Normal Form Game
A ‘‘Normal Form Game’’ (also ‘‘Matrix Form Game’’ or ‘‘Strategic Game’’) if a type of games from the Game Theory
- main idea: the players move simultaneously
- compare to Extensive Form Games where players move sequentially
In these games:
- There is a finite number $n$ of rational players: $N = { 1, 2, …, n }$
- Each player $i$ has a finite set of actions $A_i$
- Also each player $i$ has a set of possible consequences $C$, in this case it’s $C \equiv \mathbb{R}$
- The chosen alternatives form an ‘‘action profile’’ (or ‘‘strategy profile’’) $A: A_1 \times … \times A_n$
- There’s a consequence function $g_i: A \mapsto C$ that associates each alternative $a \in A_i$ with some consequence
- The preferences of agents are modeled with an utility function $u_i: A \mapsto \mathbb{R}$
’'’def:’’’ So a ‘‘normal form game’’ (or a ‘‘strategic game’’) is
- a tuple $\langle N, A, u \rangle$ where
- $N = { 1, 2, …, n }$ - set of all players
- $A = {A_1, …, A_n}$ - set of each players’ actions
- $u = {u_1, …, u_n}$ - set of utility functions that expresses preferences
Utility Function (of payoff function) $u_i$
- $u_i: A_i \mapsto \mathbb{R}$
- each agent has a preference relation $S_i$
- agents are rational: therefore $\forall a,b \in A_i: u_i(a) \geqslant u_i(b) \Rightarrow a \ S_i \ b$
- in other words, if alternative $a$ gives a better payoff than $b$, $i$ will always prefer $a$ over $b$
Representation
These games are typically represented with a pay-off matrix
- columns/rows represent actions that players can take
- each cell shows the outcome of game: it lists utilities that all players will receive
For example,
- consider this 2-player game, with two players $p_1$ and $p_2$
- each has two strategies: $A_1 = {a, b}, A_2 = {x, y}$
- the payoffs are shown in the cells of the matrix
| $p_2 \to$
$p_1 \downarrow$ | $x$ | $y$ | $a$ | $[u_1(a,x), u_2(a,x)]$ | $[u_1(a,y), u_2(a,y)]$ || $b$ | $[u_1(b,x), u_2(b,x)]$ | $[u_1(a,y), u_2(a,y)]$ | Another example - 2 players, 3 strategies
| $p_2 \to$
$p_1 \downarrow$ | $L$ | $C$ | $R$ | $T$ | (1,0) | (1,3) | (3,0) || $M$ | (0,2) | (0,1) | (3,0) || $B$ | (0,2) | (2,4) | (5,3) | - This game can be solved by Iterative Removal
- At the end the profile $(C, B)$ will be chosen
Types
There are several types of normal form games: