take the test: 0.
   • The prior belief is that if the doctor recommends the test, the test is beneficial with prob. $\mu$; if the doctor does not recommend the test, the test is beneficial with prob. $\gamma$.

</aside>

It is easy to see that the doctor has a bias towards recommendation. The interests of two players coincide iff $a=0$.

<aside> <img src="/icons/apple_red.svg" alt="/icons/apple_red.svg" width="40px" /> Babbling Equilibrium (Pooling Equilibrium)

• The messages from the sender are uninformative. First, consider the equilibrium on the path.

• The doctor recommends the test regardless of its benefits.

  • The patient’s beliefs coincide with the priors, i.e., $\mu=1/3$.

  • The patient’s payoff is lower when taking the test, i.e.,

    $$ \frac{1}{3} 5 + \frac{2}{3}(-5) = -\frac{5}{3} < 0. $$

• Next, consider the equilibrium off the path. Suppose the doctor does not recommend the test. The patient takes the test iff , i.e., $\gamma >1/2$.

  $$ \gamma 5 + (1-\gamma)(-5) = 10\gamma-5 > 0. $$

• Let’s go back to the doctor’s decision.

  • Case 1 $\gamma > 1/2$: Regardless of his nature, he gets 0 ($a+5$) from (not) recommending it. The optimal decision is not recommending the test, thus this pooling strategy profile cannot be sustained as a PBE.
  • Case 2 $\gamma < 1/2$: Regardless of his nature, he gets 0 (0) from (not) recommending it. The doctor is indifferent between two decisions. This pooling strategy profile can be sustained as a PBE.

• There is no information being transmitted from the doctor to the patient, a Pareto improvement can be made if they communicate better.

  

</aside>

<aside> <img src="/icons/apple_red.svg" alt="/icons/apple_red.svg" width="40px" /> Truth-revealing Equilibrium (Separating Equilibrium)

• The test is only recommended when it is beneficial.

• Patient’s belief is $\mu=1$ after observing a recommendation, and $\gamma=0$ after observing no recommendation.

• The patient’s optimal decision is T after observing a recommendation, and NT after observing no recommendation.

• Let’s check the doctor’s optimal decision.

  • When the test is beneficial, he recommend it iff $a+5 > 0$ (which holds since $a>0$).
  • When the test is useless, he does not recommend it iff $0>a-5$, i.e., $a<5$.

  

</aside>

In summary

• A separating equilibrium is supported iff $a<5$, i.e., the difference in the preferences between two players cannot be too large.
• Otherwise, only pooling equilibria are supported.

Example of Supply Yield Risk [2]

References

1. Felix, M.-G. (2022). EconS 424, Strategy and Game Theory. School of Economic Sciences, Washington State University. https://felixmunozgarcia.com/econs-424/
2. Lu, T. (2024). Can a Supplier’s Yield Risk Be Truthfully Communicated via Cheap Talk? Manufacturing & Service Operations Management. https://doi.org/10.1287/msom.2023.0089