Payoff Matrices

Defining outcomes for every combination of strategies.

Basic Syntax

payoff PlayerName {
    (MyStrategy, TheirStrategy): value
}

The tuple (MyStrategy, TheirStrategy) represents:

  • What I play (first position)
  • What they play (second position)

Payoff Values

Numeric Literals

payoff Alice {
    (Cooperate, Cooperate): 3
    (Cooperate, Defect): 0
    (Defect, Cooperate): 5
    (Defect, Defect): 1
}

Negative Values

payoff Player1 {
    (Swerve, Straight): -1   // Lose face
    (Straight, Straight): -10 // Crash!
}

Decimal Values

payoff Firm {
    (High, High): 4.5
    (High, Low): 2.25
}

Variables in Payoffs

Use variables for parameterized games:

var T = 5;  // Temptation
var R = 3;  // Reward
var P = 1;  // Punishment
var S = 0;  // Sucker's payoff

game PrisonersDilemma {
    players P1, P2
    strategies C, D
    
    payoff P1 {
        (C, C): R
        (C, D): S
        (D, C): T
        (D, D): P
    }
    
    payoff P2 {
        (C, C): R
        (D, C): S
        (C, D): T
        (D, D): P
    }
}

Now you can experiment by changing T, R, P, S!

Expressions in Payoffs

Payoff values can be arithmetic expressions:

var base = 10;
var bonus = 5;

payoff Firm1 {
    (Cooperate, Cooperate): base
    (Cooperate, Defect): base - bonus
    (Defect, Cooperate): base + bonus
    (Defect, Defect): base / 2
}

Supported operators:

  • + addition
  • - subtraction
  • * multiplication
  • / division

Zero-Sum Games

In zero-sum games, one player's gain is another's loss:

game MatchingPennies {
    players Matcher, Mismatcher
    strategies Heads, Tails
    
    payoff Matcher {
        (Heads, Heads): 1
        (Heads, Tails): -1
        (Tails, Heads): -1
        (Tails, Tails): 1
    }
    
    payoff Mismatcher {
        (Heads, Heads): -1
        (Heads, Tails): 1
        (Tails, Heads): 1
        (Tails, Tails): -1
    }
}

Notice: For each cell, Matcher + Mismatcher = 0.

Payoff Matrix Visualization

The payoff blocks above represent this bimatrix:

                    Mismatcher
                Heads       Tails
        ┌───────────┬───────────┐
Heads   │  (1, -1)  │  (-1, 1)  │
Matcher ├───────────┼───────────┤
Tails   │  (-1, 1)  │  (1, -1)  │
        └───────────┴───────────┘

Each cell shows (Matcher's payoff, Mismatcher's payoff).

Next Steps