Name: _________________________________________
In this lab, you will perform numerous simulations of the repeated Prisoner's Dilemma. In addition to experimenting with basic strategies such as coop, defect, and titForTat, you will define your own strategies in the form of JavaScript functions. These functions can then be incorporated into an existing Web page to allow for experimentation and comparison with the other strategies.
For each round of play, two players independently choose to cooperate or defect.The goal for each player is to accumulate as many points as they can.
- if both players cooperate, then both receive 3 points
- if both players defect, then both receive 1 point
- if one cooperates and one defects, then the defector receives 5 points
Several Prisoner's Dilemma strategies were described in the online article. The coop strategy always cooperates, the defect strategy always defects, while the randomness strategy cooperates or defects at random. The titForTat strategy cooperates in the first round, and then mimics whatever the opponent did in the previous round. It is interesting to note that the defect strategy will never "lose" to an opponent, meaning that it will never accumulate fewer points than any opponent. Similarly, coop and titForTat will never "win" against an opponent. However, the number of points accumulated in "winning" and "losing" may vary greatly.
EXERCISE 1: For each of the following pairings, predict the outcome if the two strategies were to compete in a repeated Prisoner's Dilemma. In particular, how many points would each strategy accumulate (assuming 50 rounds)?coop vs. coop defect vs. defect titForTat vs. titForTat coop vs. defect coop vs. titForTat defect vs. titForTat Which of the strategies would tend to do best against randomness? Which would do worst? Explain.
EXERCISE 2: The Web page dilemma.html contains JavaScript code for simulating the repeated Prisoner's Dilemma. You are able to select which strategies you want to compete, and then click on a button to see the simulation. The button labeled "Click for next round" will display one round at a time, while the button labeled "Click for complete simulation" will complete all 50 rounds.Load this page into the browser and use it to verify your predictions from the previous exercise.
What makes the Prisoner's Dilemma interesting is that locally optimal behavior is not necessarily globally optimal. Consider a round-robin tournament between defect and two copies of titForTat (assuming 50 rounds per contest). Even though defect outscores each opponent in head-to-head contests, it only earns 54 points in each, for a total of 108. When one copy of titForTat plays the other, however, they both earn 150 points. Adding this to the 49 points earned against defect, each copy of titForTat earns a total of 199 points. Thus, defect wins all of its battles, but loses the war!
A Prisoner's Dilemma tournament can be described nicely in ecological terms. The defect and coop strategies, because of their aggressive and passive natures, can be characterized as hawks and doves. The titForTat strategy, which is passive by nature but will become aggressive when provoked, may be characterized as a sparrow. According to this analogy, the fact that defect outscores titForTat in a head-to-head contest shows that a lone hawk dominates a lone sparrow. However, the tournament shows that two sparrows support each other and can fight off a lone hawk. Using this type of analogy, the Prisoner's Dilemma has been used by researchers as a way of explaining how cooperative behavior can evolve without consciousness. In fact, studies done on the hunting patterns of lions and the defensive behaviors of fish have supported the Prisoner's Dilemma model of cooperation.
The results of a round-robin tournament can be displayed neatly in a table. The above tournament between a hawk (defect) and two sparrows (titForTat) would appear as follows. The entries across a row contain the points earned by a strategy playing against each of the other strategies. For example, the hawk earns 54 points when it goes against sparrow1, so there is a 54 in the table at the row labeled hawk and column labeled sparrow1. Conversely, sparrow1 only earns 49 points against the hawk, so there is a 49 in the table at the row labeled sparrow1 and column labeled hawk.
| hawk | sparrow1 | sparrow2 | TOTAL | ||
| hawk | 54 | 54 | 108 | ||
| sparrow1 | 49 | 150 | 199 | ||
| sparrow2 | 49 | 150 | 199 |
EXERCISE 3: In the table below, show the results of a round-robin tournament between a hawk, a dove, and two sparrows. How does the addition of the dove affect the dynamics between the hawk and the sparrows compared to the example above?
hawk dove sparrow1 sparrow2 TOTAL hawk dove sparrow1 sparrow2
EXERCISE 4: In the table below, show the results of a round-robin tournament between a hawk, a dove, and three sparrows. How does the additional sparrow affect the results when compared to the previous exercise?
hawk dove sparrow1 sparrow2 sparrow3 TOTAL hawk dove sparrow1 sparrow2 sparrow3
The dilemma.html page includes numerous HTML elements for controlling and
displaying the results of the repeated Prisoner's Dilemma. These include text boxes
for displaying the histories (past moves) of the two strategies and their scores, as
well as buttons for controlling the simulation. There are also select boxes
that allow the user to select from the different strategies. A select box is defined
using the tags <select> and </select>. Within those
tags, the different options from which the user can select are listed using
option tags. For example, a select box for picking a Prisoner's Dilemma
strategy:
Before continuing with the following exercises, make a local copy of dilemma.html on your disk. To copy the page, first load the page in the browser and select "Source" under the "View" menu. Then, cut-and-paste the displayed source code into your text editor and save it under the name dilemma.html.
EXERCISE 5: The following function defines a new Prisoner's Dilemma strategy called titFor2Tats. The function is similar to the titForTat strategy, only more forgiving. This strategy will only defect if the opponent defected the last two times.
function titFor2Tats(myHistory, yourHistory) // Assumes: myHistory and yourHistory are strings of C's and D's (same length) // Returns: "D" if opponent defected last two times, else "C" { if (yourHistory.length >= 2 && yourHistory.charAt(yourHistory.length-1) == "D" && yourHistory.charAt(yourHistory.length-2) == "D") { return "D"; } else { return "C"; } } What would the titFor2Tats strategy do in the first round? How about the second round? Why?
Cut-and-paste a copy of the titFor2Tats function into your copy of dilemma.html. Then, add a new option to each of the select boxes in your page. Once you have done this, describe how this new strategy performs relative to titForTat. That is, for each of the following opponents, does titFor2Tats accumulate fewer or more points than titForTat would?
titFor2Tats vs. coop titFor2Tats vs. defect titFor2Tats vs. randomness Consider a round-robin tournament such as the ones you conducted in Exercises 3 and 4. Would the presence of titFor2Tats strategies be able to ward off the hawk (defect) and protect the dove (coop). If so, how many copies of titFor2Tats would be needed. If not, why not?
EXERCISE 6: Define a new Prisoner's Dilemma strategy called randomTat in your copy of dilemma.html. The function should have a 10% chance of automatically defecting at any given round; otherwise it should behave the same as titForTat (mimicking the opponent's move on the previous round). Hint: consider rolling a 10-sided die. If the roll comes up 1 (which should happen roughly 10% of the time), then the function should automatically return "D". Otherwise, it should behave the same way thattitForTatdoes.Add a new option to each of the select boxes on the page, then use the page to determine how this new strategy performs relative to titForTat. That is, for each of the following opponents, does randomTat accumulate fewer or more points than titForTat would? Note: there is some variability due to the random component of the randomTat function.
randomTat vs. coop randomTat vs. defect randomTat vs. randomness Consider a round-robin tournament such as the ones you conducted in Exercises 3 and 4. Would the presence of randomTat strategies be able to ward off the hawk (defect) and protect the dove (coop). If so, how many copies of randomTat would be needed. If not, why not? Again: there is some variability due to the random component of the randomTat function.
EXERCISE 7: Design your own Prisoner's Dilemma strategy and encode it as a JavaScript function in the page. Your function should have your last name as its name, and should follow the same basic pattern as the provided strategies. That is, it should take the two lists of histories as inputs, and should return either "C" or "D". Your function can utilize any of the features of the provided functions, or you can devise new functionality of your own. You must be extremely careful that your function behaves reasonably in all cases. That is, it should be able to handle histories of any length and always return either "C" or "D". Add a new option to each of the select boxes on the page, then use the page to test your strategy.On the last day of class, we will hold a round-robin tournament that pits all of the student strategies against each other. The strategy that earns the most points after playing all other strategies will be declared the winner, and its creator will receive a prize.
Hand in a printout of your modified dilemma.html page, attached to these sheets. Also, mail a copy of your function from Exercise 7 to the instructor.