Chapter IV: A Career Choice Problem with Information Asymmetry in Ability 70
4.5 Equilibrium
4.5.1 Pure Strategy Equilibrium
First, I prove that there is no symmetric pure strategy equilibrium in this problem.
Proposition 23 There is no symmetric pure strategy equilibrium.
6See the appendix for the derivation.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5
1 1.5 2 2.5 3 3.5 4
θ
E[u(⋅)]
Complete Information Incomplete Information
Figure 4.2: Expected Utilities
Intuitively, there must exist a cut-off such that agents with higher ability than the cut-offpursue the high-paying career with probability one since they are matched with H1 with very high probability. Notice that an agent with the cut-off ability must obtain the same expected utility whatever she does in the equilibrium. Now, assume that there is a non-degenerate interval below the cut-off such that agents with ability in the interval use pure strategy. Denote this interval by [a,b], where a < b. If an agent with ability auses the same (pure) strategy as one with ability b, her expected utility should be strictly less than that of the agent with ability b in equilibrium. However, if she uses the other (pure) strategy7, she can obtain the same expected utility as the agent with ability bsince there is no competitors with ability θ ∈ (a,b) . This is a contradiction. Hence, there is no symmetric pure strategy equilibrium.
4.5.2 Mixed Strategy Equilibrium
In this section, I examine the existence and uniqueness of the symmetric mixed strategy equilibrium.
Proposition 24 Among integrable functionsg(θ), there exists a unique symmetric mixed strategy equilibrium (s∗). This has the following form:
Agents with abilityθ ≥ Dθ pursue the high-paying career with probability one. On the other hand, agents with abilityθ <Dθpursue the high-paying career with proba- bilityg(θ)and the low-paying career with probability1−g(θ), whereg(θ)is given by:
g(θ)= − N(θ)
M(θ)− N(θ), θ ≤ θ <Dθ,
7Note that there are only two pure strategies:H andL.
where
M(θi) = ∂E[u(·)|(si(θi) =H,s∗−i(θ−i))]
∂θi and N(θi) = ∂E[u(·)|(si(θi) =L,s∗−i(θ−i))]
∂θi .
I provide the sketch of the proof of this proposition. Details are in the appendix.
Proof 2 Claim 1 The cut-offabilityDθ is strictly greater thanθ, the lower bound of the type space.
If every agent pursues the high-paying career, low ability agents can increase their expected utility by deviating from the strategy since they will be matched with L1 for sure if they participate in the other labor market. Hence, the cut-offmust be strictly greater thanθ.
For the next claim and the rest of this paper, I define a function D(·)as the difference in the expected utility between pursuing the high-paying career and the low-paying career:
D(θ) = E[u(·)|(si(θ)= H,s∗−i(θ−i))]−E[u(·)|(si(θi)= L,s∗−i(θ−i))].
Also, I introduce two functions:
P(a,b) := Z b
a
f(θ)g(θ)dθ
!
, and Q(a,b) =
Z b a
f(θ)(1−g(θ))dθ
! .
P(a,b) represents the ex-ante probability that an agent pursues the high-paying career with ability θ ∈ (a,b) Similarly, Q(a,b) is the ex-ante probability that an agent attend the labor market of the low-paying career paths with abilityθ ∈ (a,b).
Claim 2 There exists a unique value ofQ(θ,Dθ) ∈ (0,1)such that D(θ) =0.
For the lowest ability agent, the only factor to take into consideration is the prob- ability that other agents participate in the labor market of the low-paying career paths (Q(θ,Dθ)) or the probability that other agents pursue the high-paying career (1− Q(θ,Dθ))since it is not possible (probability is zero) that there is an agent whose ability is less than or equal toθ. Since D(θ)should be zero in equilibrium,Q(θ,Dθ) must have the unique value to guarantee the condition, D(θ) =0.
Claim 3 There exists a uniqueDθ <θ¯such that D(Dθ)= 0and D(θ) =0.
First, note that D( ¯θ) = u(H1) − u(L1) > 0 since she will be matched with the best career path in the labor market she attends. Therefore, the cut-offDθ should be strictly lower than the upper bound,θ. The condition D¯ ( ¯θ) > 0also implies that it is better for a high-ability to pursue the high-paying career since the probability of being matched with the career path H1is very high. However, this chance decreases as the ability gets lower. At the end, for an agent with the cut-offability Dθ, both strategies should provide the same expected utility in equilibrium. That is, D(Dθ) should be zero, and there is a unique Dθ satisfying this condition and D(θ) = 0.
In words, given an environment, Claim 2 tells us that the ex-ante probability that an agent participates in the labor market of the low-paying career paths should be fixed. This fixed probability leads us to Claim 3, which determines the cut-off ability. That is, there is a certain cut-offsuch that an agent with a higher ability than the cut-offuse the pure-strategy (pursue the high-paying career).
Claim 4 For given Dθ∗ and Q(θ,Dθ∗), there is a unique value of Q(θ, θ) satisfying D(θ) =0for eachθin(θ,Dθ).
SinceQ(θ,Dθ) andDθ are fixed,Q(θ, θ)determines the value of D(θ). For a given θ, this claim means that there is a unique probabilityg(θ) sinceQ(θ, θ)only depends ong(θ). Therefore, in equilibrium, an agent with abilityθ ∈[Dθ,θ)¯ pursues the high- paying career with probability one, and an agent with abilityθ ∈[θ,Dθ)pursues the low-paying career with probability1−g(θ) ∈(0,1).
4.5.3 Characterization
I have proved the existence of the equilibrium strategy g(θ) for each agent with ability θ. In this section, I characterize the equilibrium strategy. In particular, I show that there is a discontinuity ofg(θ) atDθ.
Proposition 25 g(θ)has a discontinuity atDθ.That is,lim
θ↑Dθ g(θ) < 1= g(Dθ).
Proof 3 Proof is in the Appendix.
This implies that the two groups, the high-ability group and the low-ability group, are strictly divided. In other words, around the cut-offability, a small difference in ability causes a significant change in agent’s behavior.