EC487 Advanced Microeconomics, Part I: Lecture 6econ.lse.ac.uk/staff/lfelli/teach/EC487 Slides Lecture 6.pdf · EC487 Advanced Microeconomics, Part I: Lecture 6 Leonardo Felli 32L.LG.04

EC487 Advanced Microeconomics, Part I:Lecture 6

Leonardo Felli

32L.LG.04

3 November, 2017

Game Theory

I It is the analysis of the strategic interaction among agents.

I This is a situation in which each agent when deciding how tobehave explicitly takes into account the decision of the otheragents that interact with him.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 2 / 54

Example: Entry Game

I Two individuals have to decide whether to sell newspapers ata given exit of the underground.

I They take this decision without observing the decision takenby the other individual.

I If only one individual decides to locate herself at the exit shewill make the highest level of profits since she will serve allclients. Let this profit be £300.

I If both individuals decide to locate themselves at the exit thenclients are equally shared (we assume newspaper prices arepre-set). Each individual’s profit is £150.

I Finally if an individual does not locate herself at the exit thanshe makes zero profits.


Example (cont’d)

I We can describe the situation with the following table:

1\2 E NE

E 150, 150 300, 0

NE 0, 300 0, 0

I Rows denote individual 1’s decisions.

I Columns denote individual 2’s decisions.

I The first number of each ordered pair denotes individual 1’sprofit, while the second number denotes individual 2’s profit.


Example (cont’d)

I Notice that predicting the outcome of this situation is fairlyeasy provided that we assume that both individuals wants tomaximize profits, in other words they are rational.

I The predicted outcome is that both individuals locatethemselves at the exit (E ,E ).

1\2 E NE

E 150, 150 300, 0

NE 0, 300 0, 0

I Notice that this conclusion can be reached without requiringeach individual to make a prediction on the behavior of theother individual.


Battle of Sexes

I This is not true in general.

I Consider for example the following situation known as battleof sexes:

1\2 B S

B 1, 2 0, 0

S 0, 0 2, 1

I In this case each individual needs to make a prediction on thebehavior of the other individual.


Coordination Game

I Consider the following simple coordination game (no conflictof interest).

I There is still a need for predictions:

1\2 M C

M 2, 2 0, 0

C 0, 0 1, 1

I Notice that we will be more confident in our prediction if theindividuals involved encounter this situation more than once.


Jargon and Definitions

I The strategic situations we described above are known asgames.

I A simple static game or game in normal (strategic) form (notime dimension) comprises three elements:

1. Set of players, economic agents:

N = {1, . . . , I}

2. For each player i ∈ N an action space, or a pure strategy spacedenoted Ai .

This is the set of choices available to each player:

A1 = {locate at the exit, do not locate at the exit}.


Definitions

I Denote: ai ∈ Ai player i ’s strategy choice;

I Then a−i = (a1, . . . , ai−1, ai+1, . . . , aI ) is the strategy profileof every player but player i .

I Therefore a = (ai , a−i ) ∈ A1 × . . .× AI = A.

I Finite games are games with finite strategy spaces (a finitenumber of strategies).


Definitions (cont’d)

3. Finally define for each player i ∈ N a payoff functionassociated with his strategy choice ai and the other players’strategy choice a−i :

ui (a1, . . . , aI ) = ui (ai , a−i ) = ui (a).

I The payoffs ui (·) is taken to be the utility representation ofplayer i ’s preferences.

I The objective of game theoretic analysis is to give predictionson the behavior of agents in strategic situations.


Rationality:

I What assumptions do we need on the players’ behavior todeliver these predictions?

I First assumption rationality (maximization of utility orpayoff).

I In our example above rationality and knowledge of own payoffis enough to deliver a prediction:

1\2 E NE

E 150, 150 300, 0

NE 0, 300 0, 0


Prisoners’ dilemma

I An other classic example of a situation in which rationalityand knowledge of own payoff is enough to deliver a predictionis the the prisoners’ dilemma game.

I This is characterized by the following normal form:

1\2 C NC

C 0, 0 4,−1

NC −1, 4 3, 3


Prisoners’ dilemma (cont’d)

I The three elements of the game are:

I N = {1, 2},

I Ai = {C ,NC},

I u1(C ,C ) = u2(C ,C ) = 0, u1(NC ,C ) = u2(C ,NC ) = −1,u1(C ,NC ) = u2(NC ,C ) = 4, u1(NC ,NC ) = u2(NC ,NC ) = 3.

1\2 C NCC 0, 0 4,−1NC −1, 4 3, 3


Prisoners’ dilemma (cont’d)

Consider:

1\2 C NC

C 0, 0 4,−1

NC −1, 4 3, 3

I Each player will choose the strategy C independently of theaction chosen by the other player.

I The predicted outcome is therefore (C ,C ). This is clearly theinefficient outcome, it is Pareto dominated by (NC ,NC ).

I The only information needed to make a prediction is the factthat players are rational and they know their own payoffs.


Knowledge of Rationality

I Consider now the following modification of the previous game:

1\2 L C R

T 0, 0 4,−1 1,−1

M −1, 4 3, 3 3, 2

B −1, 2 0, 1 4, 1

I In this case we need some extra assumptions to make aprediction on the outcome of the game.


Knowledge of Rationality (cont’d)

I Indeed:

1\2 L C R

T 0, 0 4,−1 1,−1

M −1, 4 3, 3 3, 2

B −1, 2 0, 1 4, 1

I L dominates C and R for player 2;

I if player 1 knows that player 2 is rational then he will focusonly on the first column;


Knowledge of Rationality (cont’d)

I Therefore:

1\2 L

T 0, 0

M −1, 4

B −1, 2

I In the first column T dominates M and B.

I The prediction is therefore (T , L).


Relevant Assumptions:

I The information needed to make a prediction is then:

I both players are rational;

I both players know their own and the other player’s payoff;

I player 1 knows that player 2 is rational.


Common Knowledge of Rationality

I Consider now the following game:

1\2 L C R

T 1, 0 1, 2 0, 1

B 0, 3 0, 1 2, 0

I Player 2 will never play R since R is a strictly dominatedstrategy and both players are rational and know each otherpayoffs.

I Since player 1 knows that player 2 is rational he also knowsthat R will never be played.


Common Knowledge of Rationality (cont’d)

I Notice now that:1\2 L C

T 1, 0 1, 2

B 0, 3 0, 1

I For player 2 none of the remaining strategies is strictlydominated:

I if player 2 believes that player 1 will play B then 2 will chooseL;

I while if player 2 believes that player 1 will play T then 2 willchoose C .



I However if we now assume that: player 2 knows that player 1knows that player 2 is rational

I then player 2 knows that player 1 realizes that he will neverplay R so for all intents and purposes the game is:

1\2 L C

T 1, 0 1, 2

B 0, 3 0, 1

I In this new game player 1’s strategy B is strictly dominated,therefore 1 will never choose it.

I Therefore since player 2 knows that player 1 is rational thepredicted outcome will be (T ,C ).



I The assumptions needed to make this prediction are then:

I that both players are rational;

I that both players know their own and the other player’s payoff;

I that both players know that the other player is rational;

I that player 2 knows that player 1 knows that player 2 isrational.


Necessary Assumptions:

I A set of necessary assumptions used in non-cooperative gametheory to predict an outcome are:

I rationality of the players;

I common knowledge of the rationality of the players:

I player i knows that player j is rational,I player i knows that player j knows that player i is rational,...I player i knows that player j knows that player i knows that

. . . player i is rational

I common knowledge of the structure of the game.


Is rationality a good assumption?

I Consider the following game:

1\2 L R

U 8 10 −1, 000, 000, 000, 000 9

D 7 6 6 5

I Notice that: player 2’s strategy R is strictly dominated by L.

I Since both players are rational and know that the other playeris rational then player 1 knows that player 2 will never play R.

I Therefore player 1 chooses U and the predicted outcome is(U, L).


Is rationality a good assumption? (cont’d)

I Notice however that:

1\2 L R

U 8 10 −1, 000, 000, 000, 000 9

D 7 6 6 5

I Player 1 better be absolutely sure of player 2’s rationality!

I Any mistake is extremely costly (−1, 000, 000, 000, 000).


Stronger Predictions

I We have identified a first set of behavioral assumptions thatallow us to make predictions on simple normal form games.

I These are very reasonable assumptions — the last examplenotwithstanding.

I However reasonable comes at a cost.

I In most situations these assumptions are not enough to allowus to make specific predictions.


Battel of Sexes II

I Consider once again the game:

1\2 B2 S2

B1 1, 2 0, 0

S1 0, 0 2, 1

I Rationality of the players, common knowledge of therationality of the players and common knowledge of thestructure of the game are not enough.

I We need a sharper tool for a prediction: Nash equilibrium.

I We need to impose a restriction on the beliefs that the playershave on the behavior of other players.

I We will require these beliefs to be correct in equilibrium.


Nash Equilibrium Construction:

The basic building block of a Nash equilibrium is each player’s bestresponse to the behavior of the other players.

Definition (Best Response)

Consider a general game in normal form

{N;Ai ,∀i ∈ N; ui (a), ∀i ∈ N}

The best response (reply) of a player i to the behavior of the otherplayers is player i ’s strategy choice(s) ai that maximizes i ’s utilitygiven the other players’ strategy choice a−i .

Bi (a−i ) = {ai ∈ Ai | ui (ai , a−i ) ≥ ui (a′i , a−i ),∀a′i ∈ Ai}.


Nash Equilibrium Construction (cont’d):

I This best response (correspondence) associates to any givenstrategy profile of all the other players a−i , player i ’sstrategies that maximize player i ’s payoff ui (ai , a−i ):

I A Nash equilibrium in pure strategies is a strategy profile suchthat each player’s strategy choice is a best response to thestrategy choice of the other players.


Nash Equilibrium Construction (cont’d):

Definition (Pure Strategy Nash Equilibrium)

Definition: A pure-strategy Nash equilibrium is a strategy profilea∗ = (a∗i , a

∗−i ) such that for every i ∈ N

ui (a∗i , a∗−i ) ≥ ui (ai , a

∗−i ) ∀ai ∈ Ai .

ora∗i ∈ Bi (a∗−i ) ∀i ∈ N.

Notice that according to the definition above in equilibrium thebeliefs of each player are indeed correct.


Example


1\2 L R

U 3, 2 2, 0

D 0, 0 1, 1

I Notice that:

B1(L) = {U} B1(R) = {U}

I andB2(U) = {L} B2(D) = {R}

I Hence the unique pure strategy Nash equilibrium of such agame is (U, L).


No guarantee of uniqueness:

I Consider one more time the battle of sexes game:

1\2 B2 S2

B1 1, 2 0, 0

S1 0, 0 2, 1

I Clearly:B1(B2) = {B1} B1(S2) = {S1}

I andB2(B1) = {B2} B2(S1) = {S2}

I There exist two pure strategy Nash equilibria of such a game:(B1,B2) and (S1, S2).


Best Response Correspondence

I Notice that the best response in both games above associatesa unique strategy ai to every vector of strategies a−i , the bestreply is a single-valued function.

I Indifference may lead to more than one strategy ai in the bestreply correspondence associated with a given a−i .

I The definition of Nash equilibrium is such that: whenindifferent between two strategies both strategies are part ofthe best response of a player.


Example


1\2 L C R

U 3, 2 2, 0 4, 2

M 0, 0 1, 1 5, 0

D 1, 2 2, 2 0, 3

I Notice that:

B1(L) = {U} B1(C ) = {U,D} B1(R) = {M}

I and

B2(U) = {L,R} B2(M) = {C} B2(D) = {R}

I The unique pure strategy Nash equilibrium is: (U, L).


Indifference

I The underlying behavioral assumption is that: whenindifference a player will choose the strategy that sustains theequilibrium.

I Indifference plays a big role in the characterization of theproperties of Nash equilibrium.

I In particular the whole proof of existence (Nash Theorem) willbe essentially based on indifference.


Existence

I Consider now the following game, known as matching pennies:

1\2 H2 T2

H1 1,−1 −1, 1

T1 −1, 1 1,−1

I Notice that:

B1(H2) = {H1} B1(T2) = {T1}

I andB2(H1) = {T2} B2(T1) = {H2}

I Clearly there does not exist any pure strategy Nashequilibrium of this game.


Mixed Strategies

I To be able to have predictive power in strategic situations likethe one described in matching pennies we need to extend thedefinition of strategy.

I Suppose player 1 tries to be as unpredictable as possible.

I In other words, player 1 randomizes with probability p and(1− p) between the choice H1 and the choice T1.

I Assume that if player 1 is unpredictable so is player 2.

I In other words, player 2 randomizes with probability q and(1− q) between the choice H2 and the choice T2.


Mixed Strategies (cont’d)

H2 T2

H1 1,−1 −1, 1

T1 −1, 1 1,−1p

1− p

q 1− q

I Player 2’s best reply is then obtained solving the followingproblem:

maxq

q[p u2(H1,H2) + (1− p) u2(T1,H2)

]+

+ (1− q)[p u2(H1,T2) + (1− p)u2(T1,T2)

]=

= q [p (−1) + (1− p)(1)] ++ (1− q) [p (1) + (1− p)(−1)]



I In other words:

maxq

q [1− 2 p] + (1− q) [2 p − 1]

I The solution is then:

q = 1 if p < 1/20 ≤ q ≤ 1 if p = 1/2q = 0 if p > 1/2



I Consider now player 1’s best reply. This is obtained solvingthe following problem:

maxp

p[q u1(H1,H2) + (1− q) u1(H1,T2)

]+

+ (1− p)[q u1(T1,H2) + (1− q)u1(T1,T2)

]=

= p [q (1) + (1− q)(−1)] ++ (1− p) [q (−1) + (1− q)(1)]

I In other words:

maxp

p [2 q − 1] + (1− p) [1− 2 q]

I The solution is then:

p = 1 if q > 1/20 ≤ p ≤ 1 if q = 1/2p = 0 if q < 1/2



The mixed strategy Nash equilibrium is therefore:

p =1

2and q =

1

2

Definition (Mixed Strategy)

A mixed strategy σi is a probability distribution (randomization)defined over player i ’s pure strategy space Ai (it includes purestrategies).

Let the set of possible probability distributions (mixed strategies)over Ai be ∆(Ai ).

If the game considered is finite (Ai finite with n strategies) then∆(Ai ) is the (n − 1)-dimensional simplex.



I The key assumption on mixed strategies is that each player irandomizes independently from other players: σi independentof σj for i 6= j .

I The mixed extension of the gameΓ = {N;Ai ,∀i ∈ N; ui (a), ∀i ∈ N} is the game:

Γ∆ = {N; ∆(Ai ),∀i ∈ N;Ui (σ),∀i ∈ N}

I where σi ∈ ∆(Ai ) and

Ui (σ) =∑a∈A

(σ1(a1) · . . . · σI (aI )

)ui (a1, . . . , aI )


Mixed Strategy Nash Equilibrium

Definition (Mixed Strategy Nash Equilibrium)

A mixed strategy Nash equilibrium is a mixed strategy profileσ∗ = (σ∗i , σ

∗−i ) such that for every player i ∈ N

σ∗i = arg maxσi∈∆(Ai )

Ui (σi , σ∗−i )


Battel of Sexes III

I Consider the mixed strategy Nash equilibria of the battle ofsexes game:

1\2 B2 S2

B1 1, 2 0, 0

S1 0, 0 2, 1p

1− p

q 1− q

I Define:I p player 1’s mixed strategy (the probability with which player 1

plays B1);I q player 2’s mixed strategy (the probability with which player 2

plays B2).

I Strategy spaces:

∆(p) = [0, 1] and ∆(q) = [0, 1].


Battel of Sexes III (cont’d)

I Payoffs:

U1(p, q) = p [q u1(B1,B2) + (1− q)u1(B1,S2)] +

+ (1− p) [q u1(S1,B2) + (1− q)u1(S1, S2)] =

= p q + (1− p)2(1− q)

and

U2(p, q) = q [p u2(B1,B2) + (1− p)u2(S1,B2)] +

+ (1− q) [p u2(B1,S2) + (1− p)u2(S1, S2)] =

= q 2 p + (1− q)(1− p)



I The mixed strategy best reply for player 1 is then:

p = 1 if q > 2/30 ≤ p ≤ 1 if q = 2/3p = 0 if q < 2/3

I The mixed strategy best reply for player 2 is then:

q = 1 if p > 1/30 ≤ q ≤ 1 if p = 1/3q = 0 if p < 1/3


Graphically:

-

6

q

p

(13 ,

23

)

B1(q)

B2(p)

(1, 1)

u

u

u

(0, 0) q = 1q = 2/3

p = 1

p = 1/3



I The mixed strategy Nash equilibrium of the game is (p∗, q∗)such that:

p∗ = arg maxp

p q∗ + (1− p) 2 (1− q∗)

and

q∗ = arg maxq

q 2 p∗ + (1− q)(1− p∗)

I There exist three mixed strategy Nash equilibria:

(0, 0),

(1

3,

2

3

), (1, 1).


Comments

I Notice that indifference plays a critical role in everynon-degenerate mixed strategy Nash equilibrium.

I In particular player 1’s indifference condition defines player 2’smixed strategy and viceversa:

U1(B1) = q u1(B1,B2) + (1− q)u1(B1, S2) = q =

= U1(S1) = q u1(S1,B2) + (1− q)u1(S1,S2) = 2(1− q)

U2(B2) = p u2(B1,B2) + (1− p)u2(S1,B2) = 2 p =

= U2(S2) = p u2(B1,S2) + (1− p)u2(S1, S2) = (1− p)

I orq = 2(1− q), 2p = (1− p)


Existence

The frist natural question is: Whether a Nash equilibrium in mixedstrategies exists?

At this purpose we will focus exclusively on finite games: the(pure) strategy space of each player is a finite set.

Theorem (Nash Theorem)

Every finite normal form game Γ

Γ = {N;Ai ,∀i ∈ N; ui (a),∀i ∈ N}

has a mixed strategy Nash equilibrium.

We will come back to this result.


Oddness Theorem

The next natural question to ask on Nash equilibria of normal formgames is: How many Nash equilibria are there?

A partial answer is given by the Oddness Theorem Wilson (1971).

Theorem (Oddness Theorem)

Almost all finite normal form games have a finite and odd numberof Nash equilibria.


Oddness Theorem (cont’d)

I Intuition of the result can be obtained by considering thefollowing game:

1\2 L R

U 1, 1 0, 0

D 0, 0 0, 0

I This game has two pure strategy Nash equilibria: (U, L) and(D,R).

I It has no non-degenerate mixed strategy Nash equilibrium.



I Notice that player 1’s expected payoff if he chooses U andplayer 2 randomizes with probability q on L and withprobability (1− q) on R is:

U1(U, q) = q

I Player 1’s expected payoff if he chooses D and player 2randomizes in the same way is instead:

U1(D, q) = 0

I Therefore there does not exist a value of q ∈ (0, 1) for whichplayer 1 will be indifferent between playing U and D.



I However consider the following modification of the payoff ofthe previous game:

1\2 L R

U 1, 1 0, 0

D 0, 0 ε, ε

I Let ε be an arbitrary small positive number: ε > 0.

I Now the Nash equilibria of the game are: (U, L), (D,R) andthe non-degenerate mixed strategy Nash equilibrium(

ε

1 + ε,

ε

1 + ε

)


Documents

EC487 Advanced Microeconomics, Part I: Lecture 6econ.lse.ac.uk/staff/lfelli/teach/EC487 Slides Lecture 6.pdf · EC487 Advanced Microeconomics, Part I: Lecture 6 Leonardo Felli 32L.LG.04