Talk:Signaling game

formal section corrected

I corrected the formal section on equilibrium which was badly done. (The dependence of the receiver's action on the sender's was not expressed, leading to a wrong formalisation.) Some minor edits too. The section is not as clear as it could be, but at least not wrong now. I added the possibility of mixed strategies (necessary for equilibrium sometimes), and take credit for the joke about mixed messages!

One point: I am not sure the message/action wording is the best: it seems to imply costless signalling (that different messages are equally costly). CSMR 03:46, 21 November 2005 (UTC)

I think the formal section is still wrong for the following reasons:

• Requirement 2 states that the receiver has to take action a^*(m) that maximises \sum_{t_i} \mu(t_i|m)U_R(t_i,m,a). But a^*(m) is defined as a probability distribution over the set of messages A (in my opinion this choice of notation is not felicitious). It does not make sense to say that a probability distribution maximises the above sum. It should at least be switched to the following: using the notation a^*(m)[a_j] standing for the probability for the receiver to play a_j given he has received message m<, For all m, probabilistic strategy a^*(m) for the receiver has to be a best response to m, i.e., has to maximises \sum_{t_i} \sum_{a_j} \mu(t_i|m) a^*(m)[a_j] U_R(t_j,m,a_i). Now this is feasible only if probabilistic strategy a^*(m) assignes non-zero probability to actions a that maximises \sum_{t_i} \mu(t_i|m)U_R(t_i,m,a) which is the formula has it appears now in the section. So I think there is a global confusion between a^*(m) seen has a probabilistic strategy on the one hand, and has a pure best response on the other hand, induced by the choice of notation.

• Notation U_S (t, m,a^*(m)) is undefined in requirement 3. My guess is that it means \sum_{a_i} a^*(m)[a_i] U_S(t, m,a_i). But once again, if m^* stands for a probablistic strategy, then this does not make sense, and it should be said instead that each m in the support of m^*(t) has to maximise the previous sum.

• Last and most important point I think; the Bayes formula exhibited in requirement 4 does not make sense at all. Assuming the undefined notation p(t_j) refers to the prior probability of the sender to be of type t_j, then \sum_{t_i} p(t_i) = 1 and p(t)/\sum_{t_i} p(t_i) = p(t) which do not make sense for \mu(t|m). The priors should be revised coherently with the belief of the receiver towards the sender's strategy, which in equilibrium is assumed to be m^*. Hence application of the Bayes formula should go as follows: For each message m_i if there is a t_j such that m^*(t_j)[m_i] ≠ 0 then \mu(t_k|m_i) = p(t_k)m^*(t_k)[m_i]/\sum_{t_j} p(t_j)m^*(t_j)[m_i] — Preceding unsigned comment added by Akhcourtep (talk • contribs) 16:52, 14 June 2013 (UTC)

I rewrote the formal section on Perfect Bayesian equilibrium to be more general, but with, I hope, more consistent notation and accuracy. I first tried keeping the formality at the level of the previous version, but if that is done accurately, even with just two types, signals, and response actions, it becomes unduly cumbersome for likely readers. Even as is, I wonder whether readers would get a better idea by going straight to the example of a signalling game.

--editeur24 (talk) 01:29, 8 December 2020 (UTC)

Where is Zahavi?

As far as I know, according to Dawkins, A. Zahavi was the first to propose the handicap principle in regards to Birds of Paradise and Thompson's Gazelles. Even if future biologists have expanded upon the theory, Zahavi deserves initial credit for the ideas. (q.v. handicap principle)

If nobody objects, I will update the article accordingly. — MSchmahl… 11:57, 25 December 2007 (UTC)

Yeah, sure, go ahead. Though, in way of some defence of the present state, the article is about signalling games in a fairly strict game theory sense, and I don't think Zahavi's work (as undeniably influential as it is) is as "on point" as Grafen's work. Grafen's work applies game formal game theory, and a signalling game to the problem, along the lines suggested by Zahavi. Cheers, Pete.Hurd (talk) 05:48, 26 December 2007 (UTC)

Spelling

Shouldn't we use either British English OR American English? Signalling vs. signaling etc.? —Preceding unsigned comment added by 132.231.54.1 (talk) 14:58, 10 February 2009 (UTC)

Yeah, the WP convention is to consistently apply whichever spelling was discernably used first in the history of the article. Judging from the title, I'd guess that it is US spelling that should be used throughout this article. Pete.Hurd (talk) 19:12, 10 February 2009 (UTC)

Diagram explanation

I'd love to see some explanation for the extensive form diagram, either in the caption or the main text, or both. What are X, Y, N and q? There are no S and R in the diagram, though we'd expect that from the main text. How do we read these diagrams? Sender says something, receiver reacts, from left to right... or something? Also, how do the various symbols and formalism in the main text relate to the diagram (actions, messages, types)? What are the horizontal lines, solid vs. dotted? I have been googling around trying to figure out how these diagrams work, to no avail. Someone help! Jyoshimi (talk) 16:45, 30 September 2015 (UTC)

Hi Jyoshimi, after wondering the same, I updated the description on Commons with an explanation of the image. I think that should suffice in this case as the details of the graph are not important to the article. 31.187.2.43 (talk) 22:48, 17 December 2023 (UTC)

A belated thanks for this! Jyoshimi (talk) 18:23, 12 April 2024 (UTC)

I rewrote the introduction--Dec. 7, 2020

I rewrote the introduction to add more qualitative features of signalling. I removed the notation, which isn't needed in an introduction. Also, signalling games are not restricted to two players, discrete types, no signal by the receiver, etc., so I rephrased the description as being an example of a simple signalling game. I fixed the spelling of Rubinstein and made other stylistic changes. --editeur24 (talk) 22:47, 7 December 2020 (UTC)

Why different second payoff summands for the sender if the receiver stays ?

In section Signaling game#Reputation game the payoff matrix entry is P1+P1 for the preying sane sender and D1+D1, both when the receiver stays. But it seems P1 and D1 only refer to the signal costs for the sender -which are a priori not related to the payoff summand depending on the receiver's action. Then why is there a particular M1 summand in case of monopoly -thus fixed regardless of the sender's signal- and not some fixed payoff, say C1, in case the receiver of competition -that is the receiver staying ? In other words the payoff for the sender should be the sum "signal cost [a negative number]"+"state payoff only depending on the receiver choice [positive]", which would imply P1=D1 -which is not reasonable.

I would therefore see (P1+C1, D2) in the upper left entry and (D1+C1, D2) in the middle left entry -modulo choices of notation. Please anyone correct me -i will also try to check literature for usual such game models later. Thank you. Plm203 (talk) 13:53, 24 November 2023 (UTC)