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博弈论代写代考| Sequential Games: The Theory of Moves 数学代写

博弈论代考

In this section, we engage in a little sensitivity analysis with ordinal games. Our starting point is that while the chicken dilemma modeled the essential dynamics of the Cuban Missile Crisis, it is not necessarily a good model to use for predictive purposes. Indeed, if each player tries for the Nash equilibrium that most benefits him or her, the prediction is the disastrous ( 1,1 ) outcome. As we know, that did not happen in 1962. As noted before, in Brams (1985a and 1985b), Brams considered the model obtained by allowing communication between the players, and he considered the sequential nature of the game when the players do not move simultaneously. Furthermore, we noted earlier that in the chicken dilemma model of the US federal government shutdown, while the $(1,1)$ outcome was indeed obtained, it did not last because players were
8.3 Sequential Games: The Theory of Moves
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allowed to change their strategies. We suggested that a model that allows changes of strategies in a sequential game might be more realistic. Also, if players are allowed to switch strategies, that implies there is some starting point from which initial changes might be made.
When one modifies $2 \times 2$ ordinal game models to incorporate sequential moves, a starting position, and the ability to switch strategies, one arrives at the theory of moves, a technique first proposed in Brams (1994) and slightly modified in Taylor and Pacelli (2008). We use the version in Taylor and Pacelli $(2008)$ here.
The idea is that each $2 \times 2$ ordinal payoff matrix can be used to create eight different sequential games, four when the row player has the first option to switch strategies (depending on which of the four outcomes is the initial position) and four when the column player has the first move option. Each such sequential game is modeled in extensive form, and, assuming the tree diagram is finite, one can analyze it via reverse induction to see what would actually happen in each of the eight cases. Note that the assumption that it can be analyzed by reverse induction implies we do not relax the intelligence assumption of game theory – we cont and agree on all the preference rankings in the payoff matrix. Here are the rules for creating the tree diagram. We give the rules for when the row player moves first. If the column player moves first, the rules are similar:

ALGORITHM $8.9$ Suppose you are given a $2 \times 2$ ordinal payoff matrix, and a sequential game is to be played with the row player moving first. The steps taken to set up the tree diagram for the Theory of Moves are as follows: .

1. Each player makes a simultaneous and independent choice of a strategy. The resulting outcome on the payoff matrix is the starting position.
2. The row player makes a choice either to switch to the other strategy or to stay with her Current strategy. In each case, draw a branch of the tree, and label it with the resulting outcome. For any branch that is not a terminal node, go to step 3. If every node is terminal, then stop-the tree is complete.
3. The column player then has a choice to either switch to the other strategy or to stay with his current strategy. For any branch that is not a terminal node, go to step $2 .$ If every node is terminal, then stop-the tree is complete.
The conditions for a terminal node are as follows:
4. If it is the row player’s turn to move, and the current outcome is $(4,-)$, except on the row player’s first move.
5. If it is the column player’s turn to move, and the current outcome is $(-, 4)$.
6. If either player elects to stay, except a stay on the row player’s first move.
The rules ensure that both players have at least one chance to switch strategies if they want, which is why an initial move by the row player (or whoever goes first) never ends the game. Furthermore, if one player places the other in an outcome, then presumably, he or she is happy with that outcome, but if it happens to be the first choice of the other player (the 4 preference), then – since both players are satisfied – there should be no further switches. The same holds if either player elects to stay with their current choice, as long as both players have had a chance to switch. Once the game tree is set up, the outcome of the game is determined using backwards induction.
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Sensitivity Analysis, Ordinal Games, and n-Person Games
When a game is played under simultaneous moves, the “stable points” are called Nash equilibria. For the theory of moves, we have the following definition:

DEFINITION 8.10 If an outcome of the game has the property that whenever it is the starting position, it is also the final position when the row player moves first, it is said to be a nonmyopic equilibrium when row goes first. Similarly, if it has the property that whenever it is the starting position, it is also the funal position when the column player moves first, it is said to be a non-myopic equilibrium when column goes first. If an outcome

The name stems from the idea that by analyzing the tree diagram for the sequential moves, one is looking ahead, or being “far-sighted”by anticipating what the other would do in each outcome. we denote by $C$ the cooperation strategy (swerve out of the way) and by $D$ the defection strategy (drive straight on).

8.3 顺序博弈：移动理论
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1. 每个玩家同时独立地选择策略。收益矩阵的结果是起始位置。
2. 排玩家选择切换到其他策略或保持当前策略。在每种情况下，绘制树的一个分支，并用结果标记它。对于不是终端节点的任何分支，转到步骤 3。如果每个节点都是终端，则停止树完成。
3. 纵队玩家可以选择切换到其他策略或保持当前策略。对于不是终端节点的任何分支，转到步骤 $2。$ 如果每个节点都是终端，则停止树完成。
终端节点的条件如下：
4. 如果轮到排位玩家移动，并且当前结果是$(4,-)$，排位玩家的第一个动作除外。
2.如果轮到列玩家移动，当前的结果是$(-, 4)$。
5. 如果任一玩家选择留下，除了停留在排玩家的第一步。
规则确保双方玩家至少有一次机会根据需要切换策略，这就是为什么排玩家（或先走的人）的初始移动永远不会结束游戏的原因。此外，如果一个玩家将另一个放在一个结果中，那么他或她可能对该结果感到满意，但如果它恰好是另一个玩家的首选（4 偏好），那么 – 因为两个玩家都满意- 不应该有更多的开关。如果任何一个玩家选择保留他们当前的选择，只要两个玩家都有机会切换，这同样适用。一旦建立了博弈树，博弈的结果就使用反向归纳来确定。
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敏感性分析、序数博弈和 n 人博弈
当游戏同时进行时，“稳定点”称为纳什均衡。对于移动理论，我们有以下定义：

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编码理论代写

1. 数据压缩（或信源编码
2. 前向错误更正（或信道编码
3. 加密编码
4. 线路码

复分析代考

(1) 提到复变函数 ，首先需要了解复数的基本性左和四则运算规则。怎么样计算复数的平方根， 极坐标与 $x y$ 坐标的转换，复数的模之类的。这些在高中的时候囸本上都会学过。
(2) 复变函数自然是在复平面上来研究问题，此时数学分析里面的求导数之尖的运算就会很自然的 引入到复平面里面，从而引出解析函数的定义。那/研究解析函数的性贡就是关楗所在。最关键的 地方就是所谓的Cauchy一Riemann公式，这个是判断一个函数是否是解析函数的关键所在。
(3) 明白解析函数的定义以及性质之后，就会把数学分析里面的曲线积分 $a$ 的概念引入复分析中， 定义几乎是一致的。在引入了闭曲线和曲线积分之后，就会有出现复分析中的重要的定理: Cauchy 积分公式。 这个是易分析的第一个重要定理。