The Computational Beauty of Nature — Spring 2020

Problem Set 6: The Prisoner's Dilemma

Due by class time Thursday, April 16

For these exercises, please write up your answers in the form of a PDF or Microsoft Word document.

Reading

Written Exercises

  1. Consider the following creatures produced in Karl Sims' experiments on evolving virtual creatures with genetic algorithms:

    For each creature, draw the genotype (i.e. the "nodes-and-links" structure) that would be used to represent the creature's body. You don't need to specify the neural network "brain", just draw the nodes and links for the body structure. Assume that each of the above creatures is made up of just 2 types of parts: one called "body" for the central torso and another called "segment" for the appendages. Make sure to indicate which nodes correspond to which parts in your diagrams.

  2. Suppose Alice and Bob play the Iterated Prisoner's Dilemma (that is, they play N rounds of the game against each other), using the payoff matrix given on page 215 of Complexity: A Guided Tour.

    1. Suppose Alice's strategy is "Always Defect" and Bob's strategy is "Always Defect". What will each player's average score be at the end of the N rounds?

    2. Suppose Alice's strategy is "Always Cooperate" and Bob's strategy is "Always Cooperate". What will each player's average score be at the end of the N rounds?

    3. Suppose Alice's strategy is "Always Defect" and Bob's strategy is "Always Cooperate". What will each player's average score be at the end of the N rounds?

    4. Suppose Alice's strategy is "Always Defect" and Bob's strategy is "Tit For Tat". What will each player's average score be at the end of 10 rounds?

    5. Suppose Alice's strategy is "Always Cooperate" and Bob's strategy is "Tit For Tat". What will each player's average score be at the end of 10 rounds?

  3. Assume that both Alice and Bob can remember what the other did on the previous round of the game. Alice's strategy can be represented by the table below. This is the "Always Cooperate" strategy for Alice. To represent a different strategy, the values in the "Current Round" column must be changed. Write the "Tit For Tat" strategy in this form.

    Previous Round    Current Round
  (Time t-1)         (Time t)
 Alice   Bob          Alice
   C      C             C
   C      D             C
   D      C             C
   D      D             C

  4. How many possible strategies are there when each player remembers only the previous round? How many strategies are there when the players remember the two previous rounds?

Extra Credit Problem (optional)

  1. Write the "Tit For Two Tats" strategy for Alice by filling in the rightmost column of the table below with C or D as appropriate. This strategy is like "Tit For Tat" except that it lets the first defection go unchallenged. If the opponent defects twice in a row, "Tit For Two Tats" will respond by defecting, and then resume cooperating only if the opponent does. Some rows in the table correspond to situations that can never arise with Tit for Two Tats. For these rows, place a dash (-) in the rightmost column to indicate this.

       Previous Rounds      Current Round
(Time t-2)  (Time t-1)     (Time t)
Alice Bob   Alice Bob       Alice
  C    C      C    C          
  C    C      C    D          
  C    C      D    C          
  C    C      D    D          

  C    D      C    C          
  C    D      C    D          
  C    D      D    C          
  C    D      D    D          

  D    C      C    C          
  D    C      C    D          
  D    C      D    C          
  D    C      D    D          

  D    D      C    C          
  D    D      C    D          
  D    D      D    C          
  D    D      D    D