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Unformatted text preview: Biology 1M03 TUTORIAL 3: ANIMAL BEHAVIOUR & THE PRISONER’S DILEMMA Student Package Objectives Students will be able to: !"Learn the basic concepts of game theory, in particular, the Prisoner’s Dilemma !"Use game theory to understand the evolution of cooperative behavior ! Apply Hamilton’s Law to a given scenario ! Read the Introduction in this tutorial manual ! Review sections 51.4 (pp. 1161‐1166) and 51.6 (pp. 1167‐1170) in Chapter 51 of Freeman’s Biological Science, 3rd Ed. ! Watch the YouTube videos “Honeybee Waggle Dance Experiment” “Waggle Dance” http://ca.youtube.com/watch?v=‐7ijI‐g4jHg and http://ca.youtube.com/watch?v=ywdTfEBVcSY&feature=related ! Complete the ELM pre‐tutorial quiz Preparation BACKGROUND INFORMATION INTRODUCTION: An important aspect of animal behaviour involves the interactions between individuals. When individuals behave in a way that helps others but comes at an expense to that individual, we refer to the behaviour as self‐sacrificing or altruistic behaviour. This sort of behaviour creates a problem for the theory of natural selection; if animals can “selfishly” benefit by refusing to co‐operate, and thus increase their own fitness, and this behaviour is at least partly genetic, then genes for selfish behaviour should be passed on. But in fact, many animals are observed participating in altruistic or cooperative behaviour. For example, in many bird populations, a breeding pair receives help in raising its young from other ‘helper’ birds, who help to feed the babies and defend the nest from predators. The “waggle dance”, used by honeybees to direct others to a food source, is another example. How can these observations be explained? Two theories of behaviour attempt to account for these observations: Kin Selection and Reciprocal Altruism. Kin selection theory argues that individuals help relatives, who by definition share some of their genes, because helping relatives promotes the survival of one’s own genes. According to kin selection, an individual will harbour a greater willingness to help another individual with whom he shares more genes, or greater relatedness. Inclusive fitness considers the direct fitness benefit of 1 behaviour to an individual as well the fitness benefit of the behaviour to close genetic relatives of the individual. An effective way of determining the total benefit of an action in terms of inclusive fitness is: total benefit= bself + r* brelative . The kin selection theory can be represented by a simple mathematical model known as Hamilton’s Rule. The rule states that an allele which contributes to altruistic behavior could spread throughout a population if: Where : !"r=the coefficient of relatedness !"B= the reproductive benefit to the recipient, normally measured in units of offspring produced !"C=the reproductive cost to the actor, measured in the same units This theory may therefore explain the cases of altruism where individuals are related, but what about when individuals are not related or are unaware of relatedness? Reciprocal altruism (also referred to as Reciprocity Theory) predicts that individuals will help other, non‐related individuals if they expect that they will reciprocate, or return the favour. This sort of behaviour will occur if the benefits of helping are greater than the costs. But what happens if one individual helps and the second individual does not reciprocate? This tutorial will investigate the probability of cooperative behaviour arising among unrelated individuals in the company of selfish individuals. Can individuals benefit from mutual cooperation while protecting themselves from selfish behaviour at the same time? rB > C GAME THEORY & THE PRISONER’S DILEMMA Game theory is a branch of mathematics that is often applied to economics, computer science, and the social sciences as well as to evolutionary biology. Game theory analyzes how to choose between behaviours whose cost or benefit depends on the choices of others. Thus, the interactions between two individuals are treated as if they are a game in which the “players” have a finite number of alternatives, and the outcome depends on the behaviour chosen by both individuals. In many situations, the best alternative is not simply the behaviour by which the individual stands to benefit the most; in fact, the best alternative often involves cooperation. Games where the players’ interests are in total opposition are called zero‐sum games. This means that the total benefit to all players in the game always adds to zero, such that an individual may only benefit at the expense of another individual. In biology, predator‐prey interactions are zero‐sum; what one individual stands to gain, the other individual must lose. In non‐zero‐sum games, the total sum of benefits may be greater or less than zero. A player doesn’t necessarily have to lose something for another to benefit. Game theorists often study non‐zero‐sum games because many scenarios in science 2 and social science are non‐zero‐sum. In biology, some types of parasitism provide an example of this, because a parasite may derive maximum benefit from maintaining a healthy host. A classic example of a non‐zero‐sum game is the Prisoner’s Dilemma, a hypothetical situation where two prisoners, A and B, may cooperate or betray one another. Consider the following scenario: Two prisoners are isolated in separate cells. They are both charged with a robbery and face 10 years in prison, however, the authorities do not have the evidence to convict them, and as a consequence are attempting to convince the prisoners to implicate one another in the robbery. The authorities offer both criminals the same deal: Testify against your partner and get a reduced sentence of 5 years. The prisoners have hidden the $10 000 they obtained in the robbery, so if one prisoner betrays his partner, he will be released and will not have to split the take. Thus, both criminals have an incentive to betray one another, but if they choose to cooperate with each other and remain silent, they will both go free and get half the cash. The payoff matrix for Prisoner A would therefore look like this: B Cooperate (C) Defect (D) Cooperate (C) Freedom; 10 years prison; A $5000 $0 Defect (D) Freedom; 5 years in prison; $10 000 $5000 Note that the terms “cooperate” and “defect” refer to your behaviour toward your partner Based on this matrix, what should you do? If you know that your partner is going to defect, you will be better off defecting (you save 5 years in prison, and split the money when you get out. Similarly, if you know your partner is going to cooperate, you can gain the whole $10,000 by defecting. In other words, no matter what your partner chooses to do, you are always better off defecting. However, if your partner uses the same logic, and also defects, you will both be worse off than if you had both cooperated. In studies of animal behaviour, a general payoff matrix may be created for any two “players”, where the costs (c) of performing a behaviour are weighed against the benefits (b) derived from the behaviour. The following payoff matrix is similar to the previous one, except that b and c are applied generally here, but have been replaced with specific values above. B Cooperate (C) Defect (D) Cooperate (C) b‐c ‐c A Defect (D) b 0 3 From this chart, we can see that if player A cooperates, and so does player B, then A receives the benefits of cooperation and also incurs a cost (b‐c). However, if A cooperates and B defects, then A only incurs a cost and receives no benefit (‐c). If player A decides to defect, he may receive all of the benefits of cooperation and incur no costs if B cooperates (b), or incur no costs or benefits if B also defects (0). **Note ** as long as b>c, this matrix is an example of a Prisoner’s Dilemma game. 4 TUTORIAL ACTIVITY: PLAYING THE PRISONER’S DILEMMA GAME PART 1: SINGLE INTERACTION GAME In this tutorial, you will become a “player” in a typical Prisoner’s Dilemma game. First, we must construct a payoff matrix for our game. 1. Construct a raw matrix, assigning point values for the cost and benefit of cooperating, such that b=4 points and c=2 points Raw Payoff Matrix My move Cooperate (C) Defect (D) My partner’s move Cooperate (C) Defect (D) 2. Construct an adjusted payoff matrix, which makes it easier to tally points (i.e. no negative values). Add the value of c to each cell, so all payoffs are positive or zero. This adjusted matrix shows the number of points that you will receive for your behavior over the course of the game. Normalized Matrix My move Cooperate (C) Defect (D) My partner’s move Cooperate (C) Defect (D) 3. Partner with another student. Each of you should have two cards, cooperate (C) and defect (D). Decide which card you will play, but be sure not to let your partner know your intended move. When your TA says “ready”, place your card face down on the table. When the TA says “go”, turn your card over. Record your move and the number of points received in the table below. Your goal here is not to do well compared to your partner, but to do well compared to the whole class. This is analogous to the biological situation where an animal is interacting with a few other animals directly, but is competing evolutionarily with a whole species. You Partner Move (C/D) Points 4. Record the class data in the table below. Move (C/D) Number of Total Points students C D 5 Average points (number/points) In a single‐interaction game such as this one, what is the best move in Prisoner’s Dilemma? _____ __ PART 2: MULTIPLE MOVE GAMES In some situations where we might want to study behaviour, individuals will meet and interact with the same “partner” more than once. In this type of game, players may have a different strategy than in a single‐interaction game; both players know that defecting may bring the greatest benefit, but they also learn from previous interactions and can expect further encounters as well. Think: What is the best strategy to use? Can you build trust and count on cooperation from your partner? Or are there subtle ways to cheat while still managing to elicit cooperation? MULTIPLE‐MOVES GAME: NO ASSIGNED STRATEGY In this round, you will play the game again, 10‐20 times, with the same person. The TA will dictate when to flip the cards over, so that all pairs play the same number of games. Keep track of your moves and scores in the table provided. # My move Partner’s move My points Some points to consider while playing: 1 2 !The aim is not to get more points than your 3 partner; rather, you are trying to do well 4 compared to the entire class 5 !"You may try to employ potential strategies 6 to maximize your points 7 8 !"Why is it important that players do not 9 know how long the game will last? 10 11 12 13 14 15 16 17 18 19 20 Total Points 6 Your TA will tally the class results on the board. Think: Did students who scored highest employ a strategy? What strategies were associated with lower scores? MULTIPLE‐MOVES GAME: EVOLUTIONARY STRATEGIES This time, you will be given a card with the name and a brief description of a strategy that you will play out over the next two rounds. You will play the game 20 times per round, switching partners after the first round. Your TA will tally the total number of points at the end of both rounds. Think: Which strategy do you predict will have the best outcome (i.e. will result in the most points?) Why doesn’t it matter this time if players know how many times they will play? My assigned strategy:__________________ My partner’s strategy:__________________ ROUND 1 # My move Partner’s move My points 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total Points My assigned strategy:____________________ My partner’s strategy:____________________ 7 ROUND 2 Partner’s move Total Points # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 My move My points POST‐TUTORIAL ASSIGNMENT Student Name:_________________________________ Student #:_________________ Tutorial #:____ 1. Shoaling fish, such as minnows (Phoxinus phoxinus), guppies (Poelcilia reticulate) and sticklebacks (Gasterosteus aculeatus) are known to display cooperative anti‐predator behaviour. Two fish will swim towards a predator together and pause to inspect the predator before returning back to the shoal. If two fish inspect the predator together then each will receive the benefit of inspection (information about the predator) and SHARE the cost (the risk of being eaten). However, if one if the fish stays behind while the other inspects, it will receive the benefit without the associated cost. The fish that has inspected on its own will receive the benefit, but also the FULL cost of the inspection. a) Draw a payoff matrix that depicts the four scenarios given the above parameters. Follow the format given in the Introduction section of this manual. Include descriptions of the outcome of the behavior as well as equations in terms of b and c. Also remember to define your variables. b) Based on your matrix, what would be the best move in a single interaction (cooperate or /5 /1 /1 defect)? c) What would be the best move if fish always travelled with the same partner to inspect? 8 2. You are a member of a shoal of sticklebacks and it is your turn to participate in predator inspection on behalf of the shoal. Assuming you and your partner will co‐operate in this inspection, determine whether or not this act of co‐operation will benefit your fitness based on the relatedness between you and your partner. Consider the following information: ‐ 60% of the time, one of the fish that approaches while co‐operating in predator inspection is eaten by the predator (Note: this cost will be SHARED EQUALLY between you and your partner) ‐sticklebacks only breed for one season and will lay 100 eggs in that breeding season ‐Hamilton’s Law : rB>C (to see how to calculate the co‐efficient of relatedness please see: p. 1169, Box 51.2) where: B=number of offspring your partner could potentially have in lifetime C=(chance of being eaten during inspection)*(number of offspring you can have if you survive) a) Use Hamilton’s Law to determine if it is in your best interests, in terms of inclusive fitness, to /2 /2 /2 cooperate with your brother (r=0.5) for predator inspections. Assume for the purposes of this question that this is a single interaction. b) What if your partner is your cousin (r=0.125)? 3. Briefly describe one example of, discussed in the textbook, where animals engage in co‐
operative behavior that may be influenced by game theory dynamics. Be sure to specify the name of the animal and describe the behavior. 9 4. Construct a matrix for the above behavior which outlines the costs and benefits of cooperating or defecting /5 /18 TOTAL: Submit your completed assignment (Page 9‐10) into your dropbox outside BSB 201A by 1pm the day after your scheduled tutorial. Your drop box number corresponds with your tutorial section number on your solar schedule. Please make sure that you deposit your assignment into the 1M03 dropbox. Assignments must be stapled together. Late assignments must be time‐stamped by Alastair Tracey before they are submitted. Please ensure that your name, student number, and tutorial section are noted. This information is copyright material (intellectual and academic property of Dr. J. Dushoff, Mr. A. Tracey, Ms. K. Dutchak and Ms. L. Wheeland, Department of Biology, McMaster University). It may be used only for study purposes and only by students enrolled in 2009‐2010 Fall/Winter Biology 1M03 (Biodiversity, Evolution, and Humanity). This information may not be reproduced for any other purpose, nor distributed to any person (using any type of media), who is not enrolled in the course, except by written permission of the Biology 1M03 professors and staff. References Evolutionary Game Theory”. Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/game‐evolutionary/ . [Accessed July 8, 2008]. Goldman, C. and Dennison, M. (1995). The evolution of cooperative behaviour. In Practical Studies in Evolution, Ecology & Behaviour, 6th Edition. University of Toronto Press: Toronto. Lynch, A. (1994). The Evolution of cooperative behaviour. In Tested Studies for Laboratory Teaching. Proceedings of the 15th Workshop/Conference of the Association for Biology Laboratory Education (ABLE), 15, 319‐333. 10 ...
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This note was uploaded on 04/13/2011 for the course BIO 1M03 taught by Professor Jonathanstone,jamesquinn during the Spring '11 term at McMaster University.
- Spring '11