Download - Game Theory
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
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 wondering 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
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
First-mover advantage (Coordination Games)
SALLY
Starbucks Banyan
HARRYStarbucks 2, 1 0, 0
Banyan 0, 0 1, 2
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.