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Economics 201AEconomic Theory(Fall 2009)
Extensive Games with Perfect and Imperfect Information
Topics: perfect information (OR 6.1), subgame perfection (OR 6.2), for-ward induction (OR 6.6), imperfect information (OR 11.1), mixed and be-havioral strategies (OR 11.4), sequential equilibrium (OR 12.2), Bayesianequilibrium (OR 12.3), trembling hand perfection (OR 12.4).
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A B
C D
d
b
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E
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1
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C D
ab c
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Perfect information (OR 6.1)
A finite extensive game with perfect information Γ = hN,H,P, (%i)iconsists of
— A set N of players.
— A set H of sequences (histories) where ∅ ∈ H and for any L < K
(ak)Kk=1 ∈ H =⇒ (ak)Lk=1 ∈ H.
— A player function P : H\Z → N where h ∈ Z ⊆ H if (h, a) /∈ H.
— A preference relation %i on Z for each player i ∈ N .
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Strategies, outcomes and Nash equilibrium
A strategy
si : h→ A(h) for every h ∈ H\Z such that P (h) = i.
A Nash equilibrium of Γ = hN,H,P, (%i)i is a strategy profile (s∗i )i∈Nsuch that for any i ∈ N
O(s∗) %i O(si, s∗−i) ∀si
where O(s) = (a1, ..., aK) ∈ Z such that
sP (a1,...,ak)(a1, ..., ak) = ak+1
for any 0 ≤ k < K (an outcome).
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The (reduced) strategic form
G =DN, (Si), (%0i)
Eis the strategic form of Γ = hN,H,P, (%i)i if for
each i ∈ N , Si is player i’s strategy set in Γ and %0i is defined by
s %0i s0⇔ O(s) %0i O(s0) ∀s, s0 ∈ ×i∈NSi
G =DN, (S0i), (%00i )
Eis the reduced strategic form of Γ = hN,H,P, (%i)i
if for each i ∈ N , S0i contains one member of equivalent strategies in Si,that is,
si, s0i ∈ Si are equivalent if (si, s−i) ∼0j (s0i, s−i)∀j ∈ N,
and %00i defined over ×j∈NS0j and induced by %0i.
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Subgames and subgame perfection (OR 6.2)
A subgame of Γ that follows the history h is the game Γ(h)
hN,H |h , P |h , (%i |h)iwhere for each h0 ∈ Hh
(h, h0) ∈ H,P |h (h0) = P (h, h0) and h0 %i |h h00⇔ (h, h0) %i (h, h00).
s∗ ∈ ×i∈NSi is a subgame perfect equilibrium (SPE) of Γ if
Oh(s∗i |h , s∗−i |h) %i |h Oh(si |h , s∗−i |h)
for each i ∈ N and h ∈ H\Z for which P (h) = i and for any si |h .
Thus, the equilibrium of the full game must induce on equilibrium on everysubgame.
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Backward induction and Kuhn’s theorems
Let Γ be a finite extensive game with perfect information
— Γ has a SPE (Kuhn’s theorem).
The proof is by backward induction (Zermelo, 1912) which is alsoan algorithm for calculating the set of SPE.
— Γ has a unique SPE if there is no i ∈ N such that z ∼i z0 for any
z, z0 ∈ Z.
— Γ is dominance solvable if z ∼i z0 ∃i ∈ N then z ∼j z0 ∀j ∈ N
(but elimination of weakly dominated strategies in G may eliminate theSPE in Γ).
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2
A B
C D
2,10,0
1,2
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S
C
1,0
1 12 2 2
0,2 3,1 2,4 5,3 4,6
6,5
S S S S S
C C C C C
OR 107.1 (the centipede game)
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Forward induction (OR 6.6)
• Backward induction cannot always ensure a self-enforcing equilibrium (for-ward and backward induction).
• In an extensive game with simultaneous moves, players interpret a deviationas a signal about future play.
• The concept of iterated weak dominance can be used to capture forwardand backward induction.
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Home Out
B
1,3
3,12,0
0,0
0,0
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Imperfect information (OR 11.1)
An extensive game with imperfect information
Γ = hN,H,P, fc, (Ii)i∈N, (%i)i
consists of
— a probability measure fc(· |h) on A(h) for all h such that P (h) = c
(chance determines the action taken after the history h), and
— an information partition Ii of {h ∈ H : P (h) = i} for every i ∈ N
such that
A(h) = A(h0)
whenever h, h0 ∈ Ii (an information set).
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Perfect and imperfect recall
Let Xi(h) be player i’s experience along the history h:
— all Ii encountered,
— actions ai ∈ A(Ii) taken at them, and
— the order that these events occur.
An extensive game with imperfect information has perfect recall if for eachi ∈ N
Xi(h) = Xi(h0)
whenever h, h0 ∈ Ii.
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Pure, mixed and behavioral strategies (OR 11.4)
In an extensive game hN,H,P, fc, (Ii)i∈N, (%i)i, for player i ∈ N
— a pure strategy assigns and action ai ∈ A(Ii) to each information setIi ∈ Ii,
— a mixed strategy is a probability measure over the set of pure strategies,and
— a behavioral strategy is a collection of independent probability measures(βi(Ii))Ii∈Ii.
For any σ = (σi)i∈N (mixed or behavioral) an outcome O(σ) is a prob-ability distribution over z that results from σ.
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Outcome-equivalent strategies
Two strategies (mixed or behavioral) of player i, σi and σ0i, are outcomeequivalent if
O(σi, s−i) = O(σ0i, s−i)
for every collection s−i of pure strategies.
In any finite game with perfect recall, any mixed strategy of a player hasan outcome-equivalent behavioral strategy (the converse is true for a setof games that includes all those with perfect recall).
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Strategies and beliefs (OR 12.1)
• Under imperfect information, an equilibrium should specify actions andbeliefs about the history that occurred (an assessment).
• An assessment thus consists of a profile of behavioral strategies and a beliefsystem (a probability measure for each information set).
• An assessment is sequentially rational if for each information set, the strat-egy is a best response given the beliefs.
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Consistency of the players’ beliefs:
(i) derived from strategies using Bayes’ rule
(ii) derived from some alternative strategy profile using Bayes’ rule at in-formation sets that need not be reached
(iii) all players share the same beliefs about the cause of any unexpectedevent.
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Sequential equilibrium (OR 12.2)
An assessment (β, µ) is sequentially rational if for each i ∈ N and everyIi ∈ Ii
O(β, µ |Ii) %i O((β0i, β−i), µ |Ii) for all β0i.
(β, µ) is consistent if there is a sequence ((βn, µn))∞n=1 → (β, µ) suchthat for each n:
— βn is completely (strictly) mixed and µn is derived from βn usingBayes’ rule.
(β, µ) is a sequential equilibrium if it is sequentially rational and consistent(Kreps and Wilson, 1982).
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(2,2)
(3,1) (0,0) (0,2) (1,1)
L M R
L LR R
1
2
OR 219.1
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2,2
3,1 0,2 0,2 1,1
L M R
L LR R
1
2
OR 220.1
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1,1
3,1 -2,0 2,0 -1,1
L M R
L LR R
1
2
OR 226.1
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3,3
0,1 0,0 1,0 5,1
L M R
L LR R
1
2
OR 227.1
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1 2
3
3,3,2 0,0,0 4,4,0 0,0,1
1,1,1
D d
C c
L R L R
OR 225.1 (Selten’s horse)
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Perfect Bayesian equilibrium (OR 12.3)
A Bayesian extensive game hΓ, (Θi), (pi), (ui)i is a game with observableactions where
— Γ is an extensive game of perfect information and simultaneous moves,
— Θi is a finite set of possible types of player i,
— pi is a probability distribution on Θi for which pi(θi) > 0 for allθi ∈ Θi, and
— ui : Θi × Z → R is a vNM utility function.
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Let σi(θi) be a behavioral statuary of player i of type θi and µ−i(h) be aprobability measure over Θi.
(σ, µ) is sequentially rational if for every h ∈ H\Z, i ∈ P (h) and θi ∈ Θi
O(σ, µ−i |h) %i O((σ0i, σ−i), µ−i |h) ∀σ0i
(σ, µ) is PB-consistent if for each i ∈ N µ−i(∅) = pi (correct initialbeliefs) and µ−i is derived from pi and ai ∈ A(h) via Bayes’ rule (action-determined beliefs) when possible.
(σ, µ) is a perfect Bayesian equilibrium (PBE) if it is sequentially rationaland PB-consistent.
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W
0.5
0.5
4,1
0,0
1,4
0,0
T
T
B
B
R
RL
L2,5
2,5
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0.5
4,1
0,0
T
B
R
L
L2,5
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W
0.9
0.1
-2,1
0,0
-1,0
-3,-1
f
f
r
r
w
ws
s
Beer-Quiche
f
f
r
r
-3,1
-1,0
0,0
-2,-1
S
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2
2
W
0.9
0.1
-2,1
0,0
-1,0
-3,-1
f
f
r
r
w
ws
f
r0,0
-2,-1
S
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Trembling hand perfection (OR 12.4)
Trembling hand perfect equilibrium (THP ) excludes strategies that are“unsafe” given the risk of small mistakes.
σ is a THP of a finite strategic game if ∃(σk)∞k=1 of completely mixedstrategy profiles such that
— (σk)∞k=1 converges to σ, and
— σi ∈ BRi(σk−i) for each i and all k.
A THP of a finite extensive game is a behavioral strategy profile β thatcorresponds to a THP of the agent strategic form of the game.