A strategy is dominated when another strategy yields at least as good an outcome against every possible choice by the opponent, and strictly better against at least one. A rational player should never use a dominated strategy, so such strategies can be deleted from the analysis, sometimes iteratively to shrink the game.
In a strictly competitive (zero-sum) two-person game, each player can secure a guaranteed payoff by choosing the strategy that maximizes their minimum outcome. When the maximin of one player equals the minimax of the other, this common value is the saddle point and defines the game's rational solution; where no pure saddle point exists, mixed (randomized) strategies restore one.
A player can choose not a single action but a probability distribution over actions, then let a random device pick the actual move. This protects against an opponent who could exploit any predictable pattern, and in strictly competitive games guarantees an expected value no opponent can push below the minimax level.
When players' interests are not strictly opposed, no single concept of rational solution suffices: individually rational choices can produce outcomes that both players would reject in favor of another. The theory offers competing solution notions rather than one answer, because cooperation, communication, and enforceable agreements each change what counts as rational.
If a person's preferences among risky prospects satisfy axioms such as transitivity and continuity, those preferences can be represented by a numerical utility function that is unique up to a positive linear transformation. Rational choice under uncertainty then means maximizing expected utility, not expected monetary value.
Two prisoners each do better individually by confessing regardless of the other's choice, so both confess, yet both would have been better off staying silent. The example shows that the pursuit of individually dominant strategies can lead everyone to a jointly inferior outcome, with no escape under the game's stated rules.