Combining Sequential and Simultaneous Moves

Simultaneous-move games in tree from Moves are simultaneous because players cannot

observe opponents’ decisions before making moves. EX: 2 telecom companies, both having invested $10

billion in fiberoptic network, are engaging in a price war.

GlobalDialog

High Low

CrossTalkHigh 2, 2 -10, 6

Low 6, -10 -2, -2

C moves before G, without knowing G’s moves. G moves after C, also uncertain with C’s moves. An Information set for a player contains all the nodes

such that when the player is at the information set, he cannot distinguish which node he has reached.

CHigh

Low

G

G

High

Low

HighLow

(2, 2)

(-10, 6)

(6, -10)

(-2, -2)

G’s information set

A strategy is a complete plan of action, specifying the move that a player would make at each information set at whose nodes the rules of the game specify that is it her turn to move.

Games with imperfect information are games where the player’s information sets are not singletons (unique nodes).

Battle of Sexes

SallyHarry

Harry

Starbucks

Banyan

StarbucksBanyan

StarbucksBanyan

1, 2

Harry

Starbucks Banyan

SallyStarbucks 1, 2 0, 0

Banyan 0, 0 2, 1

0, 00, 0

2, 1

Two farmers decide at the beginning of the season what crop to plant. If the season is dry only type I crop will grow. If the season is wet only type II will grow. Suppose that the probability of a dry season is 40% and 60% for the wet weather. The following table describes the Farmers‘ payoffs.

Dry Crop 1 Crop 2

Crop 1 2, 3 5, 0

Crop 2 0, 5 0, 0

Wet Crop 1 Crop 2

Crop 1 0, 0 0, 5

Crop 2 5, 0 3, 2

NatureDry 40%

Wet 60%

1

2A

A1

2

B

B

B

B

12

1212

1

2

2, 3

5, 00, 5

0, 00, 0

0, 55, 0

3, 2

When A and B both choose Crop 1, with a 40% chance (Dry) that A, B will get 2 and 3 each, and a 60% chance (Wet) that A, B will get both 0.

A’s expected payoff: 40%x2+60%x0=0.8. B’s expected payoff: 40%x3+60%x0=1.2.

1 2

1 0.8, 1.2 2, 3

2 3, 2 1.8, 1.2

Combining Sequential and Simultaneous Moves I GlobalDialog has invested $10 billion. Crosstalk is w

ondering if it should invest as well. Once his decision is made and revealed to G. Both will be engaged in a price competition.

G

High Low

CHigh 2, 2 -10, 6

Low 6, -10 -2, -2C

I

NIG High

Low

0, 14

0, 6Subgames

C

G

G

C

C

HighLow

High

Low

High

Low

HighLow

2, 2

6, -10

-10, 6

-2, -2

0, 14

0, 6

I

NI★

Subgame (Morrow, J.D.: Game Theory for Political Scientists)

It has a single initial node that is the only member of that node's information set (i.e. the initial node is in a singleton information set).

It contains all the nodes that are successors of the initial node.

It contains all the nodes that are successors of any node it contains.

If a node in a particular information set is in the subgame then all members of that information set belong to the subgame.

Subgame-Perfect EquilibriumA configuration of strategies (complete plans of action) such that their continuation in any subgame remains optimal (part of a rollback equilibrium), whether that subgame is on- or off- equilibrium. This ensures credibility of the strategies.

C has two information sets. At one, he’s choosing I/NI, and at the other he’s choosing H/L. He has 4 strategies, IH, IL, NH, NL, with the first element denoting his move at the first information set and the 2nd element at the 2nd information set.

By contrast, G has two information sets (both singletons) as well and 4 strategies, HH, HL, LH, and LL.

HH HL LH LL

IH 2, 2 2, 2 -10, 6 -10, 6

IL 6, -10 6, -10 -2, -2 -2, -2

NH 0, 14 0, 6 0, 14 0, 6

NL 0, 14 0, 6 0, 14 0, 6

(NH, LH) and (NL, LH) are both NE. (NL, LH) is the only subgame-perfect Nash

equilibrium because it requires C to choose an optimal move at the 2nd information set even it is off the equilibrium path.

Combining Sequential and Simultaneous Moves II

C and G are both deciding simultaneously if he/she should invest $10 billion.

G

I N

CI , 0

N 0, 0, 0G

H L

CH 2, 2 -10, 6

L 6, -10 -2, -2

CH

L

14

6

GH

L

14

6

G

I N

CI -2, -2 14, 0

N 0, 14 0, 0

One should be aware that this is a simplified payoff table requiring optimal moves at every subgame, and hence the equilibrium is the subgame-perfect equilibrium, not just a N.E.

Changing the Orders of Moves in a Game Games with all players having dominant

strategies Games with NOT all players having dominant

strategies

FED

Low interest rate

High interest rate

CONGRESSBudget balance 3, 4 1, 3

Budget deficit 4, 1 2, 2

F moves first

FedLow

High

Congress

Congress

Balance

Deficit

Balance

Deficit

4, 3

1, 4

3, 1

2, 2

C moves first

CongressBalance

Deficit

Fed

Fed

Low

High

Low

High

3, 4

1, 3

4, 1

2, 2

First-mover advantage (Coordination Games)

SALLY

Starbucks Banyan

HARRYStarbucks 2, 1 0, 0

Banyan 0, 0 1, 2

H first

HarryStarbucks

Banyan

Sally

Sally

Starbucks

Banyan

StarbucksBanyan

2, 1

0, 0

0, 0

1, 2

S first

SallyStarbucks

Banyan

Harry

Harry

Starbucks

Banyan

StarbucksBanyan

2, 1

0, 0

0, 0

1, 2

Second-mover advantage (Zero-sum Games, but not necessary)

Navratilova

DL CC

EvertDL 50 80

CC 90 20

E first

EvertDL

CC

Nav.

Nav.

DL

CC

DL

CC

50, 50

80, 20

90, 10

20, 80

N first

Nav.DL

CC

Evert

Evert

DL

CC

DL

CC

50, 50

10, 90

20, 80

80, 20

Homework

1. Exercise 3 and 42. Consider the example of farmers but now change the

probability of dry weather to 80%.(a) Use a payoff table to demonstrate the game.(b) Find the N.E. of the game.(c) Suppose now farmer B is able to observe A’ move but not the weather before choosing the crop she’ll grow. Describe the game with a game tree.(d) Continue on c, use a strategic form to represent the game.(e) Find the N.E. in pure strategies.