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• Theoremvon Neumann's minimax theorem

  Let $A$ be a real $m \times n$ matrix. For $k \ge 1$ let $\Delta_k = \set{p \in \R^k : p_i \ge 0 \text{ for all } i, \ p_1 + \cdots + p_k = 1}$ be the standard simplex of probability vectors. Then $$\max_{x \in \Delta_m} \min_{y \in \Delta_n} x^T A y = \min_{y \in \Delta_n} \max_{x \in \Delta_m} x^T A y.$$

• TheoremWell-ordering theorem

  Assume the axiom of choice. Then each set $A$ has a well-order.

• ConceptDifferential game

  A differential game is a game in which several players steer a common state through a control system. The state obeys a differential equation driven by all their controls. Each player has a cost or payoff that depends on the resulting trajectory, and the players have different goals. With one player it is an optimal control problem. The two-player zero-sum case leads to the Hamilton–Jacobi–Isaacs equation, and pursuit problems are the classical examples.

• ProblemHomicidal chauffeur problem

  Two players move in the plane. The pursuer $P$ is a car: it moves with constant speed $w_1 > 0$ and has minimum turning radius $R > 0$. Its position $(x_1, y_1)$ and heading $\theta$ obey $$x_1' = w_1 \sin \theta, \qquad y_1' = w_1 \cos \theta, \qquad \theta' = \frac{w_1}{R} \phi, \qquad \phi(t) \in [-1, 1].$$ The evader $E$ moves with speed at most $w_2$, where $0 < w_2 < w_1$, in any direction: its position $(x_2, y_2)$ obeys $x_2' = w_2 \sin \psi$, $y_2' = w_2 \cos \psi$ with $\psi(t)$ free. Let $\ell > 0$ be the capture radius; capture happens at the first time the distance between $P$ and $E$ is at most $\ell$. The problem is a differential game between $P$ and $E$. It asks for which parameters $w_1, w_2, R, \ell$ and initial positions the pursuer can force capture in finite time against every play of the evader. It also asks for the optimal strategies when the pursuer minimises and the evader maximises the time to capture.

• TheoremEnvelope theorem

  Let $X$ be a set, let $T \subseteq \R^k$ be open, and let $f \colon X \times T \to \R$ be a function. Let $V(t) = \sup_{x \in X} f(x, t)$ be the value function, and assume that $V$ is finite on $T$. Let $t_0 \in T$ and let $x_0 \in X$ be a maximizer at $t_0$, that is, $f(x_0, t_0) = V(t_0)$. If $V$ and $t \mapsto f(x_0, t)$ are both differentiable at $t_0$, then $$\nabla V(t_0) = \nabla_t f(x_0, t_0).$$

• ProblemMerton's portfolio problem

  Consider a market with a riskless asset with interest rate $r$ and a stock. The stock price follows the geometric Brownian motion $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$, where $\mu > r$, $\sigma > 0$ and $W$ is a Wiener process. An investor with initial wealth $x > 0$ chooses, at each time $t$, the fraction $\pi_t$ of wealth held in the stock and a consumption rate $c_t \ge 0$. Both are adapted to the information generated by $W$. The wealth then evolves by $$dX_t = \left( r X_t + \pi_t (\mu - r) X_t - c_t \right) dt + \pi_t \sigma X_t \, dW_t, \qquad X_0 = x.$$ Let $\rho > 0$ be a discount rate and let $U(c) = \dfrac{c^{1 - \gamma}}{1 - \gamma}$ with $\gamma > 0$ and $\gamma \ne 1$. The problem is to find the strategies $(\pi, c)$ with $X_t \ge 0$ for all $t$ that maximise the expected discounted utility of consumption $$E \int_0^\infty e^{-\rho t} U(c_t) \, dt.$$