Games

Why

We want to discuss interactive decision making.

Discussion

We are interested in talking about situations in which there are several agents. Each agent makes decisions that affect outcomes for itself and all other agents.

Example: rock paper scissors

Consider the game “rock-paper-scissors” in which there are two agents, or players, $A$ and $B$. Each player may choose one of the three actions Rock, Paper, Scissors. To play, each player simultaneously selects an action, and these are compared. Here, both agents have the same actions available to them.

Definition

There is a finite set of agents (or players, decision makers, controllers). Let $\mathcal{I} $ be a finite set with $\num{\mathcal{I} } = n$, the players.

In rock-paper-scissors, for example, $\mathcal{I} = \set{A, B}$. There, each player could pick one of the three actions. Define $\mathcal{A} _A = \mathcal{A} _B = \{$Rock, Paper, Scissors$\}$. We call $\mathcal{A} _A$ the actions of $A$ and $\mathcal{A} _B$ the actions of $B$.

We have a set of outcomes $\mathcal{O} = \{$A Wins, B Wins, Tie$\}$. Let $f: \mathcal{A} _A \times C_B \to \mathcal{O}$ defined by $f($Rock, Scissors$) = $A Wins

The set of players. Let $S$ be a finite set, the set of states. For $i = 1, \dots , n$, let $\set{A^p_s}_{s \in S}$ be a family of sets, the action sets by state. Define $\mathcal{A} ^i = \cup_s A^p_s$ the set of actions for player $i = 1, \dots , n$.

Let $f: S \times \prod_{i} \mathcal{A} ^i \to S$, the game dynamics or transition function.