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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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 3 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 4 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 5 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 6 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 7 / 54
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}.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 8 / 54
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 9 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 10 / 54
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
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 11 / 54
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
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 12 / 54
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
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 13 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 14 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 15 / 54
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;
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 16 / 54
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 17 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 18 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 19 / 54
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 .
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 20 / 54
Common Knowledge of Rationality (cont’d)
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 ).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 21 / 54
Common Knowledge of Rationality (cont’d)
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 22 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 23 / 54
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 24 / 54
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 25 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 26 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 27 / 54
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}.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 28 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 29 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 30 / 54
Example
I Consider the following game:
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 31 / 54
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 32 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 33 / 54
Example
I Consider the following game:
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 34 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 35 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 36 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 37 / 54
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)]
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 38 / 54
Mixed Strategies (cont’d)
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
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 39 / 54
Mixed Strategies (cont’d)
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
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 40 / 54
Mixed Strategies (cont’d)
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 41 / 54
Mixed Strategies (cont’d)
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 )
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 42 / 54
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 )
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 43 / 54
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].
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 44 / 54
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)
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 45 / 54
Battel of Sexes III (cont’d)
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
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 46 / 54
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
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 47 / 54
Battel of Sexes III (cont’d)
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 48 / 54
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)
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 49 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 50 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 51 / 54
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 52 / 54
Oddness Theorem (cont’d)
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 53 / 54
Oddness Theorem (cont’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
)
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 54 / 54
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