## 经济代写|博弈论代写Game Theory代考|The Cournot Duopoly of Quantity Competition

In this section, we discuss an economic model of competition between two competing firms who choose the quantity of their product, where the price decreases with the total quantity on the market. It is also called a Cournot duopoly named after Augustin Cournot (1838) who proposed this model, nowadays described as a game with an equilibrium. In this game, the strategy sets of the players are infinite. Finite versions of the game are dominance solvable, and in principle also the infinite version, as we describe at the end of this section.

The players I and II are two firms who choose a nonnegative quantity of producing some good up to some upper bound $M$, say, so $S_1=S_2=[0, M]$. Let $x$ and $y$ be the strategies chosen by player I and II, respectively. For simplicity there are no costs of production, and the total quantity $x+y$ is sold at a price $12-(x+y)$ per unit, which is also the firm’s profit per unit, so the payoffs $a(x, y)$ and $b(x, y)$ to players $\mathrm{I}$ and $\mathrm{II}$ are given by
payoff to I : $\quad a(x, y)=x \cdot(12-y-x)$,
payoff to II : $\quad b(x, y)=y \cdot(12-x-y)$.
$\Rightarrow$ Exercise $3.3$ describes an extension that incorporates production costs.
The game (3.6) is clearly symmetric in $x$ and $y$. Figure $3.8$ shows a finite version of this game where the players’ strategies are restricted to the four quantities 0 ,3,4 , or 6 . The payoffs are determined by (3.6). For example, for $x=3$ and $y=4$ player I gets payoff 15 and player II gets 20 . Best-response payoffs, as determined by this payoff table, are marked by boxes. The game has the unique equilibrium $(4,4)$ where each player gets payoff 16.

Figure $3.9$ shows that the game can be solved by iterated elimination of dominated strategies. In the $4 \times 4$ game, strategy 0 is dominated by 3 or 4 , and after eliminating it directly for both players one obtains the $3 \times 3$ game shown on the left in Figure 3.9. In that game, strategy 6 is dominated by 4 , and after eliminating this strategy for both players the middle $2 \times 2$ game is reached. This game has the structure of a Prisoner’s Dilemma game because 4 dominates 3 (so that 3 will be eliminated), but the final strategy pair of undominated strategies $(4,4)$ gives both players a lower payoff than the strategy pair $(3,3)$. Here the Prisoner’s Dilemma arises in an economic context: The two firms could cooperate by equally splitting the optimal “monopoly” quantity 6, but in response the other player would “defect” and choose 4 rather than 3 . If both players do this, the total quantity 8 reduces the price to 4 and both players have an overall lower payoff.

## 经济代写|博弈论代写Game Theory代考|Games without a Pure-Strategy Equilibrium

Not every game has an equilibrium in pure strategies. Figure $3.12$ shows two well-known examples. In Matching Pennies, the two players reveal a penny which can show Heads $(H)$ or Tails $(T)$. If the pennies match, then player I wins the other player’s penny, otherwise player II. No strategy pair can be stable because the losing player would always deviate.

Rock-Paper-Scissors is a $3 \times 3$ game where both players choose simultaneously one of their three strategies Rock $(R)$, Paper $(P)$, or Scissors $(S)$. Rock loses to Paper, Paper loses to Scissors, and Scissors lose to Rock, and it is a draw otherwise. No two strategies are best responses to each other. Like Matching Pennies, this is a zero-sum game because the payoffs in any cell of the table sum to zero. Unlike Matching Pennies, Rock-Paper-Scissors is symmetric. Hence, when both players play the same strategy (the cells on the diagonal), they get the same payoff, which is zero because the game is zero-sum.

The game-theoretic recommendation is to play randomly in games like Matching Pennies or Rock-Paper-Scissors that have no equilibrium, according to certain probabilities that depend on the payoffs. As we will see in Chapter 6, any finite game has an equilibrium when players are allowed to use randomized strategies.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|The Cournot Duopoly of Quantity Competition

⇒锻炼3.3描述了包含生产成本的扩展。

## 经济代写|博弈论代写Game Theory代考|Games without a Pure-Strategy Equilibrium

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