a) Consider first a two-plaer game with strategy sets $x_1=x_2=[-1,+1]$ (the real interval) and linear utility functions
$$
u_1\left(x_1, x_2\right)=a x_1+b x_2, u_2\left(x_1, x_2\right)=c x_1+d x_2
$$
where a, b, c, d are four fixed real numbers. Clearly each player has a dominating strategy (unique if a - resp $d$ is nonzero).
We are interested in configurations where a prisoner's dilemma arises, namely, the dominating strategy equilibrium is Pareto dominated. Prove that this is the case iff
$$
a \cdot c<0, \quad b \cdot d<0, \text { and } 1<\frac{b \cdot c}{a \cdot d}
$$
b) Now we have a n-person game with $X_i=[-1,+1]$, all $i=1, \ldots, n$, and Player i's utility function
$$
u_i(x)=\sum_{j=1}^n a_i^j x_j
$$
We assume $a^i \neq 0, a l 1 i=1, \ldots, n$. Denote by $x^*$ the unique dominating strategy equilibrium. Prove the equivalence of the three following statements:
i) From the prisonner's dilemma effect, $x^*$ is a Pareto-dominated outcome.
ii) There is an outcome $y \in X_N$ such that
$$
\left\{a_i^i y_i>0 \text { and } u_i(y)<0\right\} \text { all } i=1, \ldots, n
$$
This says that each player would be better off by using $y_i$ alone (i.e., while others stay put at strategy zero), but all will be worse off when they all use $y_i$.
iii) For a $11 \mathrm{p}$, q $\varepsilon \mathrm{R}_{+}^{\mathrm{n}^{ }}$(vectors with nonnegative components); the system
$$
\begin{aligned}
&\left\{\sum_{i=1}^n p_i a_i^j=q_j a_j^j\right\} \quad a l 1 ~ j=1, \ldots, n \\
& \text { implies } p=q=0 .
\end{aligned}
$$
As an example, consider the game "provision of a public good":
$$
u_i=-\alpha_i x_i+\lambda\left\{\sum_{j=1}^n x_j\right\} \quad \alpha_i>0
$$
Interpret $x_i$ as the (positive or negative) effort spent by $i$ for the common welfare, and $-\alpha_i$ as his marginal disutility for the effort. For which values of $\lambda$ do we have a prisonner's dilemma effect?