Posted on Categories:Game theory , 博弈论, 经济代写

# 经济代写|博弈论代考Game theory代写|Definition of Simple Timing Games

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## 经济代写|博弈论代考Game theory代写|Definition of Simple Timing Games

In a simple timing game, each player’s only choice is when to choose the action “stop,” and once a player stops he has no effect on future play. That is, if player $i$ has not stopped at any $\tau<t$, his action set at $t$ is
$$A_i(t)={\text { stop, don’t stop }}$$
if player $i$ has stopped at some $\tau<l$, then $A_i(t)$ is the null action “don’t move.” Few situations can be exactly described in this way, because players typically have a wider range of choices. (For example, firms typically do not simply choose a time to enter a market; they also decide on the scale of entry, the quality level, etc.) But economists often abstract away from such details to study the timing question in isolation.

We will consider only two-player timing games, and restrict our attention to the subgame-perfect equilibria. Once one player has stopped, the remaining player faces a maximization problem that is easily solved. Thus, when considering subgame-perfect equilibria, we can first “fold back” subgames where one player has stopped and then proceed to subgames where neither player has yet stopped. ${ }^4$ This allows us to express both players’ payoffs as functions of the time
$$\hat{i}=\min \left{t \mid a_i^t=\text { stop for at least one } i\right}$$
at which the first player stops (with the strategies we will consider, this minimum is well defined); if no player ever stops, we set $\hat{t}=\infty$. We describe these payoffs using the functions $L_i, F_i$, and $B_i$ : If only player $i$ stops at $\hat{t}$, then player $i$ is the “leader”; he receives $L_i(\hat{t})$, and his opponent receives “follower” payoff $F_j(\hat{t})$. If both players stop simultaneously at $\hat{t}$, the payoffs are $B_1(\hat{t})$ and $B_2(\hat{t})$. We will assume that
$$\lim {i \rightarrow+x} L_i(\hat{t})=\lim {i \rightarrow+x} F_i(\hat{t})=\lim _{i \rightarrow+x} B_i(\hat{t}),$$
which will be the case if payoffs are discounted.

## 经济代写|博弈论代考Game theory代写|The War of Attrition

A classic example of a timing game is the war of attrition, first analyzed by Maynard Smith (1974). ${ }^7$
Stationary War of Attrition
In the discrete-time version of the stationary war of attrition, two animals are fighting for a prize whose current value at any time $t=0,1, \ldots$ is $v>1$; fighting costs 1 unit per period. If one animal stops fighting in period $t$, his opponent wins the prize without incurring a fighting cost that period, and the choice of the second stopping time is irrelevant. If we introduce a per-period discount factor $\delta$, the (symmetric) payoff functions are
$$L(\hat{i})=-\left(1+\delta+\cdots+\delta^{i-1}\right)=-\frac{1}{1}-\delta^i$$
and
$$F(\hat{i})=-\left(1+\delta+\cdots+\delta^{i-1}\right)+\dot{\delta}^i v=L(\hat{t})+\dot{\delta}^i v$$
If both animals stop simultaneously, we specify that neither wins the prize, so that
$$B_1(\hat{i})=B_2(\hat{t})=L(\hat{t})$$
(Exercise 4.1 asks you to check that other specifications with $B(\hat{t})<F(\hat{t})$ yield similar conclusions when the time periods are sufficiently short.) Figure 4.3 depicts $L(\cdot)$ and $F(\cdot)$ for the continuous-time version of this game.

# 博弈论代写

## 经济代写|博弈论代考Game theory代写|Definition of Simple Timing Games

$$A_i(t)={\text { stop, don’t stop }}$$

$$\hat{i}=\min \left{t \mid a_i^t=\text { stop for at least one } i\right}$$

$$\lim {i \rightarrow+x} L_i(\hat{t})=\lim {i \rightarrow+x} F_i(\hat{t})=\lim _{i \rightarrow+x} B_i(\hat{t}),$$

## 经济代写|博弈论代考Game theory代写|The War of Attrition

$$L(\hat{i})=-\left(1+\delta+\cdots+\delta^{i-1}\right)=-\frac{1}{1}-\delta^i$$

$$F(\hat{i})=-\left(1+\delta+\cdots+\delta^{i-1}\right)+\dot{\delta}^i v=L(\hat{t})+\dot{\delta}^i v$$

$$B_1(\hat{i})=B_2(\hat{t})=L(\hat{t})$$
(练习4.1要求您在时间周期足够短的情况下，通过$B(\hat{t})<F(\hat{t})$检查其他规范是否产生类似的结论。)图4.3描述了这个游戏的连续时间版本的$L(\cdot)$和$F(\cdot)$。

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## MATLAB代写

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