Simultaneous Move Game

EC487_Advanced_Microeconomics

Definition A finite Simultaneous Move Game is a tuple $(N,A,u)$ consisting of

• a finite number of players $j\in \{1,\dots ,N\}$.
• for each $j\in \{1,\dots ,N\}$ a finite non-empty action set $A^j$, where $A={\times }_{j=1}^N{A}^j$.

Here, $A$ is the Cartesian product of all players’ action sets, representing the set of all possible action profiles. $\times$ denotes the Cartesian product operation.

• for each $j\in \{1,\dots ,N\}$ preferences represented by a utility function $u^j:A\to \mathbb{R};u=(u^j{)}_{j=1}^N$.

Here, $u^j$ assigns a real-valued utility to each action profile in $A$, representing player $j$ ‘s preferences over the outcomes of the game. While $u$ means the collection of all players’ utility functions.

Players by default move simultaneously. (See Simultaneous Move Game), which means you do not observe others’ choices, you predict.

$\begin{array}{rl}\sum _{j\in S}{u}_j& \ge v(S)\\ \sum _{j\in T}{u}_j& \ge v(T)\\ \sum _{j\in N\setminus (S\cup T)}{u}_j& \ge v(N\setminus (S\cup T))\stackrel{\text{零和}}{=}v(N)-v(S\cup T)\end{array}$