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