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### Principles ofGame Theory

Lecture 13: Bayesian Games

Administrative

- No Class Sunday
- I’ll try to post the next (last) homework assignment soon
- Last homework will be due a week from Saturday (due: Oct 5th)
- Last quiz a week from Sunday
- Quiz on Oct 6th.
- Final exam two weeks from today.

Incomplete information

- We’ve talked about imperfect information.
- What is it?
- Now we’ll consider games of incomplete information
- Scenarios when players may or may not know completely the information about other players: payoffs, strategies, preferences, etc.
- Often we talk about “types” of players
- For example: Type A players have payoffs ZZZ, and Type B players YYY.

Incomplete Information: Example 1

- Let’s return to a prisoners dilemma scenario:
- Player 1 has the standard preferences but Player 2 has either standard preferences or “nice” preferences.
- Player 2 knows her type but Player 1 does not.
- Here: Player 2’s preferences change but Player 1 still has a dominant strategy.

Incomplete Information: Example 2

- New twist: what if P1’s preferences depend on which type he’s playing?
- Player 1: be nice to those who play nice, mean to those who play mean.
- If 2 is selfish then player 1 will want to be selfish and choose D, but if player 2 is nice, player 1’s best response is to play C!

Incomplete as Imperfect

6, 6

C

The previous example can be written down as a game of imperfect information:

“Nice”

Player 2

C

2, 4

D

C

0, 0

1-p

D

Nature

4, 2

D

4, 4

C

C

p

0, 6

D

6, 0

“Normal”

Player 2

C

D

D

2, 2

Equilibria

- So what are the equilibria of the game?
- Player 2 knows his type, and plays his dominant strategy: D if selfish, C if nice.
- Player 1’s choice depends on her expectation concerning the unknown type of player 2.
- If player 2 is selfish, player 1’s best response is to play D.
- If player 2 is nice, player 1’s best response is to play C.
- Suppose player 1 attaches probability p to Player 2 being selfish, so 1-p is the probability that Player 2 is nice.
- Player 1’s expected payoff from C is 0p+6(1-p).
- Player 1’s expected payoff from D is 2p+4(1-p).
- 0p+6(1-p)= 2p+4(1-p), 6-6p=4-2p, 2=4p, p=1/2.
- Player 1’s best response is to play C if p<1/2, D otherwise.
- E.g.: if p=1/3, play C; if p=2/3, play D.

Back to probabilities

- But what is p?
- The probabilities associated with nature’s move are the subjective probabilities of the player facing the uncertainty about the other player’s type.
- Thinking about player types, two stories can be told:
- The identity of a player is known, but his preferences are unknown.
- “I know I am playing against Tom, but I do not know whether he is selfish or nice.”
- Nature whispers to Tom his type, and I, the other player, have to figure it out.
- Nature selects from a population of potential player types. I am going to play against another player, but I do not know if she is smart or dumb, forgiving or unforgiving, rich or poor, etc. Nature decides.

Example 3: Two-faced Marry

- Think about one of our coordination games:
- Marry has two types: “social” or “loner” and knows which one
- Assume Larry assigns probability p to Marry being the social type (wanting to meet Larry)
- Assume Marry also knows Larry’s estimate of p
- BIG assumption, but needed: Common Prior Assumption

Bayes-Nash Equilibria

- Bayes-Nash Equilibria (BNE) is a generalization of Nash Eq.
- First, convert the game into a game of imperfect information.
- Second, use the Nash equilibria of this imperfect information game as the solution concept.
- For Larry and Marry.
- Larry’s pure strategy choices are Local Latte or Starbucks. He can also play a mixed strategy, L with probability q.
- Marry’sstrategy is a pair, one for each “type” of Marry: the first component is for the Marry who likes company (“social” type) and the second component is for Marry the loner (“loner” type). Pure strategies for Marry are thus (L,L), (L,S), (S,L), and (S,S). Marry also has a pair of mixed strategies p1and p2indicating the probability Marry plays L if type 1 or if type 2.

Larry and Marry

Focus on pure strategies:

- Guess #1:
- Suppose Larry plays Latte for certain
- Type 1 Marry plays L. Type 2 Marry plays S:
- Marry (L,S).
- Verify:
- Does Larry maximize his payoffs by playing L against the Marrys’ pure strategy of (L,S)?
- With probability p, he gets the L,L payoff 2, and with probability 1-phe gets the L,S payoff, 0. So expected payoff from L against Marry (L,S) is 2p.
- If instead Larry played S against Marry (L,S) he would get with probability p the (L,S) payoff of 0 and with 1-p the (S,S) payoff of 1. So the expected payoff from S is 1-p.
- Therefore playing L against Marrys’ (L,S) is a best response if 2p > 1-p, or if p > 1/3.
- Does Marry playing (L,S) maximize her payoffs given Larry is playing L? Yes.
- Equilibrium #1: If p > 1/3, Larry: L; Marry (L,S)

Larry and Marry

Focus on pure strategies:

- Guess #2:
- Suppose Larry plays Starbucks (S) for certain
- “Social” Marry plays S. “Loner” Marry plays L:
- Marry (S,L).
- Verify:
- Does Larry maximize his payoffs by playing S against the Marrys’ pure strategy of (S,L)?
- With probability p, he gets the S,S payoff 1, and with probability 1-phe gets the S,L payoff, 0. So expected payoff from S against Marry (S,L) is 1-p.
- If instead Larry played L against Marry (S,L) he would get with probability p the (L,S) payoff of 0 and with 1-p the (L,L) payoff of 2. So the expected payoff from S is 2(1-p).
- Therefore playing S against Marrys’ (S,L) is a best response if p > 2(1-p), or p > 2/3.
- Does Marry playing (S,L) maximize her payoffs given Larry is playing S? Yes.
- Equilibrium #2: If p > 2/3, Larry: S; Marry (S,L)

Larry & Marry: Summary

- If p > 2/3, there are 2 pure-strategy Bayes-Nash equilibria
- Larry: L; Marry (L, S)
- Larry: S; Marry (S, L)
- If 2/3 > p > 1/3 there is just 1 pure strategy Bayes Nash equilibrium, #1 above:
- Larry plays Land the Marrys play (L,S).
- If p< 1/3 then there is no pure strategy equilibrium.

Coming up…

- Signaling and Screening
- A player must updated her beliefs (using Bayes Theorem) given the signal
- Pooling and Separating Equilibria
- In short:
- Pooling: different types take the same action (can’t tell them apart)
- Separating Equilibria: different types take different actions
- How can we design a game to get the types to reveal themselves? (optimal contract design)

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