Decision TreesMerely a graphical aid to seeing which outcome has the maximum Expected ValueMust draw out the whole tree, including all the contingent branchesTo simplify: Start collapsing decision tree to the left, until you end with only the branches representing the first choiceThe largest Expected Value predicts the first choiceContinue the process:Given the first choice, collapse back to the next decision (i.e., second choiceAnd so forthAlternatively, can just multiply probabilities along each branch, and list the product at the end of the final branch pointIf a thing can happen N ways, add up all N probabilities to estimate its likelihoodTo simplify: Start collapsing decision tree to the left, until you end with only the branches representing the first choiceThe largest Expected Value predicts the first choiceExample:Study GuideCOMM 402, Fall 2012Exam 2Lave & March, Chapter 4What is the choice process being modeled in this chapter?Decision-Making Process. - Can be mundane decisions (should I go home via Main St. or should I try the shortcut?) orcan be relatively consequential decisions (should I drop out of school for a year?)- Social scientists interested in developing and applying a model have assembled situations that involve an individual making a decision AND that the person making the decision cannot be certain about the outcome of the decisionAn individual makes a decision as if he were going through these steps:1. Examine all possible plans of action and see what possible outcomes can result from each.2. Judge how desirable each outcome is and how likely it is to occur as a result of following the particular plan of action.3. Choose the plan of action with the highest expected value.How do you calculate Expected Value (EV)? Expected Value:The average payoff you would receive if you played the game a large number of times.We use the word expected in a probability sense to designate an average amount of winning.- Value to decision maker of the various results possible from the different course of action- Probability, or likelihood that these results will actually occurExample from the book - A die is rolled; if it comes up with a “5”, you win $12; if it comes up with any other number you win nothing. Suppose we play the game 600 times. We throw the die 600 times in succession and keep track of how much wewin in total. Since there are 6 sides to a die, the chance of a “5” occurring is 1 in 6 or 1/6. In 600 throws, we would expect the“5” to come up 100 times. (1/6 times 600) and so you would win $1,200 ($12 X 100). For the other tosses you receive nothing. Thus your total expected winnings are $1200 for the entire series of tosses or $2 per game is the average payoff, or the expected value of the game.Since the outcome of playing this game can only by $0 or $12, there is no way of winning the expected value of $2 in a single game. We are using the word “expected” in a probability sense to designate an average amount of winning.Calculating Expected Value1. We shall designate all the possible outcomes of the game by using their values and use V as a symbol for them. We can use subscripts to designate different outcomes: Thus V1 is the value of outcome number 1; V3 is the value for outcome number 3, and so on. In the game last described there were two outcomes, which we designate:V1 = $12 (outcome number 1 is the occurrence of a “5”)V2 = $0 (outcome number 2 is the occurrence of any other number)2. We shall designate the probability of each outcome using P:P1 = 1/6 (the probability of outcome number 1 is equal to 1/6)P2 = 5/6 (the probability of outcome number 2 is equal to 5/6)3. If we designate expected value as EV we can now write a formula for expected value as:EV = P1 V1 + P2 V2 Or in the dice example we would have:EV = (1/6 X $12) + (5/6 X $0)EV = $2 + $0EV = $2If there are more than two outcomes, we just add more terms to the formula and we have the general expression for the expected value of an alternative having n possible outcomes.What is the numerical range of probability? Why is probability limited to these numerical values?When we talk about probability we are using the notion of relative frequency.- Ex: When we say the probability of a head appearing is 0.5, we mean that if you flipped the coin many times, a head would show up about ½ or 50% of the time.- All probabilities must be within the range from 0 to 1o 0% chance of occurring – never, something is certain not to occuro 100% chance of occurring – every time, something is certain to occurWhat are the two common mistakes people make when they apply probabilities in their decisions?Notation:P is the abbreviation for Probability. P(heads) = 0.5 means the probability of heads occurring is 0.5If you add together the probabilities of all the possible outcomes associated with one situation, they must equal 1.0 (provided the outcomes are mutually exclusive). That is to say, out of all of the things they might occur, at least one of them will occur.Common Mistakes People Make1. One of the most common errors people make when estimating probabilities is to assume that because there are two possible outcomes, the probability of each of them is ½. This works forcoins because the two outcomes are equally likely. But all two-outcome situations do not have equally likely outcomes.2. Gamblers Fallacy: you are never “due” for a certain outcomeHow do you draw a decision tree? How do you use a decision tree to choose?- We use boxes to designate events or outcomes- The lines connecting the boxes in the figure show all the possible connections, or relations, between the boxes- The number beside each line is the probability that the path will be followed- The dollar values at the right hand side of the figure show the value associated with each possible outcomeYou read the tree from left to rightThe die is rolled, and each of the outcomes branches from that point and ends in a value associated with a final outcome.A decision tree is a complete picture of the world, one that shows all possible outcomes. Each of the boxes in the tree is one possible outcome, and I that one outcome can possible lead to other outcomes, all of them will be shown branching off to the right.If all the possible branches are shown, one of them must occur. Therefore, the sum of the probabilities must equal 1.0 because the branches must be a complete description of the world.Decision TreesMerely a graphical aid