程序代写代做代考 Bayesian 1. Sets and Sequences

1. Sets and Sequences

Strategy -1-

Strategic equilibrium

A. Cournot duopoly 2

B. Strategic (Nash) equilibrium 7

C. First mover advantage 15

D. Bidding games 20

E. Bidding games with private information 22

F. Reserve prices 32

G. Sealed high-bid auction 45

UCLA Auction House http://games.jriley.sscnet.ucla.edu/

55 pages

http://games.jriley.sscnet.ucla.edu/

Strategy -2-

A. Cournot Duopoly

Two firms produce related products

Demands are linear

1 1 260 20 10q p p   , 2 1 2150 10 20q p p  

The demand price functions are as follows:

1 1
1 1 215 30

9p q q   , 1 12 1 230 1512p q q   .

The cost of production in firm 1 is 1 1 1( ) 4C q q and in firm 2 is 2 2 2( ) 7C q q

Generalization of Cournot’s model (1842)

Firm 1: Computes its marginal revenue and adjusts its output until 1 1 2 1( , ) 4MR q q MC 

21 1
1 1 1 1 1 1 215 30
( ) ( ) 9R q p q q q q q q    , 1 1( ) 4C q q , 1 1 1 1( ) ( ) ( )q R q C q  

1 2 1
1 1 215 30

9
R

MR q q
q


   


Strategy -3-

Firm 1 then chooses its best response 1q to satisfy

1 2 1
1 1 1 215 30

9 4MR MC q q
q


     



2 11 215 305 q q  

0 .

Therefore

1 11 1 2 22 4( ) 37q B q q  

Best response function

1 2( )B q is firm 1’s best (i.e. profit-maximizing) response for any choice by firm 2.

We can similarly derive firm 2’s best response function 2 2 1( )q B q

Exercise: Show that 1 12 2 1 12 4( ) 37q B q q   .

Exercise: Confirm that 1 2( , ) (30,30)q q 

Strategy -4-

Strategy -5-

The best response line for firm 1 is depicted.

Marginal profit,

1 2 1
1 1 1 215 30

5MR MC q q
q


    



is decreasing in 1q . Therefore if the marker

indicating the choices of the two firms is to the

right of the best response line, firm 1 will wish to

adjust by reducing its output.

If the marker is to the left, firm 1 will wish to

adjust by increasing its output.

Firm 1’s profit-increasing response.

Strategy -6-

The best response line for firm 2 is depicted.

Marginal profit

2 1 2
2 2 1 230 15

5MR MC q q
q


    



is decreasing in 2q .

Therefore if the marker

indicating the choices of the two firms is above

its best response line, firm 2 will wish to

adjust by reducing its output.

If the marker is below, firm 2 will wish to

adjust by increasing its output.

Firm 2’s profit-increasing response.

Strategy -7-

The two best response lines are depicted.

The arrows depict the directions of increasing

profit for the two firms in each “zone”.

Note that the adjustment process converges

to 1 2( , )q q q where

1 2 1( )B q q and 2 1 2( )B q q .

At this point the strategy if the firms

are said to be mutual best responses.

Then the stable outcome of this adjustment

process satisfies the two first order conditions

1
1 2

( , ) 0q q
q





and 2

1 2

( , ) 0q q
q





.

Arrows show directions of increasing profit for each firm

Strategy -8-

B. Strategic (Nash) equilibrium

Consider two firms competing. In the Cournot model a firm does not try to understand how its

opponent might respond. In modern economics we assume that firms have a deep understanding of

their close competitors. In particular, each firm can figure out how the other firm’s best respond to

its action. Let 2 2 1( )q B q be firm 2’s best response to firm 1’s output choice and let 1 1 2( )q B q be

firm 1’s best response to firm 2’s output choice.

The outputs 1 2( , )q q q are called mutual best responses if each is a best response to the

opponent’s action.

Strategy -9-

B. Strategic (Nash) equilibrium.

Consider two firms competing. In the Cournot model a firm does not try to understand how its

opponent might respond. In modern economics we assume that firms have a deep understanding of

their close competitors. In particular, each firm can figure out how the other firm’s best respond to

its action. Let 2 2 1( )q B q be firm 2’s best response to firm 1’s output choice and let 1 1 2( )q B q be

firm 1’s best response to firm 2’s output choice.

The outputs 1 2( , )q q q are called mutual best responses if each is a best response to the

opponent’s action.

Definition: Nash equilibrium (n players)

Let 1( ,…, )j nq q be player j’s payoff if the action vector is 1( ,…, )nq q q . The action vector q is a

Nash equilibrium action vector if each player’s action is a best response.

Nash equilibrium (2 players)

1q solves
1

1 1 2{ ( , )
q

Max q q and 2q solves
2

2 1 2{ ( , )
q

Max q q

Strategy -10-

Consider the two player example discussed above. The market clearing price of each firm depends on

the output vector 1 2( , )q q q .

1 1 2 1 1 2 1 1( , ) ( , ) ( )q q p q q q C q  

2 1 2 2 1 2 2 2( , ) ( , ) ( )q q p q q q C q  

1q is a best response to 2q if 1q solves
1

1 1 2{ ( , )
q

Max q q

2q is a best response to 1q if 2q solves
2

2 1 2{ ( , )
q

Max q q

FOC (i) 1
1 2

( , ) 0q q
q





(ii) 2

1 2

( , ) 0q q
q






These conditions are the conditions for equilibrium in the Cournot model.

Note that the modern approach is silent about how each competing firm learns the best response of

its opponents.

Strategy -11-

Group Exercise 1: Equilibrium when the firms produce the same product

In this case if the total output is 1 2q q then the market clearing price is 1 2( )p q q .

Suppose 1 2 1 2( ) 60 ( )p q q q q    , 1 1 1( ) 12C q q and 2 2 2( ) 12C q q .

Firms choose outputs simultaneously.

(a) Solve for the Nash equilibrium outputs and show that the equilibrium price is 28.

(b) If the two firms were to collude what would they do?

Exercise 2

owner of firm 2. If this simultaneous move game is going to be played 10 times what output would

you choose in the first round.

Strategy -12-

Answer to (c)

Total profit is 1 2 1 2 1 2( ( ) 12)( )p q q q q      .

Note that this is a function of the sum of the outputs 1 2x q q 

( ( ) 12) (60 12)p x x x x     .

As is readily checked, this is maximized if * 24x  . Then maximized total profit is 576. Note that any

pair 1 2( , )q q satisfying 1 2 24q q  is profit maximizing. The possible profits can therefore be

depicted as a line

1 2 576  

The total profit in the Nash Equilibrium is lower. Thus there are gains to colluding.

Give the symmetry of the problem it seems plausible that the two firms would agree to share profits

equally.

Strategy -13-

Symmetric collusion

If the model is not symmetric both firms will only collude if both gain. But it is no longer so clear how

the gains from collusion will be cleared.

Strategy -14-

The case of different costs (to be discussed in TA session)

Suppose 1 2 1 2( ) 60 ( )p q q q q    , 1 1 1( ) 12C q q and 2 2 2( ) 18C q q .

Firms choose outputs. The set of feasible profit levels is the shaded region R depicted below.

If you check you will find that the NE strategies are (18,12)
NEq  and the NE profits are

(324,144)NE  Consider the following joint profit maximization problem.

1 2 1 2{ ( ) ( ) | ( ( ), ( )) }
q

Max q q q q R    

Since marginal cost is higher for firm 2,

joint profit is maximized at
* (24,0)q 

and the profits are
* *
1 2( , ) (576,0)  

In the figure, the slope of the maximand is -1.

Points on the boundary with 2 0 

have a lower slope.

Strategy -15-

Note that Firm 1 could then pay a “bribe” to firm 2 to produce nothing. But exactly how big a bribe?

Without a theory of cooperation it is not so clear what the payment would be.

If such profit transfers are illegal, then the collusive agreement must be on how much the two firms

should produce.

Again it is not clear which point on the frontier to the North-East of the NE profit levels would be

chosen.

Strategy -16-

C. First mover advantage

In the model of strategic competition (or “game”) used thus far, firms must make simultaneous

decisions (equivalently, make decisions without knowing the decision of competitors).

Such games are called simultaneous move games.

In some environments (e.g. incumbent firm and potential entrant) the incumbent may have the

opportunity to choose its action first. This is called an alternating move game.

An example

2 1
1 1 23 3

4q p p   , 1 22 1 23 34q p p  

The demand price functions are as follows:

1 1 212 2p q q   , 2 1 212 2p q q   .

The cost of production in firm 1 is 1 1 1( ) 4C q q and in firm 2 is 2 2 2( ) 4C q q

Firm 1 moves first and chooses 1q . Firm 2 observes 1q and makes a best response.

Strategy -17-

2 1 2 2 2 2 2 2 1 2 2 2( , ) 4 (12 2 ) 4q q R C p q q q q q q        

2 1 2 28 2q q q q   .

The best response 2 2 1( )q B q satisfies the necessary condition for a maximum.

2
1 2

8 4 0q q
q


   


. Therefore 12 2 1 14( ) 2q B q q  

Remark: In a simultaneous move game, by a symmetric argument, 11 1 2 24( ) 2q B q q   .

As you may confirm, the Nash Equilibrium of this game is 8 81 2 5 5( , ) ( , )q q  .

Strategy -18-

Firm 1’s best response

In the alternating move game, firm 1 can predict what firm 2’s best response will be. So firm 1

predicts that

1
2 2 1 14

( ) 2q B q q   . (*)

Using this prediction, firm 1’s profit is

1 1 2 1 1 1 1 1 1 2 1 1 1 2 1 1 1( , ) 4 (12 2 ) 4 (12 2 ( )) 4q q R C p q q q q q q q B q q q            

1 1 2 1 18 2 ( )q q B q q  

Strategy -19-

Firm 1’s best response

In the alternating move game, firm 1 can predict what firm 2’s best response will be. So firm 1

predicts that

1
2 2 1 14

( ) 2q B q q   . (*)

Using this prediction, firm 1’s profit is

1 1 2 1 1 1 1 1 1 2 1 1 1 2 1 1 1( , ) 4 (12 2 ) 4 (12 2 ( )) 4q q R C p q q q q q q q B q q q            

1 1 2 1 18 2 ( )q q B q q  

Necessary condition for a maximum

1
1 2 1 1 1 1 2 1 2 1 1

( , ( )) 8 4 ( )) ( ) 0
d

q B q R C q B q B q q
dq

       

From (*) the last term is positive. 12 1 4( )B q
  . Therefore

1 71 1
1 2 1 1 1 1 1 1 14 4 2

( , ( )) 8 4 (2 ) 6 0
d

q B q R C q q q q
dq


          so

*
1

12 8

7 5
q  

Strategy -20-

Key insights:

1. If firm 1 were to choose the same output as in the simultaneous move game, then firm 2’s best

response would also be the same. Thus the first mover can achieve the same profit as in the

simultaneous move game.

2. But in contrast to the simultaneous move game, firm 1 correctly predicts that if it increases its

output, then firm 1 will lower its output in response. This increases firm 1’s demand price function

1 1 2( , )p q q and thus raises firm 1’s profit.

Strategy -21-

Pricing games

In many applications, short-run output choices are constrained by capacity constraints. So it is not

possible to lower a price and then sell a much larger quantity.

Thus, when studying strategic competition, economists often model firms as quantity setters.

However if firms can adjust capacity quickly then price setting strategic completion is possible.

This case will be discussed in the TA session.

The example (firms choose prices)

2 1
1 1 23 3

4q p p   , 1 22 1 23 34q p p  

The demand price functions are as follows:

1 1 212 2p q q   , 2 1 212 2p q q   .

The cost of production in firm 1 is 1 1 1( ) 4C q q and in firm 2 is 2 2 2( ) 4C q q

Strategy -22-

D. Bidding games

Sealed high-bid auction of a single item with I buyers

Buyer i has value iv . Buyer i submits a sealed bid ib . To win, buyer i must submit the high bid. If ib

is the winning bid, buyer i pays ib and receives the item. Buyer i ’s payoff is therefore i i iu v b  .

The other buyers’ payoffs are zero.

Tie breaking rule: In the event of tying high bids the winner is determine randomly from among the

tying high bidders.

Strategy -23-

Example 1: Two buyers with known values 1 200v  , 2 200v  .

Class exercise: We will run an auction.

What are the Nash Equilibrium strategies of this bidding game?

Example 2: Three buyers with known values 1 200v  , 2 200v  , 3 200v 

Group exercise: Are the bids 1 2 3200, 200, 200b b b   mutual best responses (i.e. Nash

equilibrium bidding strategies? Are there other bids that are mutual best responses?

Example 3: Two buyers with known values 1 800v  and 2 700v 

Bids restricted to be integers.

Group exercise: What are the Nash Equilibrium strategies of this bidding game?

Strategy -24-

E. Bidding games with private information

There are I bidders. Buyer i has a valuation iv that is private information. All that is known by

other buyers is that iv is continuously distributed on [ , ]  . The probability distribution is also

known:
0

Pr{ } ( ) ( )
iv

i i iv v F v f x dx    .

The probability density

function is depicted opposite.

The probability density function

The area of the shaded

region is

Strategy -25-

Bidding games with private information

There are I bidders. Each of the buyers may submit a non-negative sealed bid.

Strategy -26-

Bidding games with private information

There are I bidders. Each of the buyers may submit a non-negative sealed bid.

Allocation rule

Bidder i with bid ib loses if another bid is higher. If there are m bidders who submit the tying high

bid, the winner is selected randomly from one of these high bidders so that win probability of each

such bidder is 1/ m .

Strategy -27-

Bidding games with private information

There are I bidders. Each of the buyers may submit a non-negative sealed bid.

Allocation rule

Bidder i with bid ib loses if another bid is higher. If there are m bidders who submit the tying high

bid, the winner is selected randomly from one of these high bidders so that win probability of each

such bidder is 1/ m .

Payment rule

(i) Sealed high-bid auction

The winner pays his or her bid. Losers pay nothing.

(ii) Electronic ascending price auction

Asking price p rises steadily. A buyer exits the auction by switching his bidder light from green to

red. The asking price stops when only one light is green and the remaining buyer pays the final ask.

(iii) Sealed second-bid auction

The winner pays the highest of the losing bids (the second highest bid).

Strategy -28-

(iv) English ascending price auction

An auctioneer calls out (usually) ascending asking prices seeking bidders. To accept an ask, a buyer

raises a paddle with the buyer’s bidder number on it. The auction ends when no one accepts the

asking price and the auctioneer cries “Going once, going twice, sold!”

(v) Dutch or “clock” auction

An auctioneer starts an electronic clock at a high price. The clock then ticks down till a bidder

raises his hand (or hits a button to stop the clock.) This is the successful bidder and the price paid is

the price on the clock.

Strategy -29-

Equilibrium bidding in the electronic ascending price auction

If your light is still green when ip v , you incur a loss if you are the winner. Thus it is never

profitable to keep your light on green when the asking price ip v .

Strategy -30-

Equilibrium bidding in the electronic ascending price auction

If your light is still green when ip v , you incur a loss if you are the winner so it is never profitable

to keep your light on green when the asking price ip v .

If you switch your light to red when ip v , you win nothing. By leaving the light green, all the

other buyers may drop out before ip v so the final ask is below iv . You have thus missed out on

a possible profit. So it is never profitable to switch your light to red when ip v .

Thus buyer i ’s best response is to flip his switch when ip v .

Note that this is true regardless of the strategies of the other buyers. When this is the case, the

strategic equilibrium is called a dominant strategy equilibrium.

Strategy -31-

Equilibrium bidding in the sealed second price auction

Proposition: In the sealed second price auction it is a dominant strategy for buyer i to bid his

value iv .

Proof: Let m be the maximum of the bids of the opposing buyers.

Case 1: Change in payoff if iv m and buyer i does not bid iv

bids

Buyer i makes a loss as he pays

bids

Buyer i’s payoff is still zero

bids Buyer i’s payoff is still zero

Strategy -32-

Case 2: Change in payoff if im v and buyer i does not bid iv

Thus in every eventuality, buyer i either has the same payoff or a strictly lower payoff.

QED

bids

bids Buyer loses out on a profit

Buyer still wins and pays

Strategy -33-

F. Reserve prices

As a preliminary, consider the sale of an item to one buyer.

That buyer must win so the seller must set a minimum price (the “reserve “price) r . The single buyer

wins the item if and only if 1b r .

Buyer 1’s best response is to bid the minimum acceptable bid so 1b r .

The probability of a sale

1 1Pr{ } 1 Pr{ } 1 ( )v r v r F r      .

Therefore the expected profit of the seller is

Pr{item is sold} (1 ( )r r F r  

( ) 1 ( ) ( ) 1 ( ) ( )r F r rF r F r rf r       

Example: Uniform distribution [ , ] [0, ]   ( )
v

F v


 , hence
1

( )f v




( ) 1 ( ) ( ) 1 1
r r r

r F r rf r
  

         .

Then
* 1

2
r 

Strategy -34-

Alternative derivation: Consider the first order effects of raising the reserve price by r .

1Pr{ } ( )p v r F r  

1Pr{ }p r v r r    

11 Pr{ }p p v r r    

If 1r v r r   there is no sale

so the change in profit is r .

If 1v r r  the profit

rises by r .

The change in expected profit is therefore (1 )r p r p p    

(1 )r p r p r p      

first order

effects

second order

effect

The probability density function

Strategy -35-

The change in expected profit is

(1 )r p r p r p      

Therefore

1
p

r p p
r r

 
   

 
.

Ignoring the second order effect,

1
p

r p
r r

 
  

 

In the limit the second order effect vanishes and

1
d dp

r p
dr dr


   .

Since ( )p F r , ( ) ( )
dp

F r f r
dr

  .

( ) 1 ( )
d

rf r F r
dr


   = 0 for a maximum.

Strategy -36-

Reserve price in a two buyer sealed second price auction (so buyers bid their values)

Reminder: Probabilities

Pr{ } ( )ip v r F r  

Pr{ }ip r v r r    

1 Pr{ }ip p v r r    

The probability density function

Strategy -37-

We consider the effects on profit when the reserve price is raised from r to r r

when 1 2v v . Given symmetry, the effects of raising the reserve price when 2 1v v are the same.

Thus the total effect is doubled.

The equilibrium bid functions

Strategy -38-

No effect.

Case 1:

Strategy -39-

There are no bids so no effect.

Case 2:

Strategy -40-

The probability of this event is
2( )p .

So this is a second order effect.

Case 3:

Strategy -41-

The probability of this event is (1 )p p p  

The price paid rises from 2v to r r .

Thus the change in payment is 2r r v  and this lies between 0 and r .

Combining the first order probability and first order change in payment, this is a second order effect.

Case 4:

Strategy -42-

The probability is (1 )p p p  . The change in payment is r .

Case 5:

Strategy -43-

The probability is p p . The change in revenue is r

Case 6:

Strategy -44-

Case 5: The probability is (1 )p p p  . The change in payment is r .

Case 6: The probability is p p . The change in revenue is r

The change in expected profit is therefore (1 )p p p r p pr     

Ignoring the second order effect,

(1 )p p r p pr     .

Therefore

[(1 ) ]
p

p p r
r r

 
  

 

The bracketed expression is the marginal profit from raising r with one buyer.

Thus if *r is the unique solution to the necessary condition for a maximum with one buyer it is also

the maximizing reserve price for two buyers.

Remark: A very similar argument holds when there are three or more buyers.

Strategy -45-

F. Sealed high bid auction model

Private information: Each buyer’s value is private information.

Strategy -46-

Sealed high bid auction model

Private information: Each buyer’s value is private information.

Common knowledge: It is common knowledge that buyer i ’s value is an independent random

draw from a continuous distribution . We define

( ) Pr{ }iF v   .

This is called the cumulative distribution function (c.d.f.).

Strategy -47-

Sealed high bid auction model

Private information: Each buyer’s value is private information.

Common knowledge: It is common knowledge that buyer i ’s value is an independent random

draw from a continuous distribution . We define

( ) Pr{ }iF v   .

This is called the cumulative distribution function (c.d.f.).

The values: The values lie on an interval [0, ] .

Strategy -48-

Strategies

With private information a player’s action depends upon his private information. In the sealed high-

bid auction, a player’s private information is the value i that he places on the item for sale. His bid is

then some mapping ( )i i ib B  from every possible value (i.e. every i ) into a non-negative bid.

This mapping is the player’s bidding strategy.

Buyers with higher values have more to lose by not winning so it is natural to assume that buyers

with higher values will bid more so that ( )i iB  is a strictly increasing function.

Strategy -49-

Strategies

With private information a player’s action depends upon his private information. In the sealed high-

bid auction, a player’s private information is the value i that he places on the item for sale. His bid is

then some mapping ( )i i ib B  from every possible value (i.e. every i ) into a non-negative bid.

This mapping is the player’s bidding strategy.

Buyers with higher values have more to lose by not winning so it is natural to assume that buyers

with higher values will bid more so that ( )i iB  is a strictly increasing function.

Since we assume that each buyer’s value

is a draw from the same distribution it is

natural to assume that the equilibrium is

symmetric. ( ) ( )i i iB B 

Strategy -50-

Equilibrium Strategies

Bayesian Nash Equilibrium (BNE) strategies: With private information mutual best response

strategies are called Bayesian Nash Equilibrium strategies.

Symmetric BNE of the sealed high bid auction

If all other buyers other then buyer i use the bidding strategy ( )j jb B  then buyer i ’s best

response is to use the same strategy, i.e. ( )i ib B  .

Strategy -51-

An example: Two buyers with values uniformly distributed on [0,100] .

For the uniform distribution values are equally likely.

Therefore
25

Pr{ 25}
100

iv   ,
50

Pr{ 50}
100

iv   ,
80

Pr{ 80}
100

iv   …..

Thus the c.d.f. is ( ) Pr( )
100

i
i i iF v


   

For any guess as to the equilibrium strategy,

We can check to see if the guess is correct.

Strategy -52-

An example: Two buyers with values uniformly distributed on [0,100] .

For the uniform distribution values are equally likely.

Therefore
25

Pr{ 25}
100

i   ,
50

Pr{ 50}
100

i   ,
80

Pr{ 80}
100

i   …..

Thus the c.d.f. is ( ) Pr( )
100

i
iF v


   

For any guess as to the equilibrium strategy,

We can check to see if the guess is correct.

There are two buyers. Suppose that

buyer 2 bids according to the strategy

1
2 22

( )B   .

We need to show that buyer 1’s best response is to bid 11 12b  .

Then these strategies are mutual best responses.

Strategy -53-

Solving for buyer 1’s best response when his value is 1v

If buyer 1 bids b he has the high bid if 12 22( )B b   , i.e. 2 2b  .

Buyer 1’s win probability is therefore 2
2

( ) Pr{ 2 }
100

b
w b b   .

Buyer 1’s expected payoff is therefore

2
1 1 1 1 1

2 2
( , ) ( ) ( ) ( ) ( )

100 100

b
U v b v b w b v b vb b     

1
1 1

2
( , ) ( 2 )

100

U
v b v b


 



Therefore buyer 1’s expected gain is maximized if 11 12b v .

Then 1
2

( )j j jb B    is the equilibrium bidding strategy.

Strategy -54-

Exercise: Three buyers with values uniformly distributed

(a) Show that if buyer 2 and buyer 3 bid according to 1
2j j

b  , then buyer 1’s best response is to bid

1
1 12

b v when his value is 1v

(b) Show that for some 1
2

  , ( )j j jb B    is the equilibrium bidding strategy

Strategy -55-

Answer to (b)

The probability that buyer 1 wins with a bid of b is the joint probability that 2b b and 3b b , ie.

1 2 3( ) Pr{ } Pr{ }w b b b b b   

2 3Pr{ } Pr{ }v b v b    

2 2
2 3Pr{ } Pr{ } ( ) ( )

b b b b
v v F

   
     

2 2 3
1 1 1 1 12

1
( , ) ( ) ( ) ( )( ) ( )

b
U v b v b w b v b vb b

 
     

21
12

1
(2 3 ) 0

U
vb b

b 


  


for a maximum.

Therefore buyer 1’s best response is 21 1 13( )B v v .

Note that this is true if 2
3

  . Thus if the other buyers bid 2
3

( )j j jB   , then buyer 1’s best reponse

is to do so as well.

The problem with this approach is that it requires an inspired guess.

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