What is Game Theory?

#gametheory #gambit #strategicform #nashequilibrium

What is Game Theory?

Game Theory is the formal study of strategic interaction.
In game theory, “strategic form” (or “normal form”) and “extensive form” are ways of representing games. In a strategic setting the actions of several agents are independent. Each agent's outcome depends not only on his actions , but also the actions of his opponents.
So the goal is to "How to predict opponents play and respond optimally."

Key elements of a Game:

Players
Strategies : Strategies are a complete plan or the options of each player or in what order do players act.
Payoffs : Payoffs are how strategies translate into outcomes and what are players preferences over possible outcomes.

Normal Form Games

A normal (or strategic) form game is a triplet (N,S,u).
where,

N ={1,2,...,n}: finite set of players
Si: set of pure strategies of player i S =S1×···×Sn ;s =(s1,...,sn): set of pure strategy profiles
ui : S →R: payoff function of player i; u = (u1,...,un). Outcomes are interdependent.

Eg: Prisoner's Dilemma
Imagine two persons are arrested for a crime.There is not enough evidence to convict either.Differnt rooms, no communication.
After being caught by the police for committing a crime, the two prisoners are separately offered a deal:

If both stay silent (cooperate), they get light sentences.

If one defects (betrays the other) while the other stays silent, the defector goes free and the silent one gets a heavy sentence.

If both defect, they both get moderate sentences.
eg: To qualify as a Prisoner’s Dilemma, the payoffs must satisfy:
Temptation > Reward > Punishment > Sucker
T (Temptation to defect) = 3 → you defect while the other cooperates
R (Reward for mutual cooperation) = 2
P (Punishment for mutual defection) = 1
S (Sucker’s payoff) = 0 → you cooperate while the other defects
Higher numbers = better outcome.

where, c=cooperation and d=defect

Extensive form games

Represents a game as a tree showing the order of moves.Useful for sequential games, where players move one after another.Captures timing, information sets, and chance events.