Class Note for ECON 309 at UMass(3)
Class Note for ECON 309 at UMass(3)
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Date Created: 02/06/15
Dynamic Games under Complete Information Games under complete information can be classi ed into two cases Perfect inf0 each player knows where she is on the game tree when it is her turn to move Imperfect inf0 some players have to move without knowing the previous moves of other players We can de ne perfect and imperfect information more precisely using the concept of an information set Definition An information set of player i is the set of nodes among which player i cannot distinguish If each information set is a singleton then the game has perfect information If some information sets are not singletons the game is one of imperfect information Imperfect information Perfect information We have to impose some information set rules if we want to avoid a strange structure of games Rule 1 All the nodes in any information set must belong to the same player This violates Rule 1 Rule 2 If a node D1 is a predecessor of node D2 then D1 and D2 cannot be contained in the same information set This violates Rule 2 Rule 3 Exactly the same set of moves can be taken at each of the decision nodes in an information set This violates Rule 3 Actions versus strategies Actions are different from strategies in dynamic games under complete information An action or a move is a choice a player can make at each decision node A strategy on the other hand is a detailed set of plans for playing the game that specifies the player s move for every contingency Thus a strategy maps out a plan of actions under all eventualities Because a Nash equilibrium is de ned in terms of strategies not of actions we must translate an extensive form represented in terms of actions into its normal form representation in terms of strategies to find a Nash equilibrium Example 1 11 21 13 L R 10 01 21 13 L R 11 11 21 13 10 L L L R KL RR 1 0 10 01 0 1 2 1 13 21 1 3 Backward Induction and Incredible threats Consider the following game of perfect information 1 Player 1 chooses an action 611 from hisher feasible set 2 After observing a1 player 2 chooses an action a from hisher feasible set 3 Payoffs are given by r1ala2 and 461161 One way to approach the problem of prediction in this game is to simply derive its normal form representation and then apply the Nash equilibrium concept to it In this type of games however the Nash equilibrium may include incredible threats Selten believed that any Nash equilibrium incorporating incredible threats is a poor predictor of human behavior and thus should be eliminated Nash equilibria after eliminating those incorporating incredible threats are called subgame perfect Nash equilibria In games of perfect information subgame perfect Nash equilibria can be found by using the method of backward induction Example Entrant versus Incumbent 3825 Enter 430 4010 Stay Out 800 Enter Enter Enter SO 80 Enter SO 80 Expand 38 25 38 25 43 0 43 0 Don t 40 10 80 0 40 10 80 0 Expand There are two pure strategy Nash equilibna Don t Expand Enter Enter and Expand Stay Out Enter Are both of them reasonable 10 Consider Don t Expand Enter Enter The strategy of Enter Enter can be interpreted as a situation in which the Entrant tries to convince Incumbent to play Don t Expand by threatening that he would play Enter regardless of the choice made by Incumbent Bus is this threat really credible If Incumbent actually adopt Expand by ignoring this threat there is nothing to stop Entrant from backing away from its threat and not entering at all Thus it is an incredible threat for Incumbent if he is wise enough Example Ulysses and the Sirens You must bind me hard and fast so that I cannot stir from the spot where you will stand me and if I beg you to release me you must tighten and add to my bonds The Odyssey The problem of an incredible threat is a problem of dynamic inconsistency To avoid dynamic inconsistency involved in an incredible threat we need to impose the principle of sequential rationality equilibrium strategies should specify optimal behavior from any point in the game onward This principle is intimately related to the procedure of backward induction ll Now we will describe the method of backward induction in a general way When player 2 gets the move at the second stage of the game heshe will choose a Eargmaxr2ala2 Given the action a the solution to this problem is thus given by a function a aal Since player 1 can solve player 2 s problem as well as player 2 can player 1 should anticipate player 2 s reaction to each a so player l s problem at stage 1 amounts to 61 eargmax apaxao The pair 64305461 is called a backward induction outcome of this game whereas the pair 4304611 is called a backward induction equilibrium of this game Note that 0524 is a numbers while oczal is a function of a1 Example Entrant versus Incumbent continued Stay Out if afEXpand In this example afExpand and oca1 Thus the backward induction Enter if afDon39t equilibrium of this game is Expand Stay Out Enter while the backward induction outcome of the game is Expand Stay Out 12 Example Stackelberg duopoly There are two companies in the bottled industry Sparkling Water Co and Clear Water Inc They sell the exact same product in the same market and set how much water to put up for sale to maximize pro ts The price of water is set by the market then Sparkling Water is the industry leader so that it sets the quantity rst afterwards Clear Water chooses its quantity The total cost of each rm is TC3Q where i S C and inverse demand function is 10 Q istlO P 0 ifQgt10whereQ QCQS 7 QQ istlO Thus the pro t ofeach rm is 7r 3Qi ifQ gt10 A strategy for Sparkling Water is a real number QS while a strategy for Clear Water is a function RQS which determines QC for each level of Q3 13 Let us solve Clear Water s problem rst The rst order condition for Clear Water is 03C an Water 7Qs 2Qc 0 thus RQs 35 Qs This is the sequentially rational strategy of Clear Now Sparkling Water solves maX7 RQs Qs QS which gives Q 35 Thus the backward induction equilibrium is 35 RQS35 QS while the backward induction outcome is 35 1725 In this game the rst mover has an advantage because 715 6125 gt 71C 3063 But in other games the second mover may have an advantage Note If this game were played simultaneously then the equilibrium would be the Coumot equilibrium which is Q Q g 14 Definition A proper subgame in an extensive form game is a set of nodes that 1 begins at a noninitial decision node n that is a singleton 2 includes all the decision and terminal nodes following that decision node n and 3 does not cut any information sets The entire game is also a subgame Subgames that are not the entire game are called proper subgames Three subgames and two proper subgames No proper subgame 15 16 Definition A strategy pro le is a subgame perfect Nash equilibrium if it is a Nash equilibrium of every subgame Remark 1 A backward induction equilibrium is a Nash equilibrium which is consistent with the backward induction procedure Thus it is subgame game perfect Thus subgame perfection is a generalization of backward induction procedure 2 In simultaneous move games under complete information the entire game is the only subgame Thus the Nash equilibrium in that case is trivially subgame perfect 3 A subgame perfect Nash equilibrium is a Nash equilibrium but not vice versa 17 A pure strategy SPNE may not exist even in nite games 15 D u d U L 15 15 U R 15 15 D L 32 03 D R 24 11 Thus there are two pure strategy Nash equilibria U R d and ULd None of the pure strategy Nash equilibria are subgame perfect indeed player II will never play d when he is given a chance to move because u strictly dominates d for him whatever player I does afterwards 18 The notion of subgame perfection is powerful enough to capture the principle of sequential rationality in most cases But sometimes especially in games under imperfect information it may still fail to eliminate the equilibria that incorporate incredible threats 1 1 6 Fight Accommodate i n1 3 daQe 1111 l l 30 I Accommo 39quotCl39mb 391 391 1112 1 1 21 7 Entrant Out 0 2 0 2 There are two pure strategy NE InI Accommodate Out Fight Both of them are also SPNE since the game has no proper subgames Nonetheless Out Fight depends on an incumbent s incredible threat since Accommodate strictly dominates Fight for incumbent the entrant would not be induced to play Out by incumbent s threat to play Fight if given his move Thus the criterion of subgame perfection is of no use here to eliminate the equilibrium incorporating an incredible threat 19
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