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I've looked all over the internet but couldn't find a simple answer. My claim is that if none of the players utility functions depend on the messages being sent, then the game is a cheap talk. Consider this :

Suppose that after you graduate you have two options when applying for a position at a firm: You can either use your undergraduate diploma or pursue graduate studies for two years and receive an MA diploma and send it as part of your application. Your skill level is either High or Low. You learn your skill level but the employer does not know your skill level. The employer only knows that your it is High with probability 1/4. After you learn your type you either apply for the job with your undergraduate diploma or get an MA degree and apply with master’s diploma. Thus you have two possible messages. You already have a BA degree, so its cost is zero. The cost of MA degree is 2 if you are High type and 5 if you are Low type. The employer, observing the message you send but not your type, decides what kind of job to offer you. There are two possible jobs: job1 pays 10 and job2 pays 6. The net payoff of the employer is 10 if you are High type and employed at job1, 5 if you are High type and employed at job2, 0 if you are Low type and employed at job1, and 3 if you are Low type and employed at job2. Is this game a cheap talk game?

We can clearly see that for the employee the messages (whether choosing a MA or BA degree) will have an effect on her payoffs. Therefore I conclude that the game is not cheap talk. But I'm not sure since this kind of games are characterised under cheap talk games, what makes a game cheap talk? How I can explain whether the game above is a cheap talk or not

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Your general question is in my opinion far more interesting than the course problem, so I will answer that.

What conditions characterise “cheap-talk” games?

  1. As you have noted yourself, the talk needs to be cheap, meaning signals/messages should have no direct effect on the players' utility. Indirect effects are allowed. Even in cheap talk games it is possible that the beliefs of a player are updated by a cheap message, hence his response changes altering the outcome. Simple examples include coordination problems.

  2. A second necessary condition is that all messages are available to all types. hence there are no type specific message spaces $M_i$, only a shared message space $M$ from which all types of the sender can freely choose their message. Without this, a signal, though costless, could carry information, as certain types are incapable of sending it.

It's a cheap-talk game, so what?

If these two conditions are met then the game is a cheap-talk game. Why is that important?

Let us call a version of the cheap-talk game where we leave out the signal round the simple variant. For any equilibrium of the simple variant there is a payoff equivalent equilibrium of the cheap-talk game. Similarly, for any equilibrium of the cheap-talk game there is a payoff equivalent equilibrium of the simple variant. Hence it seems that the ability to signal did not fundamentally alter the set of equilibrium outcomes. The reason for this is that the signal did not carry any information, as it was easily imitable by other types, it was 'cheap'.

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  • $\begingroup$ How can I make a distinction between direct/indirect effect of messages on the outcomes? For instance I believe in the game I gave as an example would have a direct effect. $\endgroup$ – Xenidia May 21 '17 at 10:16
  • $\begingroup$ Not sure what you mean by "effect of messages on the outcomes". If message $m$ has a direct effect on a player's utility then the utility function has $m$ among its variables. $\endgroup$ – Giskard May 21 '17 at 10:25
  • $\begingroup$ I see, an approach towards my question would have made everything clear. $\endgroup$ – Xenidia May 21 '17 at 14:07
  • $\begingroup$ @Xenidia I am sorry but I do not understand your comment. $\endgroup$ – Giskard May 21 '17 at 15:54

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