EC487 Advanced Microeconomics, Part I: Lecture 10 Leonardo Felli - - PowerPoint PPT Presentation

EC487 Advanced Microeconomics, Part I: Lecture 10

Leonardo Felli

32L.LG.04

1 December 2017

Repeated Games

◮ This is the class of dynamic games which is best understood

in game theory.

◮ Players face in each period the same normal form stage game. ◮ Players’ payoffs are a weighted discounted average of the

payoffs players receive in every stage game.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 2 / 66

Repeated Games (cont’d)

Main point of the analysis:

◮ players’ overall payoffs depend on the present and the future

stage game payoffs,

◮ it is possible that the threat of a lower future payoff may

induce a player at present to choose a strategy different from the stage game best reply.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 3 / 66

Example: the repeated prisoner dilemma

◮ Stage game:

1\2 C D C 1, 1 −1, 2 D 2, −1 0, 0

◮ Per period payoff depends on current action: gi(at) . ◮ Players’ common discount factor δ. ◮ It is convenient to label the first period t = 0.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 4 / 66

Repeated Prisoner Dilemma (cont’d)

◮ Since we are going to compare the equilibrium payoffs for

different time horizons we need to re-normalize the payoffs so that they are comparable.

◮ The average discounted payoff for a T-periods game is:

Π = 1 − δ 1 − δT

T−1

• t=0

δtgi(at)

◮ Clearly if gi(at) = 1

Π = 1 − δ 1 − δT

T−1

• t=0

δt = 1 − δ 1 − δT 1 − δT 1 − δ

• = 1

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 5 / 66

Finitely Repeated Prisoner Dilemma

◮ Assume first that the prisoners’ dilemma game is repeated a

finite number of times.

◮ Nash equilibrium payoffs of the stage game: (0, 0). ◮ Subgame Perfect equilibrium strategies: each player chooses

action D independently of the period and the action the other player chose in the past. 1\2 C D C 1, 1 −1, 2 D 2, −1 0, 0 Proof: backward induction.

◮ Subgame Perfection seems to prevent any gain from repeated,

but finite interaction, but...

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 6 / 66

Finitely Repeated Game

◮ Consider a different finitely repeated game. ◮ Stage game:

L C R T 1, 1 5, 0 0, 0 M 0, 5 4, 4 0, 0 B 0, 0 0, 0 3, 3

◮ Nash equilibria of the stage game: (T, L) and (B, R).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 7 / 66

Finitely Repeated Game (cont’d)

Assume the game is played twice and consider the following strategies: Player 1:

◮ play M in the first period; ◮ in the second period play B if the observed outcome is (M, C); ◮ in the second period play T if the observed outcome is not

(M, C); Player 2:

◮ play C in the first period; ◮ in the second period play R if the observed outcome is (M, C); ◮ in the second period play L if the observed outcome is not

(M, C);

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 8 / 66

Finitely Repeated Game (cont’d)

Proposition

If δ ≥ 1

2 then these strategies are a subgame perfect equilibrium of

the game. L C R T 1, 1 5, 0 0, 0 M 0, 5 4, 4 0, 0 B 0, 0 0, 0 3, 3 Proof: Backward induction: in the last period the strategies prescribe a Nash equilibrium. In the first period both player 1 and player 2 conform to the strategies if and only if: 1 − δ 1 − δ2

• [4 + δ 3] = 4 + δ 3

1 + δ ≥ 1 − δ 1 − δ2

• [5 + δ] = 5 + δ

1 + δ The inequality is satisfied for δ ≥ 1

2.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 9 / 66

Infinitely Repeated Prisoner Dilemma

◮ Consider now the the infinitely repeated prisoner dilemma:

T = +∞.

◮ Stage game:

1\2 C D C 1, 1 −1, 2 D 2, −1 0, 0

Proposition

Both player choosing strategy D in every period is an SPE of the repeated game.

◮ Proof: by one deviation principle. Notice that an infinitely

repeated game is continuous at infinity.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 10 / 66

Infinitely Repeated Prisoner Dilemma (cont’d)

Proposition

The (D, D) equilibrium is the only equilibrium if we restrict players’ strategies to be history independent.

Proposition

If δ ≥ 1

2 then the following strategy profile (σA, σB) is a SPE of

the repeated game:

◮ Player i chooses C in the first period. ◮ Player i continues to choose C as long as no player has

chosen D in any previous period.

◮ Player i will choose D if a player has chosen D in the past

(for the rest of the game).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 11 / 66

Infinitely Repeated Prisoner Dilemma (cont’d)

Proof: If a player i conforms to the prescribed strategies the payoff is 1. If a player deviates in one period and conforms to the prescribed strategy from there on (one deviation principle) the continuation payoff is: (1 − δ)(2 + 0 + . . .) = (1 − δ) 2 If δ ≥ 1

2 then

1 ≥ (1 − δ) 2.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 12 / 66

Infinitely Repeated Prisoner Dilemma (cont’d)

We still need to check that in the subgame in which both players are choosing D neither player wants to deviate. However, choosing D in every period is a SPE of the entire game hence it is a SPE of the (punishment) subgames. Notice that using this type of strategies not only choosing (C, C) in every period is a SPE outcome, a big number of other SPE

• utcomes are also achievable.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 13 / 66

Infinitely Repeated Prisoner Dilemma (cont’d)

Indeed there exists a Folk Theorem.

✻ ✲ q q ❍❍❍❍❍❍❍❍ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆

(1, 1) (0, 0) (−1, 2) (2, −1) Π2 Π1

❆ ❆ ❆ ❆ ❆ ❍ ❍ ❍ ❍ q q

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 14 / 66

General repeated normal form game

Definition

Let G be a given stage game: a normal form game G =

• N, Ai, gi(at)
• Definition

Let G ∞ be the infinitely repeated game associated with the stage game above: G ∞ = {N, H, P, Ui(σ)} such that:

◮ H = ∞ t=0 At where A0 = ∅; ◮ P(h) = N for every h ∈ H − Z;

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 15 / 66

Repeated Normal form Game (cont’d)

◮ The payoffs for the game G ∞ in the case δ < 1 are:

Ui(σ) = (1 − δ)

∞

• t=0

δtgi(σt(ht))

◮ Denote ht the history known to the players at the beginning

• f period t: ht = {a0, a1, . . . , at−1}.

◮ Let Ht = At−1 to be the space of all possible period t

histories.

◮ A pure strategy for player i ∈ {1, 2} in the game G ∞ is then

the infinite sequence of mappings: {st

i }∞ t=0 such that

st

i : Ht → Ai.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 16 / 66

Repeated Normal form Game (cont’d)

In general we will allow players to mix in every possible stage game: ∆i(Ai) set of probability distributions on Ai. A behavioral mixed strategy in this environment is instead an infinite sequence of mappings: {σt

i }∞ t=0 such that

σt

i : Ht → ∆(Ai).

Notice that mixed strategies cannot depend on past mixed strategies by the opponents but only on their realizations. The payoffs for the game G ∞ in the case δ < 1 are: Ui = Eσ(1 − δ)

∞

• t=0

δtgi(σt(ht))

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 17 / 66

Repeated Normal form Game (cont’d)

◮ Notice that Eσ(·) is the expectation with respect to the

distribution over the infinite histories generated by the profile

• f mixed behavioral strategies {σt

i }∞ t=0. ◮ Notice that this specification of payoffs allows us to

reinterpret the discount factor δ as:

◮ the probability that the game will be played in the following

period, where these probabilities are assumed to be independent across periods.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 18 / 66

Repeated Normal form Game (cont’d)

We allow the players to coordinate their strategies through the use

• f a public randomizing device whose realization in period t is ωt .

Therefore a period t history for player i is: ht = {a0, . . . , at−1; ω0, . . . , ωt}.

Proposition

If α∗ is a NE strategy profile for the stage game G, then the strategy: “each player i plays α∗

i independently of the history of

play” are a NE and a SPE of the infinitely repeated game G ∞(δ).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 19 / 66

Repeated Normal form Game (cont’d)

◮ The proof that the strategies above are a Subgame Perfect

equilibrium of the game G ∞(δ) is easily obtained by using

• ne-deviation-only principle.

◮ In any given period consider the deviation in the immediate

period and then let the players continue playing the equilibrium strategies.

◮ Then any deviation cannot be profitable: it is a deviation from

the Nash equilibrium action of stage game G.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 20 / 66

Repeated Normal form Game (cont’d)

Assume now that the stage game G has n NE

• αj,∗n

j=1.

Proposition

Then, for any map j(t) from time periods into an index of the NE

• αj,∗n

j=1, the strategies:

“each player i plays αj(t),∗

i

in period t” are a SPE of the game G ∞(δ). These SPE strategies are history independent. Therefore each player’s best response in every period t is to play the stage game best response in t: today’s decision does not affect the future play.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 21 / 66

Repeated Normal form Game (cont’d)

◮ In other words, playing repeatedly the stage game G does not

reduce the set of equilibrium payoffs.

◮ To be able to move to the Folk Theorem we first need to

define an area known as the set of feasible and individually rational payoffs.

◮ Consider as an example the following battle of sexes game G:

1\2 B F B 1, 2 0, 0 F 0, 0 2, 1

◮ Assume this game is repeated an infinite number of times

G ∞(δ).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 22 / 66

Repeated Normal form Game (cont’d)

◮ We first need to define the set of feasible payoffs of G ∞. ◮ Recall that when choosing actions in every period players can

coordinate using a public randomizing device.

◮ This implies that: every payoff associated with a pure strategy

profile a can be achieved: (1, 2), (0, 0), (2, 1).

◮ It also implies that every payoff generated by any linear and

convex combination of the pure strategy profile can be achieved.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 23 / 66

Repeated Normal form Game (cont’d)

◮ In general these payoffs are all in the convex hull of the

payoffs associated with the pure strategy profiles.

◮ This is the smallest convex set that includes the payoffs

associated with the pure strategy profiles.

◮ Formally:

V = convex hull {π | πi = gi(a) ∀a ∈ A}

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 24 / 66

Repeated Normal form Game (cont’d)

Graphically:

✻ ✲ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁❅ ❅ ❅ ❅ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ q q q

(0, 0) (1, 2) (2, 1) π2 π1 V

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 25 / 66

Repeated Normal form Game (cont’d)

◮ We now define the set of individually rational payoffs of G ∞. ◮ We first need to define the minmaxing payoff of each player. ◮ The minmaxing payoff to player 1 is the lowest payoff that

player 2 can impose on player 1.

◮ Given that player 1 is rational this is the lowest payoff among

the ones that are player 1’s best reply to player 2 strategies.

◮ This payoff is a best reply for player 1 since he is trying to

achieve the best for himself, given his rationality.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 26 / 66

Repeated Normal form Game (cont’d)

◮ In other words, among the best reply payoffs for player 1,

player 2 chooses her strategy that minimizes these payoffs.

◮ In the battle of sexes game:

1\2 B F B 1, 2 0, 0 F 0, 0 2, 1

◮ the minmax payoff for player 1 is π1 = 1. ◮ the minmax payoff for player 2 is π2 = 1.

◮ In general:

πi = min

α−i

• max

αi gi(αi, α−i)

• Leonardo Felli (LSE)

EC487 Advanced Microeconomics, Part II 1 December 2017 27 / 66

Repeated Normal form Game (cont’d)

◮ Denote mi −i the profile of minimax strategies for players −i if

they minmax player i.

◮ This is the lowest payoff player i’s opponents can hold player i

to by choice of α−i.

Definition

A payoff πi for player i is individually rational if and only if: πi ≥ πi = min

α−i

• max

αi gi(αi, α−i)

• Leonardo Felli (LSE)

EC487 Advanced Microeconomics, Part II 1 December 2017 28 / 66

Repeated Normal form Game (cont’d)

Definition

We define the set of individual rational payoffs to be the set of payoffs that give to each player a payoff I = {(πi, π−i) | πi ≥ πi} The relevant set for us is the set of feasible and individually rational payoffs: V = I ∩ V

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 29 / 66

Repeated Normal form Game (16)

The region of feasible and individually rational payoffs V:

✻ ✲ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁❅ ❅ ❅ ❅ ❅ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ q

(0, 0) (1, 2) (2, 1) π2 π1

q ❅ ❅ ❅ ❅ ❅ q q q q

π2 π1

✻

(1, 1) V

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 30 / 66

Repeated Normal form Game (cont’d)

◮ Consider the following game:

1\2 L R U −2, 2 1, −2 M 1, −2 −1, 2 D 0, 1 0, 1

◮ Restricting attention to pure strategies then we obtain:

◮ m2

1 = D and π2 = 1;

◮ m1

2 ∈ {L, R} and π1 = 1 .

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 31 / 66

Repeated Normal form Game (cont’d)

◮ Consider mixed strategies: assume that player 2 randomizes

with probability q on L.

◮ Then player 1’s expected payoffs for every possible strategy

choice are: Π1(U, q) = 1 − 3q Π1(M, q) = 2q − 1 Π1(D, q) =

◮ This implies that m1 2 = q ∈

1

3, 1 2

• and Π1 = 0.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 32 / 66

Repeated Normal form Game (cont’d)

◮ Assume that player 1 randomizes with probability pU on U

and probability pM on M.

◮ Then player 2’s expected payoffs for every possible strategy

choice are: Π2(pU, pM, L) = 2 (pU − pM) + 1 − pU − pM Π2(pU, pM, R) = 2 (pM − pU) + 1 − pU − pM

◮ This implies that m2 1 = (pU, pM) =

1

2, 1 2

• and that Π2 = 0.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 33 / 66

Folk Theorem

◮ Consider a general finite normal form stage game:

G = {N; Ai, gi(a), ∀i ∈ N}

◮ and the dynamic game that consist of the infinitely repeated

play of the game G when players’ discount factor is δ: G ∞(δ).

◮ The payoffs of the infinitely repeated game are:

πi = (1 − δ)

∞

• t=0

δt gi(ai, a−i)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 34 / 66

Subgame Perfect Folk Theorem

Theorem (Subgame Perfect Folk Theorem – Fudenberg and Maskin (1986))

Consider a stage game such that dim(V) = #N, where #N denotes the number of players and V denotes the set of feasible and individually rational payoffs. Then for any v ∈ V such that (vi > πi), there exists a δv such that for every δ ≥ δv there exists a SPE of G ∞(δ) with payoff vector v.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 35 / 66

Subgame Perfect Folk Theorem (cont’d)

◮ Notice that the extra condition dim(V) = #N is not tight, in

particular the theorem can be proved when dim(V) = #N − 1.

◮ The dimensionality assumption is for example satisfied in the

following example.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 36 / 66

Subgame Perfect Folk Theorem: Example

◮ Let G be the following finite normal form game:

L R U 2, 1 0, 2 D 0, 0 −1, −1

◮ Consider the dynamic game G ∞(δ). ◮ The minmax payoff for both players in pure and mixed

strategies are: π1 = 0 π2 = 0.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 37 / 66

Subgame Perfect Folk Theorem: Example (cont’d)

◮ The set V satisfies the dimensionality assumption dim(V) = 2:

✻ ✲

(0, 2) (0, 0) (−1, −1) (2, 1) π2 π1 Π2 Π1 V

❍❍❍❍❍❍❍❍ q ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✑✑✑✑✑✑ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ q ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ q q

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 38 / 66

Subgame Perfect Folk Theorem: Proof

Proof:

◮ For simplicity we focus on the case in which there exists a

pure action profile a such that g(a) = v.

◮ Assume first that the minmax action profile mi −i for every

i ∈ N is also a pure strategy so that any deviation from minmax behavior is easy to detect.

◮ Choose v′ ∈ int(V) — recall that (vi > πi) — such that:

πi < v′

i < vi

∀i ∈ N

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 39 / 66

Subgame Perfect Folk Theorem: Proof (cont’d)

◮ Choose also an ε > 0 and a

v′(i) = (v′

1 + ε, . . . , v′ i−1 + ε, v′ i , v′ i+1 + ε, . . . , v′ I + ε)

such that: v′(i) ∈ V ∀i ∈ N.

◮ Notice that the role of the full-dimensionality assumption is to

assure that there exists a v′(i) for all i and for some ε and v′.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 40 / 66

Subgame Perfect Folk Theorem: Proof (cont’d)

◮ Once again for simplicity assume that for every i ∈ N there

exists an action profile a(i) such that g(a(i)) = v′(i).

◮ Further denote wj i = gi(mj) player i’s payoff when minmaxing

player j.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 41 / 66

Subgame Perfect Folk Theorem: Proof (cont’d)

◮ Finally, choose n such that

max

a

gi(a) + nπi < min

a gi(a) + nv′ i

• r

n > max

a

gi(a) − min

a gi(a)

v′

i − πi

.

◮ Clearly there exists an n satisfying the condition above being

the numerator bounded above and the denominator bounded below.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 42 / 66

Subgame Perfect Folk Theorem: Proof (cont’d)

◮ We label n the length of a punishment. ◮ To understand the condition above notice that for δ close to 1:

(1 − δn) ≃ (1 − δ)n.

◮ Consider now the following strategy profile.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 43 / 66

Subgame Perfect Folk Theorem: Proof (cont’d)

• 1. The play starts in Phase I.

Phase I: play the action profile a, (g(a) = v).

• 2. The play remains in Phase I so long as in each period:

◮ either the realized action is a ◮ or the realized action differs from a in two or more

components.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 44 / 66

Subgame Perfect Folk Theorem: Proof (cont’d)

• 3. If a single player j deviates from a then the play moves to

Phase IIj. Phase IIj: play mj each period.

• 4. The play stays in Phase IIj for n periods so long as in each

period:

◮ either the realized action is mj ◮ or the realized action differs from mj in two or more

components.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 45 / 66

Subgame Perfect Folk Theorem: Proof (cont’d)

• 5. After n subsequent periods in Phase IIj the play switches to

Phase IIIj. Phase IIIj: play a(j).

• 6. If during Phase IIj a single player i’s action differs from mj

i

begin Phase IIi.

• 7. The play stays in Phase IIIj so long as in each period:

◮ either the realized action is a(j) ◮ or the realized action differs from a(j) in two or more

components.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 46 / 66

Subgame Perfect Folk Theorem: Proof (cont’d)

• 8. If during Phase IIIj a single player i’s action differs from ai(j)

then begin Phase IIi.

◮ Using one-deviation-principle we check now that no player has

an incentive to deviate from the prescribed action in any subgame.

◮ Clearly each phase corresponds to a different type of proper

subgame.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 47 / 66

Subgame Perfect Folk Theorem: Proof (cont’d)

Consider Phase I.

◮ By conforming player i receives payoff vi while by deviating he

cannot receive a payoff higher than: πi = (1 − δ) max

a

gi(a) + δ

• (1 − δn) πi + δnv′

i

• ◮ Since by construction vi > v′

i for δ sufficiently close to 1:

πi < vi.

◮ Notice indeed that if δ = 0 then πi = max a

gi(a) and if δ = 1 then πi = v′

i .

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 48 / 66

Subgame Perfect Folk Theorem: Proof (cont’d)

Consider Phase IIIj, j = i.

◮ By conforming player i receives payoff v′ i + ε while by

deviating he cannot receive more than: πi = (1 − δ) max

a

gi(a) + δ

• (1 − δn) πi + δnv′

i

• ◮ Payoff πi < v′

i + ε for δ sufficiently close to 1.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 49 / 66

Subgame Perfect Folk Theorem: Proof (cont’d)

Consider Phase IIIi.

◮ By conforming player i receives payoff v′ i while by deviating he

cannot receive more than: πi = (1 − δ) max

a

gi(a) + δ

• (1 − δn) πi + δnv′

i

• ◮ Indeed we need:

v′

i > (1 − δ) max a

gi(a) + δ

• (1 − δn) πi + δnv′

i

• ◮ That can be re-written as:

(1 − δn+1)v′

i > (1 − δ) max a

gi(a) + δ (1 − δn) πi

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 50 / 66

Subgame Perfect Folk Theorem: Proof (cont’d)

◮ Using the approximation (1 − δn) ≃ (1 − δ)n we get:

(n + 1)v′

i > max a

gi(a) + δnπi

◮ Since v′ i > mina gi(a) and δ < 1 the following is a sufficient

condition for the inequality above: min

a gi(a) + nv′ i > max a

gi(a) + nπi

◮ Clearly from the definition of n for δ sufficiently close to 1

v′

i > πi

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 51 / 66

Subgame Perfect Folk Theorem: Proof (cont’d)

Consider Phase IIj, j = i.

◮ If n′ periods remaining in Phase IIj player i’s payoff by

conforming is ui =

• 1 − δn′

wj

i + δn′(v′ i + ε) ◮ while by deviating he cannot obtain more than:

πi = (1 − δ) max

a

gi(a) + δ

• (1 − δn) vi + δnv′

i

• ◮ Notice that for δ sufficiently close to 1

ui > πi

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 52 / 66

Subgame Perfect Folk Theorem: Proof (cont’d)

Finally consider Phase IIi.

◮ If n′ < n periods remain in Phase IIi player i’s payoff by

conforming is u′

i =

• 1 − δn′

πi + δn′v′

i ◮ while by deviating:

π′

i = (1 − δn) πi + δnv′ i ◮ Clearly u′ i > π′ i.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 53 / 66

Application of Repeated Games: Cartels

◮ Consider two firms repeatedly involved in a Cournot Duopoly

for an infinite number of periods.

◮ Both firms produce a perfectly homogeneous good with cost

functions: c(qi) = c qi ∀i ∈ {1, 2}.

◮ and inverse demand function:

P(q1 + q2) = a − (q1 + q2) where c < a.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 54 / 66

Application of Repeated Games: Cartels (cont’d)

◮ The two firms’ profit functions are:

Π1(q1, q2) = q1 [a − (q1 + q2) − c] Π2(q1, q2) = q2 [a − (q1 + q2) − c]

◮ The stage game equilibrium choices (q1, q2) are:

max

q1∈R+ q1 [a − (q1 + q2) − c]

max

q2∈R+ q2 [a − (q1 + q2) − c] ◮ which is the solution to the following problem:

qc

1 = 1

2 (a − qc

2 − c)

qc

2 = 1

2 (a − qc

1 − c) .

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 55 / 66

Application of Repeated Games: Cartels (cont’d)

◮ This solution is:

qc

1 = qc 2 = (a − c)

3

◮ with profits:

πc

1 = πc 2 = (a − c)2

9

◮ Consider now a single firm that is a monopolist in this market

and produces a quantity Q.

◮ This firm profit maximization problem is:

max

Q∈R+ Q [a − Q − c]

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 56 / 66

Application of Repeated Games: Cartels (cont’d)

◮ The first order conditions are then:

a − 2Q − c = 0

◮ or the monopolist quantity:

Qm = (a − c) 2 , Πm = (a − c)2 4

◮ Assume now that the two firms, without any explicit deal,

decide each to produce half of the monopolist quantity: qm = 1 2 Qm = (a − c) 4

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 57 / 66

Application of Repeated Games: Cartels (cont’d)

◮ Each firm’s profit in this case is:

πm

1 = πm 2 = (a − c)2

8

◮ Notice that clearly:

πc

i = (a − c)2

9 < πm

i

= (a − c)2 8

◮ The quantity qm does dominate qc i for both firms. ◮ However, if one of the firm, say firm 1, produces quantity

qm

1 = (a − c)

4 then firm 2 can gain by choosing a different quantity.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 58 / 66

Application of Repeated Games: Cartels (cont’d)

◮ In particular, if firm 2 chooses the quantity:

¯ q2 = (a − qm − c) 2 = 3 (a − c) 8

◮ Then firm 2’s profit is:

¯ π2 = 9 (a − c)2 64

◮ which clearly is:

¯ π2 = 9 (a − c)2 64 > πm = (a − c)2 8

◮ This is the reason why for both firms to choose (qm 1 , qm 2 ) is

not a Nash equilibrium of the Cournot model.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 59 / 66

Application of Repeated Games: Cartels (cont’d)

◮ Assume however that the two firms compete for an infinite

number of periods.

◮ Consider the following strategies: ◮ Firm 1:

◮ choose quantity qm

1 in the first period;

◮ in every subsequent period choose quantity qm

1 if the observed

• utcome in the previous period is (qm

1 , qm 2 );

◮ in every subsequent period choose quantity qc

1 if in the

previous period you observe that firm 2 chose quantity ¯ q2;

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 60 / 66

Application of Repeated Games: Cartels (cont’d)

◮ Firm 2:

◮ choose quantity qm

2 in the first period;

◮ in every subsequent period choose quantity qm

2 if the observed

• utcome in the previous period is (qm

1 , qm 2 );

◮ in every subsequent period choose quantity qc

2 if in the

previous period you observe that firm 2 chose quantity ¯ q1;

◮ Recall that the average discounted payoff of each firm is:

(1 − δ)

∞

• t=0

δtπi(t)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 61 / 66

Application of Repeated Games: Cartels (cont’d)

◮ These strategies do not require an explicit agreement between

the two firms provided each firm believes the other firm behaves this way.

◮ Question: for which δ neither firm wants to deviate from

these strategies?

◮ Consider firm i:

πm

i

≥ (1 − δ)¯ πi + δπc

i

• r

(a − c)2 8 ≥ (1 − δ)9 (a − c)2 64 + δ(a − c)2 9

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 62 / 66

Application of Repeated Games: Cartels (cont’d)

◮ which is satisfied if and only if:

δ ≥ 9 17

◮ Moreover no firm has an incentive to deviate from punishment

strategies since (qc

1, qc 2) is a Nash equilibrium of the Cournot

stage game.

◮ Therefore the cartel behaviour described by the strategies

above is a Subgame Perfect equilibrium of the infinitely repeated game if and only if δ ≥ 9/17.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 63 / 66

Subgame Perfect Folk Theorem: Comment

◮ Notice that in the theory of repeated games there does not

exists a commonly accepted theory predicting that the player will play an equilibrium whose payoff is on the Pareto-frontier

• f V.

◮ In other words nothing guarantees that the outcome will be

• n the Pareto frontier.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 64 / 66

Subgame Perfect Folk Theorem: Comment (cont’d)

Indeed:

✻ ✲

(0, 2) (0, 0) (−1, −1) (2, 1) π2 π1 Π2 Π1 V

❍❍❍❍❍❍❍❍ q ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✑✑✑✑✑✑ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ q ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ q q

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 65 / 66

Subgame Perfect Folk Theorem: Comment (cont’d)

Overall the Folk Theorem warns us to use caution when arguing that the best way of making predictions in a strategic setting is by using Nash and even Subgame Perfect equilibria.

◮ Office hours in the next two weeks:

Tuesday, Dec. 5 and 12: 11:00-13:00 am

◮ Exam scheduled for:

Thursday, January 4, 2018 at 14:30p.m.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 66 / 66