• We looked at a simultaneous game of incomplete information

• Now let's look at some common models of sequential games with incomplete information, and some famous applications

• Again, typically assume that Player 2 has private information about their “type,” that Player 1 does not know

• We needed Bayes’ Rule because we will use it to describe how rational players update their beliefs (about others’ type) when presented with new evidence (i.e., the other player makes a move)

• Players share common (prior) belief about likelihood of encountering a player-type (p)

• Informed player with private information (their type) will make a move, uninformed player will update their belief about that player’s type

• Consider our previous game in sequential form

• A signalling game where Colin moves first

• Rowena observes Colin's move (Cooperate or Defect), but does not know which type Colin is (and thus, the consequences of her own move)

• We could again find what conditions lead to valid pooling equilibria and separating equilibria

  • But I'd rather focus on applications of these famous models

• Essentially, Colin makes a move, either Cooperate or Defect

• Rowena’s prior beliefs about Colin’s type (p) get updated based on Colin’s move

• Then Rowena must decide how to respond, given her (updated) posterior belief about Colin’s type

• A possible solution for a sequential game with incomplete (asymmetric) information is known as an assessment which has two parts:

1. A behavioral strategy profile: complete plan of action for each player about how to play at each information set

2. A belief system: probability distribution over nodes in each information set

   • estimate how likely a player is a particular type, given their action

• An assessment is a Perfect Bayesian Nash Equilibrium (PBNE) if it fits both conditions:

1. Sequentially rational — i.e. credible & survives backwards induction
   • at each information set, strategies are optimal given beliefs
2. It is consistent with Bayes’ Rule
   • beliefs are updated by Bayes' Rule wherever possible

• The analog of subgame perfect Nash equilibrium, but with incomplete information

• Used to explore the economic problem of adverse selection: informed players exploit uninformed players

  • Again, one player has private information (their type), the other doesn’t know

• Let Player 1 be the “uninformed player”

• Player 2 be the “informed player” with private information about their type

• Two types of sequential games with adverse selection:

1. Screening games: uninformed player moves first

   • Will try to determine what type the other player is
   • Offer options where informed players can self-select and reveal their type

2. Signaling games: informed player moves first

   • Will try to signal their type to uninformed player
   • But “bad”-types also want to signal they are “good”-types!

• Consider the used car market

• Two types of used cars

  • “Lemon”: low-quality car
  • “Peach”: high-quality car

• Suppose you, the buyer value a peach at H and a lemon at L dollars

• You cannot tell a good car from a bad one, but believe some fraction q of cars are Peaches

• Now let’s search for Perfect Bayesian Nash equilibria (PBNE)

• Solve for conditions under which the following could be PBNE:

  1. Pooling eq. I: Both Types of Dealer Offer
  2. Pooling eq. II: Both Types of Dealer Hold
  3. Separating eq. I: Good: Offer and Bad: Don't
  4. Separating eq. II: Good: Don't and Bad: Hold

• Suppose both types of Dealer offer
• Bayes' Rule implies:

• Suppose both types of Dealer offer
• Bayes' Rule implies:

P(good|offer)=P(offer|good)P(good)P(offer)

• Suppose both types of Dealer offer
• Bayes' Rule implies:

P(good|offer)=P(offer|good)P(good)P(offer)P(good|offer)=P(offer|good)P(good)P(offer|good)P(good)+P(offer|bad)P(bad)

• Suppose both types of Dealer offer
• Bayes' Rule implies:

P(good|offer)=P(offer|good)P(good)P(offer)P(good|offer)=P(offer|good)P(good)P(offer|good)P(good)+P(offer|bad)P(bad)P(good|offer)=1×q1×q+1(1−q)

• Suppose both types of Dealer offer
• Bayes' Rule implies:

P(good|offer)=P(offer|good)P(good)P(offer)P(good|offer)=P(offer|good)P(good)P(offer|good)P(good)+P(offer|bad)P(bad)P(good|offer)=1×q1×q+1(1−q)P(good|offer)=q

• Since both dealers offer, so the probability that an offer is good is q!

• Suppose both types of Dealer offer

• The probability that an offer is good is q

• If you buy a car based on your beliefs, your expected payoff is:

E[Buy]=q(H−p)+(1−q)(L−p)

• Suppose both types of Dealer offer

• The probability that an offer is good is q

• If you buy a car based on your beliefs, your expected payoff is:

E[Buy]=q(H−p)+(1−q)(L−p)

• Sequential rationality for You: Buy if E[Buy]>0

• Suppose both types of Dealer offer

• The probability that an offer is good is q

• If you buy a car based on your beliefs, your expected payoff is:

E[Buy]=q(H−p)+(1−q)(L−p)

• Sequential rationality for You: Buy if E[Buy]>0

• Sequential rationality for Dealer: Offer if p>c

• Suppose both types of Dealer offer

• If p>c and E[Buy]>0, the following is a PBNE:

  • Behavioral strategy: {Buy, (Good: Offer, Bad: Offer)}
  • Belief system: P(good|offer)=q

• Suppose firms believe there is some amount of education y⋆ such that if:
  • a worker has less than y⋆ education, they are low ability (Group I), thus firm offers them wage of 1
  • a worker has at least y⋆ education, they are high ability (Group II), thus firm offers them wage of 2

• Suppose firms believe there is some amount of education y⋆ such that if:

  • a worker has less than y⋆ education, they are low ability (Group I)
  • a worker has at least y⋆ education, they are high ability (Group II)

• Then Group I will get no education y=0 (so long as 2−y⋆<1), and Group II will get y⋆ education (so long as 2−y⋆2>1)

• This signalling equilibrium occurs when 1<y∗<2

• If employers could only observe a worker's education level:

  • Wages only depend on level of education

• To offer a wage, employers must form beliefs about workers’ ability (given their education level)

• Consider the conditions for one separating equilibrium where:

  • high-ability workers acquire (more) education
  • low-ability workers acquire no (or less) education

• Employers offer wages:

  • wH=H
  • wL=L

• Employer beliefs:

  • P(High Ability|Education)=1
  • P(Low Ability|No Education)=1

• High-ability workers: choose to get Education if H−cH>L

• Low-ability workers: choose to get No Education if L>H−cL

• Main condition: cH<H−L<cL

• Higher education contributes nothing to productivity, but is more costly for low ability workers

  • High ability workers incur the cost of high education just to signal they are high ability
  • Low ability workers cannot afford to incur the cost, and don't get high education

• In signalling equilibrium, high ability workers are worse off by cH

• Ordeals only work for people who believe in iudicium Dei

  • Reserved for the most difficult cases with no evidence or witnesses

• What about “skeptics”?

  • Priest had to let a proportion of those undertaking ordeal fail them (even if they were innocent!)
  • Maintains an equilibrium of belief in iudicium Dei

• Known non-believers (or non-Christians) were not presented with Ordeals as an option

• (Credible) signals and reputation both encourage cooperation, but via different mechanisms

• Reputation is an ex post enforcement mechanism for cooperation via a threat

  “If you cheat, then I will sever our existing relationship in the future and tell all my friends not to trust you”

  • Folk theorem requires repeat interaction and high discount rates
  • Limited to your social network, where you can spread the word

• Signaling is an ex ante commitment via a promise

  “I, a stranger, am demonstrating to you before we establish a relationship that I am trustworthy”

• A good (credible) signal:

  • Must be costly, not free
  • But more costly for “bad” type than “good” type

• A bad (non-credible) signal:

  • May be free
  • Or may have same cost for all types