Part 2-1 Sequential game

prolog

• 出招有先後順序 The relationship of interaction between players in

the dynamic game represent at the order of move, but not simultaneous move as in static game.

• Extensive form• Nash equilibrium 是完全訊息靜態賽局解之基本概念 SPNE 是完全訊息動態賽局解之基本概念

the extensive form and the game tree

• A game tree is a “directed graph” and has a “transitive precedence relation” Kreps ＆ Wilson （ 1982 ）：

• definition of building block A node is a point in the game at which some

player or Nature takes an action, or the game ends.

A successor to node X is a node that may occur later in the game if X has been reached.

A predecessor to node X is a node must be reached before X can be reached.

A starting node is a node with no predecessors.

A end node or end point is a node with no successors. A branch is one action in a player’s action set at particular node. A path is a sequence of nodes and branches from the starting node to an end node.

deter entry

• strategic setting Players ： incumbent ： S New entry ： P Move sequence ： sequential first mover ： S, action set ：﹛擴廠 , 不擴

廠﹜ second mover ： P, action set ：﹛ Build,

don’t ﹜

• Subgame ：承襲自一特定完全訊息節點，包含所有 節點與報酬的賽局• Proper subgame （完整子賽局）：一個不等於整個 賽局的子賽局• Commitment structure ： sequential game 中，所有 proper subgame 的集合，均為此 sequential game 的承諾結構。• SPNE ： A game is in SPNE iff every subgame is in a Nash equilibrium an equilibrium in which the players always, consistently, anticipate their rivals’ decision in this way.

• Backward induction( 倒推法 ) 一種發現子賽局完美均衡的方法，其步驟是先找出所有基本子賽局的均衡，然後再將每個基本子賽局的均衡報酬代回，以簡化賽局，重複此步驟直到所有完整小賽局都被分析完為止。• Basic subgame( 基本子賽局 ) ：一個不包含任何其他完整子賽局的子賽局，即稱之。• Complex subgame( 複雜子賽局 ) ：一個包含任何其他完整子賽局的子賽局，即稱之。

node2 所包含之訊息集無法分割，整個賽局不包含任 何其他「完整子賽局」，整個賽局本身即為一子賽 局

The concept of subgame and sequential game

• Recalling 14.1 Node PA 、 Node PB 之訊息集均為

sinslelton ，之後的分支均各為一完整子賽局，因此，開始於 initial node S 的賽局，稱為複雜子賽局（ complex subgame ）。 此賽局擁有二個完整子賽局，進而構成一組子賽局完美均衡。而子賽局完美均衡策略必須是任一子賽局的 NE 故稱為 SPNE 。

• 尋找 SPNE 的方法：逆推法（ backward induction

think forward, reason backward Step1 ：找出基本子賽局（特點：已無向下 承諾結構） Step2 ：找出各基本子賽局之 NE induced game

Step3 ：找出 induced game 之 NE 持續進行，直到所有完整子賽局分析完 1. 一原賽局一序列的均衡即構成 SPNE 2. 逆推法乃是重複剔除 dominated

strategy 在 extensive game 之應用

若到達 PA ： P 廠最適選擇 “ don’t” 若到達 PB ： P 廠最適選擇 “ build” 1. S 廠最適選擇“ build” 2. SPNE ： (S,P) ： (B, (BD, DB)) Outcome ： S 建廠， P 不建

先進優勢或後進優勢？• strategic setting H’s action set ：﹛ C, B﹜ ， C ：前進新迦太基， B ：貝特河 P’s action set ： ﹛ L, C﹜ ， L ：前進拉米尼

• 先行者： H

• SPNE ：（ B, （ CL,BC ）） Outcome ： H 選 B ， P 選 C

• 先行者： P

NE and SPNE

子賽局完美均衡是納許均衡的一種精煉(refinement)

均衡的精煉，指的是納許均衡必須額外符合一些特定的判準。

SPNE ：（ Enter, （ EA, NEU ））

NE ：（ Enter, （ EA, NEU ））、（ NE, （ EP, NEU ））

• a price war is not credible threat because it is not an equilibrium in the basic subgame.

• Every subgame perfect equilibrium is a NE, but not every NE is subgame perfect. SPNE is a refinement of NE A strategy profile is a SPNE if i. it is a NE for the entire game; and ii. its relevant action rules are a NE for every subgame.

Note 1 ： A strategy profile is a perfect equilibrium if it

remains an equilibrium on all possible paths, includes not only the equilibrium path but all the other paths, which branch off into different“ subgames”. the term “perfectness” reflects the idea of sequential rationality.

Note 2 ： backward induction 不適用於無限賽 局、不完美訊息賽局（ imperfect information game ）

the centipede game

• The two-stage centipede game has just one proper subgame, and it is basic.

• There is possible that SPNE of the sequential game in extensive form may have inefficient outcome.

Going Coconutty

• SPNE ：（ Friday, Robison ）： 非均衡路徑上的選擇 Outcome ： Friday 選 ， payoff 5, Robison’s payoff 0 social dilemma

• Interpretation ：為何需要法律保障之契約機制？ 比較利益 分工 迂迴生產 生產力提升 若無契約約束， opportunism 發生， social optimum 無法達到。

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