Class Note for ECON 309 at UMass(1)
Class Note for ECON 309 at UMass(1)
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Date Created: 02/06/15
Definition Consider a strategic form game T lt N SiENH6N gt 1 A strategy s 6 Si is said to strictly dominate s 6 Si for player i if uisg si gt uisiquot s4 for all s es A strategy is said to be strictly dominated if there is at least one strategy that strictly dominates it 2 A strategy is a strictly dominant strategy for player i if it strictly dominates all his other strategies that is ui isigtuisisi for all s es and all s4 6S4 3 The strategy profile 1 is a strictly dominant strategy equilibrium if for all i e N 7 is a strictly dominant strategy for player i Assumption 1 No rational players will adopt a strictly dominated strategy Example Oil Drilling Game Narrow Wide Narrow 1414 1 16 Wide 16 1 1 1 Example Prisoner s Dilemma Confess Don t Confess Confess 3 3 0 4 Don t Confess 4 0 1 1 Example In the following game neither players have strictly dominant pure strategies Thus the concept of a strictly dominant strategy equilibrium is of no use for this game as long as pure strategies are concerned L R U 51 40 M 60 31 D 73 24 Example Boxed Pigs Two pigs one small and the other big are put into a skinner box with a special lever at one end and a food dispenser at the other At least one pig must press the lever for the food to be dispensed When the lever is pressed at a utility cost of 2 units 10 units of food are dispensed A pig that presses the lever arrives late at the dispenser One pig is large and if he gets to the dispenser first he eats 9 units the other ie a small pig will only get his leavings worth 1 unit The small pig does somewhat better if he gets there first eating 4 units of food If both pigs press the lever at the same time and then run they arrive at the food dispenser at the same time and the small pig can eat 3 units Small pig Large pig Press Wait Press 5 1 4 4 Wait 9 1 0 0 Wait is a strictly dominant strategy for the small pig but the large pig does not have a strictly dominant strategy there is no strictly dominant strategy equilibrium But there is an iterated strictly dominant strategy equilibrium which is Press Wait First if the large pig knows that the small pig is rational ie the small pig will not use a strictly dominated strategy then the large pig may discard the possibility that the small pig plays Press Once this is done the relevant game to be considered is reduced to Wait Press 4 4 Wait 0 0 Left Right Left Right Up 810 1009 Up 810 l9 Down 7 6 6 5 Down 7 6 6 5 In both games the iterated deletion of dominated strategies yields U L as the unique solution But experiments show that for the leftside game about half of individuals taking the role of player I played D and some indiViduals taking the role of player 2 played L Why Although U is better than D when player 2 is certain not to use R D is better than U when there is 1 chance that player 2 plays R In other words unless the strategy space the payoffs and rationality in the sense that players do not play a strictly dominated strategy are common knowledge there is no reason to believe that players will play the equilibrium strategies Definition Consider a strategic form game F lt N a S EN 2 u EN gt 1 A strategy s GSi is said to weakly dominate s esi for player i if L1 SS Zui S39S for all S ES with the inequality being strict for at least one 54 A strategy is said to be weakly dominated if there is at least one strategy that weakly dominates it 2 A strategy i is a weakly dominant strategy for player i if it weakly dominates all his other strategies that is Mi iSZuiSSi for all S i and all S71 6 S71 3 The strategy profile 139 39 n is a weakly dominant strategy A equilibrium if for all i e N 7 S 1 is a weakly dominant strategy for player i Example lB Consider the following variation of the oildrilling game Narro Wide Narr 14 14 0 16 ow Wide 16 0 0 0 Narrow is no longer strictly dominated by Wide But Narrow is weakly dominated by Wide So Wide Wide is a weakly dominant strategy equilibrium Example L R U 51 40 M 60 31 D 64 44 Note that U and M are weakly dominated by D for player 1 If we delete in the order of U4 LHM then we have D R If the order of deletion is Me R a U then we have D L If we delete M and U simultaneously then we have DR and DL The concept of dominance gives strong predictions of outcomes in some games such the prisoner s dilemma but it is very rare to find an equilibrium using the method of elimination or successive elimination of dominated strategies L U 11 15 M 12 11 D 20 32 D is a strictly dominated strategy for player 1 so we can eliminate it But we cannot proceed afterwards Indeed in many games no player has dominated strategies at all There are two Nash equilibria in this game though De nition Consider a strategic form game F lt N a SiieNu6N gt A strategy profile SS is a Nash equilibrium if no player would like to deviate unilaterally from it De nition Consider a strategic form game F lt N a SiieNuEN gt 1 A strategy S is player i s best response to s if uss2uss for all s e 3 When s is player i s best response to 571 we denote it by s e BRIs 2 A Nash equilibrium is a strategy profile sfs such that sf is player i s best response to Sf for all ieN Put it differently a strategy profile sfs is a Nash equilibrium if for all i L1s sf 2 uiss for all s e S Example continued L R U 11 015 M 1 11 D 2 0 3 2 The unique pure strategy Nash equilibrium is U R and M L 1 At a NE each player s equilibrium strategy is a best response to the belief that the other players will adopt their Nash equilibrium strategies 2 Nobody wants to deviate unilaterally at a Nash equilibrium Thus if you want to show a strategy profile is not a Nash equilibrium it suffices to show that there is at least one player who would like to deviate from the situation described by the strategy profile Showing a strategy profile is a Nash equilibrium however needs more work for you have to show that nobody wants to deviate unilaterally from the strategy profile Finally finding a set of Nash equilibria is the most cumbersome because you have to consider every possible strategy profiles and for each strategy profile you have to see whether it is a Nash equilibrium or not Life is hard but that s the life 3 The strictly dominant strategy equilibrium is a Nash equilibrium Because the SDSE is unique if exists it is the unique NE of the game Finally any strategy profile that does not survive iterated deletion of strictly dominated strategies cannot be a NE 4 Likewise WDSE is a NE But not all NE are WDSE 5 A NE does not have to be unique there may be more than one Nash equilibria 6 A NE may not eXist Example Cournot duopoly The profits for each airline were 190 10qi We 100 if ii lt11 6 Oi 19 100ql if qt qj gt 19 7T1 97g Recall that a 90 10qJ 20q120 The Nash equilibrium is a pair qfq such that 1 611 maximizes 751 given shrimper 1 s belief that shrimper 2 will choose 61 and 2 q maximizes 72 given shrimper 2 s belief that shrimper 1 will choose qf Hence q qb is the solution of 61 a the equations 6 0 and 6 0 SolV1ng these two equations Simultaneously we 1 2 have 654 33 The equilibrium price is then if 190 103 3 130 and 7i w 1303 100390 1 Flyme s output Airgo 5 output Example Bertrand paradox There are two shrimpers in New Haven Gump and Ky Both choose prices and sell as much as they can Consumers buy shrimps from the shrimper charging the lowest price If both shrimpers charge the same price then they split the market equally Market demand is given by D 19 10p The marginal cost of catching shrimp is c gt 0 for both shrimpers Payoff functions are then determined in the following way 1 Suppose pG lt pK Then Gump takes the entire market DG 19 10pG and DK 0 Hence 75G pG cl9 10pG and 75K 0 2 Likewise if pG gt pK then 115K pK cl 9 10pK and TEG 0 3 Finally if pG pK p then they split the market equally so 19 10P 2 nGzerp c Summarizing pG C19 10pG if pG lt pK 7 s 2 PG 019 10PG2 if PG 2 PK 0 if pG gt pK pK C19 10pK if pK lt 76 75K PK C19 1OPK2 if PK PG 0 if pK gt176 What is the Nash equilibrium of this game Note that this is a game with discontinuous payoff functions l A pair of prices 17mm such that pG gt pK gt0 does not constitute a NE If pG gt 17K gt c then Gump can be better off by slightly undercutting the price of Ky Likewise a pair of prices 19mm such that pK gt pG gtcdoes not constitute a NE Ky can be better of by slightly undercutting the price of Gump 2 A pair of prices pg 19K such that PG gt PK C does not constitute a NE If pg gt pK c then Ky can be better off by slightly increasing the price Likewise a pair of prices pg 29K such that pK gt p0 c does not constitute a NE 3 A pair of prices 1991 such that 1 p6 gt C does not constitute a NE Anyone who slightly undercuts the price of the other shrimper can be better off 4 The only possibility we need to check is therefore 17K 17G c Since there is no one who would like to deViate this is the only Nash equilibrium Hence HG 7rK 0 at the NE Although the market structure is duopoly the equilibrium outcome is the same as that of perfect competition Bertrand paradox Example 11 Hotelling s location competition model There is a continuum of consumers with the total mass of N in the linear city and consumers are uniformly distributed in the unit interval Each consumer has unit demand Suppose both shrimpers are forced to take the same price P gt C perhaps by the regulation of the government and are not allowed to change the price Shrimpers can however change the location of their stores Of course the two shrimpers can choose the same place if they wish and in that case we assume that they split the market half and half Hence the strategy of each shrimper is the choice of location Suppose Ky s location is a 6 01 and Gump s is l b 6 01 If a lt1 b then each shrimper can be better off by moving closer to the other shrimper so this cannot be a Nash equilibrium Likewise the situation in which a gt1 b cannot be a Nash equilibrium What happens if a 1 b they choose the same location Because they chose the same location they split the market half and half There are two possibilities to consider 1 2 Suppose the same location that they chose is different from the center More specifically suppose the same location is to the left of the center Then any shrimper who moves to the right of the same location absorbs more than half of the market demand and therefore can be better off Likewise if the same location that they chose is to the right of the center then a shrimper can be better off by deviating to the left of the same location The only possibility left is therefore the situation in which both shrimpers choose to locate in the center In this case no one can be better off by unilaterally deviating from the center therefore this is the unique Nash equilibrium
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