Lecture 8: Strategic Games with Incomplete Information...Strategic games with incomplete information...

Microeconomics I: Game Theory

Lecture 8:

Strategic Gameswith Incomplete Information

(see Osborne, 2009, Sect 9.1, 9.2.1)

Dr. Michael TrostDepartment of Applied Microeconomics

December 13, 2013

Dr. Michael Trost Microeconomics I: Game Theory Lecture 8 1 / 50

Strategic games with complete information

Up to now, we have examined simultaneous move games in

which players have complete information about their own and

the opponents’ payoffs being attainable in the game.

Complete information about the players’ payoffs means that

there is common knowledge about these payoffs. That is, every

player knows the players’ payoffs, every player knows that

every player knows the players’ payoffs, and so on ad infinitum.


Incomplete information about payoffs

The topic of this lecture is to examine situations of interaction in

which this information condition is not satisfied.

In the following, we analyze games in which the players do not

know completely the payoffs they might receive.


Strategic games with incomplete information

To this end, we introduce a new class of games. The so-called

class of strategic games with incomplete information.

Strategic games with incomplete information are simultaneous

move games in which the players are uncertain about the

payoffs being attainable in the game.



Strategic games with incomplete information has been studied

by Hungarian economist János Harsányi for the first time.

János Károly Harsányi (May 29, 1920 - August

9, 2000) received the Sveriges Riksbank Prize in

Economic Sciences in Memory of Alfred Nobel

in 1994 because he showed how games of

incomplete information can be analyzed,

thereby providing a theoretical foundation for a

lively field of research - the economics of

information - which focuses on strategic

situations where different agents do not know

each others’ objectives.


Example: Bank run game

To exemplify the relevance of strategic games with incomplete

information, let us consider the following simple model of

possible bank runs.


Bank runs

A bank run is a situation in which customers of a bank

withdraw their deposits due to theirs fear that the bank is

unable to repay their deposits.

Such bank run might cause a liquidity insolvency of the bank

since the bank holds usually only a small fraction of its assets in

cash. The most part of its assets is invested in securities with

long maturity which cannot (or, only with huge losses) be sold

immediately (problem of maturity transformation).


Recent example of a bank run: Northern Rock

Due to the US sub-prime crisis, insti-

tutional lenders became nervous about

lending to mortgage banks. As a con-

sequence, over the summer of 2007 Bri-

tish mortgage bank Northern Rock had

difficulties in raising funds. These dif-

ficulties led to panic among its deposi-

tors. On Friday 14 September 2007, they

queued outside its branches to with-

draw their deposits.


The case of a profitable investment

(A profitable project) Consider a bank with two depositors.

Each depositor has a deposit of e 10 in the bank. The bank

holds e 10 in cash, and e 10 are invested in a project that

yields a rate of return of 40 percent after some period of time.

The bank promises the depositors an interest rate of 20 per-

cent if they do no withdraw their deposits during this period

of time. Otherwise, only the amount of the deposit (without

interests) is paid out. In the case the bank needs cash it may

sell currently the project at the price of e 6 to another bank.


The case of a profitable investment

QUESTION: Suppose the depositors may only choose between

withdrawing their deposits or retaining their deposits. How

does the strategic game between the two depositors look in the

case the bank’s investment is profitable? What behavior does

the Nash equilibrium concept predict?

Depositor B

withdraw retain

Depositor Awithdraw 8,8 10,12

retain 12,10 12,12


The case of a false investment

(A false investment) Suppose the project in which the bank has

invested e 10 turns out to be less profitable. Indeed, the pro-

ject does not even repay the amount of investment. After com-

pletion it will have a total value of e 4. If the bank needs cash,

the project may be sold currently at the price of e 2 to another

bank.


The case of a false investment

QUESTION: Suppose the depositors may only choose between

withdrawing their deposits or retaining their deposits. How

does the strategic game between the two depositors look in the

case the bank’s investment is unprofitable? What behavior does

the Nash equilibrium concept predict?

Depositor B

withdraw retain

Depositor Awithdraw 6,6 10,4

retain 4,10 7,7


Uncertainty about the bank’s investment

(Uncertainty about the bank’s investment) Suppose the two de-

positors have no complete information about the profitability

of banks’ investment. Both depositors hold beliefs which are

representable by probability measures. It is assumed that both

believe with probability of 0.75 that the project is profitable

and with probability 0.25 that the project is unprofitable.



QUESTION: Suppose both depositors believe with probability of

0.75 that the project is profitable and with probability 0.25 that

the project is unprofitable. What behavior would we predict for

such beliefs?

Case: Profitable investment

B

withdraw retain

Awithdraw 8,8 10,12

retain 12,10 12,12

Case: Unprofitable investment

B

withdraw retain

Awithdraw 6,6 10,4

retain 4,10 7,7

0.75 0.25



Obviously, our bank run becomes a strategic game in which

both decision makers are uncertain about the payoffs the might

receive.

In the following, we set up a framework which enables us to

study such situations with incomplete information.



To model strategic games with incomplete information, the

following concepts are needed:

state space,

prior belief of a player,

information partition of a player.


State space

A state space Ω is a formal device to model the uncertainty of

the players’ about their own and the opponents’ payoffs.

A member ω of state space Ω is called a state of the world and

represents a specific resolution of the players’ uncertainty.

A subset E of state space Ω is called an event.


Prior belief

A prior pi of player i is a probability measure on Ω whose

support is equal to Ω (i.e., it attaches positive probability to each

state of the world).

A prior gives the initial belief of a player about the states of the

world.


Information

It is supposed that each state of the world is associated with

some, but maybe incomplete information to the players about

the actual state of the world.

In literature, this additional information about the actual state of

the world is often called signal.


Information partition model

The information the players have at different states of the world

is captured by an information partition model. This model has

been introduced in 1976 by Robert Aumann.

Robert Aumann (June 8, 1930) received the

Sveriges Riksbank Prize in Economic Sciences

in Memory of Alfred Nobel in 1994 “for having

enhanced our understanding of conflict and

cooperation through game-theory analysis”.


Information partition

Let Ω be a state space.

An information partition Πi of decision maker i on Ω is a system

of subsets of Ω satisfying the following three properties:

1 every element Pi ∈ Πi is non-empty.

2 for every elements Pi ,P′i ∈ Πi , if Pi 6= P ′i , then Pi ∩ P ′i = ∅.

3 it holds⋃

Pi∈ΠiPi = Ω.


Information cell

The elements of an information partition Πi are called

information cells.

The information a player i receives at state ω is the (unique) cell

Pi of information partition Πi which contains ω.

It is interpreted as the set of states player i considers possible at

state ω. Henceforth, we denote this information cell by Pi(ω).


Information cell

Consider a state space Ω, an information partition Πi of decision

maker i and two states ω, ω′ ∈ Ω of the world.

By the properties of information partition Πi , we conclude

for each ω ∈ Pi(ω), it holds ω ∈ Pi(ω).

if ω′ ∈ Pi(ω), then Pi(ω) = Pi(ω′).

if ω′ /∈ Pi(ω), then Pi(ω) ∩ Pi(ω′) = ∅.


Example: Red-green blind person

Let Ω := blue, red , green, yellow be a state space where each

state of the world describes a possible color of a point on the

screen.

The information partition of a red-green blind person is

Π := blue, red , green, yellow.

blue red green yellow


Examples: Information partitionLet Ω := ω1, ω,ω3, ω4 be a state space.

- System ω1, ω2, ω3, ω4 is an information partition of Ω

ω1 ω2 ω3 ω4

- System ω1, ω2, ω3, ω3, ω4 is not an information partition of Ω

ω1 ω2 ω3 ω4

- System ω1, ω2, ω4 is not an information partition of Ω

ω1 ω2 ω3 ω4


Bayesian game

Since the formal concepts to model players’ uncertainty about

the payoffs are now at our hands, we are ready to introduce the

class of strategic games with incomplete information.

Such games are also referred to as Bayesian games.


Bayesian game

Definition 8.1 (Bayesian game)A Bayesian game Γ := (I ,Ω, (pi )i∈I , (Πi )i∈I , (Ai )i∈I , (%i )i∈I ) consists of

a finite set I of players,

a set Ω of states of the world,

for each player i ∈ I , a prior pi on Ω,

for each player i ∈ I , an information partition Πi on Ω,

for each player i ∈ I , a set Ai of actions,

for each player i ∈ I and for each state ω ∈ Ω, a preference relation %i,ω

on ∆(×i∈IAi ) which is representable by some expected utility function

Ui (., ω) with Bernoulli function ui (., ω).


Exercise: Bank run gameEXERCISE: Determine the components of our bank run game.

set ofplayers I :=

state space Ω :=

priorbeliefs

pA :

pB :

informationpartitions ΠA :=

ΠB :=

action sets AA :=

AB :=

preferences %A:

%B :


Strategies in Bayesian Games

To solve Bayesian games, the idea of a strategy is applied.

A strategy of a player is a complete plan of actions that

describes the action the player would realize in every possible

state of the world given her information about the actual state of

the world.



Being more concretely, consider some Bayesian game Γ.

Then a strategy si of player i in Γ is a mapping that assigns an

action ai ∈ Ai to every state of the world ω ∈ Ω where the same

actions are assigned to states belonging to the same information

cell.



Consider some Bayesian game Γ.

Formally, a strategy si of player i in Γ is a mapping si : Ω→ Ai

satisfying

si(ω) = si(ω′)

for every states ω ∈ Ω and ω′ ∈ Pi(ω)



REMARK 1:

Strategies are complete plans of action. In Bayesian games, they

are mappings that specify the action of a player in every

possible state of the world.

Strategies are devices that describe how a player would behave

if a state of a world proves to be the actual one.



REMARK 2:

If ω ∈ Ω is the actual state of the world, player i receives the

information that the actual state of the world belongs to Pi(ω).

However, despite of this signal she is still unable to distinguish

between the states belonging to Pi(ω).

As a consequence of this informational fuzziness, the actions

chosen by her in these states have to be identical.



Player i ’s set of strategies is denoted by Si and the set of all

strategy profiles is denoted by S := ×i∈ISi .

Let ω ∈ Ω be a state of the world, then list

s(ω) := (si(ω))i∈I

gives the strategy profile that is chosen by the players in state ω.


Bayesian equilibrium

Definition 8.2 (Bayesian equilibrium)Let Γ := (I ,Ω, (Πi)i∈I , (pi)i∈I , (Ai)i∈I , (%i)i∈I ) be a Bayesian game.

A Bayesian equilibrium of Γ is a profile of strategies s∗ := (s∗i )i∈I

satisfying∑ω∈Ω

ui(s∗i (ω), s∗−i(ω), ω) pi(ω) ≥

∑ω∈Ω

ui(si(ω), s∗−i(ω), ω) pi(ω)

for every player i ∈ I and for every strategy si ∈ Si .


Bayesian equilibrium

A profile of strategies constitute a Bayesian equilibrium

whenever no player has incentive to change her strategy given

that the other players follow the strategies of this profile.

That is to say, a profile of strategies is a Bayesian equilibrium

whenever each strategy of this profile maximizes the expected

utility of the player subject to the condition that the other

players choose the strategies of this profile.


Bayesian equilibrium as Nash equilibrium

In the following, we characterize a Bayesian equilibrium as a

Nash equilibrium.

In order to obtain this characterization, the Bayesian game is

transformed into a strategic game. This strategic game is known

as the strategic form of the Bayesian game.

It turns out that every Bayesian equilibrium is a Nash

equilibrium of the strategic form, and vice versa.


Strategic form of a Bayesian game

Definition 8.3The strategic form of Bayesian game Γ := (I ,Ω, (Πi )i∈I , (pi )i∈I , (Ai )i∈I , (%i )i∈I )

is the strategic game Γ′ := (I , (Si )i∈I , (%′i )i∈T ) consisting of

the set I of players,

for each player i ∈ I , the set Si of strategies.

for each player i ∈ I , the preference relation %′i on S := ×i∈ISi which is

representable by utility function

U ′i (s) :=

∑ω∈Ω

ui (s(ω), ω) pi (ω)

where s ∈ S holds.


Strategic form of a Bayesian game

As stated in the previous definition, the strategic form of a

Bayesian game is a strategic game having the following

properties:

the set of players is identical to the set of players of the

Bayesian game.

the action set of each player consists of her available

strategies.

the utility function of each player is the expected utility

resulting from her Bernoulli utility and her prior belief.


Exercise: Bank run game

EXERCISE: Determine the strategic form of our bank run game.

set of

playersI :=

action

setsSA :

SB :

prefer-

ences%′A:

%′B



EXERCISE: Depict the strategic form of our bank run game.

Depositor B

Depositor A


Characterization of Bayesian equilibrium

Theorem 8.4Consider a Bayesian game Γ. A strategy profile s∗ := (s∗i )i∈I is a

Bayesian equilibrium of Γ if and only if it is a Nash equilibrium of the

strategic form of Γ.


Bayes-Nash equilibrium

The message of the previous theorem is that the Bayesian

equilibrium is representable as a Nash equilibrium of a strategic

game.

Due to this characterization the Bayesian equilibrium is also

called Bayes-Nash equilibrium.



EXERCISE: Determine the Bayesian equilibrium of our bank run

game by applying Theorem 8.4.

Depositor B

sB(profit) := withdraw

sB(loss) := withdraw

s ′B(profit) := retain

s ′B(loss) := retain

Depositor A

sA(profit) := withdraw

sA(loss) := withdraw(7.50,7.50) (10.00,10.00)

s ′A(profit) := retain

s ′A(loss) := retain(10.00,10.00) (10.75,10.75)


Rumors about the bank’s investment

(Rumors about the bank’s investment) Suppose newspapers re-

port about rumors among investors that the project in which

the bank invested fails to be profitable. Both depositors take

these rumors serious. These rumors affect their beliefs in the

way that their initial beliefs are reversed. Now, both believe

with probability 0.25 that the project is profitable and with

probability 0.75 that the project is unprofitable.


Bank run game with rumorsEXERCISE: Describe our bank run game after these rumors.

set ofplayers I :=

state space Ω :=

priorbeliefs

pA :

pB :


ΠB :=

action sets AA :=

AB :=

preferences %A:

%B :


Bank run game with rumors

EXERCISE: Depict the strategic form of our bank run game after

the rumors, and determine its Bayesian equilibrium.

Depositor B

Depositor A


Asymmetric information

(Asymmetric information about the bank’s investment) Consider

the original situation of our bank run game in which both

players initially believe that the project is profitable with pro-

bability 0.75 and the project is unprofitable with probability

0.25. However, depositor A has access to information about

the profitability of this project. Before she decides whether to

withdraw her deposit or to retain, she learns about its profi-

tability. Assume that there is no communication between de-

positors A and B .


Bank run game with asymmetric informationEXERCISE: Describe our bank run game with asymmetric information .

set ofplayers I :=

state space Ω :=

priorbeliefs

pA :

pB :


ΠB :=

action sets AA :=

AB :=

preferences %A:

%B :


Bank run game with asymmetric information

EXERCISE: Depict the strategic form of our bank run game with

asymmetric information, and determine its Bayesian

equilibrium.

Depositor B

Depositor A


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