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Prisoner's Dilemma

Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain.

Payoff Matrix: vs

Rows:	Columns:
	Confess	Silent
Confess	, Nash Equilibrium	, Nash Equilibrium
Silent	, Nash Equilibrium	, Nash Equilibrium

Blue: Player A Payoff, Red: Player B Payoff.

Cells highlighted in Green represent a Pure Nash Equilibrium.

Probability Calculation (Mixed Strategies)

Calculation Origin

When no pure Nash equilibrium exists, players randomize their strategies to make the opponent indifferent. We calculate p and q by equating expected utilities.

Result Validity:

Valid (0 to 1): If the result is between 0 and 1 (0% to 100%), a Mixed Strategy Nash Equilibrium exists.
Invalid (< 0 or > 1): If the result is negative or greater than 1, mathematically it is not a real probability. This indicates that no mixed equilibrium exists in this game (likely a pure dominant strategy exists).

Algebraic Formulas and Substitution:

q (Probability B Left):

q = (A22 - A12) / (A11 - A12 - A21 + A22)

q = ( - ) / ( - - + )

p (Probability A Top):

p = (B22 - B21) / (B11 - B21 - B12 + B22)

p = ( - ) / ( - - + )

Calculated Results

Probability p (A)

Probability q (B)

Probability of each Cell

Top-Left (p*q):

Top-Right (p*(1-q)):

Bottom-Left ((1-p)*q):

Bottom-Right ((1-p)*(1-q)):

Payoff Matrix Interpretation

1 Positive and Negative Values Selection

Values in the matrix represent the utility or benefit each player receives. A scale of -10 to 10 has been selected to represent preference intensity:

Range -10 to 10: Allows modeling a full spectrum of results, from catastrophic situations (-10) to ideal outcomes (10).
Values 1 to 10 (Positive): Represent gains, benefits, or satisfaction. The higher the number, the greater the reward.
Value 0: Represents a neutral outcome, where there is neither significant gain nor loss (indifference point).
Values -10 to -1 (Negative): Represent costs, losses, punishments, or undesirable consequences. The lower the number (more negative), the worse the outcome.

2 Nash Equilibrium (Green Cells)

Cells highlighted in green indicate a Nash Equilibrium. This means that, at that point, neither player has incentives to unilaterally change their strategy. If a player changes their decision while the other maintains theirs, their payoff would decrease or not improve. It is the "stable" state of the game where both strategies coincide optimally.

3 Maximization, Optimization, and Rationality

Rationality It is assumed that players are rational: they think logically, understand the rules, and always act to satisfy their own interests.

Maximization The fundamental goal is to maximize their own utility. Each player seeks to obtain the highest possible value in the matrix for themselves.

Optimization It is the process of choosing the "best response" to the opponent's actions. It is not just seeking the highest number, but the best possible result given the circumstances.