Part II: Strategic Interaction

• Introduction of competition• Three instruments to compete in a market (classifyaccording to the speed at which they can be altered):– In short-run: prices (Chapter 5), with rigid coststructure and product characteristics.

– In longer-run: - cost structure and product character-istics can be changed. Capacity constraint (Chapter5), quality, product design, product differentia-tion, Advertising (Chapter 7); Barrier to entry,accommodation and exit (chapter 8); Reputationand predation (Chapter 9).

– In long-run: product characteristic, cost structures,R&D (chapter 10)

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Chapter 11: Introduction to NonCooperative Game Theory

1 Introduction• “The Theory of Games and Economic Behavior”, Johnvon 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 ofhuman interactions.

• A game is being played any time people interact witheach other.

• People interact in a rational manner.• Rationality: fundamental assumption in Neoclassicaleconomic theory. But the individual needs notconsider her interactions with other individuals.

• Game theory: study of rational behavior in situationinvolving interdependence.

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Outline1. Introduction2. Games and Strategies3. 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 payofffunction is common knowledge among all the players

• Game of incomplete information - some players areuncertain about other players payoff functions

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2 Games and strategies2.1 The rules of the gameThe 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 player2 knows the rules. Player 1 knows that player 2 knowsthat player 1 knows the rules and so on and so forth.(“I know that you know, I know that you know that Iknow....”).

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

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

πi(s1, ..., sn)

• Information setDefinition A strategy for a player is a complete plan ofactions. It specifies a feasible action for the player in everycontingency in which the player might be called on to act.Definition A pure strategy is the choice by a player of agiven action with certainty.Definition A mixed strategy is when one player playsrandomly between different strategies.Remark A pure strategy is a special case of a mixedstrategy.

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2.2 Normal formThe normal-form representation of a n-player gamespecifies:• 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 specifies1. the players of the game,2.a. when each player has to move,2.b. what each player can do at each of his opportunitiesto move,2.c. what each player knows at each of the opportunitiesto move.3. The payoff received by each player for each combina-tion of moves that could be chosen by the players.

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2.4 Example: Prisoners’ Dilemma• 2 suspects are arrested and charge for a crime.• The police lack sufficient evidence to convict thesuspects, unless at least one confesses.

• Deal from the police with each suspect (separately):– if neither confesses then both will be convicted of aminor offence (= 1 month in jail);

– if both confess then both will be sentenced to jail for6 months;

– if one confesses but the other does not, then theconfessor will be released immediately, the other willbe sentenced to 9 months in jail.

1/2 not confess

not −1,−1 −9, 0confess 0,−9 −6,−6

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3 Static Game of Complete Infor-mation• Iterated elimination of strictly dominated strategiesDefinition In the normal-form game G, let s0i and s00ibe feasible strategies for player i. Strategy s0i is strictlydominated by strategy s00i if for each feasible combinationof the other players’ strategies,πi(s1, ..., si−1, s0i, si+1, ..., sn) < πi(s1, ..., si−1, s00i , si+1, ..., sn)for each s−i = (s1, ..., si−1, si+1, ..., sn).

• Nash EquilibriumDefinition 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 forthe 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 solvesmaxsi∈Si

πi(s∗1, ., s∗i−1, si, s

∗i+1, .., s

∗n).

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Proposition In the normal-form game G, if iteratedelimination of strictly dominated strategies eliminatesall but strategies (s∗1, ..., s∗n), then these strategies are theunique 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 eveningtogether 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?

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

Player1/P layer2 Heads Tails

Heads 1,−1 −1, 1Tails −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 + dpjwith 0 ≤ d ≤ b.

– Each firm maximizes its profitMaxpi

πi = (pi − c)(1− bpi + dpj)– There exists an unique Nash equilibrium

p∗1 = p∗2 =

1 + cb

2b− d

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4 Dynamic Game of CompleteInformation• Players’ payoff function are common knowledge.• Perfect information: at each move in the game theplayer with the move knows the full history of the playof the game thus far.

• Imperfect information: at some move the player withthe 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)

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4.1 Dynamic game of complete and perfectinformationTiming:1. Player 1 chooses an action a1 ∈ A12. Player 2 observes a1 and thus chooses an actiona2 ∈ A23. 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 solutionR2(a1).

– Player 1 should anticipate R2(a1), and chooses a1that maximizes π1(a1, R2(a1)). Assume there existsa unique solution a∗1.

– Backward induction outcome (a∗1, R2(a∗1))– Subgame Perfect equilibrium is (a∗1, R2(a1))

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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 nin 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 andperfect 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 withpayoff 2 to player 1 and 0 to player 2.

• Player 2 observes 1’s choice. If 1 chooses R then 2chooses L0 or R0 where L0 ends the game with 1 to eachplayer.

• Player 1 observes 2’s choice. If the earlier choices wereR and R0 then 1 chooses L00 and R00, both of which endthe game, L00 with payoffs (3,0) and R00 with (0,2).

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

4.2 Dynamic game of complete informationbut imperfect information4.2.1 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) anda∗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.

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4.3 Repeated game4.3.1 Two stage repeated game (Finite horizon)

Example: Prisoners’ Dilemma played twice

1/2 L2 R2

L1 1, 1 5, 0

R1 0, 5 4, 4Timing:1. 2 players play simultaneously,2. Then they observe the outcome of the first play beforethe 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 identicalto player 1 and 2.

• Backward induction: the second period is equivalentto a one-shot game. Nash equilibrium is (L1, L2) withpayoffs (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 thefinitely repeated game in which G is played T times, withthe outcome of all the preceding plays observed before thenext play begins. The payoffs forG(T ) are simply the sumof the payoffs from the T stages.Proposition If the stage game G has a unique Nashequilibrium then, for any finite T , the repeated gameG(T )has a unique subgame Nash outcome: the Nash equilibriumof G is played in every stage.

4.3.2 Infinite repeated game

• Credible threats about future behavior can influencecurrent behavior.

• Even if the game has a unique Nash equilibrium, there16

may be subgame-perfect outcome of the infinitelyrepeated game in which no stage’s outcome is a Nashequilibrium of G.

Example: Prisoners’ Dilemma played an infinite numberof times• δ = 1

1+r is the discount factor (r interest rate)• Present value of an infinite sequence of payoffs

π1,π2, .... is

PV =∞Xt=1

δt−1πt

• Trigger strategy:PlayRi in the first stage. In the tth stage, if the outcomeof all t − 1 preceding stages has been (R1, R2), thenplay Ri; otherwise play Li.

• Is this trigger strategy a Nash equilibrium?• Assume i has adopted this trigger strategy. What will doj?– j’s best response to Li is Lj forever.– what is j’s best response to Ri ?

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– 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 ≥ 14.

• For δ ≥ 14 the trigger strategy is a Nash equilibrium.

• It is also a subgame perfect Nash equilibrium.

• Folk theorem (Friedman (1971)): any feasible payoffsabove the “individually rational payoffs” can be sustainon average as a subgame perfect equilibrium payoff ofthe infinitely repeated game for δ → 1.

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5 Static Game of IncompleteInformation• Bayesian games• Example: sealed bid auction.• Each player knows his own payoff function but isuncertain 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 otherplayers

t−i = (t1, ..., ti−1, ti+1, ..., tn) ∈ T−i• Player i’s belief about the other players’ types:

pi(t−i/ti)

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Definition The normal-form representation of an n-players static Bayesian game specifies the players’ actionspace A1, ..., An, and their type space T1, ..., Tn, theirbeliefs 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), andis a member of the set of possible types, Ti. Player i’sbelief pi(t−i/ti) describes i’s uncertainty about the n − 1other players’ possible types, t−i, given i’s own type, ti.

• Change a game of incomplete information to a game ofimperfect information.

Harsanya (1967)’s timing:1. Nature draws a type vector t = (t1, ..., tn) where ti isdrawn 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 ichooses ai ∈ Ai.

4. Payoffs πi(a1, ..., an; ti).

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• A strategy for player i is a function si(ti) where foreach ti ∈ Ti, si(ti) specifies the chosen strategy of typeti.

• Bayes’ rulepi(t−i/ti) =

p(t−i, ti)p(ti)

Definition In a static Bayesian game, the strategiess∗ = (s∗1, ..., s

∗n) are a (pure-strategy) Bayesian Nash

equilibrium if for each player i and for each of i’s types tiin Ti solvesmaxsi∈Si

Xt−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 t02• Player 1 puts equal probabilities on the two types.

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• Normal form of the game ist2 t02

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(t02) = R.

• It is equivalent to a game where player 1 faces anopponent who played L or R with equal probability.Thus expected profit from playing U is 123 +

122 =

52 and

expected profit from playing R is 120 +124 =

42. Player 1

chooses s∗1 = U .

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6 Dynamic Game of IncompleteInformation• Perfect Bayesian Equilibrium - refinement of Bayesianequilibrium

• Signalling game, Spence (1973)

Timing:1. Player 1 chooses among three actions: L,M , R2. If player 1 chooses R then the game ends. If player 1chooses L orM , then player 2 learns that R was notchosen (but not which of L orM was chosen). Player 2then chooses between L0 or R0; then game ends.

• If complete information (if simultaneous choices)1\2 L’ R’L 2,1 0,0M 0,2 0,1R 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 themove must have a belief about which node has beenreached.R2. Sequential rationalityR3. 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 ofstrategies and beliefs satisfying Requirements 1-4.

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6.1 Signalling Game• 2 players: a Sender (S) and a receiver (R)Timing:1. Nature draws the type t1 (resp. t2) for the Senderaccording to a probability p = 0.5 (resp. 1− p = 0.5).

2. The Sender observes his type and chooses a message Lor R.

3. The Receiver observes the messages (L or R) but notthe type and then chooses an action u or d.

4. Payoffs are given by πS(t,m, a) and πR(t,m, a) wheret = {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 t2chooses 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

Maxai∈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’saction.

• Assumption:each firm’s profit is strictly concave∂2Πi(ai,aj)

∂a2i< 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 upwardsloping?

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• 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)

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