# Can I refine the set of equilibria in a signaling game to the sender-optimal outcome?

Main question: I've been reading about communication games a lot, and I'm wondering if there are good criteria to select between two separating-ish equilibria. I think of a separating equilibria as coordination equilibria among types. So, if we grant that these types successfully coordinate, why wouldn't we grant that they coordinate to a sender-optimal (in a Pareto efficient among senders sense) equilibrium? That is, suppose there is a single sequential equilibrium where all senders do strictly better than in the remaining equilibria. What arguments are there for selecting this equilibrium?

Consider the following communication game. Receiver payoffs are the second number in the pair. There are six types of senders, with payoffs given as the first element of the pairs. I will show there is a pooling equilibrium and at least two partial separations. I'm wondering what kind of techniques can be used to argue in favor of either separating equilibrium. One is sender-optimal and the other is receiver-optimal.

$$\begin{array}{l*{6}{c}r} & Action \;B & Action \;L & Action\;R& Action\:LL & Action\;RR & \\ \hline type\;B & (0,3) & (1,2) & (1,2) & (2,1)& (2,1) \\ type\;L & (0,2) & (1,3) & (1,2) & (2,0) & (2,2.25) \\ type\;R & (0,2) & (1,2) & (1,3) & (2,2.25) & (2,0) \\ type\;LL & (0,1) & (1,2) & (1,0) & (2,3) & (2,1) \\ type\;RR & (0,1) & (1,0) & (1,2) & (2,1) & (2,3) \\ type\;H & (0,0) & (1,0.9) & (1,0.9) & (2,3.1) & (2,3.1) \\ \end{array}$$

Let their be a prior distribution on types $\pi$ where $$\pi(B)=.3,\pi(L)=\pi(R)=.2, \pi(LL)=\pi(RR)=.1, \pi(H)=.1.$$

In a pooling equilibrium, the receiver will take action $B$ for expected payoff $EU_2(B) = .3(3) + .4(2) + .2(1)=1.9$, edging out $EU_2(L)= .3(2) + .2(3) + .2(2) + .1(2) + .1(.9)=1.89$.

However, there are partially separating equilibria.

Separation 1 Let types $L,LL$ "ask" for action $L$, types $R$ and $RR$ "ask" for $R$ and then $B$ and $H$ mix 50/50 between the two signals. Let the messages be $l$ and $r$ with the natural interpretation.

So $EU_2(L\mid l)\Pr(l)=.15(2) + .2(3) + .1(2) + .025(1)=1.125=EU_2(R\mid r)\Pr(r)$

So the receiver earns $2.25$ in expectation. The senders are better off too.

Separation 2 But let's consider another kind of separation. Types $R$ and $LL$ always send a message $ll$, "asking" for action $LL$. Types $L$ and $RR$ send $rr$, asking for action $RR$. Again, $B$ and $H$ randomize evenly.

Then, $EU_2(RR\mid rr)\Pr(rr)= .15(1) + .2(2.25) + .1(3) + .025(3.1) = .9775=EU_2(LL\mid ll)\Pr(ll).$ The expected payoff is 1.955 because each message is received half the time.

Responding to $rr$ with action $R$ and $ll$ with $L$ yields a lower payoff of, so the separation, being jumbled with types $L$ and $RR$ pooling, isn't useful for taking the "correct" actions $L$ or $R$ as the receiver would like.

It seems to me that this last equilibrium is more robust. There are two separating equilibrium, which require coordination. Granting that senders can coordinate, why wouldn't they coordinate in the sender-optimal way?

I'm wondering if any methods exist that would refine the equilibrium set to exclude the receiver-optimal separation. The first pooling equilibrium might be said not to be neologism proof.

Neologism proofness is defined in section 3 of this paper. Roughly, there must not be an additional (off path) message such, that if observed, the receiver could form beliefs and a rational strategy based on those beliefs such that all who sent the message are strictly better off relative to the proposed equilibrium and those who didn't weakly prefer the proposed equilibrium outcome. I'm guessing that won't work here, because you have to consider two neologisms ($ll$ and $rr$) at once to eliminate separation 1, which requires collusion essentially. But are there any other ideas?

• I'm curious how you calculate the sender's payoff here. It seems that it is the sender's ex ante payoff that you're using to judge optimality. But what's the objective distribution of sender types? Is it the same as the receiver's prior? – Herr K. Dec 2 '14 at 22:16
• Yes, ex ante. The objective is the same as the prior. – Pburg Dec 2 '14 at 22:46
• Are you interested in hearing about focal point arguments, or are you looking for some more "standard" equilibrium refinement? – Martin Van der Linden Dec 11 '14 at 22:18
• Preferably something more standard, but focal points would also be welcome. – Pburg Dec 12 '14 at 4:59
• A trivial answer is that you could just select the Pareto optimal equilibrium. Many papers do this, usually with a phrase like "focus on the sender-optimal equilibrium". A justification is in Mailath, Okuno-Fujiwara and Postlewaite (1993). A more principled approach is to add noise, so that every message is sent by every type with a positive probability. The probability is close to 1 for the intended message and close to 0 for unintended. You can take the error probability to zero and use the limit equilibrium as a refinement. Different error structure => different selected equilibrium. – Sander Heinsalu Jul 4 '17 at 7:52