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. 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

1. Two-stage game. Second-stage game is a simultaneous game2. 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 之訊息 （承諾），且此一承諾具 credible ， 故 N 之 best response 為 “ SE”

StudySE

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 and forward induction ：complement relationship

• backward induction

• 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.

• remark 2nd game （ 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.

Theft Game(the Maltese Falcon)

• By the logic of backward induction ： 2nd stage ： 1. Upper subgame ： （ guard, armed ） , payoff ：（ -5, 4 ） 2. Lower subgame ： （ don’t guard, unarmed ） , payoff ：（ -1,

0 ）

• 1st satge ：

• SPNE ：（ , unarmed ） Outcome ： GK ： fake and don’t

guard ， GG ： unarmed ， Payoff ：（ -1, 0 ） quardtdonfake ' ,

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

• a larger and more complicated game to include the commitment

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 without Union

NE ：（ don’t concede, don’t strike ）， but why are there strikes ？

• 勞方加入工會， 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 ？

Strike game with Union

• 1st stage ： Employees’ action set

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

• 2nd stage ： employeR and UTG move

simultaneously. employeR’s action set UTG’s action set

affiliate ,'tdon

conced ,NO

tdontrike ' ,S

• 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.

652employeesEV

62UTGEV

• why do employees affiliate with the UTG ？ 與工會合作進行罷工預期報酬 2.5 ，而不與工會合作，報酬為 2 罷工成為 incredible threat ，整體而言，勞工扣除工會費用後仍有利。