# 经济代写|博弈论代写Game Theory代考|ECON3503

## 经济代写|博弈论代写Game Theory代考|Investing and Betting

Assume that an investor (or bettor or gambler or simply player) is considering a financial engagement in a certain venture. Then the obvious – albeit rather vague – big question for the investor is:

• What decision should best be taken?
More specifically, the investor wants to decide whether an engagement is worthwhile at all and, if so, how much of the available capital should be invested how. Obviously, the answer depends on additional information: What is the likelihood of a success? What gain can be expected? What is the risk of a loss? etc.

The investor is thus about to participate as a player in a 2-person game with an opponent whose stratcgics and objective are not always clear or known in advance. Relevant information is not completely (or not reliably) available to the investor so that the decision must be made under uncertainties. Typical examples are gambling and betting where the success of the engagement depends on events that may or may not occur and hence on “fortune” or “chance”. But also investments in the stock market fall into this category when it is not clear in advance whether the value of a parlicular investment will rise or fall.

We are not able to answer the big question above completely but will discuss various aspects of it. Before going into further details, let us illustrate the difficulties of the subject with a classical – and seemingly paradoxical – gambling situation.

The St. Petersburg paradox. Imagine yourself as a potential player in the following game of chance.

## 经济代写|博弈论代写Game Theory代考|Proportional investing

Our general model consists of a potential investor with an initial portfolio $B$ of $b>0$ euros (or dollars or…) and an investment opportunity $A$. If things go well, an investment of size $x$ would bring a return $r x>x$. If things do not go well, the investment will return nothing.
In the analysis, we will denote the net return rate by
$$\rho=r-1 .$$
The investor is to decide what portion of $B$ should be invested. The investor believes:
(PI) Things go well with probability $p>0$ and do not go well with probability $q=1-p$.

Under the assumption (PI), the investor’s expected portfolio value after the investment $x$ is
$$B(x)=[(b-x)+r x] p+(b-x) q=[b+\rho x] p+(b-x) q$$
since an amount of size $b-x$ is not invested and therefore not at risk. The derivative is
$$B^{\prime}(x)=\rho p-q$$
So $B(x)$ is strictly increasing if $\rho>q / p$ and non-increasing otherwise. Hence, if the investor’s decision is motivated by the maximization of the expected portfolio value $B(x)$, the naive investment rule applies.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Investing and Betting

• 最好做出什么决定？
更具体地说，投资者想要决定一项参与是否值得，如果值得，应该以何种方式投资多少可用资本。显然，答案取决于其他信息：成功的可能性有多大？可以期待什么收获？损失的风险是什么？等等

## 经济代写|博弈论代写Game Theory代考|Proportional investing

$$\rho=r-1 .$$

(PI) 事情进展顺利的概率 $p>0$ 并且不太可能顺利 $q=1-p$.

$$B(x)=[(b-x)+r x] p+(b-x) q=[b+\rho x] p+(b-x) q$$

$$B^{\prime}(x)=\rho p-q$$

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