part 2 2 imbedded game n.
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Part 2-2 Imbedded game. prolog. Sometimes , it often seems that people act irrationally in some games they play, but the action is understood as rational when realize that the game is really part of a bigger game.

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Sometimes, it often seems that people act

irrationally in some games they play, but the

action is understood as rational when realize

that the game is really part of a bigger game.

The game in which people seem to act irrationally is imbedded or nested in a larger game.


Nested games(被巢覆賽局):

if a game is a part of a larger game, then equilibrium strategy in the smaller game may depend on the larger game. The smaller game is said to be nested game within the larger.

  • Imbedded games(被嵌入賽局):

if a nested game is a proper subgame of the larger game, then the nested game must be in equilibrium for the larger game to be in subgame perfect equilibrium. The smaller game is said to be imbedded in the larger.

doctoral study game
Doctoral study game

Two-stage game. Second-stage game is a

simultaneous game

2. the proper game with two pure strategy


Second-stage game is a game of imperfect information

  • This game is a coordination game. There is uncertainty as to which NE will occur.

Solution:(forward induction)

Second-mover N can infer something(N reason that the A guess that N chooses the “SE”) from a choice that firs-mover A made in the first-stage, and this inference can solve the uncertainty in the coordination.

A 在first-stage選擇 “study”,向second-stage

mover N傳遞他在second stage 將選擇


故N之 best response 為 “SE”

forward induction
Forward induction
  • 由之前歷史推論未來:鑒往知來
  • changing the rules:

When a game has an unsatisfactory outcome, the outcome may be changed via some method or arrangement of mechanism, for example, forward induction method, contract, etc.

Thus, the original game is nested or imbedded in a larger game, so that the outcome is changed from the original game.


Backward induction with forward induction

First-stage player A uses forward induction. Player takes into account what second-stage player N will infer from his choice and the influence that inference can have on N’s choice.

Player N does as in A’s anticipation by forward induction.



2ndgame (subgame)is imbedded in the larger game. The larger game is the imbedding game. Subgame perfect equilibrium in the imbedding game will require that the imbedded game is in equilibrium. The reverse is not necessarily true.


By the logic of backward induction:


1. Upper subgame:

(guard, armed), payoff:(-5, 4)

2. Lower subgame:

(don’t guard, unarmed), payoff:(-1, 0)



  • SPNE:( , unarmed)

Outcome:GK: fake and don’t guard,GG:unarmed,Payoff:(-1, 0)

the centipede solved how to solve social dilemma
the centipede solved(how to solve social dilemma)

inefficient outcome with

no commitment

player B engage a third party and post a bond of 2.

If he chooses “ grab” , he forfeits the bond; otherwise, he could be returned

counterattack revisited
Counterattack revisited

美國可以透過駐軍來改變賽局(change the rules)

此大賽局的子賽局完美均衡就是(美國派軍駐西德, 蘇聯不攻擊)

strike game the proper game without pure strategy equilibrium
Strike Game(the proper game without pure strategy equilibrium)

Strike game without Union

NE:(don’t concede, don’t strike),but

why are there strikes?



players :employer and UTG

UTG’s payoff is proportional to his reputation of toughness.

UTG’s payoffs are quite different from those of the employees.

no pure strategy NE

why do employees affiliate with the UTG?


1st stage:Employees’ action set

If employees affiliate with UTG., they give up the freedom to decide whether to strike.

  • 2ndstage:employeR and UTG move


employeR’saction set

UTG’s action set


By backward induction:

Since lower-side game in Fig15.12 has no pure strategy equilibrium it must be an equilibrium in mixed strategy.

  • Suppose, UTG call “ strike” with probability

1/2, the employers make “concede” with

probability 2/3.


why do employees affiliate with the UTG?


罷工成為incredible threat,整體而言,勞工扣除工會費用後仍有利。