Games.md

title
Games

[!SUMMARY]+

Necessary components for a game:

• Players: who are the decision makers?
  • People
  • Governments
  • Companies
  • Employees, etc
• Actions: what can the players do?
  • Enter a bid in an auction?
  • Decide to end a strike?
  • Decide to sell a stock?
  • How to vote?
• Payoffs: what motivates the players?
  • Profits?
  • Other players?

There are two standard representations of a game:

1. Normal form (aka Matrix form, Strategic form) which lists the payoffs a player gets as a function of their actions
   • As if players moved simultaneously
   • Many possible strategies
2. Extensive form
   • Players move sequentially, represented by a tree
     • Eg. chess where white player moves, then black player sees move and reacts
   • Keeps track of what each player knows when they make each decision
     • Eg. poker where each new card or bet affects the decision

Normal form

• Finite, $n$-person game $\langle N,A,u \rangle$
• Players: $N = {1, \dots, n}$ is a finite set of $n$, indexed by $i$ (generic player)
• Actions for player $i$ $A_i$
  • Potential actions $i$ can take: $a = (a_1, \dots, a_n)$
  • List of actions $i$ has taken = $A = A_1 \times \dots \times A_n$
  • $a = (a_1, \dots, a_n) \in A = A_1 \times \dots \times A_n$
• [[Utility function]] for $i: u_i: A \mapsto \mathbb{R}$ ($\mathbb{R}$ is the payoff for $i$)
  • $u = (u_1, \dots, u_n)$ is the list of [[Utility]] functions for all players

2 player games are represented as a matrix where player 1 is the "row player" and player 2 is the "column player". Hence the rows correspond to the actions $a_1 \in A_1$ and the columns correspond to $a_2 \in A_2$. The cells of the matrix show the payoff values for each player - the row player first and the column player second.

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• Index:: [[_Game Theory]]
• Related::

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