Chapter 3

Static Games of Incomplete Information

Normal-Form Representation of Static Bayesian

Games

Bayesian Games: Most popular model for describing situations with incomplete information,

developed by Harsanyi (1967).

Recall the Battle of Sexes.

Pat

Chris

Opera

Fight

Opera

2 ，1

0 ，0

Fight

0 ，0

1 ，2

Suppose that Chris and Pat are not sure of each other’s payoff.

In particular, suppose that the following situation;

Privately known by Chris

Opera

Chris

Opera

2+tc , 1

0 , 0

Fight

Pat

Fight

0 , 0

1 , 2+tp

The Battle of the Sexes with Incomplete Information

TC : the set of possible tc

Tp : the set of possible tp

Privately known by Pat

Type sets

Following Harsanyi (1967), we assume that the timing of the static game above is

as follows.

(1) Nature (God) draws a type vector t=(tc , tp) according to the prior

probability distribution p(t), which is assumed to be common

knowledge among them.

(2) Nature reveals tc to Chris but not to Pat, and reveals tp to Pat but

not to Chris.

Given her type tp, Pat computes the belief (probability)

pp(tc | tp) using Bayes’ rule: 𝑝𝑝 𝑡𝑐 |𝑡𝑝 =

𝑝(𝑡𝑐 ,𝑡𝑝 )

𝑝(𝑡𝑝 )

=σ

𝑝(𝑡𝑐 ,𝑡𝑝 )

𝑡𝑐 ∈𝑇𝑐

𝑝(𝑡𝑐 ,𝑡𝑝 )

.

Given his type tc, Chris computes the belief (probability)

pc(tp | tc) using Bayes’ rule: 𝑝𝑐 𝑡𝑝 |𝑡𝑐 =

𝑝(𝑡𝑝 ,𝑡𝑐 )

𝑝(𝑡𝑐 )

=σ

𝑝(𝑡𝑝 ,𝑡𝑐 )

𝑡𝑝 ∈𝑇𝑝

𝑝(𝑡𝑝 ,𝑡𝑐 )

.

(3) Chris and Pat simultaneously choose actions, Opera or Fight, and

then payoffs are received.

Definition The normal-form representation of an n-player

static Bayesian game specifies the players’ action spaces A1,…,An,

their type spaces T1,…,Tn,

their beliefs p1,…,pn, and their payoff functions u1,…,un.

Player i’s type, ti, is privately known by player i, and is a member

of the set of possible types Ti. Each type vector t=(t1,…,tn)

determines i’s payoff function ui(a1,…,an ; t) where (a1,…,an) is a

profile of all players’ actions.

Player i’s belief pi(t-i | ti) describes i’s uncertainty about n-1 other

players’ possible types, t-i, given i’s own type, ti.

We denote this game by G={A1,…,An ; T1,…,Tn ; p1,…,pn ; u1,…,un}.

Definition In the static Bayesian game G={A1,…,An ; T1,…,Tn ;

p1,…,pn ; u1,…,un}, a strategy for a player is a function si(ti), where for

each type ti in Ti, si(ti) specifies the action from the feasible set Ai

that type ti would choose if drawn by nature.

Definition In the static Bayesian game G={A1,…,An ; T1,…,Tn ; p1,…,pn ;

u1,…,un}, the strategies s*=(s*1,…,s*n) are (pure-strategy) Bayesian Nash

equilibrium if for each player i and for each of i’s type ti in Ti, s*i(ti)

solves

∗

∗

max σ𝑡−𝑖 ∈𝑇−𝑖 𝑢𝑖 𝑠1∗ 𝑡1 , … , 𝑠𝑖−1

𝑡𝑖−1 , 𝑎𝑖 , 𝑠𝑖+1

𝑡𝑖+1 , … , 𝑠𝑛∗ 𝑡𝑛 ; t 𝑝𝑖 (𝑡𝑖−1 |𝑡𝑖 )

𝑎𝑖 ∈𝐴𝑖

.

The Battle of the Sexes with Incomplete Information

Pat

Privately known by Chris

Opera

Opera

2+tc , 1

Fight

0 , 0

Chris

Fight

0 , 0

1 , 2+tp

Privately known by Pat

Suppose that tc and tp are independent

draws from a uniform distribution on [0,x].

In terms of the abstract static Bayesian game in normal form

G={Ac,Ap;Tc,Tp; pc,pp;uc,up};

・ the action spaces are Ac=Ap={Opera,Fight},

・ the type spaces are Tc=Tp=[0,x],

・ the beliefs are pc(tp)=pp(tc)=1/x for all tc and tp.

Task 1 Let’s construct a pure-strategy Bayesian Nash equilibrium in

which

・ Chris plays Opera if tc exceeds a critical value c, and plays Fight

otherwise;

・ Pat plays Fight if tp exceeds a critical value p, and plays Opera

otherwise.

Task 2 Find the probability that Chris plays Opera, and the probability

that Chris plays Fight in such a equilibrium.

Task 3 (an interpretation of mixed-strategy Nash equilibria)

Show that as x approaches zero, that is, as the incomplete information

disappears, those probability above (i.e., the players’ behavior in this

pure-strategy Bayesian Nash equilibrium of the incomplete information

game) approaches their behavior in the mixed-strategy Nash

equilibrium in the original game of complete information.

Static Games of Incomplete Information

Normal-Form Representation of Static Bayesian

Games

Bayesian Games: Most popular model for describing situations with incomplete information,

developed by Harsanyi (1967).

Recall the Battle of Sexes.

Pat

Chris

Opera

Fight

Opera

2 ，1

0 ，0

Fight

0 ，0

1 ，2

Suppose that Chris and Pat are not sure of each other’s payoff.

In particular, suppose that the following situation;

Privately known by Chris

Opera

Chris

Opera

2+tc , 1

0 , 0

Fight

Pat

Fight

0 , 0

1 , 2+tp

The Battle of the Sexes with Incomplete Information

TC : the set of possible tc

Tp : the set of possible tp

Privately known by Pat

Type sets

Following Harsanyi (1967), we assume that the timing of the static game above is

as follows.

(1) Nature (God) draws a type vector t=(tc , tp) according to the prior

probability distribution p(t), which is assumed to be common

knowledge among them.

(2) Nature reveals tc to Chris but not to Pat, and reveals tp to Pat but

not to Chris.

Given her type tp, Pat computes the belief (probability)

pp(tc | tp) using Bayes’ rule: 𝑝𝑝 𝑡𝑐 |𝑡𝑝 =

𝑝(𝑡𝑐 ,𝑡𝑝 )

𝑝(𝑡𝑝 )

=σ

𝑝(𝑡𝑐 ,𝑡𝑝 )

𝑡𝑐 ∈𝑇𝑐

𝑝(𝑡𝑐 ,𝑡𝑝 )

.

Given his type tc, Chris computes the belief (probability)

pc(tp | tc) using Bayes’ rule: 𝑝𝑐 𝑡𝑝 |𝑡𝑐 =

𝑝(𝑡𝑝 ,𝑡𝑐 )

𝑝(𝑡𝑐 )

=σ

𝑝(𝑡𝑝 ,𝑡𝑐 )

𝑡𝑝 ∈𝑇𝑝

𝑝(𝑡𝑝 ,𝑡𝑐 )

.

(3) Chris and Pat simultaneously choose actions, Opera or Fight, and

then payoffs are received.

Definition The normal-form representation of an n-player

static Bayesian game specifies the players’ action spaces A1,…,An,

their type spaces T1,…,Tn,

their beliefs p1,…,pn, and their payoff functions u1,…,un.

Player i’s type, ti, is privately known by player i, and is a member

of the set of possible types Ti. Each type vector t=(t1,…,tn)

determines i’s payoff function ui(a1,…,an ; t) where (a1,…,an) is a

profile of all players’ actions.

Player i’s belief pi(t-i | ti) describes i’s uncertainty about n-1 other

players’ possible types, t-i, given i’s own type, ti.

We denote this game by G={A1,…,An ; T1,…,Tn ; p1,…,pn ; u1,…,un}.

Definition In the static Bayesian game G={A1,…,An ; T1,…,Tn ;

p1,…,pn ; u1,…,un}, a strategy for a player is a function si(ti), where for

each type ti in Ti, si(ti) specifies the action from the feasible set Ai

that type ti would choose if drawn by nature.

Definition In the static Bayesian game G={A1,…,An ; T1,…,Tn ; p1,…,pn ;

u1,…,un}, the strategies s*=(s*1,…,s*n) are (pure-strategy) Bayesian Nash

equilibrium if for each player i and for each of i’s type ti in Ti, s*i(ti)

solves

∗

∗

max σ𝑡−𝑖 ∈𝑇−𝑖 𝑢𝑖 𝑠1∗ 𝑡1 , … , 𝑠𝑖−1

𝑡𝑖−1 , 𝑎𝑖 , 𝑠𝑖+1

𝑡𝑖+1 , … , 𝑠𝑛∗ 𝑡𝑛 ; t 𝑝𝑖 (𝑡𝑖−1 |𝑡𝑖 )

𝑎𝑖 ∈𝐴𝑖

.

The Battle of the Sexes with Incomplete Information

Pat

Privately known by Chris

Opera

Opera

2+tc , 1

Fight

0 , 0

Chris

Fight

0 , 0

1 , 2+tp

Privately known by Pat

Suppose that tc and tp are independent

draws from a uniform distribution on [0,x].

In terms of the abstract static Bayesian game in normal form

G={Ac,Ap;Tc,Tp; pc,pp;uc,up};

・ the action spaces are Ac=Ap={Opera,Fight},

・ the type spaces are Tc=Tp=[0,x],

・ the beliefs are pc(tp)=pp(tc)=1/x for all tc and tp.

Task 1 Let’s construct a pure-strategy Bayesian Nash equilibrium in

which

・ Chris plays Opera if tc exceeds a critical value c, and plays Fight

otherwise;

・ Pat plays Fight if tp exceeds a critical value p, and plays Opera

otherwise.

Task 2 Find the probability that Chris plays Opera, and the probability

that Chris plays Fight in such a equilibrium.

Task 3 (an interpretation of mixed-strategy Nash equilibria)

Show that as x approaches zero, that is, as the incomplete information

disappears, those probability above (i.e., the players’ behavior in this

pure-strategy Bayesian Nash equilibrium of the incomplete information

game) approaches their behavior in the mixed-strategy Nash

equilibrium in the original game of complete information.

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