<aside> <img src="/icons/anchor_blue.svg" alt="/icons/anchor_blue.svg" width="40px" /> Signlaing Game

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The Market of Lemons (Akerlof, 1970) [3]

<aside> Sequence of events

1. Nature moves first determining the type of the car seller:
   • with prob $p$ the car is of good quality (worth $G$ to the buyer and $g$ to the seller),
   • with prob $1-p$ the car is a lemon (worth $L$ to the buyer and $l$ to the seller).
   • This information is only known by the worker.
   • Assumption: $G>g, L>l, G>L, g>l$.
2. The seller (he) decides whether to acquire the certificate as a signal.
   • The H (L) seller incurs cost (-).
3. Consumers make purchase decisions. </aside>

Example of Education (Simplified Model) [1]

<aside> Sequence of events

1. Nature moves first determining the type of the worker:

   • with prob $1/3$ the woker has a high (H) productivity,
   • with prob $2/3$ the worker has a low (L) productivity.
   • This information is only known by the worker.

2. The worker (he) decides whether to acquire some education as a signal about his productivity.

   • The H (L) worker incurs cost -4 (-7).

3. The firm (she) decides whether to hire the worker as a manager (M) or a cashier (C).

   • The worker receive a pay 10 (4) as a M (C).
   • The firm has a prior belief pair $(\mu,\gamma)$ on where she is.

   

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Education is useless here, but it is a pain to go through.

<aside> Separating Equilibrium ($E^H, NE^L$)

• The posterior belief: $\mu=1$, $\gamma=0$.
• After observing $E$, the firm knows the worker is high-type since $\mu=1$, the optimal decision is $M$. Similary, the optimal decision after observing $NE$ is $C’$.
• Let’s go back and check the worker’s decision. For a H-worker, his payoff is 6 (4) when choosing E (NE), thus his optimal decision is E. For a L-worker, his payoff is 3 (4) when choosing E (NE).
• Therefore, this equilibrium holds. </aside>

<aside> Separating Equilibrium ($NE^H, E^L$)

• This equilibrium cannot hold since the L-worker has incentives to deviate to $NE^L$. </aside>

<aside> Pooling Equilibrium ($NE^H, NE^L$)

• The posterior belief is $\gamma=1/3$, and $\mu$ remains the same, i.e., prior belief.
• First, consider the firm’s decision on the equilibrium path. Her payoff is $10/3$ (4) when choosing M’ (C’). The optimal decision is C’.
• Next, check the firm’s decision off the equilibrium path. Her payoff is $10\mu$ (4) when choosing M (C).
• Now, we can check the worker’s optimal decision. There are two cases.
• Case $\mu > 2/5$:
  • For a H-worker, his payoff is 6 (4) when choosing E (NE). Thus, H-worker has incentives to deviate to $E^H$.
  • There is no such a pooling equilibrium in this case.
• Case $\mu < 2/5$:
  • For a H-worker, his payoff is 0 (4) when choosing E (NE). For a L-worker, his payoff is -3 (4) when choosing E (NE).
  • Therefore, this equilibrium holds in this case. </aside>

<aside> Pooling Equilibrium ($E^H, E^L$)

• This equilibrium cannot hold since the H-worker has incentives to deviate to $NE^H$. </aside>

In summary, a pooling equilibrium $(NE^H, NE^L)$ holds if $\mu < 2/5$, and a seperating equilibrium $(E^H, NE^L)$ always exists.

onsumers make purchase decisions.

Example of Education (Spece, 1973) [3]

Example of Crowdfunding [2]

<aside> Sequence of events

1. Nature moves first determining the type of the creator:
   • fixed cost $S_H$ for H-type,
   • $S_L (\leq S_H)$ for L-type.
   • This information is only known by the creator.

2. The creator (she) decides the reward price $p$ and the funding target $C$, i.e., $(C, p)$.

3. Backers decide whether to back the project.
   • Backers have homogenous valuations $v_H$ and $v_L (<v_H)$ for the H-type and L-type products.
   • $1-\alpha$ of backers are informed, they can discern different creators.
   • Uninformed backers hold common prior belief $b’\in(0,1)$ of the prob. that a product is of H-type. Their posterior belief is denoted as $b’’$.
   • The total number of backers $N\sim [0,\bar{N}]$.
   • Assumption: $S_i < \bar{N} v_i$, i.e., success is possible. </aside>

<aside> Case of Full Information

A creator of type $i\in \{H, L\}$ sets target $C=S_i$ and price $p=v_i$.

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The study of case of full information is helpful for us to study the separating equilibrium. Under separating equilibrium, the L-type prefers to reveal her true type, i.e., adopts the optimal strategy under full information. By contrast, the H-type’s optimal strategy must satisfy (IC), i.e., it is no profitable for the L-type to mimic.

<aside> Separating Equilbrium

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