Game Theory 8 – chap12. Random events and incomplete information

4. Bayesian Games
4.1 Random Events and Incomplete Information (Chapter 24)
4.2 Bayesian Games: Bayesian Nash Equilibrium (Chapter 26)
4.3 Applications: Lemons, Auctions, and Information Aggregation (Chapter 27)
– Markets and Lemons, Provision of Public Goods*, Auctions

So far, we assumed the players knows which game is to be played; in particular,

each player i knows \(u_{i}(a_{i}, a_{j})\) and \(u_{j}(a_{i}, a_{j})\) for every \((a_{i}, a_{j})\)

However, this assumption is not in general satisfied: players are not completely informed about the payoffs in most situations. Then we say information is incomplete. In the remaining two parts, we relax the assumption of (CI) and study how to analyze the games.

Introduction

One example of incomplete information is the next:

EXAMPLE 4.1 (FIRST-PRICE AUCTIONS). Suppose an indivisible goods is auctioned to two bidders. Each bidder i knows his own valuation \(\nu_{i}\) but does not know \(\nu_{j}\). For each combination of bids \((b_{1}, b_{2})\), bidder 1 knows his own payoff

\(u_{1}(b_{1}, b_{2})\) = \(\begin{cases} & \nu_{1} – b_{1} \text{ if } b_{1} > b_{2}\\ & \text{ 0 otherwise, } \end{cases}\)

but is uncertain about bidder 2’s payoff

\(u_{2}(b_{1}, b_{2})\) = \(\begin{cases} & \nu_{2} – b_{1} \text{ if } b_{2} > b_{1}\\ & \text{ 0 otherwise, } \end{cases}\)

because \(\nu_{2}\) is bidder 2’s private information.

Incomplete information pervades a variety of (almost all) economic environments:

(1) bargaining: each party does not know the other’s willingness-to-pay

B: Buyer S: Seller
\(u_{B}(P) = v_{B} – P, u_{s}(P) = P – v_{s}\)

(2) market competition: firms do not know the cost of their competitiors

\(u_{1}(q_{1}, q_{2}) = p \cdot q_{1} – c_{1}(q_{1}), p \cdot q_{2} – c_{2}(q_{2})\)

(3) contracting: employers do not know he level of skills of employees

\(\pi\): company’s profit
\(\pi(w, hire) = \nu (e, t) – w)\)

(4) public good provision: the planner does not know the exact value of the public good to a society

\(w = \sum_{i=1}^{n}\nu_{i} – c\)

For now on, we tackle on incomplete information

As before, we subdivide the class of games into two:

(1) static games with incomplete info, or (static) Bayesian games, and
(2) dynamic games with incomplete info, or dynamic Bayesian games

info \ player’s moves	simultaneous	sequential
complete	Part I: Normal-Form Games ISD, Nash Eqbm	Part II: Extensive-Form Games Subgame Perfect Eqbm
incomplete	Part III: Static Bayesian Games Bayesian Nash Eqbm	Part IV: Dynamic Bayesian Games Perfect Bayesian Eqbm

The solution concept of (1) and (2) is a generalization of Nash eqbm and SPE, respectively

4. Bayesian Games
4.1 Random Events and Incomplete Information (Chapter 24)

We start with one example that formalizes the key idea of Bayesian games:

EXAMPLE 4.2 (A GUESSING GAME, EXERCISE 2 ON PAGE 333). Suppose Andy and Brian play a guessing game

• Two slips of paper: one is black and the other is white
• Each player has one of the slips pinned to his back
• Andy can see the slip on Brian’s back, but Brian see neither his own nor Andy’s
• Andy first chooses between Y and N. If he selects Y, then the game ends with a payoff of 0 for Brian. Andy obtains 10 if Andy’s slip is black and -10 if white.
• If he selects N, then Brian chooses between Y and N, ending the game. If Brian says Y and Brian’s slip is black, then he obtains 10 and Andy obtains 0. If Brian chooses Y but his slip is white, then he obtains -10 and Andy obtains 0.
• If Brian chooses N, then both obtain 0

Question: How to model Andy’s private information (or type)?

The idea of JOHN HARSANYI (1967, 1968) is to introduce a move by Nature, which transforms

a problem of incomplete info into the problem of imperfect info

Figure: The extensive-form representation of Example 4.2

In this example, Nature’s move is observed only by Andy (= player 1).

EXAMPLE 4.3 (THE GIFT GAME ON PAGE 327).

Observe that Nature’s move results in two different info sets. P1’s strategy therefore must specify what he intends to do if he is a friend and if he is an enemy.

In this sense, a player’s strategy is type-contingent in Bayesian games

In the example, the players’ possible strategies are

\(S_{1} = {N^{F}N^{E}, G^{F}G^{E}, N^{F}G^{E}, G^{F}N^{E}}\) and \(S_{2}\) = {A, R}

It is an incomplete information game, but we can change it to imperfect information through the harsanyi idea.
There are two equilibrium notions we have learned so far: NE and SPE.
Since there is no proper subgame, the prediction of NE and SPE is the same.

Bayesian Normal Form

Notice that in the gift game, both SPE and NE give us the same prediction

To find an equilibrium, we first put the game in the normal form

The procedure is the same as before, except

\((s_{1}, s_{2})\) = one path w/o Nature’s move but \((s_{1}, s_{2})\) ≠ one path with Nature’s move

As a result, a strategy profile \((s_{1}, s_{2})\) does not pin down players’ payoff

1 \ 2	A	R
\(G^{F}G^{E}\)	1, 2p-1	-1, 0
\(G^{F}N^{E}\)	p, p	-p, 0
\(N^{F}G^{E}\)	1-p, p-1	p-1, 0
\(N^{F}N^{E}\)	0, 0	0, 0

The payoffs are averaged according to Nature’s probability distribution

For example, \((N^{F}G^{E}, A)\): Depending on Nature’s move, the players obtain either (0, 0) or (1, -1)

The expected payoff from \((N^{F}G^{E}, A)\) is

\(u_{1}\) = 1∙(1-p) = 1-p
\(u_{2}\) = -1∙(1-p) = p-1

EXAMPLE 4.4.

1 \ 2	C	D
A	x, 9	3, 6
B	6, 0	6, 9

x = \(\begin{cases} & \text{ 12 with probability 2/3 } \\ & \text{ 0 with probability 1/3 } \end{cases}\)

Players 1 and 2 play the above normal-form game

The realization of x is player 1’s private information

Player 2 knows only that x = 12 with probability 2/3 and x = 0 with probability 1/3

Nature’s move in this example determines which normal-form game is to be played

• Reference: Chang-Koo Chi, (28/50) Game Theory and Applications 8 – Random events and incomplete information, Jul 10, 2020, https://youtu.be/b-AA4E2cqmM