Combining Sequential and Simultaneous Moves Simultaneousmove games in

Combining Sequential and Simultaneous Moves

Simultaneous-move games in tree from n n 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. Cross. Talk High Low Global. Dialog High Low 2, 2 -10, 6 6, -10 -2, -2

G’s information set C High Low n n n G G High Low (2, 2) (-10, 6) (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.

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

n Battle of Sexes Starbucks Sally Banyan Harry Starbucks 1, 2 Banyan 0, 0 Starbucks 0, 0 Banyan 2, 1 Harry Sally Starbucks Banyan Starbucks 1, 2 0, 0 Banyan 0, 0 2, 1

Dry n 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. 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

A Dry 40% Nature Wet 60% A 1 2 1 2 B B 1 2 1 2 2, 3 5, 0 0, 5 0, 0 0, 5 5, 0 3, 2

n n n 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%x 2+60%x 0=0. 8. B’s expected payoff: 40%x 3+60%x 0=1. 2. 1 2 1 0. 8, 1. 2 2, 3 2 3, 2 1. 8, 1. 2

Combining Sequential and Simultaneous Moves I n Global. Dialog 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 C I C NI G High Low High 2, 2 -10, 6 Low 6, -10 -2, -2 0, 14 0, 6 Subgames

G C NI High Low I G C C High Low 0, 14 0, 6 2, 2 6, -10, 6 -2, -2 ★

n 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.

n Subgame-Perfect Equilibrium A 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 2 nd element at the 2 nd information set. n By contrast, G has two information sets (both singletons) as well and 4 strategies, HH, HL, LH, and LL. n

HH HL LH LL IH 2, 2 -10, 6 IL 6, -10 -2, -2 NH 0, 14 0, 6 NL 0, 14 0, 6

(NH, LH) and (NL, LH) are both NE. n (NL, LH) is the only subgame-perfect Nash equilibrium because it requires C to choose an optimal move at the 2 nd information set even it is off the equilibrium path. n

Combining Sequential and Simultaneous Moves II n C and G are both deciding simultaneously if he/she should invest $10 billion. 14 H G C L 6 I N I , 0 C G N 0, 0, 0 G H 14 L 6 C H L H 2, 2 -10, 6 L 6, -10 -2, -2

G C I N I -2, -2 0, 14 N 14, 0 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 n n Games with all players having dominant strategies Games with NOT all players having dominant strategies FED Low interest High interest rate CONGRESS Budget balance 3, 4 1, 3 Budget deficit 4, 1 2, 2

n F moves first Fed Congress Low High Congress Balance 4, 3 Deficit 1, 4 Balance 3, 1 Deficit 2, 2

n C moves first Congress Fed Balance Deficit Fed Low 3, 4 High 1, 3 Low 4, 1 High 2, 2

n First-mover advantage (Coordination Games) SALLY Starbucks Banyan Starbucks 2, 1 0, 0 Banyan 0, 0 1, 2 HARRY

n H first Harry Sally Starbucks Banyan Sally Starbucks 2, 1 Banyan 0, 0 Starbucks 0, 0 Banyan 1, 2

n S first Sally Harry Starbucks Banyan Starbucks 2, 1 Banyan 0, 0 Starbucks 0, 0 Harry Banyan 1, 2

n Second-mover advantage (Zero-sum Games, but not necessary) Navratilova Evert DL CC DL 50 80 CC 90 20

n E first Evert Nav. DL CC Nav. DL 50, 50 CC 80, 20 DL 90, 10 CC 20, 80

n N first Nav. Evert DL CC Evert DL 50, 50 CC 10, 90 DL 20, 80 CC 80, 20

Homework 1. 2. Exercise 3 and 4 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.