3.Simultaneous-Move Games
We now want to study the central question of game theory:how should a game be played.That is,what should we expect about the strategies that will be played in a game.We will start from the simplist types of games,tho of simultaneous-move games in which all players move only once and at the same time.We will…rst study simutaneous games with complete information,and then study simutaneous games with incompete information,where each player’s payo¤s may be known only by the player.
工伤保险怎么报销Dominant strategy and dominated strategy
One quite convincing way of reaching a prediction in a game is the idea of domi-nance.Let’s start from a simple example.
The Prisoner’s dilemma.Two individuals are arrested for allegedly commiting a crime.The district attorny(DA)does not have enough evidenc to convict them. The two suspects are put in two parate jail cells,and are told the following:if one confess and another does not confess,then the confesd suspect will be rewarded with a light ntence of1year,while the unconfesd prisoner will be ntenced to 10years;if both confess,then both will be ntenced to5years;if neither confess, then the DA will still have enough evidence to convict them for a less rious crime, and ntence each of th
em to jail for2years.The game is depicted below:
Prisoner2
DC C
DC-2,-2-10,-1
Prisoner1
C-1,-10-5,-5
热气球制作Each player has two possible strategies:DC or C.What strategy will each player choo?Notice that strategy C is best for each player regardless of what the other
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player’s strategy is.In this ca,we should expect that each player will choo C. In this game,C is said to be dominant strategy for each player.That is,a palyer’s dominant strategy is a strategy that is best for the player regardless of what other players do.The game is called the prisoner’s dilemma be
cau both players could have achieved higher payo¤s if they both choo DC,but this outcome will not be achieved when each player eks to maximize his own payo¤. De…nition.A strategy s i2S i is a strictly dominant strategy for player i in game
¡N=[I;f S i g;f u i(¢)g]if for all s0i=s i;we have
u i(s i;s¡i)>u i(s0i;s¡i)
for all s¡i2S¡i:
A related concept is dominated strategy.Consider the following game:
Player2
L R
酱香茄子U1,-1-1,1
Player1M-1,11,-1
D-2,5-3,-2
Neither player in the game has a dominant strategy.But no matter what2does,1 does better playing U(or M)than playing D.Thus a rational player should not play D.D in this ca is called a strictly dominated strategy.Formally, De…nition.A strategy s i2S i is called a strictly dominated strategy for player i in game¡N=[I;f S i g;f u i(¢)g]if there is another strategy s0i2S i such that
u i(s0i;s¡i)>u i(s i;s¡i)
for all s¡i2S¡i:In this ca we say that strategy s0i strictly dominates s i:
We should expect a player not to play a strictly dominated strategy.
Thus,a strictly dominant strategy for player i is a strategy that strictly dominates all other player i’s strategies.
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沥青延度试验
Still another relevant concept is weakly dominated strategy.Consider the following game:
Player2
L R
U5,14,0
Player1M6,03,1
D6,44,4
1’s payo¤from D is at least as high as from M,whether2choos L or R,and is strictly higher if2choos R.In this ca,we say that M is a weakly dominated strategy for player1.In this ca,we say that M is a weakly dominated strategy for player1.Similarly,1’s payo¤from D is at least as high as from U,whether2choos L or R,and is strictly higher if2choos L.Thus U is also a weakly dominated strategy. Formally,
De…nition.A strategy s i2S i is called a weakly dominated strategy for player i in game¡N=[I;f S i g;f u i(¢)g]if there is another strategy s0i2S i such that
u i(s0i;s¡i)¸u i(s i;s¡i)
for all s¡i2S¡i with inequality holds for some s¡i:In this ca we say that strategy s0i weakly dominates s i:A strategy is called a weakly dominant strategy for player i if it weakly dominates each of i’s other strategies.
In the game above,D is a weakly dominant strategy.
The cond-price aled bid auction(the vickrey auction):Each bidder submits a aled bid.The bidder who bids the highest gets the object,and pays the price equal to the cond-highest bid.Show that each bidder has a weakly dominant strategy: bidding his true valuation.
Proof.Let bidder i’s true valuation for the object be v i;and his bid b i:We show b i=v i is a weakly dominant strategy for i.First consider any b i>v i:If b i is not the winning bid,or if b i wins and the cond highest bid b(2);is less than v i;then八子补肾胶囊
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bidding b i>v i results in the same payo¤to i as if he bids b i=v i:But if b i wins and b(2)>v i;then i will loss the amount equal to b(2)¡v i;while i could have avoided this
loss by bidding v i:Thus b i=v i weakly dominates b i>v i:Next consider any b i<v i: If b i is the winning bid,or if the highest bid is large than v i;than b i<v i result in the same payo¤to i as if i bids b i=v i:But if the highest bid is higher than b i but lower than v i;then bidding b i<v i yields zero payo¤to i while bidding b i=v i would have yielded positive payo¤to i.Thus b i=v i weakly dominates b i<v i:Thus b i=v i is a weakly dominant strategy.
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Unlike the ca of strictly dominant strategy,it is less clear that a rational player should not choo a weakly dominated strategy.In the above example,if1is certain that2will choo L,then playing M is as good as playing D.So we need some additional consideration if a weakly dominated strategy is to be eliminated.
The logic behind the idea that a strictly dominated strategy should not be played can be extended in the following way.After a strictly dominated strategy is elim-inated,we can consider what remains as a new game and again look for strictly dominated stragies in the new game.We can then eliminate any strictly dominted strategy in this game,and then start yet another new game,and so on.This process of achieving a predicition about what should not be played in game is called iterated deletion of strictly dominated strategies.Consider the next game,a modi…ed version of the prisoner’s dilemma game,called the DA’s Brother.
隐面人
Prisoner2
DC C
DC0,-2-10,-1
刀郎西海情歌歌词Prisoner1
C-1,-10-5,-5
Now prisoner1has no dominant strategy:if2plays DC,1’s best respon is DC; and if2plays C,1’s best respon is C.But DC is a dominated strategy for2,and if we eliminate this,then it becomes clear that1should play C.Thus the unique predicted outcome is still(C,C).
So far,we have considered only games with pure strategies.But the ideas can be easily extended to games allowing mixed strategies.加班费英文
De…nition.A strategy¾i24(S i)is strictly dominated for player i in game¡N= [I;f4(S i)g;f u i(¢)g]if there exists another strategy¾0i24(S i)such that for all ¾¡i2Q j=i4(S j)u i(¾0i;¾
¡i)>u i(¾i;¾¡i):
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