Introduction
Nash equilibrium concept assumes that each player knows the other
players’ equilibrium behavior
Microeconomics II
Dominance and Rationalizability
This is problematic at least in one-shot games and when there are
multiple equilibria
Levent Koçkesen
What happens if we relax and use only rationality and common knowledge
of rationality?
Koç University
◮
◮
◮
Levent Koçkesen (Koç University)
Dominance
1 / 26
Dominant Strategy Equilibrium
Iterated Elimination of Dominated Strategies
Rationalizability
Levent Koçkesen (Koç University)
Dominance
2 / 26
Dominant Strategies
Consider Prisoners’ Dilemma
Definition (Dominant Strategies)
Player 2
Player 1
C
N
C
−5, −5
−6, 0
Let G = (N, (Si )i∈N , (ui )i∈N ) be a strategic form game. A pure strategy si ∈ Si
strictly dominates another pure strategy s′i ∈ Si if
N
0, −6
−1, −1
ui (si , s−i ) > ui (s′i , s−i )
C is optimal irrespective of what the other player does
We call such a strategy a dominant strategy
ui (si , s−i ) ≥ ui (s′i , s−i )
If every player has a dominant strategy we call the corresponding strategy
profile dominant strategy equilibrium
for all s−i ∈ S−i
and
It requires only the assumption that players are rational
ui (si , s−i ) > ui (s′i , s−i )
But it usually does not exist
Levent Koçkesen (Koç University)
for all s−i ∈ S−i
si ∈ Si weakly dominates s′i ∈ Si if
for some s−i ∈ S−i
A pure strategy si ∈ Si is strictly dominant if it strictly dominates every s′i ∈ Si . It
is called weakly dominant if it weakly dominates every s′i ∈ Si .
Dominance
3 / 26
Levent Koçkesen (Koç University)
Dominance
4 / 26
Dominant Strategy Equilibrium
Proposition
Let G be a strategic form game. Ds (G) ⊆ Dw (G) ⊆ N(G)
Definition
A pure strategy profile s∗ ∈ S is a strictly (weakly) dominant strategy equilibrium
of G = (N, (Si ), (ui )) if s∗i is a strictly (weakly) dominant strategy for each
player i ∈ N . The set of strictly (weakly) dominant strategy equilibria is denoted
Ds (G) (Dw (G)).
Proof.
Exercise
A reasonable solution concept
It only demands the players to be rational
H
L
H
10, 10
15, 2
L
2, 15
5, 5
H
L
H
10, 10
15, 5
L strictly dominates H
L weakly dominates H
Ds (G) = {(L, L)}
Dw (G) = {(L, L)}
Levent Koçkesen (Koç University)
Dominance
It does not require them to know that the others are rational
L
5, 15
5, 5
But it does not exist in many interesting games, e.g., Battle of the Sexes
B
S
B
2, 1
0, 0
S
0, 0
1, 2
Dw (BoS) = 0/
5 / 26
Levent Koçkesen (Koç University)
Dominance
6 / 26
Player 2 (he)
T
Player 1 (she) M
B
Consider the following game
T
M
B
L
3, 0
0, 0
1, 1
R
2, 1
3, 1
1, 0
L
3, 0
0, 0
1, 1
R
2, 1
3, 1
1, 0
B is never optimal: T always does better
◮ B is strictly dominated
◮ If player 1 is rational, then she should never play B
If player 2 knows that player 1 is rational, then he knows that she will not
play B
Therefore, if player 2 is rational he should never play L
If player 1 knows that
Is there a dominant strategy for any of the players?
There is no dominant strategy equilibrium for this game
So, what can we say about this game?
◮
◮
player 2 is rational
player 2 knows that player 1 is rational
then she knows that player 2 will not play L
and therefore should never play T
The unique outcome that survives this elimination process is (M, R)
This process is known as Iterated Elimination of Dominated Strategies
Levent Koçkesen (Koç University)
Dominance
7 / 26
Levent Koçkesen (Koç University)
Dominance
8 / 26
Iterated Elimination of Dominated Strategies
Iterated Elimination of Dominated Strategies
Consider the following game
In order to reach a unique outcome we used the following assumptions:
1
2
3
Everybody is rational
Everybody knows that everybody is rational
Player 1 knows that everybody knows that everybody is rational
In more complicated games it may take more than this
R
0, 1
3, 1
1, 0
Is there a dominated strategy for any of the players?
In the limit we have common knowledge of rationality
◮
L
3, 0
0, 0
1, 1
T
M
B
everybody is rational, everybody knows that everybody is rational,
everybody knows that everybody knows that everybody is rational, and so
ad infinitum.
B is strictly dominated by σ1 = (1/2, 1/2, 0)
It is possible that a strategy is not strictly dominated by any pure strategy,
yet it is strictly dominated
Let us introduce these concepts a little more formally
Levent Koçkesen (Koç University)
Dominance
9 / 26
Dominated Strategies
Levent Koçkesen (Koç University)
Dominance
10 / 26
Iterated Elimination of Dominated Strategies
Common knowledge of rationality justifies eliminating dominated
strategies iteratively
Definition
Let G = (N, (Si ), (ui )) be a strategic form game. A pure strategy si ∈ Si is
strictly dominated if there is a mixed strategy σi ∈ Σi such that
Ui (σi , s−i ) > Ui (si , s−i )
This procedure is known as Iterated Elimination of Dominated Strategies
If every strategy eliminated is a strictly dominated strategy
◮
for all s−i ∈ S−i .
Iterated Elimination of Strictly Dominated Strategies
If at least one strategy eliminated is a weakly dominated strategy
si is weakly dominated if there is a mixed strategy σi ∈ Σi such that
Ui (σi , s−i ) ≥ Ui (si , s−i )
◮
for all s−i ∈ S−i .
Iterated Elimination of Weakly Dominated Strategies
Guessing Game
Pick a number between 1 and 99
while
Ui (σi , s−i ) > Ui (si , s−i )
You win if your number is closest to 2/3 of the average
for some s−i ∈ S−i .
What do you pick?
Levent Koçkesen (Koç University)
Dominance
11 / 26
Levent Koçkesen (Koç University)
Dominance
12 / 26
Never Best Response
You might object to using mixed strategies to eliminate strategies
Let us instead agree to eliminate only those strategies that are never
optimal regardless of the belief that the player might hold about the play of
the others
◮
Let us allow a player to believe that others might be coordinating their
strategy choices
Let G = (N, (Si ), (ui )) be a strategic form game. A pure strategy si ∈ Si is a
never best response if, and only if, it is strictly dominated.
Theorem (Minkowski’s Theorem)
We call such strategies never best response
Let X, Y ⊂ Rn be non-empty convex sets with disjoint interiors. Then there
exists a hyperplane that separates them, i.e., there exists µ ∈ Rn such that
µ 6= 0 and
Definition
Let G = (N, (Si ), (ui )) be a strategic form game. A pure strategy si ∈ Si is a
never best response if there is no µ−i ∈ ∆(S−i ) such that
Ui (si , µ−i ) ≥ Ui (s′i , µ−i )
Proposition
µ · x ≥ µ · y,
∀x ∈ X, ∀y ∈ Y
for all s′i ∈ Si .
It turns out that eliminating strictly dominated strategies is equivalent to
eliminating never best responses
Levent Koçkesen (Koç University)
Dominance
13 / 26
Consider again
Dominance
14 / 26
Proof
T
M
B
L
3, 0
0, 0
1, 1
B is strictly dominated
Suppose that s′i is strictly dominated by σi . Then, Ui (σi , s−i ) > Ui (s′i , s−i ), ∀s−i ,
i.e.,
R
0, 1
3, 1
1, 0
∑ σi (si )Ui (si , s−i ) > Ui(s′i , s−i ), ∀s−i
si
This, in turn, implies that for any µ−i ∈ ∆(S−i )
∑ µ−i (s−i ) ∑ σi (si )Ui (si , s−i ) > ∑ µ−i (s−i )Ui(s′i , s−i )
B is a never best response
s−i
U1 (., µ−1 )
U1 (., R)
3
Levent Koçkesen (Koç University)
M
U1 (M, µ−1 )
si
s−i
or
U1 (T, µ−1 )
3
∑ σi (si )Ui(si , µ−i ) > Ui(s′i , µ−i )
si
2
(1/2, 1/2, 0)
Therefore, for any µ−i ∈ ∆(S−i ) there is an si ∈ supp(σi ) such that
3/2
B
1
U1 (B, µ−1 )
1
Ui (si , µ−i ) > Ui (s′i , µ−i ),
T
1
2
3
0
U1 (., L)
1
µ−1
i.e., s′i is a never best response.
∀µ−i ∈ ∆(S−i ), ∃s′i : Ui (s′i , µ−i ) > Ui (B, µ−i )
∃σi : Ui (σi , s−i ) > Ui (B, s−i ), ∀s−i
Levent Koçkesen (Koç University)
2/3 1/2 1/3
Dominance
15 / 26
Levent Koçkesen (Koç University)
Dominance
16 / 26
Proof (continued)
Proof (continued)
Let y = Ui (s′i ) and xj = (yj + ε, y−j ), j = 1, 2, . . . , m with ε > 0. Note that y ∈ Ui
and xj ∈ Uid , ∀j. Therefore,
Let m = |S−i | and for any σi ∈ ∆(Si ) define:
Ui (σi ) = Ui (σi , s−i )s−i ∈S−i : payoff vectors corresponding to σi
λ · (xj − y) = λj ε ≥ 0,
Suppose, for contradiction, that s′i is not strictly dominated and let
Ui = {Ui (σi ) : σi ∈ ∆(Si )} : all possible payoff vectors
Uid = x ∈ Rm : x ≥ Ui (s′i ) : vectors that dominate Ui (s′i )
µ−i · Ui (s′i ) − Ui(σi ) ≥ 0,
Both Ui and Uid are non-empty, convex sets with disjoint interiors. (They
intersect only at Ui (s′i )). By Minkowski’s theorem there exists a hyperplane that
separates Ui and Uid , i.e., there exists λ ∈ Rm such that λ 6= 0 and
λ · (x − y) ≥ 0,
Levent Koçkesen (Koç University)
∀j = 1, 2, . . . , m
which implies that λj ≥ 0, ∀j = 1, 2, . . . , m, with at least one λk > 0. Therefore,
normalizing if necessary, there exists µ−i ∈ ∆(S−i ) such that
∀x ∈ Uid , ∀y ∈ Ui
∀σi ∈ ∆(Si )
or
∑ µ−i(s−i )
Ui (s′i , s−i ) − Ui(si , s−i ) ≥ 0,
∀si ∈ Si
s−i
In other words, s′i is not a never best response.
Dominance
17 / 26
Levent Koçkesen (Koç University)
Dominance
18 / 26
Iterated Elimination of Strictly Dominated Strategies
Consider the following and suppose
that B is not strictly dominated
T
M
B
L
3, 0
0, 0
u1 (B, L), 1
Definition
Let G = (N, (Si ), (ui )) be a strategic form game. Let Xi (0) = Si and define Xi (t)
as the set of pure strategies that are not strictly dominated in Xi (t − 1), i.e.,
U1 (., R)
R
0, 1
3, 1
u1 (B, L), 0
5
3
Since B is not dominated it must
lie to the northeast of the line
connecting M and T
Xi (t) = {si ∈ Xi (t − 1) : ∄σi ∈ ∆ (Xi (t − 1))
Uid
4
such that Ui (σi , s−i ) > Ui (si , s−i ) , ∀s−i ∈ X−i (t − 1)} ,
M
B
µ(L)U1 (., L) + µ(R)U1 (., R) = c
2
Ui
T
for all t = 1, 2, . . . . Let Xi = ∞
t=0 Xi (t). The set of outcomes that survives
iterated elimination of strictly dominated strategies in game G is
X(G) = ×i∈N Xi
1
T
1
2
3
4
5
U1 (., L)
Previous proposition implies that we could define Xi (t) as
Xi (t) = {si ∈ Xi (t − 1) : ∃µ−i ∈ ∆(X−i (t − 1)) s.t.
Ui (si , µ−i ) ≥ Ui (s′i , µ−i ), ∀s′i ∈ Xi (t − 1)}
Levent Koçkesen (Koç University)
Dominance
19 / 26
Levent Koçkesen (Koç University)
Dominance
20 / 26
Rationalizability
Rationalizability
It has been independently developed by
◮
◮
Bernheim (1984,Econometrica)
Pearce (1984, Econometrica)
Definition
Let G = (N, (Si ), (ui )) be a strategic form game. Let Ri (0) = Si and define
It is based on three basic premises:
1
2
Agents view their opponents’ choices as uncertain events
Agents are rational
⋆
3
Ri (t) = si ∈ Ri (t − 1) : ∃µ−i ∈ ×j6=i ∆(Rj (t − 1)), such that
Ui (si , µ−i ) ≥ Ui (s′i , µ−i ) for all s′i ∈ Ri (t − 1)
they act optimally given their beliefs about these events
Rationality and preferences are common knowledge
for t = 1, 2, . . . . The set of rationalizable strategies for player i is
Suppose there are two players: A and B
Ri =
Rationality of player A implies that she should not play a strategy which is
not a best response to some belief she might have about B’s behavior
Furthermore, when forming her beliefs, A should be consistent with B’s
rationality, i.e., her beliefs should put zero probability on actions of B
which are not best response (for B) to some beliefs held by B regarding
A’s behavior
∞
\
Ri (t) .
t=0
and the set of rationalizable outcomes of G is
R(G) = ×i∈N Ri .
... and so on
Levent Koçkesen (Koç University)
Dominance
21 / 26
Rationalizability
Levent Koçkesen (Koç University)
Dominance
22 / 26
Rationalizability, Nash Equilibrium, and IESDS
Proposition
Let G be a finite strategic form game and σ∗ ∈ N(G). Then, supp(σ∗ ) ⊆ R(G),
where supp(σ∗ ) = ×i∈N supp(σ∗i ).
Note that we do not allow players to believe that other players’ strategies
maybe correlated
◮
This is what distinguishes rationalizability and iterated elimination of strictly
dominated strategies
Some allow correlation and call it correlated rationalizability
◮
Proof.
Exercise
Proposition
For any finite strategic form game G, if an outcome is rationalizable, then it
survives IESDS: R(G) ⊆ X(G).
This is equivalent to iterated elimination of strictly dominated strategies
R is non-empty if Si is compact for all i
Proof.
Exercise
Proposition
For any finite two-player strategic form game G, R(G) = X(G).
Levent Koçkesen (Koç University)
Dominance
23 / 26
Levent Koçkesen (Koç University)
Dominance
24 / 26
Iterated Elimination of Weakly Dominated Strategies
IEWDS and Nash Equilibrium
Defined in a manner similar to IESDS
Proposition
Let G be a finite strategic form game. If IEWDS results in a unique outcome,
then this outcome must be a Nash equilibrium of G.
Order of elimination does not matter in IESDS
It matters in IEWDS
U
M
D
L
3, 1
4, 0
4, 4
R
2, 0
1, 1
2, 4
Proof.
Exercise
Is every outcome that survives IEWDS a Nash equilibrium? No.
Start with U
Could a Nash equilibrium involve a weakly dominated strategy? Yes.
Start with M
Levent Koçkesen (Koç University)
Dominance
25 / 26
Levent Koçkesen (Koç University)
Dominance
26 / 26