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# 经济代写|博弈论代考Game theory代写|ECON40010 Pure strategies, mixed strategies and behavioral strategies

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## 经济代写|博弈论代考Game theory代写|Pure strategies, mixed strategies and behavioral strategies

A pure strategy of player $i$ in $G$ is a map $s_i$ that assigns to each $J \in \mathscr{I}i$ an action $a \in A(J)$. We can think of $s_i$ as an element of $A_i:=\prod{J \in \mathscr{I}_j} A(J)$. A mixed strategy $\sigma_i$ of a player in an extensive form game is a distribution over pure strategies. Given a $J \in \mathscr{I}_i$, we denote by $\sigma_i(\cdot \mid J)$ the distribution over $A(J)$ given by $\sigma_i$, conditioned on the experience $X(J)$. That is, $\sigma_i(a \mid J)$ is the probability, if we choose $s_i$ according to $\sigma_i$, that $s_i(J)=a$ conditioned on $s_i\left(J^1\right)=b^1, \ldots, s_i\left(J^k\right)=b^k$, where $\left(\left(J^1, b^1\right),\left(J^2, b^2\right), \ldots,\left(J^k, b^k\right)\right)$ is the unique experience that terminates in $J$. Of course, it could be that this conditional probability is not well defined, in the case that $s_i\left(J^1\right)=b^1, \ldots, s_i\left(J^k\right)=b^k$ occurs with zero probability.

Recall that $A_i=\prod_{J \in \mathscr{I}i} A(J)$ is the product of all action sets available to player $i$. A behavioral strategy $\sigma_i$ of player $i$ is a product distribution on $A_i$ : $$\sigma_i=\prod{J \in \mathscr{I}i} \sigma_i(\cdot \mid J)$$ where $\sigma_i(\cdot \mid J)$ is a distribution on $A(J)$. Note that $\sigma_c$, the chance player’s distribution, is a behavioral strategy. Note also that each element of $\prod{J \in \mathscr{I}_i} A(J)$ can be identified with a function that assigns to each element $J \in \mathscr{I}_i$ an element of $A(J)$. Therefore, by our definition of behavioral strategies, every behavioral strategy is a mixed strategy.

Given a strategy profile $\sigma$ of either pure, mixed or behavioral strategies (or even a mixture of these), we can define a distribution over the terminal histories $Z$ by choosing a random pure strategy for each player (including the chance player), and following the game path to its terminal history $z$. A player’s utility for $\sigma$ is $u_i(\sigma)=\mathbb{E}\left[u_i(z)\right]$, her expected utility at this randomly picked terminal history.

## 经济代写|博弈论代考Game theory代写|Belief systems and assessments

Let $G=\left(N, A, H, \mathscr{I}, P, \sigma_c,\left{u_i\right}\right)$ be a finite extensive form game. A belief system ${\mu(\cdot \mid J)}_{J \in \mathscr{I}}$ is a collection of probability measures, with $\mu(\cdot \mid J)$ a probability measure over $J$.

Fix a mixed strategy profile $\sigma$, and for a history $h=\left(a^1, \ldots, a^k\right)$ denote, as above, by $\mathbb{P}\sigma[h]$ the probability of the event that the path of play includes $h$. We likewise denote by $\mathbb{P}\sigma[J]$ the probability that an information set $J$ is visited. We say that $\mu(\cdot \mid J)$ is derived from $\sigma$ if for any history $h \in J$ such that $\mathbb{P}\sigma[J]>0$ it holds that $\mu(h \mid J)=\mathbb{P}\sigma[h \mid J]$. For $J$ with $\mathbb{P}\sigma[J]=0$, $\mu(\cdot \mid J)$ can take any value. An assessment is a pair $(\sigma, \mu)$ such that ${ }^{10} \mu$ is derived from $\sigma$. Recall that we say that $\sigma$ is completely mixed if for all $i, J \in \mathscr{I}_i$ and $a \in A(J)$ it holds that $\sigma_i(a \mid J)>0$. That is, in every information set every allowed action has positive probability. In this case $\mathbb{P}\sigma[J]>0$ for every information set $J$, and so there is only one belief system $\mu$ that is derived from $\sigma$. Hence for completely mixed $\sigma$ there is only one assessment $(\sigma, \mu)$.

# 博弈论代写

## 经济代写|博弈论代考Game theory代写|Belief systems and assessments

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## MATLAB代写

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