9. Signalling Games

What is a signalling game?

• We introduce a subclass of dynamic game with incomplete information

• There are two players, sender and receiver, and the sender has private information the receiver doesn’t know

• The sender sends a message m, and the receiver decides an action based on this signal

• The message is a function which maps types into actions, m : Θ → A, whereas an action is a function which maps messages into actions, a : M → A

  • The number of messages is greater than the number of types

What are the two types of equilibria that signalling games are concerned with?

• Separating equilibria and pooling equilibria

What is a separating equilibria?

• A PBE (m*, a*, µ) is a separating equilibrium when when m(θ) ̸= m(θ′) for each θ, θ′ ∈ Θ

• In this game, the receiver is perfectly able to infer the type from the message that they send because all types send different messages

• Different types choose different functions

What is a pooling equilibria?

• A PBE (m*, a*, µ) is a pooling equilibrium when m(θ) = m(θ′), for each θ, θ′ ∈ Θ

• The receiver can’t infer what type the other player is because different types send the same message

• If a receiver can’t form a system of beliefs, they are unable to best respond, which is difficult because off-path beliefs are hard to define

How does the sender play in the Beer-Quiche Game

• Person 1 has two strategies, (Beer if strong, quiche if weak) and (Quiche if strong, and beer if weak)

• The best responses of the receiver will depend on beliefs attached to the information sets hT2 and hT1

• Let µ_Q = P(t_s = S|m = Q) and µ_B = P(t_s = S|m = B)

• If sender chooses (Beer, Quiche), then it must be that µ_Q = 0 and µ_B = 1

• If sender chooses (Quiche, Beer), then it must be that µ_Q = 1 and µ_B = 0

• µ_Q(Quiche,Beer) = P(S and Q) / (P(S)P(Q|S) + P(W )P(Q|W ) = 1

How do Pooling Equilibria games work?

• Every type picks the same option (Beer)

• P(Observing Beer) = 1, but we can just condition the probability on nature; if 0.6 are strong and 0.4 are weak, then P(B given strong) = P(Strong) = 0.6; nature becomes the basis for beliefs

What is the extra all-important step to the Beer-Quiche game?

• Because off-path events never occur, one can choose a PBE with beliefs in the off-path nodes which are completely arbitrary and don’t make much sense

• A receiver might respond to the choice beer (which occurs off-path) by fighting, because it assumes that the selection of Beer must come from a weak type because µ_B < ½ , but this is only , ½ because it never occurs in the scenario where the sender chooses (Quiche, Quiche)

• We see that a weak type has no incentive to deviate in this equilibrium, but the strong type does; if the strong type deviated, the receiver would know they are strong because weak will never deviate, and thus they could gain

What is the Cho-Kreps Intuitive Criterion

• A criterion designed to refine PBE by restricting off-path beliefs

• A perfect Bayesian equilibrium σ* fails the intuitive criterion if there exists a (deviating) action a₁ and type θ

• (i) π₁(σ*; θ′) > max_{µ,a2∈BR(µ,a1)} π₁(a₁, a₂) for all θ′ ̸= θ

• (ii) π₁(σ*; θ) < min_{a2∈BR(θ,a1)} π₁(a₁, a₂)

• The best response is the maximiser of expected utility

What is Spence’s education game?

• Education can be seen as a signal from workers to potential employers

• Employers can decide to use this signal to determine wages

• If the cost of education is correlated to potential productivity, then potential separating equilibria might arise

Modelling Spence’s student game

• Some students take EC220

• The population of students is divided in two groups: Exceptional and Decent students

• There are 60% of Exceptional students in the cohort for whom the cost of achieving a grade g in EC220 is g_E/2 , while for decent students is 2g_D

• The cost of effort to not take EC220 is normalised to 0 (as if g = 0)

• In the future, the productivity of exceptional students is 2 and the productivity of decent students is 1

  • Exceptional students have a lower cost of effort because they are better

Wage-setting in the student game

• Assume all Exceptional students achieve a grade g_E and the decent students achieve g_D

• Crucially, employers cannot observe productivity but only the student’s grade in EC220

• If g_D ̸= g_E , a firm will pay w_E = 2 and w_D = 1, their productivity. If g_D = g_E , the firm cannot distinguish and will only pay the average productivity: w¯ = 0.6 · 2 + 0.4 · 1 = 1.6

What do students do in separating equilibrium in the education game?

• Decent students, given that their wage will be one and the cost to obtain the module is 2 should not take the module, because they receive a payoff of -1 by doing so rather than 0; aim for g_D = 0

• Exceptional students need to exert some cost such that their grade g_E > 0

• The cost is sufficiently high that decent students don’t want to incur the cost required to get a higher wage, and the cost should be low enough that the exceptional students aren’t incentivised to mimic the decent students

What do students do in pooling equilibrium in the education game?

• The employer plays an average wage on path because all students have the same signal

• If g ̸= g¯, assume that the deviator is always a Decent student: offer low wage w = 1

• If g > g¯, assume the deviator is an Exceptional student: offer w = 2