The inverse demand function is continuous over [0,+∞[, strictly decreasing and concave on [0, S]

step 1 Suppose the inverse demand function is given and its cost function is given. Draw the marginal r

The inverse demand function is continuous over [0,+∞[,...

The inverse demand function is continuous over $[0,+\infty[$, strictly decreasing and concave on $[0, S]$, and zero after $S$. The two cost functions $c_i, i=1,2$ are continuous on $[0,+\infty]$ and such that: $$ \begin{aligned} & c_i(0)=0 \\ & c_i\left(x_i\right) \geq 1 \end{aligned} $$ a) Compute the guaranteed utility level to each player. b) Show that the best-reply correspondences $\mathrm{BR}_i$ are nonincreasing in the following sense: $$ \left\{x_j<y_j, x_i \in B R_i\left(x_j\right), y_i \in B R_i\left(y_j\right)\right\} \Rightarrow\left\{y_i \leq x_i\right\} $$ c) Deduce the existence of a Nash equilibrium by picking two single-valued selections $b_1, b_2$ of $B_1, B_2$ respectively and showing that $b_1 \circ b_2$ has a fixed point on $[0, \mathrm{~s}]$.

The inverse demand function is continuous over $[0,+\infty[$, strictly decreasing and concave on $[0, S]$, and zero after $S$. The two cost functions $c_i, i=1,2$ are continuous on $[0,+\infty]$ and such that:
$$
\begin{aligned}
& c_i(0)=0 \\
& c_i\left(x_i\right) \geq 1
\end{aligned}
$$
a) Compute the guaranteed utility level to each player.
b) Show that the best-reply correspondences $\mathrm{BR}_i$ are nonincreasing in the following sense:
$$
\left\{x_j<y_j, x_i \in B R_i\left(x_j\right), y_i \in B R_i\left(y_j\right)\right\} \Rightarrow\left\{y_i \leq x_i\right\}
$$
c) Deduce the existence of a Nash equilibrium by picking two single-valued selections $b_1, b_2$ of $B_1, B_2$ respectively and showing that $b_1 \circ b_2$ has a fixed point on $[0, \mathrm{~s}]$.

Key Concepts

Fixed Point Theorem

The fixed point theorem is a fundamental mathematical tool used to show the existence of equilibria. It states that under certain conditions—a function mapping a compact convex set into itself—there exists at least one point that is mapped to itself. In game theory, this theorem is applied to the composition of best-reply functions, ensuring that a fixed point in the strategy space corresponds to a Nash equilibrium.

Nash Equilibrium

A Nash equilibrium is a set of strategies where no player can unilaterally deviate to improve their payoff. In this context, showing that such an equilibrium exists involves demonstrating that players' best replies converge and that a consistent strategy pair can be found, often leveraging fixed point theorems. It represents a stable state of the game where every player's choice is optimal given the choices of others.

Best-Reply Correspondence

A best-reply correspondence maps each strategy of one player to the set of optimal responses (best replies) of the other player. The property of it being nonincreasing in this context implies that if one player increases their strategy, the opponent's optimal response decreases or does not increase. This monotonicity aids in proving convergence to equilibrium and is key in demonstrating the existence of a Nash equilibrium.

Guaranteed Utility Level

The guaranteed utility level refers to a player's minimum utility that can be assured regardless of the opponent's strategies. It is calculated based on the structure of the payoffs derived from the inverse demand and cost functions. This concept is important in strategic scenarios as it helps players decide on strategies that secure a baseline outcome or safety level in competitive environments.

Inverse Demand Function

An inverse demand function expresses price as a function of quantity. In the context of economics and game theory, its properties—such as continuity, monotonicity (strictly decreasing), and concavity—affect the behavior of market players. Continuity ensures there are no abrupt jumps in price, while the decreasing nature reflects that higher quantities lead to lower market price. Concavity can further imply diminishing marginal returns, crucial for analyzing equilibria in competitive settings.

Cost Functions

Cost functions represent the cost incurred by players when producing or supplying goods. Their properties—continuity and lower bounds—ensure that costs are assigned in a well-behaved manner without sudden jumps, and that there is a minimum cost threshold. These functions are central to calculating the players' utilities and in determining best responses, making them essential in game theoretical analysis.

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