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1

What are Static Games of Incomplete Information?

Static Game: All players move once and “simultaneously.”
Incomplete Information: Some players do not know something about the game.

2

What are Bayesian games?

Introducing Nature as an additional player, a Bayesian game can be interpreted as a game of complete but imperfect information:
• Nature moves first, choosing a type profile t according to a commonly known distribution.
• Each player i only observes her own type tᵢ.
-> Imperfect information about Nature’s move.
= type of player i is determined by random draw and type is i's private information BUT distribution of type profiles is common knowledge.

3

What is the timing of a Bayesian game?

1) Nature draws a type profile t = (t₁, ..., tₙ) ∈ T according to the prior probability distribution p : T -> [0, 1].
2) The players privately learn their types; each i learns tᵢ
3) The players simultaneously choose their actions; each i chooses aᵢ
4) The players receive their payoffs; each i receives
uᵢ(a₁, ..., aₙ; t₁, ..., tₙ).

4

What is Bayesian updating?

When actions are chosen, each player i
• knows her own type ti but does not know t₋ᵢ;
• knows how (tᵢ, t₋ᵢ) was determined!
Thus the probability of the other player's type is updated by a probability distribution conditional on tᵢ.

5

Which is the probability distribution denoting i's belief about the other player?

It is the probability distribution of t₋ᵢ conditional on tᵢ:
p(t₋ᵢ | tᵢ) = p(t₋ᵢ, tᵢ) / p(tᵢ)

6

What does a strategy for player i specify in the Bayesian Nash Equilibrium?

A strategy for player i specifies an action for every possible type of player i, even though player i knows his type when actions are chosen.
Remark: Since the other players do not know i’s type, their optimal strategies depend on what actions they expect of various types of player i, and these beliefs are captured by i’s strategy.

7

What is first-price sealed-bid auction?

• Bidders submit their bids in sealed envelopes. (-> simultaneous moves)
• The highest bidder gets the object and pays the price she bid.
• The other bidders get nothing and pay nothing.
• In case of a tie, the winner is determined randomly.

8

What is second-price sealed-bid auction?

• Bidders submit their bids in sealed envelopes. ( -> simultaneous moves)
• The highest bidder gets the object and pays the price of the second-highest bidder.
• The other bidders get nothing and pay nothing.
• In case of a tie, the winner is determined randomly.