ECO290E: Game Theory
Lecture 11: Static Games of Incomplete Information
Review of Lecture 10
If a stage game is played infinitely many times, then cooperative outcomes different from static Nash
equilibrium can be sustained as long as a discount factor is large enough.
When a stage game has multiple Nash equilibria,
cooperative outcomes may be sustained even if the game is played finitely many times.
⇒ Long-term relationships make cooperation possible!
Games of Incomplete Information
In a game of incomplete information, at least one player is uncertain about a thing that other player knows.
Some of the players possess private information, or there exists information asymmetry at the beginning of the game.
For example, a firm in an oligopoly may not know the cost of the rival firm, a bidder participating an auction does not usually know her competitors’ valuations.
Following Harsanyi (1967), we can translate a game of
incomplete information into a Bayesian game whose Nash equilibrium is called Bayesian Nash equilibrium.
Formulation of Bayesian Games
To express asymmetric information, Bayesian game introduces an artificial player called Nature.
Expect for the existence of Nature, the game is identical to the usual static game.
What is Bayesian Game?
1. Nature draws a type vector t, according to a prior probability distribution p(t).
2. Nature reveals i’s type to player i, but NOT to any other player.
3. The players simultaneously choose actions.
4. Payoffs are received.
Remarks
A strategy for a player is a complete action plan, which specifies her action for all possible types.
Strategies for each player are constructed from types and actions, i.e., mapping from type to action.
Given this definition of a strategy in a Bayesian game, Nash equilibrium is defined straightforwardly, which is named as a Bayesian Nash equilibrium (BNE).
Instead of considering strategies, it is often easier to derive BNE by solving optimal action for each type. (see next slide)
Maximization in Strategy ⇔ Action
Maximizing the last line is identical to maximizing the part inside of the brackets (for all i’s types).
A strategy is optimal if and only if the specified action is optimal for each of her possible type.
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Cournot Game with Unknown Cost
Firm 1’s marginal cost is constant (c), while firm 2’s marginal cost is private information:
high (h) with probability θ, or
low (l) with probability 1 - θ.
Firm 1’s strategy is a quantity choice, but firm 2’s strategy must be a complete action plan, which specifies her
quantity choice for each possible marginal cost.
Assume each firm tries to maximize an expected profit given this information structure of the game.
How to Derive Equilibrium
Instead of focusing on optimal strategy, let us consider optimal actions for different types (of player 2).
Equilibrium actions can be derived by the following three maximization problems:
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Bayesian Nash Equilibrium
Firm 2 will produce more (/less) than she would in the complete information case with high (/low) cost.
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Market for Lemon
A seller and a buyer meet and decide whether to trade a used car (T) or not (N) for given price p.
Trade occurs if and only if both choose T.
The used car can be
Peach (good): with probability 1/2
Lemon (bad): with probability 1/2
Only seller knows the actual type (value) of the car.
Value of the lemon for the buyer (/seller) is 1000 (/0); the value of the peach for the buyer (/seller) is 3000 (/1500).
Remarks
What are strategies for each player?
Bayesian Nash equilibrium varies as price changes (or, games are different when prices are different):
Only lemon is traded. (if p < 1000)
Neither types are traded. (if 1000 < p < 1500)
Both types are traded. (if 1500 < p < 2000)
Neither types are traded. (if 2000 < p)
Trade is always profitable since the buyer’s value is higher the seller’s one for both types of the car.
Exchange Game
There are k envelopes, each of which contains a card inside with different integer from 1 to k.
Each of two individuals randomly receives an envelope and sees the card inside whose number indicates the size of a prize she may obtain.
They are given the following option simultaneously:
If both wish to exchange, then prizes are exchanged.
Otherwise, each individual receives her own prize.
⇒ What is a Bayesian Nash equilibrium?
No Exchange Result
When k = 2, no uncertainty for each player.
One player receives 1, the other receives 2.
No exchange realizes in a Bayesian equilibrium.
What happens when k is more than 2?
Players have private information, and have to guess the number (type) of other player.
The player with highest possible number (= k) doesn’t have incentive to exchange.
Given this equilibrium strategy, type k – 1 doesn’t have
Further Exercises
Solve the following modifications of the Cournot model with private cost:
1. Both firms have privately informed cost.
2. The cost is uniformly distributed (instead of binary).
Consider our example of market for lemon. What
happens if the seller’s value of peach becomes higher than 2000 while other prices remain the same?
