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Combining Sequential and Simultaneous Moves

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.

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Combining Sequential and Simultaneous Moves

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  1. Combining Sequential and Simultaneous Moves

  2. 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.

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

  4. 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).

  5. Battle of Sexes 1, 2 Starbucks Harry Starbucks Banyan 0, 0 Sally 0, 0 Banyan Starbucks Harry Banyan 2, 1

  6. 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.

  7. 2, 3 1 B 2 1 5, 0 A 0, 5 1 2 B Dry 40% 2 0, 0 Nature 0, 0 1 B Wet 60% 1 2 0, 5 A 2 1 5, 0 B 2 3, 2

  8. 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.

  9. 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. I C NI 0, 14 High G Subgames Low 0, 6

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

  11. 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.

  12. 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.

  13. 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.

  14. (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.

  15. Combining Sequential and Simultaneous Moves II • C and G are both deciding simultaneously if he/she should invest $10 billion. 14 H C L 6 14 H G L 6

  16. 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.

  17. Changing the Orders of Moves in a Game • Games with all players having dominant strategies • Games with NOT all players having dominant strategies

  18. 4, 3 Balance Congress 1, 4 Deficit Low Fed High 3, 1 Balance Congress Deficit 2, 2 F moves first

  19. 3, 4 Low Fed 1, 3 High Balance Congress Deficit 4, 1 Low Fed High 2, 2 C moves first

  20. First-mover advantage (Coordination Games)

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

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

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

  24. 50, 50 DL Nav. 80, 20 CC DL Evert CC 90, 10 DL Nav. CC 20, 80 E first

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

  26. Homework • 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.

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