TOC
1. Introduction
1.1 Motivating Example: Auction
Suppose a seller wants to sell a product among a group of buyers. Each buyer is willing to pay for the product, where is buyer ’s private information, i.e. only buyer knows its valuation , but not all other buyers or the seller. In order to sell the product, the seller runs an auction (first-price auction, second-price auction, etc.). Each buyer must bid for the product in order to be tha winner. The question is: what should each buyer bid?
In previous lectures, games are of complete information, i.e., each player’s payoff function is common knowledge among all players. In the auction example, each player’s payoff function is no longer common knowledge. This is an example of incomplete information games, in which at least one player is uncertain about another player’s payoff function.
Games of incomplete information are also called Bayesian games. There are two types of Bayesian games: static and dynamic.
1.2 Cournot Com. under Asymmetric Information
Consider the Cournot duopoly model with an inverse demand function , where and . Firm 1’s cost function is and firm 2’s cost function is
where and .
Different from the standard Cournot model, the information is asymmetric:
- Firm 1’s cost function is known by both firms ⇒ is common knowledge
- Firm 2’s cost function is completely known by itself, but not known by firm 1 ⇒ is not common knowledge. Firm 1 only knows the distribution of firm 2’s marginal cost, i.e., with probability and with probability
Naturally, firm 2 may want to choose a different (and presumably lower) quantity if its marginal cost is high than if it is low. Firm 1 should rationally anticipate that firm 2 may tailer its quantity to its cost in this way. Let and denote firm ’s quantity choices when its marginal cost is and respectively, and let denote firm 1’s single choice of quantity.
If firm 2’s cost is (), it will choose to solve
Since firm 1 knows that firm 2’s marginal cost is with probability of and anticipated firm 2 to choose depending on its cost, firm 1 chooses to solve
The (interior) first-order conditions (or best response functions) for the firms are
Thus, the equilibrium of this game is , where
We know ⇒ firm 2 produces less when its marginal cost increases.
Firm 2 has two payoff functions
Firm 1 has only one payoff function
Firm 2 knows firm 1’s payoff function, while firm 1 does not kown firm 2’s payoff functions but only knows the probability distribution. This is an example of (static) Bayesian games.
2. Static Bayesian Games
2.1 Definition and Representation
Consider a general static Bayesian game:
- Let player ’s possible payoff function be , where is player ’s action and is called player ’s type, which belongs to a set of possible types (or type spaces).
- Player ’s type is his private information, and each type corresponds to a different payoff function of player .
- Let be the types of other players and be the set of all .
- Player is uncertain about other players’ types, but only knows the probability distribution on , which is ’s belief about other players’ types, given ’s knowledge of his own .
The normal-form representation of an n-player static Bayesian game specifies players’
- action spaces
- type spaces
- beliefs
- payoff functions
We denote this game by
In the Cournot game with asymmetric information
- and
- , , and ;
- Payoff functions are
Timing
The timing of a static Bayesian game:
- Nature draws a type vector , where ;
- Nature reveals to player , but not to any other players;
- The players simultaneously choose actions, player choosing ;
- Payoffs are received.
By introducing the frictional moves by nature in previous two points, we have described a game of incomplete information as a game of imperfect information.
Bayes’ Rule
Assume that the nature draws according to the prior probability distribution , which is common knowledge. Then the (posterior) belief can be computed by Bayes’ rule
Remarks
- There are games in which player has private information not only about his or her own payoff function but also about another player’s payoff function. We write player ’s payoff function as . (interdependent).
- We typically assume that players’ types are independent (otherwise, correlated), i.e., does not depend on , which can be denoted by . But is still derived from the prior distribution .
2.2 Strategy and Bayesian NE
Strategy Definition: In the static Bayesian game , a strategy for player is a function , i.e., . For given type , gives an action of player . Player ’s strategy space is the set of all functions from into .
In the previous example, is a strategy for firm 2, while is a strategy for frim 1.
Bayesian NE Definition: In the static Bayesian game , the strategy profile are a (pure-strategy) Bayesian Nash equilibrium if for each player and for each of ’s types , solves
In a general finite static Bayesian game (finite players, finite actions, and finite types), a Bayesian Nash equilibrium exists, perhaps in mixed strategies. In a Bayesian Nash equilibrium, each player’s strategy is a best reponse to other players’ strategies. In other words, no players wants to change his or her strategy unilaterally given other players’ equilibrium strategies, even if the change involves only one action by one type. A Bayesian Nash equilibrium is simply a Nash equilibrium in a Bayesian game.
In the Cournot game with asymmetric information, the strategies are a Bayesian Nash equilibrium since neither firm 1 nor firm 2 wants to deviate from its equilibrium strategy.
3. Applications
3.1 App1. A trading game
Suppose a seller can produce a proudct at a cost of ; A buyer wants to buy the good, and is willing to pay . The buyer can also purchase the good from other places, where the valuation is his private information. The seller knows the distribution of the valuation for the outside option is either or , each with a probability of and , respectively. The price of the good is , which is exogenous and independent of where the buyer makes a purchase. All and are common konwledge among both players.
The seller decides whether to produce the good, and the buyer simultaneously deicdes whether to order the good from the seller. If the seller produce the good, its payoff is . If the buyer makes a purchase, and otherwise. If the seller does not produce the good, its payoff is zero regardless of the buyer’s choice. The buyer’s payoff is if he buys from the seller, and otherwise. The extensive-form representation of the game is as following
Normal-form representation of the game:
- Action spaces: and ;
- Type spaces: and ;
- Beliefs:
- the buyer’s belief on the seller’s type is 1 on type 1,
- the seller’s belief on the buyer’s types is 2/3 on 10 and 1/3 on 14;
- Payoffs are shown in above figure
Strategy spaces: and . BN means that the buyer with outside option 10 chooses to buy and with outside option 14 chooses not to buy.
Alternatively, we can use the following bi-matrix to represent the game
For example, consider the outcome :
- the buyer with type 10 receives , and with type 14 receives
- the seller’s expected payoff is .
In particular, we can consider two types of the buyer as two players and we can solve the Bayesian Nash equilibira in the above normal-form representation of the game.
We first find out the best response functions for each of the “three players” (the seller and each type of the buyer)
There are two Bayesian Nash equilibria: and
3.2 App2. Mixed Strategies Revisited
Consider the game of battle of the sexes
There are three possible Nash equilibria: (O, O), (F, F) and (O+F, O+). In the mixed-strategy Nash equilibrium,
- the husband plays Opera with probability and Football with probability
- the wife plays Opera with probability and Football with probability .
Suppose the couple are uncertain about the payoffs for each other. Consider the following payoff matrix
Here is privately known by the wife while is privately known by the husband. Assume that and are independently drawn from a uniform distribution on , where .
The normal-form representation of the static Bayesian game is :
- The husband believes that (the wife believes that ) is uniformly distributed on
- and are given before
We can construct a Bayesian Nash equilibrium , where
Note that are two critical values, which need to be determined. In the Bayesian Nash equilibrium, the husband will choose Football if exceeds the critical value and choose Opera otherwise.
Given the wife’s strategy, the husband’s expected payoffs of choosing Opera and Football are
and
Thus, choosing Opera is optimal iff
Similarly, given the husband’s strategy, the wife’s expected payoffs of playing Opera and Football are
and
Thus, choosing Football is optimal iff
Solving (1) and (2) simultaneously, we obtain and satisfy
thus, .
In equilibrium, the husband plays Opera with probability and Football with probability , while the wife plays Football with probability and Opera with probability , where
when , we get that . As the incomplete information disappears, the players’ behavior in this pure-strategy Bayesian Nash equilibrium approaches their behavior in the mixed-strategy Nash equilirbium in the original game of complete information.
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