EC487 Advanced Microeconomics, Part I: Lecture 7 Leonardo Felli - - PowerPoint PPT Presentation

EC487 Advanced Microeconomics, Part I: Lecture 7

Leonardo Felli

32L.LG.04

10 November, 2017

Nash Theorem

Recall that we are focussing exclusively on finite games: Ai is a finite set for every i ∈ N.

Theorem (Nash Theorem)

Every finite normal form game Γ Γ = {N; Ai, ∀i ∈ N; ui(a), ∀i ∈ N} has a mixed strategy Nash equilibrium.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 2 / 59

General Theorem

We will prove the Nash Theorem as an immediate consequence of a somewhat more general/special existence theorem.

Theorem

Consider an N-players game Λ = {N; Di, ∀i ∈ N; vi(d), ∀i ∈ N} where:

◮ Di is a compact, convex subset of an Euclidean space; ◮ vi(di, d−i) is continuous and quasiconcave in d.

Then Λ has a Nash equilibrium in pure strategies.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 3 / 59

Proof of General Theorem

Recall that:

Definition

The function vi(di, d−i) is quasiconcave if and only if the upper level set {di|vi(di, d−i) ≥ k} is a convex set for every scalar k. An alternative definition of quasiconcavity is:

Definition

The function vi(di, d−i) is quasiconcave if and only if for every pair di ∈ Di, ˆ di ∈ Di and α ∈ [0, 1] vi(αdi + (1 − α) ˆ di, d−i) ≥ min{vi(di, d−i), vi( ˆ di, d−i)}.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 4 / 59

Proof of General Theorem (cont’d)

The key tool we will use is.

Theorem (Kakutani’s Fixed Point Theorem)

Let X be a compact, convex and non-empty set in Rn and F : X ⇒ X a correspondence that satisfies the following properties:

◮ non-empty; ◮ convex valued; ◮ upper-hemi-continuous.

Then there exists a vector x∗ ∈ X such that: x∗ ∈ F(x∗)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 5 / 59

Proof of General Theorem (cont’d)

For each player i ≤ N define the best reply correspondence φi as: φi(d) =

• di ∈ Di|vi(di, d−i) ≥ vi(d′

i , d−i), ∀d′ i ∈ Di

• Notice that:

◮ φi(d) is nonempty since Di is compact and vi(d) is

continuous (Weierstrass Theorem).

◮ φi(d) is convex-valued since Di is compact and vi(d) is

quasiconcave in di.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 6 / 59

Proof of General Theorem (cont’d)

◮ φi(d) is upper-hemi-continuous:

Consider the sequence {dh} where dh ∈ D1 × · · · × DN and assume that dh → d Consider also the sequence {dh

i } where dh i ∈ Di and assume

that dh

i → di

Assume that dh

i ∈ φi(dh),

∀h ≥ 1

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 7 / 59

Proof of General Theorem (cont’d)

Notice that by definition of the correspondence φi(·) for every ˆ di ∈ Di we have vi(dh

i , dh −i) ≥ vi( ˆ

di, dh

−i)

By continuity of vi(d) we then have vi(di, d−i) ≥ vi( ˆ di, d−i) In other words di ∈ φi(d)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 8 / 59

Proof of General Theorem (cont’d)

◮ Now define the correspondence

φ : D1 × · · · × DN → D1 × · · · × DN as φ(d) = φ1(d) × · · · × φN(d)

◮ Notice then that φ(d) satisfies all the assumptions of

Kakutani Fixed Point Theorem: The set D1 × · · · × DN is a compact, convex and non-empty subset of an Euclidean space, since each Di is a compact, convex and non-empty subset of an euclidean space.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 9 / 59

Proof of General Theorem (cont’d)

The correspondence φ(d) is non-empty, convex valued and upper-hemi-continuous since each φi(d) is non-empty, convex valued and upper-hemi-continuous.

◮ Therefore there exists a fixed point d such that

d ∈ φ(d)

• r there exists a strategy profile d such that each element of it

is a best reply to the strategy profile itself.

◮ In other words, there exists a pure strategy Nash equilibrium

• f the game Λ.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 10 / 59

Proof of Nash Theorem

Consider now the finite normal form game Γ: Γ = {N; Ai, ∀i ∈ N; ui(a), ∀i ∈ N} Its mixed extension is the game Γ∆: Γ∆ = {N; ∆(Ai), ∀i ∈ N; Ui(σ), ∀i ∈ N} where, if Ai contains n strategies, ∆(Ai) is the (n − 1)-dimensional simplex, σi ∈ ∆(Ai) and Ui(σ) =

• a∈A
• σ1(a1) · . . . · σI(aI)
• ui(a1, . . . , aI)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 11 / 59

Proof of Nash Theorem (cont’d)

Notice now that the mixed extension game Γ∆ satisfies all the assumption of the general theorem considered above.

◮ ∆(Ai) is a compact, convex subset of an Euclidean space; ◮ Ui(σ) is a continuous and quasiconcave (linear) function.

Hence by the theorem above there exists a pure strategy Nash equilibrium of the game Γ∆ that is a mixed strategy Nash equilibrium of the game Γ.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 12 / 59

A Voluntary Contribution Game

Consider an economy in which each of N individuals is allocated a share of the endowment of the private good ω = (ω1, . . . , ωN) We assume that each individual voluntarily contributes an amount zi of his private good for the (collectively run) production of the public good: y = g  

N

• j=1

zj   We are going to take each individual’s contribution decision to be the one predicted by the Nash equilibrium of this symultaneous move contribution game.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 13 / 59

A Voluntary Contribution Game (cont’d)

In other words, the amount of the private good zi that individual i contributes is the solution to the following best reply problem for i: max

{xi, zi}

Ui(xi, y) s.t. xi + zi ≤ ωi y = g  

N

• j=1

zj   The first order conditions of this problem leads to the following marginal condition: ∂Ui/∂y ∂Ui/∂xi = 1 g′(z), ∀i = 1, . . . , N

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 14 / 59

A Voluntary Contribution Game (cont’d)

The Nash equilibrium of the voluntary contribution game is then defined by the following set of marginal conditions: ∂Ui/∂y ∂Ui/∂xi = 1 g′(z), ∀i = 1, . . . , N Recall that the Pareto efficient allocation is the one defined by the Bowen-Lindahl-Samuelson condition:

N

• i=1

∂Ui/∂y ∂Ui/∂xi = 1 g′(z) Clearly, the allocation generated by this contribution game is Pareto inefficient: agent i contributes up to the point where the marginal cost of the public good (MRT) is equal to i’s marginal rate of substitution between y and z.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 15 / 59

A Voluntary Contribution Game (cont’d)

To highlight the nature of the inefficiency consider the following parametric characterization: Ui(xi, y) = xi +v(y), v′(·) > 0, v

′′(·) < 0, v(0) = 0, v′(0) = +∞

g(z) = z. Player i’s best reply is then the solution to the problem: max

{xi, zi}

xi + v(y) s.t. xi + zi ≤ ωi y =

N

• j=1

zj

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 16 / 59

A Voluntary Contribution Game (cont’d)

Notice first that monotonicity of each player’s utility function implies xi + zi = ωi Player i’s best reply is then the solution to the problem: max

zi

ωi − zi + v  

N

• j=1

zj   The first order conditions are then: v′  

N

• j=1

zj   = 1

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 17 / 59

A Voluntary Contribution Game (cont’d)

There exist potentially lots of Nash Equilibria of this game. Assume that ωi = ω and consider the symmetric Nash equilibrium such that zi = z for every i ≤ N. Each player Nash equilibrium contribution z∗ is then such that: v′ (N z∗) = 1 Consider now the symmetric Pareto efficient allocation in this environment.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 18 / 59

A Voluntary Contribution Game (cont’d)

The general central planner problem, for λi > 0 for all i ≤ N, is: max

{xi, zi} N

• i=1

λi[xi + v(y)] s.t.

N

• i=1

(xi + zi) ≤ N ω y =

N

• i=1

zi

• r

max

{xi} N

• i=1

λi

• xi + v
• N ω −

N

• i=1

xi

• Leonardo Felli (LSE)

EC487 Advanced Microeconomics, Part I 10 November, 2017 19 / 59

A Voluntary Contribution Game (cont’d)

The first order conditions are, for every i ≤ N:

N

• j=1

λj v′

• N ω −

N

• i=1

xi

• = λi
• r

v′ N

• i=1

zi

• =

λi N

j=1 λj

In the symmetric case case λi = λ and zi = z∗∗ the Pareto efficient allocation is such that: v′(N z∗∗) = 1 N

• f course in the case z∗∗ is such that v(N z∗∗) − z∗∗ ≥ 0.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 20 / 59

A Voluntary Contribution Game (cont’d)

Concavity of v(·) implies that the Nash equilibrium of the contribution game is characterized by under-provision of the public good z∗ < z∗∗: v′ (N z∗) = 1, v′(N z∗∗) = 1 N Assume now that to fundraise for the public good a new voluntary contribution game is designed introducing a minimum threshold. In other words, the public good y will be produced only if a given minimum threshold of total contributions ¯ Z is reached. In other case the contribution is returned to the player.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 21 / 59

A Voluntary Contribution Game (cont’d)

Let ¯ z = ¯ Z N Consider the symmetric Nash equilibrium of this new contribution game and assume that all other players but player i contributed z. Each player’s payoff is now: πi(xi, zi, z) =

• ω − zi + v(zi + (N − 1) z)

if

zi+(N−1) z N

≥ ¯ z ω if

zi+(N−1) z N

< ¯ z Then player i’s best reply is such that: zi(z) = ¯ Z − (N − 1) z if v( ¯ Z) ≥ ( ¯ Z − (N − 1) z) [0, Z − (N − 1) z) if v( ¯ Z) ≤ ( ¯ Z − (N − 1) z)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 22 / 59

A Voluntary Contribution Game (cont’d)

Set now ¯ Z = N z∗∗ There exist a Nash equilibrium of the voluntary contribution game with a minimum threshold where zi = z∗∗, ∀i ≤ N Notice however that there exist also many other equilibria, in particular there exists a Nash equilibrium of the voluntary contribution game with a minimum threshold where zi = 0, ∀i ≤ N

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 23 / 59

Monopoly

Consider now the pricing behavior of a profit maximizing monopolist: a firm that is the only producer of a commodity. Let the aggregate demand of this commodity at price p be x(p) assumed to be continuous, strictly decreasing and such that x(p) > 0. Assume that there exists a price ¯ p < +∞ such that x(p) = 0 for every p ≥ ¯ p. Assume that the monopolist knows x(p) and is endowed with a technology characterized by the cost function c(q).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 24 / 59

Monopoly (cont’d)

The monopolist’s problem is then: max

p

p x(p) − c(x(p)) Equivalent formulation in terms of quantity choice q is derived using the inverse demand function P(·) = x−1(·): max

q≥0

P(q) q − c(q) We focus on this (equivalent) formulation and assume that: P(·) and c(·) are twice continuously differentiable, P(0) > c′(0) and there exists a unique output qc such that P(qc) = c′(qc).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 25 / 59

Monopoly (cont’d)

The solution to the monopolist’s problem qm satisfies the following necessary first order conditions: P′(qm) qm + P(qm) ≤ c′(qm) with equality if qm > 0 The left-hand-side is known as the marginal revenue and it is equal to the derivative of the revenue function R(q) = P(q) q. The right-hand-side is the familiar marginal cost.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 26 / 59

Monopoly (cont’d)

Since P(0) > c′(0) the necessary first order conditions can only be satisfied at qm > 0. Therefore the monopolist’s optimal quantity choice is the one that sets marginal revenue equal to marginal cost: P′(qm) qm + P(qm) = c′(qm) In the typical case P′(q) < 0 we obtain that: P(qm) > c′(qm) The price under monopoly exceeds marginal cost.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 27 / 59

Monopoly (cont’d)

Correspondingly: qm < qc A reduction in the quantity sold by the monopolist allows him to increase the price charged on the remaining sales. The effect on profits is captured by the term P′(qm) qm. The welfare loss, known as the deadweight loss of monopoly is measured by the change in surplus: qc

qm

• P(s) − c′(s)
• ds > 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 28 / 59

Monopoly (cont’d)

Notice that the deadweight loss is absent in the special case of a perfectly elastic demand: P′(q) = 0 for all q. In this case the monopolist sells the same quantity as a perfectly competitive firm: qm = qc. In other words, competitive firms perceive demand as perfectly elastic: the price is given in the market.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 29 / 59

Monopoly (cont’d)

Consider now the special case of a monopolist whose technology is fully described by the cost function that exhibits constant returns to scale with no fixed costs: c(qi) = c qi ∀i ∈ {1, 2}. The consumer behavior is summarized by the following linear inverse demand function: P(q) = a − q if q ≤ a if q ≥ a We assume for viability of the economy c < a.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 30 / 59

Monopoly (cont’d)

The monopolist’s problem is now: max

{q}

Π(q) = q [p(q) − c] = q [a − q − c] The solution to this problem characterizes the monopolist’s quantity: qm = (a − c) 2 This is half of the perfectly competitive quantity: qc = (a − c)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 31 / 59

Duopoly

We distinguish between different types of duopoly (oligopoly) depending on:

◮ whether the firms involved in the market compete in quantity

• r in prices, and

◮ whether they decide their strategy simultaneously or

sequentially.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 32 / 59

Duopoly (cont’d)

Consider a Cournot Duopoly (Cournot 1838) model in which two firms compete in their choice of the quantity produced and take these decisions simultaneously. Assume that both firms produce a perfectly homogeneous good. The consumers are then indifferent on whether they consume the good produced by one firm or the other. Therefore N = {1, 2} and Ai = R+ is the set of positive quantities denoted q1 and q2.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 33 / 59

Duopoly (cont’d)

To be able to define the payoff function for each firm we need to specify the technology of the firm and the demand function in the market. Assume that the firms’ technologies are identical and fully described by the cost function that exhibits constant returns to scale with no fixed costs: c(qi) = c qi ∀i ∈ {1, 2}. The consumer behavior is summarized by the inverse demand function faced by both firms: P(q1 + q2) = a − (q1 + q2) if q1 + q2 ≤ a if q1 + q2 ≥ a

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 34 / 59

Duopoly (cont’d)

We assume for viability of the economy c < a. Firm i’s profit (payoff) function is then: Πi(q1, q2) = qi [P(q1 + q2) − c] = = qi [a − (q1 + q2) − c] ∀i ∈ {1, 2}. A pair of quantities (q∗

1, q∗ 2) is a Nash equilibrium of the Cournot

duopoly game if and only if: q∗

i = arg max qi∈R+ Πi(qi, q∗ j )

i = j, i, j ∈ {1, 2}.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 35 / 59

Duopoly (cont’d)

To identify the best reply of each firm we need to solve the following problem: Ri(qj) = arg max

qi∈R+ qi [a − (q1 + q2) − c] .

If qj < a − c this is characterized by the following set of necessary and sufficient conditions: a − 2qi − qj − c = 0 Which gives us the following pair of best reply functions: Ri(qj) = 1 2 (a − qj − c) i = j, i, j ∈ {1, 2}

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 36 / 59

Duopoly (cont’d)

These two (linear) best reply functions are represented in the following graph:

✻ ✲

q2 q1

❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❍❍❍❍❍❍❍❍❍❍❍❍ q

a − c qm

2 = (a−c) 2

a − c qm

1 = (a−c) 2

R1(q2) R2(q1) (0, 0)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 37 / 59

Duopoly (cont’d)

The Nash equilibrium of the Cournot game is then the solution to the following problem: q∗

1 = 1

2 (a − q∗

2 − c)

q∗

2 = 1

2 (a − q∗

1 − c) .

This solution is unique and is: q∗

1 = q∗ 2 = (a − c)

3 .

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 38 / 59

Duopoly (cont’d)

The Nash equilibrium is the intersection of the best reply functions:

✻ ✲

q2 q1

❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ q ❍❍❍❍❍❍❍❍❍❍❍❍

(q∗

1, q∗ 2)

q

a − c qm

2 = (a−c) 2

a − c qm

1 = (a−c) 2

R1(q2) R2(q1) (0, 0)

✻

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 39 / 59

Duopoly (cont’d)

An alternative way: iterated elimination of strictly dominated strategies. We proceed in the following way:

◮ The monopolist quantity qm = (a − c)

2 strictly dominates any higher quantity for each player.

◮ Given that quantities in excess of qm are never chosen by a

firm then quantity Ri(qm) strictly dominates any quantity below it. . . .

◮ The only quantity left to choose for each player is then q∗ i .

Notice that this process does not converge to a unique point if instead of a Cournot duopoly we consider a Cournot oligopoly in which three firms compete.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 40 / 59

Duopoly (cont’d)

The iterated elimination of strictly dominated strategies:

✻ ✲

q2 q1

❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ q ❍❍❍❍❍❍❍❍❍❍❍❍

(q∗

1, q∗ 2)

q

a − c qm

2 = (a−c) 2

a − c qm

1 = (a−c) 2

R1(q2) R2(q1) (0, 0)

✻

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R2(qm

1 )

................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R1(qm

2 )

✰ ❄

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 41 / 59

Duopoly (cont’d)

A set of sufficient conditions that guarantee existence of a Nash equilibrium of a Cournot duopoly game are that the profit function

• f each firm is quasi-concave and twice differentiable.

Quasi-concavity is checked by establishing that ∂2Π1 ∂q2

1

≤ 0 whenever ∂Π1 ∂q1 = 0 and ∂2Π2 ∂q2

2

≤ 0 whenever ∂Π2 ∂q2 = 0 These conditions imply: 2 P′(q1 +q2)+qiP′′(q1 +q2)−c′′(qi) ≤ 0 (P′ ≤ 0 then P′′ ≤ 0 sufficient condition)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 42 / 59

Duopoly (cont’d)

Consider a Bertrand Duopoly (Bertrand 1883) model in which two firms compete in their choice of prices and take these decisions simultaneously. Assume for simplicity that the consumer demands only one unit of the homogeneous good produced by either firm. Let v be the utility that the consumer derives from this unit of a good. A consumer will buy the good from the firm that charges the lowest price pi if v ≥ pi. When the price is the same he will buy the good from firm 2.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 43 / 59

Duopoly (cont’d)

Then the game is defined by N = {1, 2} and Ai = R+ the set of all positive prices pi. Assume that firm i produces this unit of the good at a cost ci and c1 > c2. where v > c1. The payoffs to each firm are then: Π1(p1, p2) = p1 − c1 if p1 < p2 if p1 ≥ p2

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 44 / 59

Duopoly (cont’d)

and Π2(p1, p2) = p2 − c2 if p2 ≤ p1 if p2 > p1 The best reply for firm 1 is then: p1 ≥ p2 if p2 ≤ c1 ∅ (p1 = p2 − ǫ) if p2 > c1 while for firm 2 is instead: p2 > p1 if p1 < c2 p2 ≥ p1 if p1 = c2 p2 = p1 if p1 > c2

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 45 / 59

Duopoly (cont’d)

The Nash equilibria of the Bertrand duopoly game are then:

◮ a choice of prices:

p2 = p1 = p∗ ∀c2 ≤ p∗ ≤ c1.

◮ in equilibrium the consumer buys the unit of the good from

firm 2. Notice the multiplicity of equilibria in terms of price choices: a whole interval of prices can be equilibria of this game.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 46 / 59

Duopoly (cont’d)

The best replies are represented in the following graph.

✻ ✲

p1 p2 c1

q

• .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c2 45◦ . . . . . . . . . . . . . . . . . . . . . . . . . . .

• Leonardo Felli (LSE)

EC487 Advanced Microeconomics, Part I 10 November, 2017 47 / 59

Dynamic Games with Perfect Information

◮ These are games in which time plays an explicit role. ◮ Consider the following example:

◮ Two parties are trying to share two indivisible units of a good

yielding positive utility.

◮ Suppose that player 1 makes a take-it-or-leave-it offer to player

2.

◮ Player 2 after having observed the offer decides whether to

accept or reject it.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 48 / 59

Extensive Form Games with Perfect Information (cont’d)

◮ Each unit generates one utile. ◮ If no agreement is reached than the two units of the good are

not allocated to any individual.

◮ Extensive form of the game = detailed description of the

sequential structure of the players’ decision problem.

◮ We consider the case in which all players have perfect

information at every instant of time.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 49 / 59

An alternative description is provided by the following tree:

❝❝❝❝❝❝❝❝❝❝❝ ❜ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ r r ✁ ✁ ✁ ✁ ✁ ✔ ✔ ✔ ✔ ✔ ❚ ❚ ❚ ❚ ❚ ✔ ✔ ✔ ✔ ✔ ❚ ❚ ❚ ❚ ❚ r ❙ ❙ ❙ ❙ ❙

1 2 2 2 (2, 0) (1, 1) (0, 2) y y y n n n (2, 0) (0, 0) (1, 1) (0, 0) (0, 2) (0, 0)

r r r r r r

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 50 / 59

Formal description of an extensive form game

• 1. The set of players:

N = {1, 2}

• 2. The set of histories of the game, (each element of a history is

an action): H =

• ∅, (2, 0), (1, 1), (0, 2), [(2, 0), y], [(2, 0), n],

[(1, 1), y], [(1, 1), n], [(0, 2), y], [(0, 2), n]

• the (sub)set of terminal histories:

Z =

• [(2, 0), y], [(2, 0), n], [(1, 1), y],

[(1, 1), n], [(0, 2), y], [(0, 2), n]

• Leonardo Felli (LSE)

EC487 Advanced Microeconomics, Part I 10 November, 2017 51 / 59

Formal description of an extensive form game (cont’d)

• 3. The player function that determines which player chooses an

action after each non-terminal history: P(∅) = 1, P(2, 0) = P(1, 1) = P(0, 2) = 2.

• 4. The players’ payoffs associated to each terminal history:

u1([(2, 0), y]) = 2 u2([(2, 0), y]) = 0 u1([(2, 0), n]) = 0 u2([(2, 0), n]) = 0 u1([(1, 1), y]) = 1 u2([(1, 1), y]) = 1 u1([(1, 1), n]) = 0 u2([(1, 1), n]) = 0 u1([(0, 2), y]) = 0 u2([(0, 2), y]) = 2 u1([(0, 2), n]) = 0 u2([(0, 2), n]) = 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 52 / 59

Strategy

Definition

A strategy in this type of game is a fully contingent plan. This plan specifies the actions chosen by the player for every history after which the player is called upon to play. The sense in which this plan is complete is that the player should be able to pass the plan to his lawyer so that the lawyer does not need to contact the player any more whatever happens.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 53 / 59

Recall the game tree:

❝❝❝❝❝❝❝❝❝❝❝ ❜ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ r r ✁ ✁ ✁ ✁ ✁ ✔ ✔ ✔ ✔ ✔ ❚ ❚ ❚ ❚ ❚ ✔ ✔ ✔ ✔ ✔ ❚ ❚ ❚ ❚ ❚ r ❙ ❙ ❙ ❙ ❙

1 2 2 2 (2, 0) (1, 1) (0, 2) y y y n n n (2, 0) (0, 0) (1, 1) (0, 0) (0, 2) (0, 0)

r r r r r r

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 54 / 59

Example of strategy:

◮ In our example above for player 1:

P1 = {∅} and s1 = s(∅) ∈ {(2, 0), (1, 1), (0, 2)}

◮ While for player 2:

P2 = {(2, 0), (1, 1), (0, 2)} and s2 = {s2(2, 0), s2(1, 1), s2(0, 2)} where s2 ∈ {y, n}.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 55 / 59

An other example of an extensive form game

❜ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ❅ ❅ ❅ ❅ ❧❧❧❧❧ ❧❧❧❧❧ r r r r r r

1 1 2 A B C D E F (2, 1) (3, 1) (0, 1) (0, 0)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 56 / 59

An other example of an extensive form game (cont’d)

◮ Notice that 1’s strategies are (A, E), (A, F), (B, E) and

(B, F) therefore a strategy s1 has to specify s1(B, C) after a history that will never be reached if s1 is followed.

◮ Two strategy profiles s and s′ are outcome equivalent if and

• nly if they lead to the same terminal node.

◮ Denote o(s) the terminal history that results when each player

follows strategy si: o(s) ∈ Z.

◮ In the example above:

ui[o((B, E), C)] = ui[o((B, F), C)] ∀i = 1, 2.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 57 / 59

Mixed strategies:

◮ In an extensive form game a mixed strategy is a probability

distribution over the set of (pure) strategies.

◮ Mixed strategies do not add much in extensive form games of

perfect information.

◮ Recall that randomization requires indifference. ◮ Since in a game of perfect information decisions are taken

sequentially randomization of other players rarely leads to indifference.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 58 / 59

Nash Equilibrium

Definition

A Nash Equilibrium is a strategy profile s∗ such that for every player i ∈ N: ui(o(s∗

i , s∗ −i)) ≥ ui(o(si, s∗ −i))

∀si The strategy s∗

i is the best reply to s∗ −i for every player i ∈ N.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 59 / 59