6 ideas
In any finite two-person zero-sum game, if players may randomize over their pure strategies, the maximum payoff one player can guarantee equals the minimum loss the other can hold them to. This common number is the value of the game. Each side can secure it no matter what the opponent does, so the game has a determinate solution even though neither player can see the other's move.
A mixed strategy chooses among actions by a deliberate probability distribution. Its purpose is not variety for its own sake. When an opponent could exploit any predictable pattern, randomizing makes your choice impossible to anticipate, even for someone who knows exactly which distribution you are using.
In a simplified poker model, the optimal strategy requires sometimes betting high with a weak hand, and doing so at a specific calculated frequency. Bluffing does two jobs: it can win pots outright, and it keeps opponents unsure, so they will call your strong hands. Bluffing therefore follows from the mathematics of hidden information rather than being a psychological quirk.
Suppose a person's preferences over risky lotteries are complete, transitive, continuous and consistent under compounding. Then a numerical utility function exists, and choosing by maximizing its expected value reproduces exactly those preferences. This utility is unique only up to the choice of zero point and unit. Utility thus becomes a measurable quantity, built from choices between gambles rather than from introspection.
Crusoe alone on an island solves an ordinary optimization problem, because every variable affecting his outcome is under his control or set by nature. In a social economy, each person's result depends on variables other people control, and those people are reasoning about him in turn. Economic behavior is therefore a game, and the standard calculus of maximization is the wrong tool for it.
In cooperative games with many players, the solution is a set of possible payoff divisions, not a single outcome. Within the set, no division is dominated by another member of the set, meaning no coalition could effectively enforce something better for itself. Every division outside the set is dominated by some division inside it. Such a set represents a stable social norm, and a single game can support several different ones, each corresponding to a different accepted standard of behavior.