Chapter 6 Extensive Games, perfect info

1
Chapter 6Extensive Games, perfect info

• Detailed description of the sequential structure
  of strategic situations
• as opposed to Strategic Games
• Players perfectly informed of occurred events
• Initially decisions are not made at the same
  time, no randomness

2
Example 91.1 / Def. 89.1 Extensive game Players
to share two objects

• Conventional definition with trees as primitives

3

• Definition with players actions as primitives
• G (N, H, P, (?i))
• A Set of Players
• N 1,2
• A Set of possible histories (sequences, finite or
  infinite)
• H ?, (2, 0), (1, 1), (0, 2), ((2, 0), yes),
  ((2, 0), no), ((1, 1), yes), ((1, 0), no), ((0,
  2), yes), ((0, 2), no)
• Terminal histories
• Z ((2, 0), yes), ((2, 0), no), ((1, 1), yes),
  ((1, 1), no), ((0, 2), yes), ((0, 2), no)

4

• A player function that assigns a player to each
  non terminal history
• P(?) 1 and P(h) 2 for every non terminal h ?
  ?
• A preference relation for each player on Z
• ?i ((2, 0), yes) gt1 ((1, 1), yes) gt1 ((0, 2),
  yes) 1 ((2, 0), no) 1 ((1, 1) 1 yes) 1((1,
  1), no) and ((2, 0), yes) gt2 ((1, 1), yes) gt2
  ((0, 2), yes) 2 ((2, 0), no) 2 ((1, 1) 2 yes)
  2((1, 1), no)

5
Def. 92.1Strategies

• A strategy of player i is a function that assigns
  an action to each nonterminal history
• Even for histories that, if strategy is followed,
  are never reached
• Player 1 below has AE, AF, BE, BF
• The outcome O(s) of strategy profile s (si)i?N
  yields the terminal history when each player i
  follows si

1
B
A
2
d
D
C
1
E
c
F
a
b
6
Def. 93.1Nash Equilibrium

• Nash Equilibrium for an extensive game with
  perfect info is a strategy profile s such that
  for every player i?N we have
• O(s-i, si) ?i O(s-i, si) for every strategy si
  of player i
• (If other players follows s you would better
  follow s too... )
• Alternatively it is the Nash Equilibrium of a
  strategic game derived from the extensive game

7
Equivalent strategic games
1
B
A
2
d
D
C
1
E
c
F
b
a
EquivalentStrategic Game
Extensive Game
EquivalentStrategic GameReduced form
8
Example 95.2
1
B
A
2
R
L
1, 2
2, 1
0, 0

• Given that player 2 chooses L it is optimal for
  player 1 to choose B
• The Nash equilibrium (B,L) lacks plausibility
  since P2 wouldnt choose L after A.

9
Def. 97.1 Subgame

• ?(h) (N, Hh, Ph, (?i)h) is the subgame to
  ? (N, H, P, (?i)) that follows the history h

h
?(h)
10
Def. 97.2 Subgame Perfect Equlibrium

• A subgame perfect equilibrium is a strategy
  profile s such that for any history h the
  strategy profile sh is a Nash equlibrium of the
  subgame ?(h)

OR?
11
Example 95.2 again
1
B
A
2
R
L
1, 2
0, 0
?(A)
2, 1

• (B,L) is a Nash equilibrium
• Is (B,L) a subgame perfect equilibrium?
• The strategy profile sh (B,L)A in the
  subgame ?(A) is for instance no Nash Equilibrium
• Player 2 wouldnt chose L given that player 1 has
  chosen A

12
Prop. 99.2Kuhns Theorem

• Every finite extensive game with perfect info has
  a subgame perfect equilibrium.
• E.g chess is draw once a position is repeated
  three times gt chess is finite

13
Two Extensions to Extensive Games with perfect
info

• Exogenous uncertainty
• The Player function P(h) has a probability that
  chance determines the action after the history h
• Definition of a subgame perfect equilibrium and
  Kuhns theorem still OK
• Simultaneous moves
• The Player function P(h) assigns a set of players
  that make choices after the history h

14
6.5.1 The Chain Store Game

• Multitude of Nash equilibria
• Every terminal history which the outcome in any
  period is either Out or (In,C)
• Intuitively unappealing for small K
• Unique Subgame Perfect Equilibrium
• Always (In, C)
• Not that appealing for large K

k
Out
In
CS
C
F
5 ,1
2, 2
0, 0
15
Ex. 110.1 BoS with an outside option

• Elimination of dominated actions yields (B, B)
• Interpretation
• BB gt1 Book gt1 SS gt1 BS 1 SB
• Player 2 knows that if player 1 selects concert
  he would choose Bach otherwise he would better
  stay home reading the book
• Thus player 1 can select B knowing that player 2
  also selects B

1
Concert
Book
2, 2
16
Ex. 111.1 Burning money

• Elimination of dominated actions yields (0B, BB)
• Interpretation
• P2 thinks that if P1 spends D then he wants to go
  Bach otherwise he would loose compared to not
  spending D gt P2 chooses B if P1 chooses D
• P1 knows this and can expect a payoff of 2 by
  choosing DB
• P2 knows that the rationality of P1 choosing 0 is
  that he expects to gain better than 2 (by
  choosing DB)
• Thus P1 can choose 0B and gain 3
• Authors think that this example is implausible

1
D
0