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 characteristics can be changed. Capacity constraint (Chapter
5), quality, product design, product differentiation, 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
π i(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 π 1, ..., π n
• Let denote this game by G = {S1, ..., Sn; π 1, ..., π n}
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 combination 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
not
−1, −1 −9, 0
conf ess 0, −9
7
conf ess
−6, −6
3 Static Game of Complete Information
• Iterated elimination of strictly dominated strategies
Definition In the normal-form game G, let s0i and s00i
be feasible strategies for player i. Strategy s0i is strictly
dominated by strategy s00i if for each feasible combination
of the other players’ strategies,
π i(s1, ..., si−1, s0i, si+1, ..., sn) < π i(s1, ..., si−1, s00i , 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):
π i(s∗1, ., s∗i−1, s∗i , s∗i+1, .., s∗n) ≥ π i(s∗1, ., s∗i−1, si, s∗i+1, .., s∗n)
for every feasible strategy si in Si; that is, s∗i solves
maxπ i(s∗1, ., s∗i−1, si, s∗i+1, .., s∗n).
si ∈Si
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.
W if e / 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
P layer1/P layer2 Heads T ails
1, −1
Heads
−1, 1
T ails
−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
Maxπ i = (pi − c)(1 − bpi + dpj )
pi
– There exists an unique Nash equilibrium
1 + cb
∗
∗
p1 = p2 =
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
of 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 π 1(a1, a2) and π 2(a1, a2)
• Examples: Stackelberg’s model of duopoly; Rubinstein’s bargaining game....
• Backwards induction
– player 2 chooses a1 that maximizes π 2(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 π 1(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 information 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 π i(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 π i(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
of 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
of 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
of times
1
• δ = 1+r
is the discount factor (r interest rate)
• Present value of an infinite sequence of payoffs
π 1, π 2, .... is
∞
X
PV =
δ t−1π t
t=1
• Trigger strategy:
Play Ri in the first stage. In the tth stage, if the outcome
of 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
δ
P Vcheat = 5 +
1−δ
– if j cooperates, present value from cooperating is
4
P Vcoop =
1−δ
– Thus, Rj is optimal if and only if P Vcoop ≥ P Vcheat.
⇒
1
δ≥ .
4
• For δ ≥ 14 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
on 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
π i(a1, ..., an; ti)
where ti is the type, ti ∈ Ti.
• Example: Ti = {t1i, t2i}; two payoffs are π i(a1, ..., an; t1i)
and π i(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 nplayers static Bayesian game specifies the players’ action
space A1, ..., An, and their type space T1, ..., Tn, their
beliefs p1, ..., pn, and their payoff functions π 1, ..., π n.
Player i’s type, ti, is privately known by player i,
determines player i’s payoff function, π i(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
other 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 π i(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
p(t−i, ti)
pi(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
X
max
π i(s∗1(t1), .., si, s∗i+1(ti+1), ., s∗n(tn); t)pi(t−i/ti)
si ∈Si
t−i ∈T−i
Example: two-player, simultaneous move game
• 2 players 1 and 2
• Set of strategies: A1 = {Up, Down}, A2 =
{Lef t, Right}
• Player 1 has only one type
• Player 2 has 2 types: t2 and t02
• Player 1 puts equal probabilities on the two types.
21
• Normal form of the game is
t2
1\2 L
R
t02
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 an
opponent who played L or R with equal probability.
Thus expected profit from playing U is 12 3 + 12 2 = 52 and
expected profit from playing R is 12 0 + 12 4 = 42 . 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
or 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 π S (t, m, a) and π R(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
MaxΠi(ai, aj )
ai ∈Ai
• The FOC are
∂Πi(ai, aj )
= 0 ⇒ ai(aj ) = Ri(aj )
∂ai
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
∂ 2 Πi (ai ,aj )
< 0 for i 6= j and i, j = 1, 2. Thus the SOC
∂a2i
(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 )
= 0,
∂ai
2 i
∂ Π (Ri (aj ),aj ) ∂Ri (aj )
∂ 2 Πi (Ri (aj ),aj )
=0
∂ 2 ai
∂aj +
∂ai ∂aj
• Thus
∂ 2 Πi (Ri (aj ),aj )
∂ai ∂aj
∂Ri (aj )
=
−
∂aj
∂ 2 Πi (Ri (aj ),aj )
∂ 2 ai
∂Ri (aj )
∂ 2 Πi (Ri (aj ),aj )
the sign( ∂aj ) = sign( ∂ai∂aj )
If
∂ 2 Πi (ai ,aj )
∂ai ∂aj
< 0 Strategic substitutes
If
∂ 2 Πi (ai ,aj )
∂ai ∂aj
> 0 Strategic complements (prices)
27
(quantities)