Rational Agent¶

Definition¶

A decision problem consists of a set of actions $A$ , a set of outcomes $X$ , a description of how each action leads to an outcome, and the decision maker's preference relation $\succsim$ over $X$ , where $x\succsim y$ means " $x$ is at least as good as $y$ ". The preference relation is assumed to be

complete: for all $x,y\in X$ $X$ , $x\succsim y$ or $y\succsim x$ (or both);
transitive: $x\succsim y$ and $y\succsim z$ imply $x\succsim z$ $z$ .

A payoff function $v\colon X\to\mathbb{R}$ represents $\succsim$ if $v(x)\ge v(y)$ if and only if $x\succsim y$ $y$ . Writing $v(a)$ for the payoff of the outcome that action $a$ produces, a rational agent (rational player) knows the actions, the outcomes, how actions lead to outcomes and his or her own preferences, and chooses an action $a^*\in A$ with

v(a^*)\ge v(a)\quad\text{for every } a\in A.

When actions lead to outcomes only with some probability, rationality means maximizing expected payoff.

Interpretation¶

Rationality is a modelling assumption about consistency, not about selfishness or money. The payoff may include altruism, fairness or effort costs; the assumption is only that choices maximize whatever the payoff measures. It turns the choice into an optimization model, which is what makes behaviour predictable.

Graph of the payoff v(a) = 12a − a² for study hours a from 0 to 10, rising to its maximum v(6) = 36 at a* = 6, with the whole curve on or below the level 36.

The figure draws the study-hours example. The rational choice is an action whose payoff is at least that of every other action: the whole graph of $v(a)=12a-a^2$ on $[0,10]$ lies on or below the line $v=36$ $36$ , which it touches only at $a^*=6$ $6$ . Replacing $v$ by $\mathrm{e}^{v(a)}$ would change the heights but not the location of the maximum, so the choice is the same.

Completeness rules out "cannot compare"; transitivity rules out preference cycles, for which no best element exists.

In a game, the outcome also depends on the other players, so a rational player maximizes given a belief about their choices. Rationality alone does not say which belief is reasonable; that needs further assumptions such as common knowledge of rationality or the correct beliefs built into Nash equilibrium.

Key Results¶

On a finite set $X$ , a complete and transitive preference relation can always be represented by a payoff function.
If $v$ represents $\succsim$ and $f$ is strictly increasing, then $f\circ v$ represents the same preferences and gives the same choice under certainty. Under uncertainty only positive affine transformations $\alpha v+\beta$ $β$ , $\alpha>0$ $0$ , are guaranteed to preserve choices (see expected payoff maximization).
With $A$ finite and nonempty an optimal action always exists. With $A$ infinite it may not: on $A=[0,1)$ with $v(a)=a$ the supremum $1$ is not attained.
In a game, a rational player's choice given a belief about the others is a best response to that belief.

Examples¶

Continuous action. A student chooses study hours $a\in[0,10]$ with payoff $v(a)=12a-a^2$ (benefit minus effort cost). Since $v'(a)=12-2a$ is zero at $a=6$ and $v''=-2<0$ $0$ , the rational choice is $a^*=6$ with $v(6)=36$ $36$ . The payoff $\mathrm{e}^{v(a)}$ represents the same preferences and gives the same $a^*$ .

Cycle. Suppose $x\succ y$ $y$ , $y\succ z$ and $z\succ x$ (strict preferences). Whichever of the three outcomes is chosen, another is strictly preferred, so no choice is optimal. Transitivity excludes this.

Firm. A firm that believes its rival will produce $20$ units chooses its own output to maximize profit given that belief; in a Cournot duopoly this choice is its best response.

Prerequisites¶

Sources¶

Game Theory · Week 1

Aliases: rational player, rational decision maker, rationality, rational actor Keywords: decision problem, preference relation, complete and transitive preferences, payoff function v(a), homo economicus Symbols: alpha, beta, element of, real numbers

Last updated: 27 October 2026 11:52