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Basics on Game Theory. For Industrial Economics (According to Shy’s Plan). The Reasons of Game Theory. What GT is the study of strategic interaction involving decisions among multiple actors Why Economic and political world is made of rules, actors and strategies - PowerPoint PPT Presentation
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Basics on Game Theory
For Industrial Economics
(According to Shy’s Plan)
The Reasons of Game Theory
• What– GT is the study of strategic interaction involving decisions
among multiple actors
• Why– Economic and political world is made of rules, actors and
strategies– GT is the right frame for studying competition
• What for– What are the alternatives of a game?– Can the behavior of other actors be predicted?– How to design the optimal strategies
What is a Game
• Normal Form Games– Players– Strategies– Payoffs
• Extensive Form Games
Concepts
• Actions
• Outcomes
• Payoffs
Concepts• In general:
– Players i:1,...,n– Ai: Set of actions of player i.– ai: One action of player‘s set Ai
– (a1, ... , an): Outcome i(a1, ... , an): Payoff for player i, according to
the actions of other players
• Summarizing:
, ,
, ,
The Search for the Solution of a Game: Elimination of Strictly Dominated Strategies:
P2
P1
Left Center
Up 1,0 1,2 0,1
0,3 0,1 2,0
Right
Down
1,0 1,2
0,3 0,1
1,0 1,2
P1Up
Down
P2Left Center
P1Up
P2Left Center
Dominant Actions (and one remark on notation)
• The payoff for player i, depends on his move and on the other’s moves (a1,a2,…, ai,…, an):
• We represent it in the form:
• An action a~ is dominant for one player i, if:
Nash Equilibrium
• If the solution of the game is unique, it is a Nash Equilibrium.
• Example
• Definition of NE:
P2
P1
Left Center
Up 0,4 4,0 5,3
4,0 0,4 5,3
3,5 3,5 6,6
Right
Middle
Down
Games with Multiple Equilibria
• The Battle of the Sexes I
He
She
Opera Football
Opera
Football
2,1 0,0
0,0 1,2
Games without NE
• The Battle of the Sexes II
He
She
Opera Football
Opera
Football
2,0 0,2
0,1 1,0
The Best Response Function
He
She
Opera Football
Opera
Football
2,1 0,0
0,0 1,2
Pareto Efficient Outcomes
• The Prisoners’ Dilemma
P2
P1
Defect Cooperate
Defect
Cooperate
-1,-1 -9,0
0,-9 -6,-6
Some Questions
• What is a normal form game?
• What is a strictly dominated strategy?
• What is a NE in a normal form game?
• What are the advantages and the shortcomings of GT in the prediction of the strategic behavior?
Exercise
• In the next game in normal form, which strategies survive to the elimination of strictly dominated strategies?
P2
P1
Left Center
Up 2,0 1,1 4,2
3,4 1,2 2,3
1,3 0,2 3,0
Right
Middle
Down
Extensive Form Games11
LL RR22 22
L´L´ R´R´ L´L´ R´R´
Payoffs P(1):Payoffs P(1):
Payoffs P(2):Payoffs P(2):
33
11
11
22
22
11
00
00
Characteristics:Characteristics:
1) Moves occur in sequence1) Moves occur in sequence
2) All the previous moves are observed before choose the next one2) All the previous moves are observed before choose the next one
3) Payoffs are common knowledge among the players3) Payoffs are common knowledge among the players
Backward Induction and NE
11LL RR
(2,0)(2,0) 22
(1,1)(1,1)
L´L´ R´R´11
L´´L´´ R´´R´´
(3,0)(3,0) (0,2)(0,2)
Induction:Induction:1. 1. Step 3.Step 3. P1 chooses L´´ with P1 chooses L´´ with uu11 = 3 instead of R´´ with = 3 instead of R´´ with uu22 = 0 = 0
2. 2. Step 2.Step 2. P2 anticipates that if the game reaches level 3, then P1 P2 anticipates that if the game reaches level 3, then P1 chooses L´´ therefore chooses L´´ therefore uu22 = 0. P2 chooses L´ with = 0. P2 chooses L´ with uu22 = 1. = 1.
3. 3. Step 1.Step 1. P1 anticipates that if the game reaches level 2, then P2 chooses P1 anticipates that if the game reaches level 2, then P2 chooses L´ and therefore L´ and therefore uu11 = 1. Then, P1 chooses L with = 1. Then, P1 chooses L with uu11 = 2. = 2.
Strategies in Extensive Form Games
One strategy is a complete plan of actions specifying a feasible action for each One strategy is a complete plan of actions specifying a feasible action for each move in each contingency for which he can be called upon to act.move in each contingency for which he can be called upon to act.
11
LL RR22 22
L´L´ R´R´ L´L´ R´R´
Payoffs P(1):Payoffs P(1):
Payoffs P(2):Payoffs P(2):
33
11
11
22
22
11
00
00 P2 has 2 actions A{L,R} but 4 strategies.P2 has 2 actions A{L,R} but 4 strategies.
Strategy 1Strategy 1: If P1 plays L, then play L´, if P1 plays R, then play L´: : If P1 plays L, then play L´, if P1 plays R, then play L´: (L´,L´).(L´,L´).
Strategy 2Strategy 2: If P1 plays L, then play L´, if P1 plays R, then play R´: : If P1 plays L, then play L´, if P1 plays R, then play R´: (L´,R´).(L´,R´).
Strategy 3Strategy 3: If P1 plays L, then play R´, if P1 plays R, then play L´: : If P1 plays L, then play R´, if P1 plays R, then play L´: (R´,L´).(R´,L´).
Strategy 4Strategy 4: If P1 plays L, then play R´, if P1 plays R, then play R´: : If P1 plays L, then play R´, if P1 plays R, then play R´: (R´,R´).(R´,R´).
P1 has 2 actions A{L,R}P1 has 2 actions A{L,R}
SSP1P1 coincides with A{L,R} coincides with A{L,R}
NE of Extensive Form Games
Which strategy is the NE of the game?Which strategy is the NE of the game?
11
LL RR22 22
L´L´ R´R´ L´L´ R´R´
Payoffs P(1):Payoffs P(1):
Payoffs P(2):Payoffs P(2):
33
11
11
22
22
11
00
00
Strategy 1Strategy 1: : (L´,L´)(L´,L´)
Strategy 2Strategy 2: : (L´,R´)(L´,R´)
Strategy 3:Strategy 3: (R´,L´)(R´,L´)
Strategy 4Strategy 4: : (R´,R´)(R´,R´)
Normal Form from Extensive Form
11
LL RR22 22
L´L´ R´R´ L´L´ R´R´
Payoffs P(1):Payoffs P(1):
Payoffs P(2):Payoffs P(2):
33
11
11
22
22
11
00
00
P2P2
(L´,L´)(L´,L´) (L´,R´)(L´,R´) (R´,L´)(R´,L´) (R´,R´)(R´,R´)
3,1 3,1 1,2 1,2(L(L))
P1P12,1 0,0 2,1 0,0
(R)(R)
Subgame Perfect Nash E.
Definition: Definition: A A NENE is is Subgame PerfectSubgame Perfect if the strategies of the players constitute a NE if the strategies of the players constitute a NE in each subgame.in each subgame.
Algorithm for Identifying a SPNE: Algorithm for Identifying a SPNE:
Identify all the smaller subgames having Identify all the smaller subgames having terminal nodes in the original tree. terminal nodes in the original tree.
Replace each subgame for the payoffs of one Replace each subgame for the payoffs of one of the NE.of the NE.
The initial nodes of the subgame are now the The initial nodes of the subgame are now the terminal nodes of the new truncated tree.terminal nodes of the new truncated tree.
Subgame Perfect Nash E.
Example:Example:
11
LL RR
22 22
L´L´ R´R´ L´L´ R´R´
Payoffs P(1):Payoffs P(1):
Payoffs P(2):Payoffs P(2):
33
11
11
22
22
11
00
00
SPNE = ( , )SPNE = ( , )
Subgame 1Subgame 1 Subgame 2Subgame 2
R`R` L`L`
Between 1 and 2, P1 prefers Between 1 and 2, P1 prefers to play R. to play R.
NE and Subgame Perfect NE
Subgame Perfect Nash Equilibrium vs. Simple NESubgame Perfect Nash Equilibrium vs. Simple NE
SPNE is more powerful than NE, for solving Imperfect Information Games:
11
LL RR
22 22
L´L´ R´R´ L´L´ R´R´
Payoffs P(1):Payoffs P(1):
Payoffs P(2):Payoffs P(2):
33
11
11
22
22
11
00
00
SPNE = (R`,L`)SPNE = (R`,L`)
Backward Backward Induction = (R,L`)Induction = (R,L`)
