Chapter 11 Game Theory and the Tools of Strategic Business Analysis.
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Transcript of Chapter 11 Game Theory and the Tools of Strategic Business Analysis.
Chapter 11
Game Theory and the Tools of Strategic Business Analysis
Game Theory
0 Game theory applied to economics by John Von Neuman and Oskar Morgenstern
0 Game theory allows us to analyze different social and economic situations
Games of Strategy Defined
0 Interaction between agents can be represented by a game, when the rewards to each depends on his actions as well as those of the other player
0 A game is comprised of 0 Number of players0 Order to play0 Choices0 Chance 0 Information 0 Utility
3
Representing Games
0 Game tree0 Visual depiction0 Extensive form game0 Rules0 Payoffs
4
Game Types
0 Game of perfect information0 Player – knows prior choices
0All other players
0 Game of imperfect information0 Player – doesn’t know prior choices
5
Strategy
0 A player’s strategy is a plan of action for each of the other player’s possible actions
Game of perfect information
7Player 2 (Toshiba) knows whether player 1 (IBM) moved to the left or to the right. Therefore, player 2 knows at which of two nodes it is located
1
2 3
IBM
ToshibaToshiba
UNIXDOS
UNIXDOS UNIXDOS
600200
100100
100100
200600
In extensive form
Strategies
0 IBM: 0 DOS or UNIX
0 Toshiba0 DOS if DOS and UNIX if UNIX0 UNIX if DOS and DOS if UNIX0 DOS if DOS and DOS if UNIX0 UNIX if DOS and UNIX if UNIX
9
Toshiba
(DOS | DOS,DOS | UNIX)
(DOS | DOS,UNIX | UNIX)
(UNIX | DOS,UNIX | UNIX)
(UNIX | DOS,DOS | UNIX)
IBMDOS 600, 200 600, 200 100, 100 100, 100
UNIX 100, 100 200, 600 200, 600 100, 100
Game of perfect informationIn normal form
Game of imperfect information
0 Assume instead Toshiba doesn’t know what IBM chooses0 The two firms move at the same time
0 Imperfect information0 Need to modify the game accordingly
Game of imperfect information
11
Toshiba does not know whether IBM moved to the left or to the right, i.e., whether it is located at node 2 or node 3.
1
2 3
IBM
ToshibaToshiba
UNIXDOS
UNIXDOS UNIXDOS
600200
100100
100100
200600
In extensive form
Information set
Toshiba’s strategies:• DOS• UNIX
12
Toshiba
DOS UNIX
IBMDOS 600, 200 100, 100
UNIX 100, 100 200, 600
Game of imperfect informationIn normal form
Another example: Matching Pennies
13
Player 2
Heads Tails
Player 1Heads - 1, +1 +1 - 1
Tails +1 - 1 - 1, +1
Extensive form of the game of matching pennies
14
Child 2 does not know whether child 1 chose heads or tails. Therefore, child 2’s information set contains two nodes.
Child 1
Child 2Child 2
Tails Heads
TailsHeads TailsHeads
- 1+1
+1- 1
+1- 1
-1+1
Equilibrium for GamesNash Equilibrium
0 Equilibrium 0 state/ outcome0 Set of strategies0 Players – don’t want to change behavior 0 Given - behavior of other players
0 Noncooperative games0 No possibility of communication or binding
commitments
15
Nash Equilibria
chosen is *s when i player to payoff
i player of choicestrategy
choicesstrategy ofarray -
i
),...,(
),...,(*
**1
**1
n
*i
n
ss
s
sss
16
ii
nini
n
Ss
ssssss
sss
in all for
If
mequilibriu Nash a is -
ii
ˆ
),...,ˆ,...,(),...,,...,(
),...,(***
1***
1
**1
17
Toshiba
DOS UNIX
IBMDOS 600, 200 100, 100
UNIX 100, 100 200, 600
Nash Equilibrium: Toshiba-IBMimperfect Info game
The strategy pair DOS DOS is a Nash equilibrium. Are there any other equilibria?
Dominant Strategy Equilibria
0 Strategy A dominates strategy B if0 A gives a higher payoff than B 0 No matter what opposing players do
0 Dominant strategy0 Best for a player0 No matter what opposing players do
0 Dominant-strategy equilibrium0 All players - dominant strategies
18
Oligopoly Game
19
General Motors
High price Low price
FordHigh price 500, 500 100, 700
Low price 700, 100 300, 300
0 Ford has a dominant strategy to price low 0 If GM prices high, Ford is better of pricing low0 If GM prices low, Ford is better of pricing low
Oligopoly Game
20
General Motors
High price Low price
FordHigh price 500, 500 100, 700
Low price 700, 100 300, 300
0 Similarly for GM0 The Nash equilibrium is Price low, Price low
The Prisoners’ Dilemma0 Two people committed a crime and are being interrogated
separately.0 The are offered the following terms:
0 If both confessed, each spends 8 years in jail.0 If both remained silent, each spends 1 year in jail.0 If only one confessed, he will be set free while the other spends
20 years in jail.
Prisoners’ Dilemma
22
Prisoner 2
confess silent
Prisoner 1Confess 8, 8 0, 20
Silent 20, 0 1, 1
0 Numbers represent years in jail0 Each has a dominant strategy to confess0 Silent is a dominated strategy0 Nash equilibrium: Confess Confess
Prisoners’ Dilemma
0 Each player has a dominant strategy0 Equilibrium is Pareto dominated
23
Elimination of Dominated Strategies
0 Dominated strategy0 Strategy dominated by another strategy
0 We can solve games by eliminating dominated strategies
0 If elimination of dominated strategies results in a unique outcome, the game is said to be dominance solvable
24
25
(a) Eliminating dominated strategies
Player 2
1 2 3
Player 11 2, 0 2, 4 0, 2
2 0, 6 0, 2 4, 0
(b) One step of elimination
Player 2
1 2
Player 11 2, 0 2, 4
2 0, 6 0, 2
(c ) Two steps of elimination
Player 2
1 2
Player 1 1 2, 0 2, 4
26
(a) Eliminated dominated strategies
Player 2
1 2 3
Player 11 20, 0 10, 1 4, -4
2 20, 2 10, 0 2, -2
(b) Reduced game eliminating column 3 first
Player 2
1 2
Player 11 20, 0 10, 1
2 20, 2 10, 0
Games with Many Equilibria
0 Coordination game0 Players - common interest: equilibrium0 For multiple equilibria
0Preferences - differ 0 At equilibrium: players - no change
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Toshiba
DOS UNIX
IBMDOS 600, 200 100, 100
UNIX 100, 100 200, 600
Games with Many Equilibria
The strategy pair DOS DOS is a Nash equilibrium as well as UNIX, UNIX
Normal Form of Matching Numbers: coordination game with ten Nash equilibria
29
Player 2
1 2 3 4 5 6 7 8 9 10
Player 1
1 1, 1 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0
2 0, 0 2, 2 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0
3 0, 0 0, 0 3, 3 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0
4 0, 0 0, 0 0, 0 4, 4 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0
5 0, 0 0, 0 0, 0 0, 0 5, 5 0, 0 0, 0 0, 0 0, 0 0, 0
6 0, 0 0, 0 0, 0 0, 0 0, 0 6, 6 0, 0 0, 0 0, 0 0, 0
7 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 7, 7 0, 0 0, 0 0, 0
8 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 8, 8 0, 0 0, 0
9 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 9, 9 0, 0
10 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 10, 10
Table 11.12
A game with no equilibria in pure strategies
30
General 2
Retreat Attack
General 1 Retreat 5, 8 6, 6
Attack 8, 0 2, 3
The “I Want to Be Like Mike” Game
31
Dave
Wear red Wear blue
Michael Wear red (-1, 2) (2, -2)
Wear blue (1, -1) (-2, 1)
Credible Threats
0 An equilibrium refinement:0 Analyzing games in normal form may result in equilibria
that are less satisfactory0 These equilibria are supported by a non credible threat0 They can be eliminated by solving the game in extensive
form using backward induction0 This approach gives us an equilibrium that involve a
credible threat0 We refer to this equilibrium as a sub-game perfect Nash
equilibrium.
32
33
Toshiba
(DOS | DOS,DOS | UNIX)
(DOS | DOS,UNIX | UNIX)
(UNIX | DOS,UNIX | UNIX)
(UNIX | DOS,DOS | UNIX)
IBMDOS 600, 200 600, 200 100, 100 100, 100
UNIX 100, 100 200, 600 200, 600 100, 100
Non credible threats: IBM-ToshibaIn normal form
0 Three Nash equilibria0 Some involve non credible threats.0 Example IBM playing UNIX and Toshiba playing UNIX
regardless:0 Toshiba’s threat is non credible
Backward induction
34
1
2 3
IBM
ToshibaToshiba
UNIXDOS
UNIXDOS UNIXDOS
600200
100100
100100
200600
Subgame perfect Nash Equilibrium
0 Subgame perfect Nash equilibrium is0 IBM: DOS0 Toshiba: if DOS play DOS and if UNIX play UNIX
0 Toshiba’s threat is credible0 In the interest of Toshiba to execute its threat
Rotten kid game
0 The kid either goes to Aunt Sophie’s house or refuses to go
0 If the kid refuses, the parent has to decide whether to punish him or relent
36
Player 2 (a parent)
(punish if the kid refuses)
(relent if the kid refuses)
Player 1(a difficult
child)
Left(go to Aunt Sophie’s House)
1, 1 1, 1
Right(refuse to go to Aunt Sophie’s House)
-1, -1 2, 0
Rotten kid game in extensive form
37
• The sub game perfect Nash equilibrium is: Refuse and Relent if refuse• The other Nash equilibrium, Go and Punish if refuse, relies on a non
credible threat by the parent
Kid
Parent
RefuseGo to Aunt Sophie’s House
Relent if refuse
Punish if refuse
-1-1
20
11
1
2