EC487 Advanced Microeconomics, Part I: Lecture 6 Leonardo Felli - - PowerPoint PPT Presentation

EC487 Advanced Microeconomics, Part I: Lecture 6

Leonardo Felli

32L.LG.04

3 November, 2017

Game Theory

◮ It is the analysis of the strategic interaction among agents. ◮ This is a situation in which each agent when deciding how to

behave explicitly takes into account the decision of the other agents that interact with him.

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

Example: Entry Game

◮ Two individuals have to decide whether to sell newspapers at

a given exit of the underground.

◮ They take this decision without observing the decision taken

by the other individual.

◮ If only one individual decides to locate herself at the exit she

will make the highest level of profits since she will serve all

• clients. Let this profit be £300.

◮ If both individuals decide to locate themselves at the exit then

clients are equally shared (we assume newspaper prices are pre-set). Each individual’s profit is £150.

◮ Finally if an individual does not locate herself at the exit than

she makes zero profits.

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

Example (cont’d)

◮ We can describe the situation with the following table:

1\2 E NE E 150, 150 300, 0 NE 0, 300 0, 0

◮ Rows denote individual 1’s decisions. ◮ Columns denote individual 2’s decisions. ◮ The first number of each ordered pair denotes individual 1’s

profit, 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)

◮ Notice that predicting the outcome of this situation is fairly

easy provided that we assume that both individuals wants to maximize profits, in other words they are rational.

◮ The predicted outcome is that both individuals locate

themselves at the exit (E, E). 1\2 E NE E 150, 150 300, 0 NE 0, 300 0, 0

◮ Notice that this conclusion can be reached without requiring

each individual to make a prediction on the behavior of the

• ther individual.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 5 / 54

Battle of Sexes

◮ This is not true in general. ◮ Consider for example the following situation known as battle

• f sexes:

1\2 B S B 1, 2 0, 0 S 0, 0 2, 1

◮ In this case each individual needs to make a prediction on the

behavior of the other individual.

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Coordination Game

◮ Consider the following simple coordination game (no conflict

• f interest).

◮ There is still a need for predictions:

1\2 M C M 2, 2 0, 0 C 0, 0 1, 1

◮ Notice that we will be more confident in our prediction if the

individuals involved encounter this situation more than once.

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Jargon and Definitions

◮ The strategic situations we described above are known as

games.

◮ A simple static game or game in normal (strategic) form (no

time 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 space

denoted Ai. This is the set of choices available to each player: A1 = {locate at the exit, do not locate at the exit}.

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Definitions

◮ Denote: ai ∈ Ai player i’s strategy choice; ◮ Then a−i = (a1, . . . , ai−1, ai+1, . . . , aI) is the strategy profile

• f every player but player i.

◮ Therefore a = (ai, a−i) ∈ A1 × . . . × AI = A. ◮ Finite games are games with finite strategy spaces (a finite

number of strategies).

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Definitions (cont’d)

• 3. Finally define for each player i ∈ N a payoff function

associated with his strategy choice ai and the other players’ strategy choice a−i: ui(a1, . . . , aI) = ui(ai, a−i) = ui(a).

◮ The payoffs ui(·) is taken to be the utility representation of

player i’s preferences.

◮ The objective of game theoretic analysis is to give predictions

• n the behavior of agents in strategic situations.

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Rationality:

◮ What assumptions do we need on the players’ behavior to

deliver these predictions?

◮ First assumption rationality (maximization of utility or

payoff).

◮ In our example above rationality and knowledge of own payoff

is 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

◮ An other classic example of a situation in which rationality

and knowledge of own payoff is enough to deliver a prediction is the the prisoners’ dilemma game.

◮ This is characterized by the following normal form:

1\2 C NC C 0, 0 4, −1 NC −1, 4 3, 3

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Prisoners’ dilemma (cont’d)

◮ The three elements of the game are:

◮ N = {1, 2}, ◮ Ai = {C, NC}, ◮ 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 NC C 0, 0 4, −1 NC −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

◮ Each player will choose the strategy C independently of the

action chosen by the other player.

◮ The predicted outcome is therefore (C, C). This is clearly the

inefficient outcome, it is Pareto dominated by (NC, NC).

◮ The only information needed to make a prediction is the fact

that 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

◮ 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

◮ In this case we need some extra assumptions to make a

prediction on the outcome of the game.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 15 / 54

Knowledge of Rationality (cont’d)

◮ 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

◮ L dominates C and R for player 2; ◮ if player 1 knows that player 2 is rational then he will focus

• nly on the first column;

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 16 / 54

Knowledge of Rationality (cont’d)

◮ Therefore:

1\2 L T 0, 0 M −1, 4 B −1, 2

◮ In the first column T dominates M and B. ◮ The prediction is therefore (T, L).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 17 / 54

Relevant Assumptions:

◮ The information needed to make a prediction is then: ◮ both players are rational; ◮ both players know their own and the other player’s payoff; ◮ player 1 knows that player 2 is rational.

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Common Knowledge of Rationality

◮ Consider now the following game:

1\2 L C R T 1, 0 1, 2 0, 1 B 0, 3 0, 1 2, 0

◮ Player 2 will never play R since R is a strictly dominated

strategy and both players are rational and know each other payoffs.

◮ Since player 1 knows that player 2 is rational he also knows

that R will never be played.

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Common Knowledge of Rationality (cont’d)

◮ Notice now that:

1\2 L C T 1, 0 1, 2 B 0, 3 0, 1

◮ For player 2 none of the remaining strategies is strictly

dominated:

◮ if player 2 believes that player 1 will play B then 2 will choose

L;

◮ while if player 2 believes that player 1 will play T then 2 will

choose C.

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Common Knowledge of Rationality (cont’d)

◮ However if we now assume that: player 2 knows that player 1

knows that player 2 is rational

◮ then player 2 knows that player 1 realizes that he will never

play R so for all intents and purposes the game is: 1\2 L C T 1, 0 1, 2 B 0, 3 0, 1

◮ In this new game player 1’s strategy B is strictly dominated,

therefore 1 will never choose it.

◮ Therefore since player 2 knows that player 1 is rational the

predicted outcome will be (T, C).

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Common Knowledge of Rationality (cont’d)

◮ The assumptions needed to make this prediction are then:

◮ that both players are rational; ◮ that both players know their own and the other player’s payoff; ◮ that both players know that the other player is rational; ◮ that player 2 knows that player 1 knows that player 2 is

rational.

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Necessary Assumptions:

◮ A set of necessary assumptions used in non-cooperative game

theory to predict an outcome are:

◮ rationality of the players; ◮ common knowledge of the rationality of the players:

◮ player i knows that player j is rational, ◮ player i knows that player j knows that player i is rational,

. . .

◮ player i knows that player j knows that player i knows that

. . . player i is rational

◮ common knowledge of the structure of the game.

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Is rationality a good assumption?

◮ Consider the following game:

1\2 L R U 8 10 −1, 000, 000, 000, 000 9 D 7 6 6 5

◮ Notice that: player 2’s strategy R is strictly dominated by L. ◮ Since both players are rational and know that the other player

is rational then player 1 knows that player 2 will never play R.

◮ Therefore player 1 chooses U and the predicted outcome is

(U, L).

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Is rationality a good assumption? (cont’d)

◮ Notice however that:

1\2 L R U 8 10 −1, 000, 000, 000, 000 9 D 7 6 6 5

◮ Player 1 better be absolutely sure of player 2’s rationality! ◮ Any mistake is extremely costly (−1, 000, 000, 000, 000).

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Stronger Predictions

◮ We have identified a first set of behavioral assumptions that

allow us to make predictions on simple normal form games.

◮ These are very reasonable assumptions — the last example

notwithstanding.

◮ However reasonable comes at a cost. ◮ In most situations these assumptions are not enough to allow

us to make specific predictions.

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Battel of Sexes II

◮ Consider once again the game:

1\2 B2 S2 B1 1, 2 0, 0 S1 0, 0 2, 1

◮ Rationality of the players, common knowledge of the

rationality of the players and common knowledge of the structure of the game are not enough.

◮ We need a sharper tool for a prediction: Nash equilibrium. ◮ We need to impose a restriction on the beliefs that the players

have on the behavior of other players.

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

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Nash Equilibrium Construction:

The basic building block of a Nash equilibrium is each player’s best response 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 other players is player i’s strategy choice(s) ai that maximizes i’s utility given 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):

◮ This best response (correspondence) associates to any given

strategy profile of all the other players a−i, player i’s strategies that maximize player i’s payoff ui(ai, a−i):

◮ A Nash equilibrium in pure strategies is a strategy profile such

that each player’s strategy choice is a best response to the strategy choice of the other players.

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Nash Equilibrium Construction (cont’d):

Definition (Pure Strategy Nash Equilibrium)

Definition: A pure-strategy Nash equilibrium is a strategy profile a∗ = (a∗

i , a∗ −i) such that for every i ∈ N

ui(a∗

i , a∗ −i) ≥ ui(ai, a∗ −i)

∀ai ∈ Ai.

• r

a∗

i ∈ Bi(a∗ −i)

∀i ∈ N. Notice that according to the definition above in equilibrium the beliefs of each player are indeed correct.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 30 / 54

Example

◮ Consider the following game:

1\2 L R U 3, 2 2, 0 D 0, 0 1, 1

◮ Notice that:

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

◮ and

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

◮ Hence the unique pure strategy Nash equilibrium of such a

game is (U, L).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 31 / 54

No guarantee of uniqueness:

◮ Consider one more time the battle of sexes game:

1\2 B2 S2 B1 1, 2 0, 0 S1 0, 0 2, 1

◮ Clearly:

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

◮ and

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

◮ 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

◮ Notice that the best response in both games above associates

a unique strategy ai to every vector of strategies a−i, the best reply is a single-valued function.

◮ Indifference may lead to more than one strategy ai in the best

reply correspondence associated with a given a−i.

◮ The definition of Nash equilibrium is such that: when

indifferent between two strategies both strategies are part of the best response of a player.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 33 / 54

Example

◮ 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

◮ Notice that:

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

◮ and

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

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

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 34 / 54

Indifference

◮ The underlying behavioral assumption is that: when

indifference a player will choose the strategy that sustains the equilibrium.

◮ Indifference plays a big role in the characterization of the

properties of Nash equilibrium.

◮ In particular the whole proof of existence (Nash Theorem) will

be essentially based on indifference.

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Existence

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

1\2 H2 T2 H1 1, −1 −1, 1 T1 −1, 1 1, −1

◮ Notice that:

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

◮ and

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

◮ Clearly there does not exist any pure strategy Nash

equilibrium of this game.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 36 / 54

Mixed Strategies

◮ To be able to have predictive power in strategic situations like

the one described in matching pennies we need to extend the definition of strategy.

◮ Suppose player 1 tries to be as unpredictable as possible. ◮ In other words, player 1 randomizes with probability p and

(1 − p) between the choice H1 and the choice T1.

◮ Assume that if player 1 is unpredictable so is player 2. ◮ In other words, player 2 randomizes with probability q and

(1 − q) between the choice H2 and the choice T2.

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Mixed Strategies (cont’d)

H2 T2 H1 1, −1 −1, 1 T1 −1, 1 1, −1 p 1 − p q 1 − q

◮ Player 2’s best reply is then obtained solving the following

problem: max

q

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)

◮ In other words:

max

q

q [1 − 2 p] + (1 − q) [2 p − 1]

◮ The solution is then:

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

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 39 / 54

Mixed Strategies (cont’d)

◮ Consider now player 1’s best reply. This is obtained solving

the following problem: max

p

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)]

◮ In other words:

max

p

p [2 q − 1] + (1 − p) [1 − 2 q]

◮ The solution is then:

p = 1 if q > 1/2 0 ≤ p ≤ 1 if q = 1/2 p = 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 2 and 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 pure strategies). Let the set of possible probability distributions (mixed strategies)

• ver 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)

◮ The key assumption on mixed strategies is that each player i

randomizes independently from other players: σi independent

• f σj for i = j.

◮ The mixed extension of the game

Γ = {N; Ai, ∀i ∈ N; ui(a), ∀i ∈ N} is the game: Γ∆ = {N; ∆(Ai), ∀i ∈ N; Ui(σ), ∀i ∈ N}

◮ 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 profile σ∗ = (σ∗

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

σ∗

i = arg

max

σi∈∆(Ai) Ui(σi, σ∗ −i)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 43 / 54

Battel of Sexes III

◮ Consider the mixed strategy Nash equilibria of the battle of

sexes game: 1\2 B2 S2 B1 1, 2 0, 0 S1 0, 0 2, 1 p 1 − p q 1 − q

◮ Define:

◮ p player 1’s mixed strategy (the probability with which player 1

plays B1);

◮ q player 2’s mixed strategy (the probability with which player 2

plays B2).

◮ 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)

◮ 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)

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

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

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

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

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 46 / 54

Graphically:

✲ ✻

q p 1

3, 2 3

• B1(q)

B2(p) (1, 1)

✉ ✉ ✉

(0, 0) q = 1 q = 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)

◮ The mixed strategy Nash equilibrium of the game is (p∗, q∗)

such that: p∗ = arg max

p

p q∗ + (1 − p) 2 (1 − q∗) and q∗ = arg max

q

q 2 p∗ + (1 − q)(1 − p∗)

◮ 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

◮ Notice that indifference plays a critical role in every

non-degenerate mixed strategy Nash equilibrium.

◮ In particular player 1’s indifference condition defines player 2’s

mixed 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)

◮ or

q = 2(1 − q), 2p = (1 − p)

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Existence

The frist natural question is: Whether a Nash equilibrium in mixed strategies 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 form games 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 number

• f Nash equilibria.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 51 / 54

Oddness Theorem (cont’d)

◮ Intuition of the result can be obtained by considering the

following game: 1\2 L R U 1, 1 0, 0 D 0, 0 0, 0

◮ This game has two pure strategy Nash equilibria: (U, L) and

(D, R).

◮ 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)

◮ Notice that player 1’s expected payoff if he chooses U and

player 2 randomizes with probability q on L and with probability (1 − q) on R is: U1(U, q) = q

◮ Player 1’s expected payoff if he chooses D and player 2

randomizes in the same way is instead: U1(D, q) = 0

◮ Therefore there does not exist a value of q ∈ (0, 1) for which

player 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)

◮ However consider the following modification of the payoff of

the previous game: 1\2 L R U 1, 1 0, 0 D 0, 0 ε, ε

◮ Let ε be an arbitrary small positive number: ε > 0. ◮ Now the Nash equilibria of the game are: (U, L), (D, R) and

the 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