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Combining Sequential and Simultaneous Moves

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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 HighLow CrossTalk High2, 2-10, 6 Low6, -10-2, -2

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

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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).

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Battle of Sexes Sally Harry Starbucks Banyan Starbucks Banyan Starbucks Banyan 1, 2 Harry StarbucksBanyan Sally Starbucks1, 20, 0 Banyan0, 02, 1 0, 0 2, 1

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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. DryCrop 1Crop 2 Crop 12, 35, 0 Crop 20, 50, 0 WetCrop 1Crop 2 Crop 10, 00, 5 Crop 25, 03, 2

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

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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= , 1.22, 3 23, 21.8, 1.2

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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 HighLow C High2, 2-10, 6 Low6, -10-2, -2 C I NI G High Low 0, 14 0, 6 Subgames

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

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

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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.

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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.

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HHHLLHLL IH2, 2 -10, 6 IL6, , -2 NH0, 140, 60, 140, 6 NL0, 140, 60, 140, 6

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(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.

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Combining Sequential and Simultaneous Moves II C and G are both deciding simultaneously if he/she should invest $10 billion. G IN C I, 0 N0,0, 0 G HL C H2, 2-10, 6 L6, -10-2, -2 C H L 14 6 G H L 6

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G IN C I-2, -214, 0 N0, 140, 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.

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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 CONGRESS Budget balance3, 41, 3 Budget deficit4, 12, 2

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

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

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First-mover advantage (Coordination Games) SALLY StarbucksBanyan HARRY Starbucks2, 10, 0 Banyan0, 01, 2

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

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

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Second-mover advantage (Zero-sum Games, but not necessary) Navratilova DLCC Evert DL5080 CC9020

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

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

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Homework 1. Exercise 3 and 4 2. 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.

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