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# 博弈论代写代考| Mixed-Strategy Constant-Sum Games 数学代写

## 博弈论代考

In this section, we will begin our study of two-person constant-sum games that are not strictly
determined. However, we begin by solving just one more strictly determined game: it is the Antares
vs. Bellatrix game that was the infinite game in Example $6.5$ of Section $6.1$. It will turn out that,
despite being strictly determined and despite being infinite, the ideas behind its solution will be
useful for solving $2 \times 2$ games that are not strictly determined. Once we understand how to solve
$2 \times 2$ games, we will build up to larger games by various techniques until we are finally able to
solve any two-person constant-sum game of any size using linear programming.
6.3.1 Antares vs. Bellatrix
Recall the folloping example from Section 6.1: Antares Industries and Bellatrix, Inc., are the only
two companies that sell a certain computer chip, the Pentangle 5.1. There is an estimated market
for 400,000 of these chips in the year 2021 . Each company must set its prices. Production costs
dictate that the price must be at least $\$ 80$per chip, while industry regulations dictate that the set by Bellatrix. Based on reliable market studies of which both companies are aware, it has been determined that the percentage of the total market which goes to Antares depends on$x$and$y$and is reasonably approximated by the function given by$E(x, y)=330-3 x-2.7 y+0.03 x y$. nd$y$that each company should choose, if their goal is to obtain the re? of this game, we observed that this can be regarded as a two-person rategies for each company are the prices they set ($x$and$y$). The values in the real interval [80, 120], so each player has an infinite number are chosen, then Antares receives a payoff of$E(x, y)$while Bellatrix$E(x, y)$. These numbers are percentages of the total market, so they sum vected. company chooses a price$x$); the column player (Bellatrix) choose a wooses a price$y$); and entering$E(x, y)$at the point in the matrix where$y$meet (We encourage the reader to review his or her solutions to tion$6.1$, which were finite versions of this game.) Then, we search for (see Exercises 13 and 14 in Section$6.2$). What are the prices$x$and$y$that each company should choose, if their goal is to obtain the largest-possible market share? In our earlier discussion of this game, we observed that this can be regarded as a two-person constant-sum game. The strategies for each company are the prices they set$(x$and$y)$. The values of these variables must be in the real interval$[80,120]$, so each player has an infinite number of strategies. Once$x$and$y$are chosen, then Antares receives a payoff of$E(x, y)$while Bellatrix receives a payoff of$100-E(x, y)$If the game were finite, we would construct the payoff matrix by letting the row player (Antares) the constant 100 , as expected. choose a row (that is, the company chooses a price$x$); the column player (Bellatrix) choose a column (that is, Bellatrix chooses a price$y$); and entering$E(x, y)$at the point in the matrix where the row$x$and the column$y$meet (We encourage the reader to review his or her solutions to Exercises 6,7 , and 8 in Section$6.1$, which were finite versions of this game.) Then, we search for a saddle point in this matrix (see Exercises 13 and 14 in Section 6.2). In our case, since we have infinitely many choices for$x$and$y$, it does not make sense to construct a matrix, but it should be clear that the function$E(x, y)$serves the exact same purpose as the payoff matrix – it gives, for each choice$(x, y)$, the payoff$E(x, y)$to the row player Antares. The attentive reader will recognize that the graph of$E(x, y)$is a surface. So for such infinite, continuous games, this surface replaces the payoff matrix. Of course, we are only concerned with that part of the surface which is defined for$x \in[80,120]$and similarly for$y$. Now, the argument that a saddle point in the payoff matrix is the same thing as a Nash equilib- rium, which we gave in Proposition$6.11$of Section$6.2$, is actually still valid when applied to this surface. A choice of$(x, y)$for which$E(x, y)$is simultaneously a minimum in its “row” (being in a row means holding$y$fixed) and a maximum in its “column” (which means holding$x$fixed) will represent an outcome of the game where neither player stands to gain by a unilateral change in$E(x, y)$in the relevant ranges for$x$and$y$.$E(x, y)$in the relevant ranges for$x$and$y$. Finally, the reader should recall Chapter 1 , where we studied surfaces of the form$F(x, y)=A-B x-C y+D x y$. We saw that any function of this form was called a saddle surface precisely because it could always be converted to standard form$F(x, y)=D(x-p)(y-q)+v$, which is a surface with a saddle point at the location$(x, y)=(p, q)$and where$F(p, q)=v$. Thus, to solve the Antares vs. Bellatrix infinite game, we must put$E(x, y)$into standard form. By the techniques we discussed in Chapter 1 , the standard form of$E(x, y)$is$E(x, y)=.03(x-90)(y-100)+60 .$we discussed in Chapter 1 , the standard form of$E(x, y)$is $$E(x, y)=03(x-00)(y-100)+60$$ Therefore, there is a saddle point at$(x, y)=(90,100)$and$E(00,100)=60 .$Notice that the Therefore, there is a saddle point at$(x, y)=(90,100)$and$E(90,100)=60$. Notice that the saddle point is located in the required range of prices$80 \leq x, y \leq 120$. Therefore, the Antares saddle point is located in the required range of prices$80 \leq x, y \leq 120 .$Therefore, the Antares vs. Bellatrix game is strictly determined (even though it is infinite). Antares should set its price at$\$90 ;$ Bellatrix should set its price at $\$ 100$. If they both follow this recommendation, the value of the game is$v=60$. Thus, Antares will have$60 \%$of the market, and Bellatrix will have$40 \%$. Since this is a Nash equilibrium, if either player were to unilaterally deviate from this strategy, we might expect the payoff in market shares to the other player to possibly improve. But actually, in this game, that does not happen. For example, if Antares sets its price at the optimum choice of$\$90$ per chip, then by evaluating (6.2), it is clear that $E(90, y)=60$ for all $y$. So in this case,
if Bellatrix changes its price unilaterally, it won’t affect the payoff to Antares at all. Similarly, if
Bellatrix sets its price at the optimum of $\$ 100$per chip, a unilateral change in price by Antares will not affect its payoff since$E(x, 100)=60$for any$x$. (Not all saddle points behave this way, but if the game is one where the payoffs are given by a saddle surface function of crossed type, We remark in passing that we can also find the revenue for each company Assuming a total market of 400,000 chips, the total revenue for Antares is$\$400,000\left(\frac{E}{\operatorname{tcog}}\right) x=\$ 400,000(0.6)(90)=\$21,600,000 ;$ and the revenue for Bellatrix is $\$ 400,000\left(\frac{100-E}{100}\right) y=\$400,000(0.4)(100)=$
$\$ 16,000,000$. Remember, however, that the goal of each company is to maximize market shares, not revenue, since we observed previously a game with revenues as payoffs would be variable-sum. 在本节中，我们将开始研究不严格的两人常数和博弈 决定。然而，我们首先解决一个更严格的游戏：它是 Antares vs. Bellatrix 游戏，即$6.1$的示例$6.5$中的无限游戏。事实证明， 尽管被严格确定并且尽管是无限的，但其解决方案背后的想法将是 对于解决没有严格确定的$2 \times 2$游戏很有用。一旦我们了解如何解决$2 \times 2$游戏，我们将通过各种技术构建更大的游戏，直到我们最终能够 使用线性规划解决任何规模的任何两人常数和游戏。 6.3.1 Antares 与 Bellatrix 回想第 6.1 节中的以下示例： Antares Industries 和 Bellatrix, Inc. 是唯一的 两家销售某种计算机芯片 Pentangle 5.1 的公司。估计有市场 在 2021 年购买 400,000 个这样的芯片。每个公司都必须设定价格。生产成本 规定价格必须至少为每芯片$\$80$，而行业法规规定

$E(x, y)=330-3 x-2.7 y+0.03 x y$。
nd $y$ 每个公司应该选择，如果他们的目标是获得

$E(x, y)$。这些数字是整个市场的百分比，所以它们总和

$y$ meet（我们鼓励读者回顾他或她的解决方案
$6.1$，这是这个游戏的有限版本。）然后，我们搜索