How to denote this in game theoretic notation

Question

I'm writing a paper that demonstrates that linguists can use the concepts in game theory to infer what interlocutors naturally infer when the literal meaning of their utterances doesn't ostensibly express the conveyed meaning.

I use the following example to demonstrate that linguists can use game theory to work out the inferences I mentioned in the previous paragraph.

...Grice’s famous reference letter example, wherein a professor writes a testimonial for one of his former students who plans to submit it as a part of his application for a mathematics job. The professor’s letter reads,

Dear Sir,

Mr.Brown's command of English is excellent; his attendance at tutorials has been regular.

Yours, etc.

Manifestly, the professor intends to discourage the student's prospective employer from hiring the student, despite commending him in his letter.

I used the following reasoning to deduce the meaning the professor intended to convey.

Let us assume, that the communicators have common knowledge of the facts that, for the most part,

1. people prefer not to say bad things about other people;
2. people more often believe what another person states than they disbelieve what another person states; and
3. people desire not to write letters that do not effect the state of affairs that they desire.

Thus,

• Let w denote the utility of writing letters, regardless of their consequences

• Let d denote the utility of saying bad things about other people

• Let x denote the probability the addressee believes what the addressor communicates Such that, d < w < n < 0, and that x > 0, x < 1, x > y

Consider

• That an intentionally vapid message that succeeds has a lower utility than no action, so the sender could not have intended to communicate a vapid message.

• If the sender wished to convey a good message, he would have experienced more utility by sending a good message than by sending a vapid one; thus, he could not have sent a vapid message to convey a good message.

• The sender did not communicate a good message, a bad message, or no message at all, so we can eliminate all the remaining cells in the columns labelled persuade, dissuade, and do nothing.

• Only the cell dissuade-vapid remains.

So using a few elements of common knowledge we have deduced that given that the sender sent a vapid message, the only rational explanation for his action is that he desired to convey his unfavourable opinion of the applicant.

I'd like to illustrate that we could more precisely represent the above reasoning with game theory's formal tools than with ordinary language. However, I'm not familiar with its system of notation.

How would game theorists notate that reasoning?

(I'll cite this stack and the answering user).

Answer 1

Suggestion: Some formalisms/ideas that are based on your treatment and that you can consider as additions. You can also check the References, where you can find more mathematical details. I choosed not to introduce too many variables and mathematical notation to prove your case (payoff matrices for example), since your conclusions follow from logical arguments, which can be however expressed in a more game-theoretic context (as you wanted), as I propose below.

The above situation can be formalized in the framework of a signaling game with two players, the sender (S) and the receiver (R) (see Wikipedia for the formalisms).

• Sender: Professor.
  Set of possible messages $M=\{\text{positive, negative, vapid}\}$
  Types $T=\{\text{bad student, good student}\}$
• Receiver: Employer
  Actions $A=\{\text{hire, not hire}\}$

The game is played as follows

1. The Professor observes his type (i.e. whether the student is good or bad) and sends a message to the prospective employer,
2. The prospective Employer observes the message of the Professor and takes an action out of the set of his possible actions (hire, no hire) that maximizes his expected payoff given the professor’s message.

As in general signaling games, the main question is whether in the above situation the professor has an incentive to signal an honest message. In this specific case, our intention is to prove that sending a vapid message is a dominant strategy in the case that the student is bad, or in other words that the professor would never send a negative message to convey that the student is bad.

In this game the Professor and the Employer both have coinciding interests. That is, they both want that the student will be hired in the case that he is good and they both want that the student will not be hired in the case his a bad. This fact simplifies the calculation of the equilibrium of the game.

The Professor and the Employer will receive payoffs dependent on

• professor’s type $t$ (good or bad student),
• message $m$ chosen by the professor (positive, negative, vapid),
• action $a(m)$ chosen by the employer (hire, not hire).

Concerning the payoffs and the strategies of the players, following considerations apply

1. Assume that Professors systematically give false information about students in their recommendation letters. That would lead the prospective Employers to gradually ignore these letters and base their decision completely upon their personal impression. Therefore, we have a good reason to assume that Professor’s do not give false information.

2. The Professor incurs a very high cost if he sends a negative letter. Although a negative letter conveys unequivocally the message that the student is bad and he should not be hired and thus increases Professor’s gains, these gains are offset by the fact that he has to say negative things about another person (the student), probably hurt her feelings or be accused for not judging him fairly. This assumption is crucial, since it is the reason we will subsequently conclude that the Professor never chooses to send this message. But, how will he convey that the student is bad, if he never sends a negative message?

3. The Employer can follow the above reasoning, and thus ascribes zero probability to the event that the Professor will send a negative letter, even in the case that the student is bad! More specifically, the employers beliefs concerning Professor’s type can be modelled as follows: $$\begin{matrix} & Bad & Good \\ Positive & 0 & 1 \\ Negative & 1 & 0 \\ Vapid & x & y \end{matrix}$$ where $x>y$. (Actually, in this step we implicitly assume that there is a tacit agreement between Professor’s and prospective employers, that the Professor will send a vapid message if the student is bad. Since, this is exactly what we want to prove, this point can be omitted, or can be slightly altered to fit as conclusion.)

4. Based on the above beliefs, the actions of the Employer are uniquely determined. He should hire the student whenever he receives a positive message, not hire the student when he receives a negative message and follow a mixed strategy when he receives a vapid message.

5. Inferring the Employer’s strategy, the Professor maximizes his own payoff, by sending following messages:

   • Positive message whenever the student is good (and never a vapid message in that case),
   • Vapid message whenever the student is bad (since, as argued above, the cost of sending a negative message is very high).

   But, since the prospective employer can infer Professor’s strategy he can actually be certain that when he receives a vapid message, that means that the student is bad, that is, his beliefs are updated as follows $$\begin{matrix} & Bad & Good \\ Positive & 0 & 1 \\ Negative & 1 & 0 \\ Vapid & \approx 1 & \approx 0 \end{matrix}$$

The conclusion is the following

• whenever the Professor observes that the student is bad he will send a vapid message and the employer should not hire a student upon receiving a vapid message and
• whenever the Professor observes that the student is good he will send a positive message and the employer should hire a student upon receiving a positive message.

The above strategies constitute a perfect Bayesian equilibrium of the signaling game. Thus – interpreting the equilibrium – we can expect that the behavior of the players will match the above description in most of the cases that this interaction will take place.

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References:

1. Credible Signaling in the Letter of Recommendation
2. Personnel Selection as Signaling Game
3. Signaling games, Wikipedia

Answer 2

Addendum: Wikipedia has a somewhat decent page on signaling games, http://en.wikipedia.org/wiki/Signaling_game and I think the applications to Philosophy are closely related to your work, in particular this book: http://www.amazon.com/Convention-Philosophical-Study-David-Lewis/dp/0631232575/

Probably this is not the answer you want to hear. It is hard to see how you could use game theory here because you assume that probability that the receiver believes the sender's message is exogenous. But in game theory this must be endogenous.

In game theory there is a vary large literature on signalling games (and cheap talk) but we never assume that there is some given probability that a statement is accepted by its literal meaning. To illustrate, consider a model where the applicant is B or G (state of the world, $\omega$) with probability $p$ and $1-p$. The employer wants to recruit the applicant if and only if the applicant is G. The recommender sends a signal $s$, which is chosen from some finite set ($\{P,D,V,N\}$) in your case, where $N$ is no action). The utility of the recommender (the sender) may depends on action of the receiver, the signal and the state of the world, $u_S(\omega,s,a)$ -- if the signal does not affect the utility of the sender, we call this a "cheap talk" model. The utility of the receiver depends only on the state of the world and his action (but not on the signal itself!), $u_R(\omega,a)$.

If the receiver does not care about the signal itself why does the signal may matter? Well, we assume the sender observers the state of the world. So the strategy of the send is a function from the state into a signal $S_s:\{ B, G \} \rightarrow \{P,D,V,N\}$. More generally we could allow for mixed strategies where the sender can randomize when choosing his signal but here to keep things simple, let's focus on pure strategies.

We assume that (in equilibrium) the receiver knows the strategy the sender uses. The receiver than computes the expected payoff of choosing an action using the posterior beliefs generated by the senders' strategy and the realization of the signal.

From example if the sender sends signal V if B and P if G. Then the sender signal is perfectly informative. The receiver knows whether the state is B or G and chooses accordingly.

From example if the sender always sends signal V. Then the sender signal is not informative. The receiver thinks the state is B or G with prob. p and 1-p (posterior beliefs are equal to prior beliefs).

From example if the sender sends signal V 2/3 of the times when the state is B and sends P 1/3 of the times when the state is B and always sends P when the state is G. Then the sender signal is somewhat informative. When the signal is P, the receiver thinks the state is G with prob: p /( p + 1/3(1-p)) and B with prob ((1/3)(1-p)) /( p + 1/3(1-p)).

The receiver then picks the action that maximizes his expected payoff. The sender knowing the best-response of the receiver given the sender's strategy, chooses a strategy to maximize his (the sender) expected payoff.

What is interesting in your example that you are not going to find in cheap talk games is that you assume the sender may have preferences over the signals that are not related to the outcome they induce that's a nice feature.

Anywyay, one can not analyze your model entirely here. I suggest reading more about signalling games, cheap talk, sender receiver games. Joel Watson's book (for undergrads) maybe a good source, Part IV: Information, 29) Job-Market Signaling and Reputation.

Answer 3

Although your treatment is comprehensive, I would like to make some comments about your analysis, that (in my opinion) should be taken under consideration before an effort to formulate your analysis in game theory notation.

1. Your assumptions 1. and 2. can be justified by adopting - very generally speaking - a common prior assumption (cpa), which is very commonly used in economics (and why not in linguistics). The cpa states that all agents, at some time in the past, agreed upon the probabilities of each possible subsequent state of the world. Then the differences in their probability assesments are due to private information. So, it is not absurd to assume that all agents agree that the probability of a professor being truthfull (i.e. that his type is truthfull instead of treacherous) is, say $x$.
2. In your 3rd assumption "people desire not to write letters that do not effect the state of affairs that they desire", the highlighted part is not clear to me.

Now, let's turn to the points you list under "Consider":

• That an intentionally ... communicate a vapid message. In my opinion, one should me more cautious in the comparison between the utility of a "vapid message" and "no action". No action, is not observed by the job-interviewer. So, a vapid message conveys more information - for example "be carefull", or "it is just ok to hire him" - whereas no action is not even observed, since the job-interviewer will never be informed about whether the applicant asked a professor for a recommendation letter and the professor denied it.
• If the sender wished ... to convey a good message. I agree with that, but then in the same logic you should also develop the point that If the sender wished to convey a bad message, he would have experienced more utility by sending a bad message than by sending a vapid one; thus, he could not have sent a vapid message to convey a bad message. That would leave us, with no rational explanation why he sent a vapid message, other than he wanted to convey a vapid message (see previous point above)! I believe that excluding this point, show bias towards the conclusion that a vapid message intends to dissuade, in the analysis.
• Based on the above, my opinion is, that a vapid letter, indeed intends to convey a vapid message! This could not have been conveyed by taking no action, thus I disagree with the first consideration. In this case the vapid message has more utility than no action. However, the job-interviewer is not enthusiastic hiring a person that has a vapid recommendation.

I believe that in your analysis, is latent the fact, that the professor knows that the student will read the recommendation. Thus, you want to make the point that he intentionally uses a moderate language (vapid message) in order not to dissatisfy the student but also to warn the job-interviewer. This fact, is actually true, especially in modern enterpreneurs, where for example the characterization "to our help" instead of "to our best help" means that he did not help at all. So, it is important to mention somewhere in your analysis, that the message should be concealed from the student, so that he is not dissatisfied or sad (if we assume that he is allowed to read and that the professor cares about his feelings).

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Suggestion: You could simplify a little bit the payoffs in the matrix, so that it is clear what everyone takes at any possible strategy profile. Moreover, the strategies of the job-interviewer "hire", "no hire" are not really mentioned in the analysis. That is necessary in order to apply a game-theoretic point of view, as you intend.