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Imperfect Information��

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Roman Sheremeta, Ph.D.

Professor, Weatherhead School of Management

Case Western Reserve University

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Outline�

  • Review
  • Imperfect Information
  • Entry Game with Imperfect Information
  • Assured Destruction game

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Review: �Extensive-Form Games

  • The extensive-form representation of a game specifies:
    • Who moves when and what action choices are available?
    • What do players know when they move?
    • What payoffs players receive for each combination of moves?

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  • In the extensive-from game, a strategy for a player is a complete plan of actions

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Extensive-Form Games �with Perfect Information

  • Perfect information: All previous moves are observed before the next move is chosen and each player knows Who has moved Where before she makes a decision
    • Player 2 makes her choice after observing player 1’s choice

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  • Imperfect information: A player may not know Who has moved Where before making a decision
    • Player 2 makes her choice at the same time as player 1 does

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Normal-Form Games�

  • Simultaneous move games
    • Each player does not know Who has moved Where before making a decision

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Player 2

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C

D

Player 1

A

3 , 4

5 , 2

B

1 , 1

0 , 6

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Extensive-Form Games�

  • We can also use an extensive-form game to represent a normal-form game by connecting the two decision nodes with the dashed line
    • The dash line signifies that Player 2 cannot tell which choice Player 1 has made

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Player 1

Player 2

C

D

3, 4

5, 2

A

B

Player 2

1, 1

0, 6

C

D

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Sequential-Move Matching Pennies: Extensive-Form (Perfect Information)

  • Each of the two players has a penny
    • Player 1 first chooses either Head or Tail
    • After observing Player 1’s choice, Player 2 chooses either Head or Tail
    • If two pennies match then Player 2 wins Player 1’s penny
    • Otherwise, Player 1 wins Player 2’s penny

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Player 1

Player 2

H

T

-1, 1

1, -1

H

T

Player 2

1, -1

-1, 1

H’

T’

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Sequential-Move Matching Pennies: Extensive-Form (Imperfect Information)

  • Each of the two players has a penny
    • Player 2 chooses either Head or Tail, without observing Player 1’s choice

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Player 1

Player 2

H

T

-1, 1

1, -1

H

T

Player 2

1, -1

-1, 1

H

T

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Information Set�

  • DEFINITION: An information set for a player is a collection of nodes satisfying the following:
    • (1) The same player has the move at every node in the information set
    • (2) When the play of the game reaches a node in the information set, the player with the move does not know which node in the information set has (or has not) been reached
    • (3) The player has the same set of feasible actions at each node in the information set

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Information Set: �Illustration

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Player 1

L

R

Player 2

L’

R’

2, 2, 3

Player 2

L’’

R’’

3

L’

R’

3

L’

R’

3

L”

R”

3

L”’

R”’

1, 2, 0

3, 1, 2

2, 2, 1

2, 2, 1

0, 1, 1

1, 1, 2

1, 1, 1

Two information sets for player 2

Three information sets for player 3

One information set for player 1

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Information Set:�Perfect vs Imperfect Information

  • A dynamic game in which every information set contains exactly one node is called a game of perfect information

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  • A dynamic game in which some information sets contain more than one node is called a game of imperfect information

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Strategy and Payoff�

  • A strategy for a player is a complete plan of actions, specifying what the player does at each of her information sets
  • What are the available strategies for player 1?
    • H and T
  • What are the available strategies for player 2?
    • H and T

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Player 1

Player 2

H

T

-1, 1

1, -1

H

T

Player 2

1, -1

-1, 1

H

T

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Entry Game�With Imperfect Information

  • HP is debating whether to enter (E) a new market or stay out (O), where the market is dominated by its rival, Dell

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  • After HP enters the market, Dell and HP simultaneously decide whether to play tough (T) or accommodate (A)

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HP

E

O

Dell

A

T

0, 5

1, 2

HP

A

T

0,-3

HP

A

T

-2,-1

-3,1

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Entry Game�With Imperfect Information

  • What are the available strategies for Dell?
    • T(if E) and A(if E)

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  • What are the available strategies for HP?
    • EA, ET, OA, OT

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HP

E

O

Dell

A

T

0, 5

1, 2

HP

A

T

0,-3

HP

A

T

-2,-1

-3,1

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Entry Game�With Imperfect Information

  • To find all Nash equilibria we need to construct a normal-form game first
  • Nash equilibria are
    • (EA, A)
    • (OT, T)
    • (OA, T)
  • Are all these three NE credible?

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Dell

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T

A

HP

ET

-2 ,-1

0 ,-3

EA

-3 , 1

1 , 2

OT

0 , 5

0 , 5

OA

0 , 5

0 , 5

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Entry Game�With Imperfect Information

  • What are SPNE?
    • How many subgames are in this game?
    • What are NE in the subgame?

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Subgame 1

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Dell

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T

A

HP

T

-2 ,-1

0 ,-3

A

-3 , 1

1 , 2

HP

E

O

Dell

A

T

0, 5

1, 2

HP

A

T

0,-3

HP

A

T

-2,-1

-3,1

Subgame 1

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Entry Game�With Imperfect Information

  • If (T, T) is chosen in the post-entry subgame, what is the optimal strategy of HP?
    • O, implying SPNE of (OT, T)
  • If (A, A) is chosen in the post-entry subgame, what is the optimal strategy of HP?
    • E, implying SPNE of (EA, A)

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Dell

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T

A

HP

T

-2 ,-1

0 ,-3

A

-3 , 1

1 , 2

HP

E

O

Dell

A

T

0, 5

1, 2

HP

A

T

0,-3

HP

A

T

-2,-1

-3,1

Subgame 2

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Entry Game�With Imperfect Information

  • NE are:
    • (EA, A), (OT, T), and (OA, T)

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  • SPNE are:
    • (EA, A) and (OT, T)

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Dell

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T

A

HP

ET

-2 ,-1

0 ,-3

EA

-3 , 1

1 , 2

OT

0 , 5

0 , 5

OA

0 , 5

0 , 5

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Assured Destruction Game�

  • Two superpowers are engaged in a provocative incident
  • The game starts with superpower 1’s choice:
    • (Peace) resulting in the payoffs (0,0)
    • (Annex) take over another country's territory
  • Following Annexation by superpower 1, superpower 2 chooses:
    • (Ignore) causing it to lose face and result in payoffs (1,-1)
    • (Escalate) proceed an atomic confrontation situation
  • Upon choosing Escalate, the two superpowers play the following simultaneous move game:
    • If both choose to retreat (R) then they suffer a small loss (pride) and payoffs are (-0.5,-0.5)
    • If either chooses doomsday (D) then the world is destroyed and payoffs are (-K,-K), where K is very large number

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Assured Destruction Game�

  • The extensive-form representation of this game is the following

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1

Peace

Annex

0, 0

2

Ignore

Escalate

1, -1

1

2

R

D

-0.5, -0.5

-K, -K

R

D

R

D

2

-K, -K

-K, -K

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Assured Destruction Game�

  • What are player 1’s and 2’s strategies?

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  • For player 1:
    • PR, PD, AR, AD

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  • For player 2:
    • IR, ID, ER, ED

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1

Peace

Annex

0, 0

2

Ignore

Escalate

1, -1

1

2

R

D

-0.5, -0.5

-K, -K

R

D

R

D

2

-K, -K

-K, -K

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Assured Destruction Game�

  • How many subgames?

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Subgame 1

Subgame 2

Not a Subgame

1

Peace

Annex

0, 0

2

Ignore

Escalate

1, -1

1

2

R

D

-0.5, -0.5

-K, -K

R

D

R

D

2

-K, -K

-K, -K

Subgame 3

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Assured Destruction Game�

  • What are the SPNE?

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  • No Attack:
    • (PR, ER)

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  • No War:
    • (AD, ID)

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1

Peace

Annex

0, 0

2

Ignore

Escalate

1, -1

1

2

R

D

-0.5, -0.5

-K, -K

R

D

R

D

2

-K, -K

-K, -K

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R

D

1

R

-0.5,-0.5

-K,-K

D

-K,-K

-K,-K

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Was Putin Crazy to Attack Ukraine?�

  • Putin was not crazy!
    • He is a rational player who knew that upon annexation of Crimea from Ukraine, neither the US nor the EU would choose to confront him in any serious manner (as it was in the case of annexation of Ossetia from Georgia and Pridnestrovie in Moldova)

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  • What did Putin lose?
    • Unfortunately, not much…
    • He’s viewed as a hero in Russia (partly because of propaganda)
    • The sanctions are fairly limited and are not going to last forever

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  • Why No Attack equilibrium did not happen?
    • Because, Putin was rationally expecting very little escalation from the US and the EU.

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Obama and Putin�

  • Conversation between Obama and Putin

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Stackelberg Models of Duopoly�

  • Next Time!

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Thank you!

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Roman Sheremeta, Ph.D.

Professor, Weatherhead School of Management

Case Western Reserve University

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References�

  • Watson, J. (2013). Strategy: An Introduction to Game Theory (3rd Edition). Publisher: W. W. Norton & Company. (Chapter 15)

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