slide chapter 2-1 - essential...

Essential Microeconomics -1-

© John Riley March 18, 2013

STRATEGIC EQUILIBRIUM

One shot games

Strictly and weakly dominated strategies 5

Nash equilibrium 15

Common knowledge 21

Existence of NE 27

Correlated mixed strategies 36



One-shot games

Example: Two player economic game

Player , 1,2i i = is the manager of firm i.

Each player submits the price of the firm’s product for the next week to be posted on the web.

Each player sets a high price Hp or a low price Lp .

Let iA be the set of possible actions then { , }i H LA p p= .

If the information is posted on the web only after both players have submitted a price, then it does not

really matter who moves first or whether the players move simultaneously. The games are strategically

equivalent. Such games are called simultaneous move games. It is this class of games that we consider

first.



Simultaneous move games

Let I be the set of players, so that in the example {1,2}=I . A list of the actions taken in the game

by each player is called an outcome profile. Thus, in the example, an outcome profile is an action

vector 1 2( , )a a a= where i ia A∈ . We write the set of possible outcome profiles as 1 2A A A= × .

Associated with each outcome profile is a payoff ( ),iu a i∈I .

In two-player simultaneous move games it is often

convenient to represent the game in “matrix” or “normal”

form. Each of the possible payoff pairs 1 2( ), ( )u a u a is

listed in a cell in a matrix with the payoff to player 1

listed first.1

1 A game of this type is called a prisoner’s dilemma game. In such a game both prisoners have the opportunity to keep quiet and get a moderate sentence. However, if one “squeals” he will be set free while the other prisoner will be severely punished. The catch is that if both squeal the sentence will be longer than if both keep quiet.

Player 2

Hp Lp

Player 1 Hp 4,4 1,8

Lp 8,1 2,2



In the example, the payoffs in the game are chosen to

Reflect a typical situation facing two competing firms.

Both would prefer to set a high price rather than both set

a low price and thus achieve the payoff vector

( , ) (4,4)H Hu p p = rather than ( , ) (2,2)L Lu p p = .

However if one sets a low price and the other sets a high

price, the firm with the high price loses almost all of its

market share. The payoff vectors are

( , ) (8,1)L Hu p p = and ( , ) (1,8)H Lu p p = .

*

Player 2

Hp Lp

Player 1 Hp 4,4 1,8

Lp 8,1 2,2



In the example, the payoffs in the game are chosen to

Reflect a typical situation facing two competing firms.

Both would prefer to set a high price rather than both set

a low price and thus achieve the payoff vector

( , ) (4,4)H Hu p p = rather than ( , ) (2,2)L Lu p p = .

However if one sets a low price and the other sets a high

price, the firm with the high price loses almost all of its

market share. The payoff vectors are

( , ) (8,1)L Hu p p = and ( , ) (1,8)H Lu p p = .

Strictly dominated strategy

Note that player 1’s low price payoff vector is strictly greater than his high price payoff vector. If this

is the case we say that the low price strategy strictly dominates the high price strategy.

Same true for player 2. Eliminating the strictly dominated strategy yields a unique strategy profile

( , )L Lp p .

Player 2

Hp Lp

Player 1 Hp 4,4 1,8

Lp 8,1 2,2



Weakly dominated strategy

Now player 1’s low price payoff vector is greater than his

high price payoff vector, and strictly greater for one action.

If this is the case we say that the low price strategy

weakly dominates the high price strategy.

Same true for player 2. Eliminating the weakly dominated

strategies, the only remaining strategy is for each to choose

the low price.

Player 2

Hp Lp

Player 1 Hp 4,4 1,8

Lp 8,1 2,2



Elimination of dominated strategies

For an I player game the set of feasible outcome profiles is

1 1... {( ,... ) | , }I I i iA A A a a a A i= × × ≡ ∈ ∈I .2

Define 1 1 1( ,..., , ,..., )i i i Ia a a a a− − +≡ to be a feasible action of player i’s competitors. Using the product

notation, the set of such feasible actions is 1 1 1 1... ...

I

i i i I jjj j

A A A A A A− − + =≠

≡ × × × × × = × .

* 2 This is called the Cartesian product set or simply the product set of the action sets of the I players.



Elimination of dominated strategies

For an I player game the set of feasible outcome profiles is

1 1... {( ,... ) | , }I I i iA A A a a a A i= × × ≡ ∈ ∈I .3

Define 1 1 1( ,..., , ,..., )i i i Ia a a a a− − +≡ to be a feasible action of player i’s competitors. Using the product

notation, the set of such feasible actions is 1 1 1 1... ...

I

i i i I jjj j

A A A A A A− − + =≠

≡ × × × × × = × .

An action i ia A∈ is said to be strictly dominated if there is some alternative action i ia A∈ that yields a

strictly higher payoff regardless of all the other players’ actions.

An action ia is said to be weakly dominated if ia yields a payoff that is at least as high regardless of

other’s actions and a strictly higher payoff for at least one action of his opponents.

Definition: Strictly and Weakly dominated action

Player i’s action ia is strictly dominated by ia if ( , ) ( , ),i i i i i i i iu a a u a a a A− − − −> ∀ ∈ .

Player i’s action ia is weakly dominated by ia if ( , ) ( , ),i i i i i i i iu a a u a a a A− − − −≥ ∀ ∈ and the inequality

is strict for some i ia A− −∈ . 3 This is called the Cartesian product set or simply the product set of the action sets of the I players.



We now modify the game and assume that there are three possible prices, , andH M Lp p p . The

payoff matrix is shown below.

Player 2

Hp Mp Lp

Hp 100,100 30,150 -40,90

Player 1 Mp 150, 30 50, 50 5,60

Lp 90,-40 60,5 10,10

For each player the one strictly dominated strategy is choosing the high price. We eliminate this

strategy.



After elimination of the dominated strategy we have the following payoff matrix.

Player 2

Mp Lp

Player 1 Mp 50,50 5,60

Lp 60,5 10,10

Payoff matrix after deletion of dominated strategy

Now choosing the low price strictly dominates choosing the medium price and so Lp is the unique

price surviving the successive (or “iterated”) elimination of strictly dominated strategies.



Pure and Mixed strategies

Thus far we have considered strategies that involve choosing a single action. This is called a pure

strategy. More generally, a player may have an interest in randomizing over his actions. Such a

strategy is called a mixed strategy.4 Given the finite action set 1{ ,..., }ii i imA a a= , a mixed strategy

assigns a probability to each of the possible actions. Player i’s strategy set iS is then the set of all

probability measures in

1

( ) { | 0, 1}im

i jj

A π π π=

∆ = ≥ =∑

4 If a player places strictly positive probability on every action is his action set, the strategy is said to be totally mixed.



As the following example shows, we can sometimes eliminate an action because it is dominated by a

mixed strategy.

player 2

Left Right

player 1

0 Top -1,6 -1,8 23 Middle -2,1 4,0

13 Bottom 4,2 -8,1

Table 9.1-3: Top is a Dominated strategy

Player 1 chooses from the action set 1 { , , }A Top Middle Bottom= and player 2 from action set

2 { , }A Left Right=

Thus Top is strictly dominated by this mixed strategy.

Exercise: What strategies survive the iterated elimination of dominated strategies?



Definition: Strict (weak) dominant strategy equilibrium

The outcome profile 1( ,..., )Ia a is a strict (weak) dominant strategy equilibrium if for all i∈I , every

other action in player i’s action set iA is strictly (weakly) dominated.

Definition: Iterated weak dominance equilibrium

The outcome profile 1( ,..., )Ia a is an iteratively weakly dominant strategy equilibrium if for all i∈I ,

every other action in player i’s action set iA is iteratively weakly dominated.



Is the weak dominance equilibrium a good predictor of how a game will be played?

Consider the following example.

Player 2

a b c

a 4,4 1,5 0,2

Player 1 b 5,1 2,2 3,1

c 2,0 1,3 3,3

Table 9.1-4: Equilibrium in weakly dominated strategies

What if the two players can talk first?



Mutual best response equilibrium (Nash Equilibrium)

Consider the following thought experiment for a game to be played by Alex and Bev. Before the game

is played, each is invited to submit a proposed strategy. These proposed strategies are then revealed to

the players. Alex can ask himself the two following questions.

1. Will it be a best response for me to do what I said I would do if Bev does what she said she

would do?

2. Will it be Bev’s best response to do what she said she would do if I do what I said I would do?

If the answer to both of these questions is yes, then the announced strategies are called mutual best

responses.

With more than two players, exactly the same thought experiment is possible.

Strategies that are mutual best response are called Nash equilibrium (NE) strategies.5

5 The mutual best response equilibrium concept was first used formally by Augustin Cournot (1842) in his analysis of duopoly. However it was

named after John Nash who was the first to apply a fixed point theorem to establish a general existence theorem for simultaneous move games.



Example: Partnership game.

Two players have an equal share in a partnership.

Let ia be the effort level of player i.

Total revenue is 1 2( ) 12R a a a= .

The cost to player i is 3( )i i iC a a= .

For simplicity we will assume that {1,2,3}iA = . Each player gets a 50% share of total revenue. Neither

the effort level ia nor the cost ( )i iC a is observed by his partner (player i− ) . Thus the two players play

a simultaneous move game.

Payoffs 312( , ) ( , )i i i i i iu a a R a a a− −= −



Player 2

2 1a = 2 2a = 2 3a =

1 1a = 5,5 11,4 17,-9

Player 1 1 2a = 4,11 16,16 28,9

1 3a = -9,17 9,28 27,27

Table 9.1-5: Partnership game

Class Exercise: What are the NE strategies?



Partnership game where the payoffs are continuous functions

Player i’s payoff is 312( , ) ( ) ( ) 6i i i i i i i iu a a R a C a a a a− −= − = − .

We look for an equilibrium in pure strategies. For any ia− , player i’s set of best responses is

( ) arg { ( , )}i

i i i iaBR a Max u a a− −=



Partnership game where the payoffs are continuous functions

Player i’s payoff is 312( , ) ( ) ( ) 6i i i i i i i iu a a R a C a a a a− −= − = − .

We look for an equilibrium in pure strategies. For any ia− , player i’s set of best responses is

( ) arg { ( , )}i

i i i iaBR a Max u a a− −=

First Order Conditions

26 3 0ii i

i

u a aa −

∂= − =

∂, 1,2i = .

Solving, the unique best response is ( ) 2i i i ia BR a a− −= = .

Therefore if 1 2( , )a a is a NE, 1 22a a= and 2 12a a= . Solving, 1 2( , ) (2,2)a a = .



Formally, for any strategy profile 1 1( ) ... ( )I Is S A A∈ = ∆ × ×∆ let ( )i iBR s− be the set of strategies

for player i that maximize player i’s payoff. We can then define a “best response mapping” as follows:

1 1( ) ( ( ),..., ( ))I IBR s BR s BR s− −= .

If for each i ( )i i is BR s−∈ the strategies are mutual best responses, that is, the strategy profile is a NE.

Definition: Nash Equilibrium

For a simultaneous move game played by players 1,.., I , the strategy profile 1( ,..., )Is s s= is a Nash

equilibrium if the strategies are mutual best responses. That is, for each i∈I and all i ia A∈ ,

( , ) ( , )i i i i i iu s s u a s− −≥ .



Common knowledge of the game

Typically the outcome of a game depends on the actions taken by all the participants. Thus player i’s

choice requires him to form beliefs about the possible actions that will be taken by his competitors. To

do so, player i must know the other players’ actions sets and their preferences over these actions. Thus

we assume that the game being played (I, A, u) is known by all the players.

But even this is not enough. Player i must also know that the other players know the game being

played. And he must also know that that others know that he knows that others know the game being

played and so on. Players who have all these higher orders of knowledge of the game are said to have

common knowledge of the game.



Common knowledge of the game

Example:

Alex and Bev both have a coin. Each writes down either Heads or Tails. (These are the players’

simultaneous actions.) The coins are then tossed. Each coin will come up Heads with probability 1/3.

If the players’ actions and the two coins all coincide they each receive $1000. Otherwise they get

nothing.

Class Exercise: NE if players have common knowledge

NE without the common knowledge assumption

Alex and Bev correctly believe that both coins will come up Heads with probability 1/3.

Alex incorrectly believes that Bev believes that Alex’s coin will come up Heads with probability 1.

Bev incorrectly believes that Alex believes that Bev’s coin will come up Heads with probability 1.

Class exercise: NE without common knowledge

HINT: What is the implications if Alex believes that Bev’s coin will come up Heads with

probability 1.Then what will Bev do?



Common knowledge of rationality

A rational player is one who makes choices consistent with his preferences. Just as with the

parameters of the game, there are higher orders of rationality. It is not enough for player i to know that

other players choose according to their preferences. He must also know that the other players know

that he is rational as well. And he must also know that others know that he knows that they are rational

and so on. Players who have all these higher orders of rationality are said to have common knowledge

of rationality.



Common knowledge of the equilibrium strategy profile

It is important to note that appealing to the NE as a solution concept requires an even stronger

common knowledge assumption: knowledge that the other players are playing their equilibrium

strategies. This is an especially strong assumption if there are multiple equilibria. It is far from clear

how beliefs can converge on any particular equilibrium. In some environments this issue can be

ameliorated by pre-play communication. Suppose that there are three equilibria but one has some

“nice” property. For example one equilibrium might be Pareto preferred over the others or one might

be seen as fairer than the others. Discussion of the “nice” property of an equilibrium might well

persuade all players that it will be played.



Common knowledge when there is a unique NE

Even if there is a unique NE its predictive power relies on the common knowledge assumption.

For without it, a player may believe that some of his opponents are cooperating rather than

independently pursuing their self interests. Given such a belief, the player left out in the cold will

typically not want to play his NE strategy.

Example: Simultaneous move three-player matching game.

Each player must choose either Heads or Tails. Player 3’s choice determines which of the two matrixes

shown below is the payoff matrix.

player 2 player 2

Heads Tails Heads Tails

player 1 Heads 2,2,2 1,0,4

player 1 Heads 0,0,0 -1,3,3

Tails 0,1,1 0,2,1 Tails 0,0,0 0,0,0

player 3 chooses Heads player 3 chooses Tails



Note that player 3’s strictly dominant strategy is to

choose Heads so we delete this strategy.

It is readily checked that after deletion of player 3’s

strictly dominated strategy the unique NE strategy

profile is for player 1 and 2 to both choose Heads.

However note that when player 1 chooses Heads, players 2 and 2 can do better than in the NE if they

can agree to both play Tails since each gets a payoff of 3. But if they do collude, player 1’s best

response is no longer Heads.

Thus choosing Heads is a best response for player 3 only given that it is common knowledge that the

strategy profile is for everyone to choose Heads.

player 2

Heads Tails

player 1 Heads 2,2,2 1,0,4

Tails 0,1,1 0,2,1



Existence of Nash Equilibrium

For games with finite strategy sets we can appeal directly to Kakutani’s fixed point theorem.

Kakutani’s Fixed Point Theorem

If nS R⊂ is compact and convex and if φ is an upper hemi-continuous correspondence from S to S

such that for all x S∈ the set ( )xφ is nonempty and convex, then φ has a fixed point.

The mapping : S Sφ → is depicted below:

The correspondence is said to have a fixed point if for some , ( )s S s sφ∈ ∈ .

Fig 9.1-1: Fixed point



Finite games

Consider any game ( , , )A uI where the strategy sets are finite.

A strategy for player i∈I is a probability measure in ( )iA∆ , the set of all probability measures on

player i’s feasible actions.

Then the strategy set 1( ) ... ( )IS A A= ∆ × ×∆ is compact and convex.

**



Finite games





For each i∈I and any strategy profile 1( ,..., )Is s s= , let *( )i iBR s− be the set of pure strategies that are

best responses for player i.

Since the game is finite, *( )i iBR s− is non-empty and finite.

Since all the elements of *( )i iBR s− have the same payoff, all probabilistic mixtures of these actions also

have the same payoff and are therefore best responses as well.

*



Finite games





For each i∈I and any strategy profile 1( ,..., )Is s s= , let *( )i iBR s− be the set of pure strategies that are

best responses for player i.

Since the game is finite, *( )i iBR s− is non-empty and finite.

Since all the elements of *( )i iBR s− have the same payoff, all probabilistic mixtures of these actions also

have the same payoff and are therefore best responses as well.

Let ( )i iBR s− be the full set of best responses, pure and mixed. Since the mixed strategies are all the

convex combinations of the pure strategies, ( )i iBR s− is compact and convex. Then we define the

compact and convex response mapping

1 1( ) ( ( ).,,,, ( ))I Is BR s BR sφ − −= .



Appealing to the Theorem of the maximum1, the best response mapping is upper hemi-

continuous. Then the mapping : S Sφ → satisfies all the requirements of Kakutani’s theorem.

Therefore we have the following result first proved by Nash (1950).

Proposition 9.1-1: Existence of Equilibrium

In a game with finite action sets, if players can choose either pure or mixed strategies, there exists a

NE.

_________ 1See Appendix C



Existence when the action sets are compact and convex subsets of n

A best response is the solution of a maximization problem. If the feasible set of a player is a closed

convex subset of some Euclidean space, we can exploit the tools of calculus to characterize the set of

best responses. Thus it is very often preferable to model games in which players make choices over

compact, convex subsets of Euclidean space rather than finite action sets. As we shall now show, under

such an assumption there is very often a NE in pure strategies.



Suppose that player i∈I has an action set niA ⊂ that is compact and convex. Suppose also

that each player’s utility function ( )iu a is continuous. Then for each ia− there is a solution to the

maximum problem

( , )i i

i ia AMaxu a a−∈

.

The set of all such solutions is the best response mapping ( )i iBR a− and so this mapping is non-empty.

Appealing to the Theorem of the maximum, the best response mapping ( )i iBR a− is also upper hemi-

continuous.

Since we are considering only pure strategies, we write the strategy set as 1 ... IA A A= × × and

define

1 1( ( ),..., ( ))I IBR a BR aφ − −= .

The best response functions are upper hemi-continuous therefore : A Aφ → is upper hemi-continuous.

This is still not quite enough. Kakutani’s fixed point theorem requires that the mapping φ be convex

valued. Hence we have the following theorem.



Proposition 9.1-2: Existence of an equilibrium in pure strategies

Let ( , , )A uI be a game in which niA R⊂ , i∈I is compact and convex and u is continuous. If the

best responses sets ( ) ,i i iBR a A i− ⊂ ∈I are convex, there exists a NE in pure strategies.

A special case in which the best responses sets are convex is when the best response mapping is

single-valued.

__________________

1This is the case, for example, if ( , ),i i iu a a i− ∈I is a strictly quasi-concave function of ia



More generally, we have the following corollary.

Corollary 9.1-3: Existence of an equilibrium in pure strategies

Let ( , , )A uI be a game in which niA R⊂ , i∈I is compact and u is continuous. If (i) the best

response mapping is single-valued or (ii) ( , )i i iu a a− i∈I is quasi-concave in ia , there exists a NE in

pure strategies.

Proof: Suppose ( , )i i iu a a− is quasi-concave in ia . Suppose that 0ia and 1 ( )i i ia BR a−∈ . Then

0 1( , ) ( , )i i i i i iu a a u a a− −= . Given the quasi-concavity if ( , )i i iu a a− it follows that for all convex

combinations iaλ , 0( , ) ( , )i i i i i iu a a u a aλ− −≥ . Hence all convex combinations must be best responses as

well.

Q.E.D.



Correlated mixed strategies

Consider the simple two-player simultaneous move pricing game below.

Player 2 (Bev) Probabilities

High Middle High Middle

Player 1

(Alex)

High 100,100 40,140 a b

Middle 140,40 30,30 c d

Table 9.1-7: Correlated equilibrium

NE in pure strategies ( , )High Middle (40,140)NEu = ( , )Middle High (140,40)NEu =

NE in mixed strategies Each player chooses High with probability ½ (52,52)NEu = .

Note that the average payoff is higher in the asymmetric equilibria.

Better to mix over equilibria.

Players gain by finding some way of randomizing over which NE strategy to play. Then both can

achieve an expected payoff of 90.



Public randomization device

The local newspaper provides the players with a simple “correlation device” that achieves this

goal. Each day the winning lottery number is announced. The players then agree that if the winning

number is odd, it will be player 1 that will choose Middle and player 2 High. If the winning number is

even the choices will be switched. If each player believes that the other player will follow the

agreement, then his best response is to follow the agreement as well. The correlated strategies are

therefore NE strategies.



Private randomization device

More generally a correlation device instructs the players how to play. The players design a

software program to send them messages with predetermined probabilities. Each receives a private

message.

Player 2 (Bev) Probabilities

High Middle High Middle

Player 1

(Alex)

High 100,100 40,140 a b

Middle 140,40 30,30 c d

Table 9.1-7: Correlated equilibrium

Consider Table 9.1-7. With probability a player 1 receives a message to choose High and player 2 does

also. With probability b the correlation device recommends High to the row player and Middle to the

column player and so on. Of course the probabilities must add to 1 so 1a b c d+ + + = . Thus using the

lottery as a correlation device means agreeing to choose 1 12 2( , , , ) (0, , ,0)a b c d = .

Consider symmetric correlated equilibria with 0d = so 12 (1 )b c a= = − .



Need to show that it is a NE for both players to follow the private recommendation that they receive.

Suppose that the column player follows his recommendation.

Case (i) The row player receives the recommendation Middle.

He knows that the column player has received a recommendation to play High (since d = 0) so Middle

is his best response.

Case (ii) The row player receives the recommendation Middle.

If the row player receives the recommendation High, he knows that the column player has received the

recommendation High with probability / ( )a a b+ (because of Bayes’ rule) and Middle with probability

/ ( )b a b+ . If he plays High his expected payoff is

1( ) 100 40 40 1601 1

R a b aU Ha b a b a a

= + = ++ + + +

, since 12

aa b ++ = .

If he chooses Middle his expected payoff is

1( ) 140 30 30 2501 1

R a b aU Ma b a b a a

= + = ++ + + +

.



Thus accepting the recommendation is a best response if

1 1 10( ) ( ) 40 160 30 250 (1 9 ) 01 1 1 1 1

R R a aU H U M aa a a a a

− = + − − = − ≤+ + + + +

That is, 1 / 9a ≤ .

Given the symmetry of the game it follows that the strategies recommended by the correlation device

are NE strategies as long as. 1/ 9a ≤

**




1 1 10( ) ( ) 40 160 30 250 (1 9 ) 01 1 1 1 1


− = + − − = − ≤+ + + + +

That is, 1 / 9a ≤ .



From Table 9.1-7, the expected value of the game to each player is

1 12 2100 (1 )40 (1 )140 90 10a a a a+ − + − = + .

Thus expected payoffs are maximized by setting 1 / 9a = and b = c = 12 (1 )a− = 4/9.

*




1 1 10( ) ( ) 40 160 30 250 (1 9 ) 01 1 1 1 1


− = + − − = − ≤+ + + + +

That is, 1 / 9a ≤ .



From Table 9.1-7, the expected value of the game to each player is

1 12 2100 (1 )40 (1 )140 90 10a a a a+ − + − = + .

Thus expected payoffs are maximized by setting 1 / 9a = and b = c = 12 (1 )a− = 4/9.

Note that the set of correlated equilibria includes the three uncorrelated equilibria. For the two pure

strategy equilibria, set b or d =1. For the mixed strategy equilibrium, set 2 1 / 25a p= = ,

(1 ) 4 / 25b c p p= = − = and 2(1 ) 16 / 25d p= − = .

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