清华大学博弈论3
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每一博弈树有一个决策点为该博弈的初始点 每一博弈树有一个决策点为该博弈的初始点。 Every tree has one decision node that is the game’s initial node, the starting point of the game.
每一终结点和参与者的一组结果（收益）相关联。 每 终结点和参与者的 组结果（收益）相关联 A terminal node is associate with a set of (payoffs) y ) for the p players y taking g outcomes (p part in.
决策树与博弈树 Decision Trees vs. Game Trees
决策树表示在一个中立的环境下，单个决策者 的所有相继决策点。 的所有相继决策点 A decision tree show all the successive decision points, points or nodes, nodes for a single decision maker in a neutral environment. 博弈树就是 个博弈当中所有参与者共用的决 博弈树就是一个博弈当中所有参与者共用的决 策树。 G Game t trees are just j t joint j i t decision d i i trees t for all of the players in a game.
A Move or A Strategy? Both B th Both Both
Strategy
Ann
A move, NOT a strategy
Slide 14
招术还是策略 Moves or Strategies?
安在出第一招时选择“ 安在出第 招时选择“Go G ”，可能已经考虑到了假如她一开始 ” 可能已经考虑到了假如她 开始 就选择了“Stop”，她在出第二招时应该做什么。 Ann’s choice of “Go” at the first move may be influenced b her by h consideration id ti of f what h t she h would ld have h t to do d at t her second move if she were to choose “Stop” originally instead. 安可能想选“Go G ”，但有很小的概率她的手颤抖选了“ 但有很小的概率她的手颤抖选了“Stop St ”， 此时刻画安的后续选择就变得重要了。 Ann may intend to press the “Go” key, but there is a small probability p obabilit that her he hand may ma tremble t emble and press p ess the “Stop” key instead in which it is important to specify how Ann will follow up. 均衡的稳定性——如果参与者的选择受到小的干扰将会发生什么？ Stability of equilibrium – what would happen if players’ choices were subjected to small disturbances?
Slide 12
策略 Strategies
参与者在某一节点上的单一行动称为一招。 A single action taken by a player at a node is called a move. 参与者的 个策略是描述该参与者在其每个决 参与者的一个策略是描述该参与者在其每个决 策点上所采取行动（招术）的完整计划（套 路） 路）。 Strategies for each player are complete plans l that th t describe d ib actions(moves) ti ( ) at t each of the player’s decision nodes.
序贯博弈 G Games with ith Sequential S ti l Moves
第3章 Chapter 3
内容提要 wk.baidu.comutline
描述博弈：博弈树 Game trees 求解博弈：反转 Rollback 扩展
增加参与者 Adding more players 增加行动 Adding more moves
Slide 3
序贯博弈 Sequential-move Games
无论何时采取行动，参与者都需要考虑他们当 前的行动会如何影响未来的行动，包括对手和 自己的行动 自己的行动。 Whenever actions are taken, players need to think about how their current actions will influence future actions, themselves both for their rivals and for themselves. 参与者在计算未来后果的基础上，决定现在如 何出招。 Players thus decide their current moves on the basis of calculations of future consequences.
有限和无限博弈 Finite and Infinite games
终结点并不是所有博弈必需的；一些博弈理论 上可以永远进行下去——无限博弈。 无限博弈 An end point is not a necessary feature of all games; some may in principle go on forever – infinite games. 我们考虑的大多数应用为有限博弈 我们考虑的大多数应用为有限博弈。 But most applications that we will consider id are finite fi it games.
证据 Evidences de ces
Slide 2
序贯博弈 Sequential-move Games
序贯博弈要求一个有严格博弈顺序的策略环境。 Sequential-move Sequential move games entail strategic situations in which there is a strict order of play. p y 参与者按顺序出招，并且知道在他们之前的那 些参与者都干了什么。 参与者都干了什么 Players take turns making their moves, and they know what players who have gone before them have done.
Slide 6
博弈树：一个例子 Game Trees: An Example
ANN Branches BOB Stop ANN Go 1 2 3 Up Down DEB High Low (10,7,1,1) (2,7,4,1) ( , , , ) (1,-2,3,0) (1.3,2,-11,3) (0,-2.718,0,0) (0, 8,0,0) Terminal nodes (6,3,4,0) (2,8,-1,2) node节点 branch分支
Good 50% NATURE Risky Bad 50%
Root CHRIS (Initial node) Safe
(3,5,3,1) (ANN,BOB,CHRIS,DEB)
Go back to the table
Slide 7
博弈树 Game Trees
博弈树由节点和分支组成。节点之间由 分支相连。 分支相连 Game trees consist of nodes and branches.Nodes are connected to one another by branches.
Slide 8
博弈树 Game Trees
节点有两类 Nodes come in two types:
每个决策点与在该节点采取某一行动的参与者相关 联。 A decision node is associated with the player who chooses an action at that node.
Slide 4
博弈树 Game Trees
博弈树，或博弈的扩展型，是用来表示和分析 序贯博弈的图形方法。 A game tree, or the extensive form of a game is a g g graphical p technique q for displaying and analyzing sequentialmove games. 它表明了所有参与者能够采取的所有可能的行 动，指出了博弈的所有可能结果。 It illustrate all of the possible actions that can be taken by all of the players and indicate all of the possible outcomes from the game. Slide 5
Slide 13
招术还是策略 Moves or Strategies?
S th See the graph h
Who? Bob Ch i Chris Deb Ann
What to play? 1 Ri k Risky Low “Choose stop, and then if Bob chooses 1, , Choose down” Go
不确定性与“自然之手” 不确定性与 自然之手 Uncertainty and “Nature’s Moves”
将自然作为参与者，使得我们可以将不确定性 将自然作为参与者 使得我们可以将不确定性 引入博弈当中，提供了允许所发生的事情不在 任何一个实际参与者掌控之中的机制。 任何一个实际参与者掌控之中的机制 Use of the player Nature allows us to introduce uncertainty in a game and gives us a mechanism to allow things to happen that outside the control of any pp y of the actual players. 引入自然的选择 意味着决策者需要确定预期 引入自然的选择，意味着决策者需要确定预期 收益。 The inclusion of a choice by Nature means players l need d to d determine i the h Slide 11 expected payoffs.
Slide 9
博弈树 Game Trees
分支代表从任一决策点出发的可能采取的行动。 分支代表从任 决策点出发的可能采取的行动 The branches represent the possible actions that t at can ca be taken ta e from o any a y decision dec s o node. ode 每一分支都从博弈树上的一个决策点指向另一个决策 点或终结点。 E hb Each branch hl leads d f from a d decision i i node d on th the tree either to another decision node, or to a terminal node. 在任何一个博弈中，从每一个决策点出发，至少要有 一个分支；不过，仅允许有一个分支指向任何一个决 策点。 策点 In any game, there must be at least one branch leading from each decision node. However, every decision node can have only Slide 10 one branch leading to it.
每一终结点和参与者的一组结果（收益）相关联。 每 终结点和参与者的 组结果（收益）相关联 A terminal node is associate with a set of (payoffs) y ) for the p players y taking g outcomes (p part in.
决策树与博弈树 Decision Trees vs. Game Trees
决策树表示在一个中立的环境下，单个决策者 的所有相继决策点。 的所有相继决策点 A decision tree show all the successive decision points, points or nodes, nodes for a single decision maker in a neutral environment. 博弈树就是 个博弈当中所有参与者共用的决 博弈树就是一个博弈当中所有参与者共用的决 策树。 G Game t trees are just j t joint j i t decision d i i trees t for all of the players in a game.
A Move or A Strategy? Both B th Both Both
Strategy
Ann
A move, NOT a strategy
Slide 14
招术还是策略 Moves or Strategies?
安在出第一招时选择“ 安在出第 招时选择“Go G ”，可能已经考虑到了假如她一开始 ” 可能已经考虑到了假如她 开始 就选择了“Stop”，她在出第二招时应该做什么。 Ann’s choice of “Go” at the first move may be influenced b her by h consideration id ti of f what h t she h would ld have h t to do d at t her second move if she were to choose “Stop” originally instead. 安可能想选“Go G ”，但有很小的概率她的手颤抖选了“ 但有很小的概率她的手颤抖选了“Stop St ”， 此时刻画安的后续选择就变得重要了。 Ann may intend to press the “Go” key, but there is a small probability p obabilit that her he hand may ma tremble t emble and press p ess the “Stop” key instead in which it is important to specify how Ann will follow up. 均衡的稳定性——如果参与者的选择受到小的干扰将会发生什么？ Stability of equilibrium – what would happen if players’ choices were subjected to small disturbances?
Slide 12
策略 Strategies
参与者在某一节点上的单一行动称为一招。 A single action taken by a player at a node is called a move. 参与者的 个策略是描述该参与者在其每个决 参与者的一个策略是描述该参与者在其每个决 策点上所采取行动（招术）的完整计划（套 路） 路）。 Strategies for each player are complete plans l that th t describe d ib actions(moves) ti ( ) at t each of the player’s decision nodes.
序贯博弈 G Games with ith Sequential S ti l Moves
第3章 Chapter 3
内容提要 wk.baidu.comutline
描述博弈：博弈树 Game trees 求解博弈：反转 Rollback 扩展
增加参与者 Adding more players 增加行动 Adding more moves
Slide 3
序贯博弈 Sequential-move Games
无论何时采取行动，参与者都需要考虑他们当 前的行动会如何影响未来的行动，包括对手和 自己的行动 自己的行动。 Whenever actions are taken, players need to think about how their current actions will influence future actions, themselves both for their rivals and for themselves. 参与者在计算未来后果的基础上，决定现在如 何出招。 Players thus decide their current moves on the basis of calculations of future consequences.
有限和无限博弈 Finite and Infinite games
终结点并不是所有博弈必需的；一些博弈理论 上可以永远进行下去——无限博弈。 无限博弈 An end point is not a necessary feature of all games; some may in principle go on forever – infinite games. 我们考虑的大多数应用为有限博弈 我们考虑的大多数应用为有限博弈。 But most applications that we will consider id are finite fi it games.
证据 Evidences de ces
Slide 2
序贯博弈 Sequential-move Games
序贯博弈要求一个有严格博弈顺序的策略环境。 Sequential-move Sequential move games entail strategic situations in which there is a strict order of play. p y 参与者按顺序出招，并且知道在他们之前的那 些参与者都干了什么。 参与者都干了什么 Players take turns making their moves, and they know what players who have gone before them have done.
Slide 6
博弈树：一个例子 Game Trees: An Example
ANN Branches BOB Stop ANN Go 1 2 3 Up Down DEB High Low (10,7,1,1) (2,7,4,1) ( , , , ) (1,-2,3,0) (1.3,2,-11,3) (0,-2.718,0,0) (0, 8,0,0) Terminal nodes (6,3,4,0) (2,8,-1,2) node节点 branch分支
Good 50% NATURE Risky Bad 50%
Root CHRIS (Initial node) Safe
(3,5,3,1) (ANN,BOB,CHRIS,DEB)
Go back to the table
Slide 7
博弈树 Game Trees
博弈树由节点和分支组成。节点之间由 分支相连。 分支相连 Game trees consist of nodes and branches.Nodes are connected to one another by branches.
Slide 8
博弈树 Game Trees
节点有两类 Nodes come in two types:
每个决策点与在该节点采取某一行动的参与者相关 联。 A decision node is associated with the player who chooses an action at that node.
Slide 4
博弈树 Game Trees
博弈树，或博弈的扩展型，是用来表示和分析 序贯博弈的图形方法。 A game tree, or the extensive form of a game is a g g graphical p technique q for displaying and analyzing sequentialmove games. 它表明了所有参与者能够采取的所有可能的行 动，指出了博弈的所有可能结果。 It illustrate all of the possible actions that can be taken by all of the players and indicate all of the possible outcomes from the game. Slide 5
Slide 13
招术还是策略 Moves or Strategies?
S th See the graph h
Who? Bob Ch i Chris Deb Ann
What to play? 1 Ri k Risky Low “Choose stop, and then if Bob chooses 1, , Choose down” Go
不确定性与“自然之手” 不确定性与 自然之手 Uncertainty and “Nature’s Moves”
将自然作为参与者，使得我们可以将不确定性 将自然作为参与者 使得我们可以将不确定性 引入博弈当中，提供了允许所发生的事情不在 任何一个实际参与者掌控之中的机制。 任何一个实际参与者掌控之中的机制 Use of the player Nature allows us to introduce uncertainty in a game and gives us a mechanism to allow things to happen that outside the control of any pp y of the actual players. 引入自然的选择 意味着决策者需要确定预期 引入自然的选择，意味着决策者需要确定预期 收益。 The inclusion of a choice by Nature means players l need d to d determine i the h Slide 11 expected payoffs.
Slide 9
博弈树 Game Trees
分支代表从任一决策点出发的可能采取的行动。 分支代表从任 决策点出发的可能采取的行动 The branches represent the possible actions that t at can ca be taken ta e from o any a y decision dec s o node. ode 每一分支都从博弈树上的一个决策点指向另一个决策 点或终结点。 E hb Each branch hl leads d f from a d decision i i node d on th the tree either to another decision node, or to a terminal node. 在任何一个博弈中，从每一个决策点出发，至少要有 一个分支；不过，仅允许有一个分支指向任何一个决 策点。 策点 In any game, there must be at least one branch leading from each decision node. However, every decision node can have only Slide 10 one branch leading to it.