Part II: Strategic Interaction Introduction of competition Three - - PDF document

Part II: Strategic Interaction

• Introduction of competition
• Three instruments to compete in a market (classify

according to the speed at which they can be altered): – In short-run: prices (Chapter 5), with rigid cost structure and product characteristics. – In longer-run: - cost structure and product character- istics can be changed. Capacity constraint (Chapter 5), quality, product design, product differentia- tion, Advertising (Chapter 7); Barrier to entry, accommodation and exit (chapter 8); Reputation and predation (Chapter 9). – In long-run: product characteristic, cost structures, R&D (chapter 10) 1

Chapter 11: Introduction to Non Cooperative Game Theory 1 Introduction

• “The Theory of Games and Economic Behavior”, John

von Neumann and Oskar Morgenstern, 1944.

• Two distinct possible approaches:

– The strategic and non-cooperative approach. – The cooperative approach.

• “Games”: scientific metaphor for a wider range of

human interactions.

• A game is being played any time people interact with

each other.

• People interact in a rational manner.
• Rationality: fundamental assumption in Neoclassical

economic theory. But the individual needs not consider her interactions with other individuals.

• Game theory: study of rational behavior in situation

involving interdependence. 2

Outline

• 1. Introduction
• 2. Games and Strategies
• 3. Static games of complete information

– Nash Equilibrium

• 4. Dynamic games of complete information

– Subgame Perfect Nash Equilibrium

• 5. Static games of incomplete information

– Bayesian Nash Equilibrium

• 6. Dynamic games of incomplete information.

– Subgame Perfect Bayesian Equilibrium

• 7. Reaction functions
• Game of complete information - each player’s payoff

function is common knowledge among all the players

• Game of incomplete information - some players are

uncertain about other players payoff functions 3

2 Games and strategies

2.1 The rules of the game

The rules must tell us

• who can do what, when they can do it,
• who gets how much when the game is over.

Essential elements of a game:

• players (who); strategies (what); information; timing

(when); payoffs (how much) 2 principal representations of the rules of the game:

• The normal or strategic form;
• The extensive form (tree).

Assumption: there is common knowledge. Player 1 knows the rules. Player 1 knows that player 2 knows the rules. Player 1 knows that player 2 knows that player 1 knows the rules and so on and so forth. (“I know that you know, I know that you know that I know....”). 4

• Players in the game: n players (firms) i = 1, 2, ..., n
• Set of strategies (or actions) available to each player

si ∈ Si

• (s1, ..., sn) is the combination of strategies
• Payoff associated with any strategy combination

πi(s1, ..., sn)

• Information set

Definition A strategy for a player is a complete plan of

• actions. It specifies a feasible action for the player in every

contingency in which the player might be called on to act. Definition A pure strategy is the choice by a player of a given action with certainty. Definition A mixed strategy is when one player plays randomly between different strategies. Remark A pure strategy is a special case of a mixed strategy. 5

2.2 Normal form

The normal-form representation of a n-player game specifies:

• The players’ strategies space S1, ..., Sn
• and their payoff functions π1, ..., πn
• Let denote this game by G = {S1, ..., Sn; π1, ..., πn}

2.3 Extensive form (Tree of the game)

The extensive-form representation of a game specifies

• 1. the players of the game,

2.a. when each player has to move, 2.b. what each player can do at each of his opportunities to move, 2.c. what each player knows at each of the opportunities to move.

• 3. The payoff received by each player for each combina-

tion of moves that could be chosen by the players. 6

2.4 Example: Prisoners’ Dilemma

• 2 suspects are arrested and charge for a crime.
• The police lack sufficient evidence to convict the

suspects, unless at least one confesses.

• Deal from the police with each suspect (separately):

– if neither confesses then both will be convicted of a minor offence (= 1 month in jail); – if both confess then both will be sentenced to jail for 6 months; – if one confesses but the other does not, then the confessor will be released immediately, the other will be sentenced to 9 months in jail.

1/2 not confess not −1, −1 −9, 0 confess 0, −9 −6, −6

7

3 Static Game of Complete Infor- mation

• Iterated elimination of strictly dominated strategies

Definition In the normal-form game G, let s0

i and s00 i

be feasible strategies for player i. Strategy s0

i is strictly

dominated by strategy s00

i if for each feasible combination

• f the other players’ strategies,

πi(s1, ..., si−1, s0

i, si+1, ..., sn) < πi(s1, ..., si−1, s00 i , si+1, ..., sn)

for each s−i = (s1, ..., si−1, si+1, ..., sn).

• Nash Equilibrium

Definition In the normal-form game G, the strategies

(s∗

1, ..., s∗ n) are a Nash Equilibrium if, for each player i,

s∗

i is player i’s best response to the strategies specified for

the n − 1 other players, (s∗

1, ., s∗ i−1, s∗ i+1, .., s∗ n):

πi(s∗

1, ., s∗ i−1, s∗ i, s∗ i+1, .., s∗ n) ≥ πi(s∗ 1, ., s∗ i−1, si, s∗ i+1, .., s∗ n)

for every feasible strategy si in Si; that is, s∗

i solves

max

si∈Siπi(s∗ 1, ., s∗ i−1, si, s∗ i+1, .., s∗ n).

8

Proposition In the normal-form game G, if iterated elimination of strictly dominated strategies eliminates all but strategies (s∗

1, ..., s∗ n), then these strategies are the

unique Nash equilibrium of the game. Proposition In the normal-form game G, if the strategies

(s∗

1, ..., s∗ n) are a Nash equilibrium, then they survive

iterated elimination of strictly dominated strategies. More examples:

• 1. The battle of the sexes

– 2 players: a wife and her husband – Strategies space: {Opera , Soccer game} – Payoffs: both players would rather spend the evening together than apart, but the woman prefers the opera, her husband the soccer game.

Wife / Husband Opera Soccer game Opera 2, 1 0, 0 Soccer game 0, 0 1, 2

– What are the equilibria? 9

• 2. Matching pennies

– 2 players: player 1 and 2 – Strategies space: {Tails, Heads} – Payoffs

Player1/Player2 Heads Tails Heads 1, −1 −1, 1 Tails −1, 1 1, −1

• 3. Price competition with differentiated goods

– 2 players: firm 1 and 2 – strategies si = pi for i = 1, 2 – c: unit cost – Demand for firm i is qi = Di(pi, pj) = 1 − bpi + dpj with 0 ≤ d ≤ b. – Each firm maximizes its profit

Max

pi

πi = (pi − c)(1 − bpi + dpj)

– There exists an unique Nash equilibrium

p∗

1 = p∗ 2 = 1 + cb

2b − d

10

4 Dynamic Game of Complete Information

• Players’ payoff function are common knowledge.
• Perfect information: at each move in the game the

player with the move knows the full history of the play

• f the game thus far.
• Imperfect information: at some move the player with

the move does not know the history of the game.

• Central issue of dynamic games: credibility.
• Subgame Perfect Nash equilibrium (Selten, 1965):

refinement of Nash equilibrium for dynamic game.

• Backward induction argument, Kuhn’s algorithm

(Kuhn, 1953) 11

4.1 Dynamic game of complete and perfect information

Timing:

• 1. Player 1 chooses an action a1 ∈ A1
• 2. Player 2 observes a1 and thus chooses an action

a2 ∈ A2

• 3. Payoffs are π1(a1, a2) and π2(a1, a2)
• Examples: Stackelberg’s model of duopoly; Rubin-

stein’s bargaining game....

• Backwards induction

– player 2 chooses a1 that maximizes π2(a1, a2). Assume that for each a1 there exists a unique solution

R2(a1).

– Player 1 should anticipate R2(a1), and chooses a1 that maximizes π1(a1, R2(a1)). Assume there exists a unique solution a∗

1.

– Backward induction outcome (a∗

1, R2(a∗ 1))

– Subgame Perfect equilibrium is (a∗

1, R2(a1))

12

Definition A subgame in an extensive-form game – begins at a decision node n that is a singleton informa- tion set (but not the first decision node), – includes all the decision and terminal nodes following n in the game tree and, – does not cut any information set. Definition A NE is subgame-perfect if the players’ strategies constitute a Nash equilibrium in every subgame. Definition In the two-stage game of complete and perfect information, the backward-induction outcome is

(a∗

1, R2(a∗ 1)) but the subgame-perfect Nash equilibrium

is (a∗

1, R2(a1)).

Example 1:

• Player 1 chooses L or R, where L ends the game with

payoff 2 to player 1 and 0 to player 2.

• Player 2 observes 1’s choice. If 1 chooses R then 2

chooses L0 or R0 where L0 ends the game with 1 to each player.

• Player 1 observes 2’s choice. If the earlier choices were

R and R0 then 1 chooses L00 and R00, both of which end

the game, L00 with payoffs (3,0) and R00 with (0,2). 13

• How many subgames?
• What is the backward-induction outcome?
• What is the subgame-perfect Nash equilibrium?

4.2 Dynamic game of complete information but imperfect information

4.2.1 Two-stage game Timing:

• 1. Players 1 and 2 simult. choose a1 ∈ A1 and a2 ∈ A2.
• 2. Players 3 and 4 observe the outcome and then simult.

choose a3 ∈ A3 and a4 ∈ A4.

• 3. Payoffs are πi(a1, a2, a3, a4) for i = 1, 2, 3, 4.

If there exists a NE for players 3 and 4 a∗

3(a1, a2) and

a∗

4(a1, a2), then the timing is

• 1 and 2 simult. choose actions a1 ∈ A1 and a2 ∈ A2.
• Payoffs are πi(a1, a2, a∗

3(a1, a2), a∗ 4(a1, a2)) for i = 1, 2.

• If there exists a unique Nash equilibrium (a∗

1, a∗ 2), then

(a∗

1, a∗ 2, a∗ 3(a1, a2), a∗ 4(a1, a2))) is a Subgame Perfect

Nash equilibrium. 14

4.3 Repeated game

4.3.1 Two stage repeated game (Finite horizon) Example: Prisoners’ Dilemma played twice

1/2 L2 R2 L1 1, 1 5, 0 R1 0, 5 4, 4

Timing:

• 1. 2 players play simultaneously,
• 2. Then they observe the outcome of the first play before

the second play begins,

• 3. third they play simultaneously a second time.
• Assumption: there is no discounting
• Identical to previous game: players 3 and 4 are identical

to player 1 and 2.

• Backward induction: the second period is equivalent

to a one-shot game. Nash equilibrium is (L1, L2) with payoffs (1,1)

• What is the first period payoff bi-matrix?

15

1/2 L2 R2 L1 1 + 1, 1 + 1 5 + 1, 0 + 1 R1 0 + 1, 5 + 1 4 + 1, 4 + 1

• The unique subgame perfect Nash equilibrium is

(L1, L2) with payoffs (2,2). Definition Given a stage game G, let G(T) denote the finitely repeated game in which G is played T times, with the outcome of all the preceding plays observed before the next play begins. The payoffs for G(T) are simply the sum

• f the payoffs from the T stages.

Proposition If the stage game G has a unique Nash equilibrium then, for any finite T, the repeated game G(T) has a unique subgame Nash outcome: the Nash equilibrium

• f G is played in every stage.

4.3.2 Infinite repeated game

• Credible threats about future behavior can influence

current behavior.

• Even if the game has a unique Nash equilibrium, there

16

may be subgame-perfect outcome of the infinitely repeated game in which no stage’s outcome is a Nash equilibrium of G. Example: Prisoners’ Dilemma played an infinite number

• f times
• δ =

1 1+r is the discount factor (r interest rate)

• Present value of an infinite sequence of payoffs

π1, π2, .... is PV =

∞

X

t=1

δt−1πt

• Trigger strategy:

Play Ri in the first stage. In the tth stage, if the outcome

• f all t − 1 preceding stages has been (R1, R2), then

play Ri; otherwise play Li.

• Is this trigger strategy a Nash equilibrium?
• Assume i has adopted this trigger strategy. What will do

j?

– j’s best response to Li is Lj forever. – what is j’s best response to Ri ? 17

– if j cheats, present value from cheating is

PVcheat = 5 + δ 1 − δ

– if j cooperates, present value from cooperating is

PVcoop = 4 1 − δ

– Thus, Rj is optimal if and only if PVcoop ≥ PVcheat.

⇒ δ ≥ 1 4.

• For δ ≥ 1

4 the trigger strategy is a Nash equilibrium.

• It is also a subgame perfect Nash equilibrium.
• Folk theorem (Friedman (1971)): any feasible payoffs

above the “individually rational payoffs” can be sustain

• n average as a subgame perfect equilibrium payoff of

the infinitely repeated game for δ → 1. 18

5 Static Game of Incomplete Information

• Bayesian games
• Example: sealed bid auction.
• Each player knows his own payoff function but is

uncertain about the other players’ payoff functions. Payoff of i is

πi(a1, ..., an; ti)

where ti is the type, ti ∈ Ti.

• Example: Ti = {t1i, t2i}; two payoffs are πi(a1, ..., an; t1i)

and πi(a1, ..., ant2i).

• Player i may be uncertain about the types of the other

players

t−i = (t1, ..., ti−1, ti+1, ..., tn) ∈ T−i

• Player i’s belief about the other players’ types:

pi(t−i/ti)

19

Definition The normal-form representation of an n- players static Bayesian game specifies the players’ action space A1, ..., An, and their type space T1, ..., Tn, their beliefs p1, ..., pn, and their payoff functions π1, ..., πn. Player i’s type, ti, is privately known by player i, determines player i’s payoff function, πi(a1, ..., an; ti), and is a member of the set of possible types, Ti. Player i’s belief pi(t−i/ti) describes i’s uncertainty about the n − 1

• ther players’ possible types, t−i, given i’s own type, ti.
• Change a game of incomplete information to a game of

imperfect information. Harsanya (1967)’s timing:

• 1. Nature draws a type vector t = (t1, ..., tn) where ti is

drawn from the set of possible types Ti.

• 2. Nature reveals ti to player i but not to any other player;
• 3. The players simultaneously choose strategies; player i

chooses ai ∈ Ai.

• 4. Payoffs πi(a1, ..., an; ti).

20

• A strategy for player i is a function si(ti) where for

each ti ∈ Ti, si(ti) specifies the chosen strategy of type

ti.

• Bayes’ rule

pi(t−i/ti) = p(t−i, ti) p(ti)

Definition In a static Bayesian game, the strategies

s∗ = (s∗

1, ..., s∗ n) are a (pure-strategy) Bayesian Nash

equilibrium if for each player i and for each of i’s types ti in Ti solves

max

si∈Si

X

t−i∈T−i

πi(s∗

1(t1), .., si, s∗ i+1(ti+1), ., s∗ n(tn); t)pi(t−i/ti)

Example: two-player, simultaneous move game

• 2 players 1 and 2
• Set of strategies: A1 = {Up, Down}, A2 =

{Left, Right}

• Player 1 has only one type
• Player 2 has 2 types: t2 and t0

2

• Player 1 puts equal probabilities on the two types.

21

• Normal form of the game is

t2 t0

2

1\2 L R L R U 3, 1 2, 0 3, 0 2, 1 D 0, 1 4, 0 0, 0 4, 1

• Player 1’s payoff depends only on the chosen actions,

and not on player 2’s type.

• What is the Bayesian equilibrium?
• Each type of player 2 has a dominant strategy: s∗

2(t2) =

L and s∗

2(t0 2) = R.

• It is equivalent to a game where player 1 faces an
• pponent who played L or R with equal probability.

Thus expected profit from playing U is 1

23 + 1 22 = 5 2 and

expected profit from playing R is 1

20 + 1 24 = 4

• 2. Player 1

chooses s∗

1 = U.

22

6 Dynamic Game of Incomplete Information

• Perfect Bayesian Equilibrium - refinement of Bayesian

equilibrium

• Signalling game, Spence (1973)

Timing:

• 1. Player 1 chooses among three actions: L, M, R
• 2. If player 1 chooses R then the game ends. If player 1

chooses L or M, then player 2 learns that R was not chosen (but not which of L or M was chosen). Player 2 then chooses between L0 or R0; then game ends.

• If complete information (if simultaneous choices)

1\2 L’

R’

L

2,1 0,0 M 0,2 0,1 R 1,3 1,3

• 2 pure strategy Nash equilibria (L, L0) and (R, R0)

23

• But (R, R0) depends on a non credible threat.

Requirements:

• R1. Belief: at each information set, the player with the

move must have a belief about which node has been reached.

• R2. Sequential rationality
• R3. At the information set on the equilibrium path,

beliefs are determined by Bayes’ rule and the players’ equilibrium strategies.

• R4. At the information set off the equilibrium path,

beliefs are determined by Bayes’ rule and the players’ equilibrium strategies where possible. Definition A perfect Bayesian equilibrium consists of strategies and beliefs satisfying Requirements 1-4. 24

6.1 Signalling Game

• 2 players: a Sender (S) and a receiver (R)

Timing:

• 1. Nature draws the type t1 (resp. t2) for the Sender

according to a probability p = 0.5 (resp. 1 − p = 0.5).

• 2. The Sender observes his type and chooses a message L
• r R.
• 3. The Receiver observes the messages (L or R) but not

the type and then chooses an action u or d.

• 4. Payoffs are given by πS(t, m, a) and πR(t, m, a) where

t = {t1, t2}, m = {L, R}, a = {u, d}.

2 kinds of equilibrium:

• Pooling (for example (L, L))
• Separating (for example: type t1 chooses L, and type t2

chooses R)

• (semi-separating equilibrium)
• one pooling Perfect Bayesian equilibrium {(L, L),

(u, d), µ = 0.5, out of equilibrium q ≤ 2/3}

• one separating PBE {(R, L), (u, u), µ = 0}

25

7 Reaction functions

• 2 firms: 1 and 2
• simultaneous-move game
• Strategies can be prices, quantities....
• Each firm maximizes its profit

Max

ai∈Ai Πi(ai, aj)

• The FOC are

∂Πi(ai, aj) ∂ai = 0 ⇒ ai(aj) = Ri(aj)

for i 6= j and i, j = 1, 2

• where Ri(aj) is the best response function of i to j’s

action.

• Assumption:each firm’s profit is strictly concave

∂2Πi(ai,aj) ∂a2

i

< 0 for i 6= j and i, j = 1, 2. Thus the SOC

(local maximum) are satisfied.

• A Nash equilibrium is (a∗

i, a∗ j) such that a∗ i = Ri(a∗ j)

and a∗

j = Rj(a∗ i).

• Are best response functions downward or upward

sloping? 26

• Let’s differentiate

∂Πi(Ri(aj), aj) ∂ai = 0,

∂2Πi(Ri(aj),aj) ∂2ai ∂Ri(aj) ∂aj

+ ∂2Πi(Ri(aj),aj)

∂ai∂aj

= 0

∂Ri(aj) ∂aj

= −

∂2Πi(Ri(aj),aj) ∂ai∂aj ∂2Πi(Ri(aj),aj) ∂2ai

• Thus the sign(∂Ri(aj)

∂aj ) = sign(∂2Πi(Ri(aj),aj) ∂ai∂aj

)

If ∂2Πi(ai,aj)

∂ai∂aj

< 0 Strategic substitutes

(quantities) If ∂2Πi(ai,aj)

∂ai∂aj

> 0 Strategic complements (prices)

27