utility function

Definition

Let $O$ be a set of outcomes and let $\succsim$ be a complete, reflexive, and transitive preference relation over $O$. A function $u : O \to \mathbb{R}$ is called a utility function representing $\succsim$ if for all $x, y \in O$, $$x \succsim y \iff u(x) \geq u(y)$$

Definition (lotteries)

Let $\succsim_i$ be a preference relation for player $i$ over the set of lotteries $\mathcal{L}$. A utility function $u_i$ representing the preferences of player $i$ is a real-valued function defined over $\mathcal{L}$ satisfying $$u_i(L_1) \geq u_i(L_2) \iff L_1 \succsim_i L_2 \quad \forall L_1, L_2 \in \mathcal{L}$$

Notes

Utility function $u$ is a function associating each outcome $x$ with a real number $u(x)$ in a way such that the more an outcome is preferred, the larger the real number associated with it.

If the set of outcomes is finite, any complete, reflexive, and transivite preference relation can be easily represented by a utility function.

See also

• Linear utility function

References

1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, p. 11.
2. https://math.ucr.edu/home/baez/games/games_6.html