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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. CrossTalk High Low GlobalDialog High Low 2, 2 -10, 6 6, -10 -2, -2 G’s information set C High Low G G High Low 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. Games with imperfect information are games where the player’s information sets are not singletons (unique nodes). Battle of Sexes Starbucks Sally Banyan Harry 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 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 A Dry 40% Nature Wet 60% A 1 2 1 2 1 2 B B B B 1 2 1 2 1 2 2, 3 5, 0 0, 5 0, 0 0, 0 0, 5 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 C I C NI G High Low 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 High Low High Low 0, 14 0, 6 2, 2 6, -10 -10, 6 -2, -2 ★ 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 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 offequilibrium. 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 C I N H G L 0, 14 6 N ,0 0, 0 C H 14 L 6 G 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 Games with all players having dominant strategies Games with NOT all players having dominant strategies FED Low interest High interest rate rate CONGRESS Budget balance 3, 4 1, 3 Budget deficit 4, 1 2, 2 F moves first Congress Low Balance 4, 3 Deficit 1, 4 Balance 3, 1 Deficit 2, 2 Fed High Congress C moves first Congress Fed Balance Low High 1, 3 Low 4, 1 High 2, 2 Deficit Fed 3, 4 First-mover advantage (Coordination Games) SALLY Starbucks Banyan Starbucks 2, 1 0, 0 Banyan 0, 0 1, 2 HARRY H first Harry Sally Starbucks Banyan Sally Starbucks 2, 1 Banyan 0, 0 Starbucks 0, 0 Banyan 1, 2 S first Sally Harry Starbucks Banyan Starbucks 2, 1 Banyan 0, 0 Starbucks 0, 0 Harry Banyan 1, 2 Second-mover advantage (Zero-sum Games, but not necessary) Navratilova DL CC DL 50 80 CC 90 20 Evert E first Evert Nav. DL CC Nav. DL 50, 50 CC 80, 20 DL 90, 10 CC 20, 80 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.