Lecture Notes on Expected Value and Utility

09.09.2024 \| System 1 and 2 ============================ [Example]: You are a financial advisor, and you are advising Mr.Lee on investing for his retirement. He plans to invest \$100 from each month's pay cheque, until he retires 10 years from now. Mr. Lee has to choose between 2 different investm...

09.09.2024 \| System 1 and 2 ============================ [Example]: You are a financial advisor, and you are advising Mr.Lee on investing for his retirement. He plans to invest \$100 from each month's pay cheque, until he retires 10 years from now. Mr. Lee has to choose between 2 different investment plans. Which plan would maximize Mr.Lee's expected Value? Plan A is a risky plan that has a 15% chance of failing, which means Mr.Lee's investment will simply be gone; a 40% chance of breaking even (he'll get back whatever he put in, but no more); and a 45% chance of giving him his investment back, plus a nice 20% profit. Plan B is safer: it has a 15% chance of losing Mr.lee just half of his investment; a 20% chance of breakin even; and a 65% chance of giving him his investment back, plus a reasonable 8% profit. If you were advising Mr.Lee, which plan would you suggest for him if he want to maximize the expected return on investment? **System 1**: really useful; most reliable - [Example]: Linda is 31 years old, single, outspoken, and very bring. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. Which is more probable? 1. 2. It's quik at recognizing represnetative descriptions - but it ignores other important information. System 1 is bad at this. **Errors of Reasoning**: - - - - **Cognitive Biases**: unconscious mental tendencies, usually rooted in intuition, that lead us into errors of judgement or decision - - - - - 09.16.2024 \| Memos =================== **Argument**: set of claims in which some of those claims are offered as reasons in support of another Example: Syringes are on sale → We should buy now From Decision, to Argument, to Impact - - - - **Memos** (memorandum): a short note designating something to be remembered - - - - ### Structure of a Memo 1. - - - - 2. - - - - - - - - - - - - - - [4 Cs of Memo Writing]: 1. 2. - - - 3. - - - - - 4. Common Writing Mistakes: 1. 2. - - - 3. - - - - - - - - - - - 09.23.2024 \| Expected Value and Expected Utility =================================================  When there are 2 clear paths... Choose the one that provides the better outcome - - Situations Involving Probability: - - - - Goal: - - - Note: - - - [Example]: What is the expected value of a lottery ticket if the prize is \$1,000,000 and if I tell you there is a 50% chance you'll win each time? (The key: look at hte value and the likelihood of the 2 possibilities) - - What is the expected value of a lottery ticket if the prize is \$1,000,000 and if I tell you there is a 25% chance (1 in 4 chances) you'll win? - What is the expected value of a lottery ticket if the prize is \$1,000,000 and if I tell you there is a 0.0001% chance (1 in a million cance) you'll win? - **Expected Value**: what the average value of some action would be if repeated many, many times - - - - **Gambler's Fallacy**: an error in thinking that occurs when an individual believes that a random event (e.g., flipping heads or tails) is more or less likely to happen given precious events (e.g., previous flips of a coin). - To calculate EV (or EU) when dealing with probability, ask: 1. - 2. - 3. - 4. - [Example]: The odds: - - - Start with 1, then subtract the probability already accounted for: [1 − (0.1 + 0.01 + 0.001) = 0.889]{.math.inline} (or 88.9% that you lose Now we can add this to the equation so that the probabilities equal 100% [(0.1 × \$10) + (0.01 × \$100) + (0.001 × \$1000) + (0.889 × \$0) = 3]{.math.inline} [Example]: Should you buy a new \$5000 Customer Management System to help your sales team manage accounts? Based on industry states, you know the following: - - - You also know your sales last year were 50k and you have no reason to believe your results from the purchase would be different from other companies [*EV* = (0.5 × \$5000) + (0.25 × \$0) + (0.25 ×  − \$5000) − \$5000]{.math.inline} [*EV* =  − 3750]{.math.inline} ∴ The expected value of investing in the new system is -\$3750. The expected value tells you what to expect, in the long run, if this decision were made many, many times under these conditions. [Example]: You friend wants to start a new business, cleaning offices on the weekend. He wants a small \$100/month investment to partner with you so that the two of you will work together, cleainng. He'll use your money to help buy the supplies, find the contracts, and generally hand the business end of things. At the end of the month, he'll split hte profits with you - if there are any. He has doen research and honestly estimates that there's a 60% chance the project will go well, and your share of the profits will be \$2000. There's a 10% chance it will go very well, and your share of hte profits will be \$4000. Otherwise, he expects to just break even leaving no profit to split. Calculate the Expected Value taking his offer. [*EV* = (0.6 × 2000) + (0.1 × 4000) + (0.3 × 0) − 100]{.math.inline} [*EV* = (1200) + (400) + (0) − 100]{.math.inline} [*EV* = \$1500]{.math.inline} To make a decision choose the option that provides a higher expected value. That's how to maximize expected value. So.. we have a clear method to decide how to maximize money in the long run. But, beware of false precision and false certainty. It's not all about the money: - - - - - **Expected Utility**: uses the same math as expected value, but is a measure of utility (i.e., preference satisfaction) - - - - - - - -  [Example]: Your friend wants to start a new nonprofit, providing tutoring for high school students from disadvantaged backgrounds. She has asked you to volunteer - every Sunday morning for the next 3 months. You like the idea of helping kids in need, and to be honest you're also aware that it could also look good on your resume. You talk to a bunch of people to gather data on this kind of opportunity. You figure there's a 20% chance you just won't like it - it will be a small line on your resume, but that's all. How good is that? Maybe 2 out of 10. There's a 60% chance you'll really enjoy it, and that's enough for a 6 out of 10 for you. Finally, there's a 20% chance that you'll like it AND that (some day) an employer ends up noticing it on your resume and it helps you get the job. 10 out of 10 [*EU* = (0.2 × 2) + (0.6 × 6) + (0.2 × 10)]{.math.inline} [*EU* = 0.4 + 3.6 + 2]{.math.inline} [*EU* = 6]{.math.inline} 09.30.2024 \| Claims and Basic Sources of Justification ======================================================= Example 1: Review ----------------- Imagine an investment with 3 possible outcomes. There's a 10% chance you'll LOSE the \$1000 you put in. There's an 80% chance you'll earn a \$100 profit, and a 10% chance that you'll earn a \$300 profit. What is the expected value of making the investment? [(0.1 × 0) + (0.8 × 1100) + (0.1 × 1300) − 1000 = 10]{.math.inline} or [(0.1 ×  − 1000) + (0.8 × 100) + (0.1 × 300) = 10]{.math.inline} Example 2: Review ----------------- Suppose Tina likes chocolate ice cream better than raspberry, but she likes vanilla the best. If she's trying to maximize utility, which of the following offers should she accept? In other words, which option maximizes utility? To know that, you have to work out the expected utility for each option. - - - **Answer**: Not enough information #### Why do we need estimated value and estimated utility? So we can look at potential possibilities for decisions we make ; can't predict what will happen (the future) Arguments ---------  ### Two Steps to Understanding Arguments As critical thinkers, we can approach arguments others have already created in two distinct steps and ask ourselves: 1. - - - 2. - - - ### Analysis: The Claims in Arguments Two central parts: premise(s) and conclusion(s) - - - ### Embedded vs Asserted Claims E.g., "If I don't study, I will fail." To identify the asserted claim ask yourself... What claim is the speaker making when they utter this statement? **What is Being Asserted**: - - - - **You will often be asked to "Number the Asserted Claims"** How many asserted claims?  ### What roles do claims play? Premises and Conclusions - - - - - ### Why Premises Matter - - From Analysis to Evaluation: How Can We Evaluate Premises? ---------------------------------------------------------- ### Claims and Beliefs - - - ### Acceptable Claims - - - - ### Precision Many claims are typically only either true or false Some claims (though false) are close enough to the truth to be acceptable in certain contexts - - - - ### Certainty "Truth" is a big word and invites us to think we need to be absolutely certain. - - #### What makes claims acceptable? One great way to justify a claim: Make an argument! → Problem → The premises of every argument are themselves claims ### Basic Sources of Justidication - - - **Common Ground** **Burden of Proof** ### BSJ 1: Definitions and Logical Truths Some claims are true by **definition** - Some claims are true by logical **necessity** - There do a lot of work as background assumptions (unstated premise) ### BSJ 2: Your Senses Rule of thumb regading your own senses? - Factors inspiring doubt? 1. 2. 3. To "doubt" does nto mean to conclude your own eyes and experiences are wrong **Pareidolia:** the imagined perception of a pattern or meaning where it does not actually exist ### BSJ 3: Eyewitness Testimony Generally, it is reasonable to take other people's word about what they've seen & experienced Unless: 1. - - 2. - ### BSJ 4: Common Sense Our own shared understanding of how the world works. **Folk Psychology**: the body of knowledge that allows us to attribute to other people moods, beliefs, desires, intentions, memories, etc - ### BSJ 5: Expert Testimony **Expert**: someone whose training and/or experience make them more knowledgeable and make their judgment more reliable \*\*Expertise is specific to a field ### BSJ 6: Scientific Studies **Scientific Studies**: the way that our knowledge of the world progresses - -  Memo 2 ====== Background ---------- You are the Financial and Operations Manager for the Spirit of the Children Society, a non-profit society that supports Indigenous families in British Columbia. One part of your job is to oversee fundraising. For one of your major fundraising events this year you are thinking of hosting a concert. But this would come with a significant up-front cost, of \$11,000. Your boss, Adwoa Nsiah, the Executive Director of Spirit of Children Society, is skeptical that hosting an event with such a high cost could be profitable. But having done some research, you are confident that, for events of this calibre, there is a chance that you could bring in as much as \$110,000 in donations. While not common, it does happen about 1% of the time! Most often, about 50% of the time, these types of events bring in about \$60,000 in donations. Even when the event doesn\'t go as well as you hope, about 25% of the time such events are still able to bring in \$11,000 in donations, which would mean you'd break even. Only rarely do they bring in less/ about 9% of the time, they only muster \$5000 in donations and the rest of the time, due to unforeseen circumstances, they end up not bringing anything in at all. Instructions ------------ Your task for this assignment is write a memo to Adwoa Nsiah, and argue why you think it *is* worth hosting the event nad ask them to let you know their final decision as soon as possible. To do so you will need to calculate the expected value of the event based on the information you have. In your explanatory paragraph you will need to include two things: 1. 2. Memo Entry ---------- To: Adwoa Nsiah, Executive Director From: Mikaellah Maglaya, Financial and Operations Manager Date: September 30, 2024 Subject: Expected Value of Fundraising Event For this year's fundraiser, we are proposing to host a concert. I believe that this concert is worth holding because we will see profit. With this initial cost of \$11 000, the expected value of this event is \$24 950. The expected value takes the probabilities of all possible outcomes and provides a value that is comparable to the initial cost. For this concert the expected value is much greater than the initial fee. Therefore, it is very likely that we will see profit from this fundraiser. Please let me know you final decision regarding the fundraiser as soon as possible. 10.07.2024 \| Diagramming Arguments =================================== Argument Analysis ----------------- 1. - - 2. - - - **Argument**: a set of claims in which some of those claims are offered as reasons in support of another (in the fact of doubt or disagreement) - - - - - - - - Argument Reconstruction ----------------------- - - - - - - - Principles of Recontruction --------------------------- **Principle of Fidelity** (faithfulness): reconstruct the argument so that it captures the argument the arguer is actually trying to make **Principle of Charity**: reconstruct the argument so that it is as strong as possible (given what is actually there) Why follow these principles? - - - - - 1. 2. Hidden Parts ------------ - - - - - - - - Argument Diagrams ----------------- We can clarify the structure of an argument by diagramming its parts... - -  More complex: joint premises - - - - - - - - Layered Arguments ----------------- - - - Bridge Principles & Deductive Logic =================================== Logic ----- ### Strong & Weak Arguments An argument can be weak in two ways: 1. - 2. - ### Logic - - - - - - The Idea of Logical Support --------------------------- ### Premise Quality The quality of the **premises** of an argument is a separate issue from the quality of its **logic** [Example]: - - - ### Logic Quality - - - Notes: - - - Remember: - - - Leaves you in a weak position when you encounter someone who doesn\'t believe the conclusion. Types of Claims and the Logical Gap Between Them ------------------------------------------------ Logic and Differing Types of Claims: Some claims cannot follow logically from certain other claims 2 Main Kinds of Claims: Descriptive Claims and Normative Claims ### Descriptive Claims **Descriptive Claims**: claims that describe how the world IS - - - - Example: - - - ### Normative Claims **Normative Claims**: claims regarding what OUGHT to be the case, what the world SHOULD be like 2 Main Categories of Normative Claims: ethical and prudential Examples: Ethical Claims - - Examples: Prudential Claims - - Kinds of Claims: With 2 Kinds of Claims: - - - - How to get from descriptive premises to normative conclusions: - - - [Example]: Descriptive Condition Normative Element - - - - - - Joint premises NEED each other to prove a conclusion. ### Evaluating Normative Claims **Normative** claims can't be evaluated as easily as descriptive claims cna. - - ### Supporting Normative Claims - - - ### Arguments Vary Arguments differ in terms of: - - - **Remember**: - - - - Intro to Logical Styles ----------------------- ### Different Kinds of Reasoning/Logic **Deductive Logic/Reasoning**: - - **Inductive Reasoning**: - -  [Example]: Modus's 2017 sales were higher than 2016's. And 2016 sales were higher than 2015. So, 2017 sales must be higher than 2015's. - If Modus wants to sell their new drug in the US, they need FDA approval. They do want to sell in the US. So they need FDA approval. - Two key types of deductive argument: 1. 2. Arguments that start by saying that IF one thing is true, THEN another must be true. - - - OR: - - Deductive Logic and Conditional Arguments ----------------------------------------- ### 2 Kinds of Conditions The stuff directly following the "IF" in a conditional claim is the "antedecent." It is the **condition** - 2 Types of Conditions: 1. 2. ### Necessary and Sufficient Conditions 1. - - 2. - - 3. - 4. - 5. - ### Deductive Conditional Arguments - -  **Moduc Ponens** (MP): affirming the antecedent - - - **Fallacy: Affirming the Consequent** - - **Moduc Tollens** (MT): denying the consequent - - - **Fallacy: Denying the Antecedent** - - ### Two Words of Caution 1. - - 2. - - ### Conditional Arguments and Bridges Bridge principles often (not always) take the form of conditional claims. - - A **bridge principle** states the descriptive CONDITIONS under which some normative conclusion will hold. Bridges do not have to follow the "if-then" format - - - Summary ------- - - - - [Valid Deductive Arguments]: Modus Ponens = affirming the antecedent Modus Tollens = denying the consequent Valid or Invalid: [^1^]{.math.inline}If you want to get good grades this semester, [^2^]{.math.inline}you will need to study over reading week. [^3^]{.math.inline}You studied over reading week. So,[^4^]{.math.inline}you will get good grades. Invalid: Affirming the Consequent. If A, then B. B. Therefore A. [Example]: Make a Conditional Argument Write out a **Modus Tollens Argument**, where the antecedent is "there is a fire" and the consequent is "there is oxygen present." If *P*, then *Q* If there is a fire, there is no oxygen present. --------------------- ------------------------------------------------- Not *Q.* There is no oxygen present. Therefore, not *P.* Therefore, there is no fire. 11.04.2024 \| Inductive Logic ============================= **2 Kinds of Logic:** 1. 2. -  **Inductive Logic/Reasoning**: - - - [Consider these Arguments]: - - - Commonality: they all have examples Generalizations --------------- - - **Generalization Pattern 1**: - - - - - #### Polls and Surveys Of the surveyed 1030 Canadians, 40% have tried marijuana at least once. - - [^1^]{.math.inline}40% of our sample has tried marijuana. [^2^]{.math.inline}[40%] [±]{.math.inline} [X of Canadians have tried marijuana.] [±]{.math.inline} → margin of error Strength of logical support? Confidence level \*\*Generally 1200-1500 is a good sample size Important assumption missing: [^3^]{.math.inline}Our sample was random **Generalization Pattern 2**: - - - - - **Evaluating Generalizations**: 1. 2. What's wring with the following argument? - - Ensuring the sample is representative: In addition to ensuring the sampel is large enough, you should ensure it is random/representative. - - Simple Induction ---------------- "Are you sure you want to open a Subway franchise on Yonge Street?" "Pretty sure! Why do you ask?" [^1^]{.math.inline}Well, Subway franchises on Yonge street are not successful.[^2^]{.math.inline}Your business would be a Subway franchise on Yonge.[^3^]{.math.inline}So, [your business is not going to be successful]. Subway franchises on Yonge street are not successful. Your business would be a Subway franchise on Yonge. So, your business is not going to be successful. #### **General Simple Induction** [^1^]{.math.inline}Proportion X of group G has characteristic C. [^2^]{.math.inline}Individual I is a member of G. [^3^]{.math.inline}[The probability that I has C is X]. #### **Evaluating Simple Induction** Is I a representative member of G? - Is G made up of sub-groups with very different proportions of X? - **Generalization or Simple Induction?** 1. 2. 3. 4. ### **From Generalization to Simple Induction** Sometimes we generalize to a conclusion about a group, and then use simple induction to get to a further conclusion about an individual. - -  Interference to Best Explanation -------------------------------- The things we observe often leave us needing to determine the causes behind them - "How did that happen?" - - - **Interference**: the word for the mental process of moving logically form one idea to another [Example]: Recall the HIVMind Test kit example involving potentially re-packaged demo kits - What could have caused that? 1. 2. 3. 4. 5. #### **Rule of Thumb for Inference to Best Explanation**: BEST Sales of your test kits are down significantly this quarter Maybe there are rumours that your products are contaminated. **B**: Which explanation fits best with our ***Background knowledge*** **E**: Which explanation has the msot ***Explanatory power***? **S**: Which explanation is ***Simple*** - does not require us to come up with weird stories? **T**: Which explanation is ***Testable***? - does it result in a prediction that, if it comes true, could support or refute your explanation? [Example]: 1. 2. 3. 4. 5. The observed phenomenon and the best explanation for it serve as joint premises for a conclusion about what's going on. #### Evaluating IBE 1. 2. 3. 4. 5. Is O really happening? Or just something similar to O? Are there other explanations we didn't consider? Does E really fulfil the BEST criteria better tahn the other options? The observed phenomenon and the best explanation for it drive as a join premises for a conclusion about what's going on. Question: Do you have to use all of the BEST criteria? No, but you should try to use all of them **What's Missing?** - - Combined Arguments ------------------  [Example]: 1. 2. "Twitter is not an awful place. In fact, Twitter draws people in like it is honey, and everyone likes honey. Plus, if a place is awful, people avoid it. But people don't avoid twitter. And, since a lot of people use Twitter, you probably use Twitter, and you would not use something awful." Claim 2 = analogy Summary ------- **Deductive Argument Types:** - - **Inductive Argument Types**: - - - - 11.11.2024 \| Fallacies & Bias ============================== Recap ----- **Kinds of Logic & Argument Types:** 1. - - - 2. - - - - - Errors in Reasoning ------------------- **2 Classic Categories**: - - In group settings: groupthink ### Argument Fallacies Flawed types of argumetns so common that we've given them names. 2 Categories: 1. 2. - - - **2 Fallacies We Already Know**: 1. 2. - - - **Argumentative Fallacies:** - - - - - - - - #### Argument from Tradition **Argument from Tradition**: any argument that moves from teh premise that[^1^]{.math.inline}some group has done or believed something for long time to the conclusion that[^2^]{.math.inline}[this is what should be done or believed] - [Example]: [^1^]{.math.inline}Our CEO has always also been Chairman of the Board of Directors.[^2^]{.math.inline}Thus, [we should keep doing it this way].  [Example]: As a new junior accountant, you've noticed that Modus measures sales and expense figures by showing a variance against budget. This seems less than ideal. You tell your colleague, Amar, that you're going to suggest measuring this month's sales and expenses against the sales and expenses from the best month in the last year. This seems like a more meaningful measure. But then... Amar says, "Don't waste your time. We've been measuring against budge forever. That's just how we do things. We've done it that way forever. So, we shouldn't change it now." #### False Dilemma **False Dilemma**: any argument that starts from a false premise that there are only two options, claims (or implies) that one of those options is clearly bad and moves to the conclusion that the other one must be accepted. - [Example]: [^1^]{.math.inline}We can either sweep the problem with the contaminated test kits under the rug or discontinue its production.[^2^]{.math.inline}Discontinuing our most successful product is not an option.[^3^]{.math.inline}So, [we need to sweep the problem under the rug]. [Example]: You've heard that Modus is considering expanding, borrowing \$4 million to invest in a new production facility for veterinary test kits. You tell your colleague, Tony, about why Modus should not expand right now. "Look," you say, "the time just isn't right. Interest rates are high, and there's huge uncertainty about the relevant regulatory framework, given the state of Amercian politics right now." But then... Tony says to you, "Look, we need to invest now. Either we expend or we admitthat we're just a small time player. It's either go big or go home! I say go big!" #### Post Hoc Ergo Propter Hoc **Post Hoc Ergo Propter Hoc**: any argument that moves directly from the premise that one thing happened before another thing to the conclusion that the first of those caused the second (without considering alternative explanations) [Example]: The coach yelled at the kids and they did better. Thus, yelling at the kids improved performance.  **What's Missing**: We have considered a number of explanations for claim 1. Claim 2 performs better than all other explanation son the B.E.S.T criteria. So, Claim 2 is the best explanation for Claim 1. [Example]**:** Mia Babie, Modus's marketing co-op student, recently animated the Modus logo on the company's homepage. The next week, visits to the site had increased by about 12%. Thrilled, she reported the results to her boss who seemd less than certain her logo was the cause of the increase. - - "After this, therefore because of this" Or just "the Post Hoc error" - - - **Confusing Correlation with Causation**: - - - - - - - - - ### Cognitive Biases **Cognitive Biases**: subconscious mental tendencies, usually rooted in intuition, that leads us into errors of judgement or decision - \*\*Note: these biases are not "biases" in the racist/sexist sense, etc **Patterns of Thinking Help Us To**: - - - These are powerful, pervasive and resist educational efforts To some extent, they are "hard-wired" #### Framing Effect **Framing Effect**: when conclusion are reached based on how a situation is described, or "framed" [Example]: Modus's current insurance for its small fleet of comapny cars comes with a \$1000 deductible. The ofs of a crash each year are 5%. Jairpreet, a co-op student from TRSM has done the research and found a zero-deductible policy. It would cost \$200/month more than teh current policy. She considers 2 ways of presenting her findings: 1. 2. #### Anchoring **Anchoring**: when we attach too much importance to a single piece of information (especially when stated early in decision-making process) #### Confirmation Bias [Example]: Ishani (VP, Marketing) is worried that sales are likely to suffer next quarter. She has a feeling that hospitals and clinics are feeling the pinch from recent budget cuts and are going to be cutting back on the number of test kits they order. So: At the next big industry conference she attends, she keeps her ears open at the networking breaks and reception, listening carefully for hospital reps talking about budget cuts, or for sales reps from other companies talking about reduced orders. - - #### Survivor Bias **Survivor Bias**: focusing only on the successful people, things, or organizations because those are most visible [Example]: Rajay (CEO): Sarah, do you really think it's wise for us to acquire IntelliPharm, instead of just continuing to buy chemicals from them? Sarah Strong (VP, Business Development): Look, Rajay... I've done an analysis of the companies in that very specialized field. There are really only 6 of them now. They're very profitable. And most of them have shown double-digit growth over the last 5 years. Rajay (CEO): Careful, now. A decade ago, there were 2 dozen companies in that field. Now there are only 6. By focusing on the surviving firms, you're looking at a very biased sample. #### General Advice for Fighting Cognitive Biases Be modest about your own abilities. Do the "numbers" Use formal tools: - - - #### Cognitive Biases: Specific Approaches - - - - - - - ### Groupthink **Other People**: To combat biases, you can also ask other people: - - **How to Help Your Team Avoid Biases & Groupthink?** 2 Interrelated Keys: 1. 2. #### Procedures **Procedures**: have explicit procedures in place that must be followed - mechanisms to wake up system 2 - [Example]: Healthcare Surgeons don't just "try harder" to account for all sponges at the end of surgery. - - - **Procedures for Fighting Groupthink**: - - - - **Conduct a Premortem**: - - - #### Culture **Culture**: shared beleifs and behaviours regarding how decisions should be made, including who should be involved and what processes should be followed Some Ways to Help Combat Erroneous Reasoning -------------------------------------------- Follows the formal process that are there to help guard against biases and groupthink. [Example]: - - - Bosses & majorities must never punish people for challenging their ideas 11.18.2024 \| Compound Probability and Insurance ================================================ EV/EU Review ------------ To calculate EV (or EU) when dealing with probability, ask: 1. - 2. - 3. - 4. - [*EV* = (*p*~1~ × *V*~1~) + (*p*~2~ × *V*~2~)+ \.... + (*p*~*n*~ × *V*~*n*~) − *fixed* *cost*]{.math.inline} #### Example: You're thinking about purchasing last minute tickets to an event for you and your best friend, but unfortunately you cannot get a hold of your friend to confirm their availability. Since, tickets are about to sell out, you have to decide whether to buy them or not. Since you know your best friend well, you suspect an 85% chance they are free and would come. The ticekts cost \$75, but your friend would only reimburse you for their ticket if they're free and able to come. You also would not be able to offload the tickets in the event your friend cannot come - you are stuck with them. Further, if they don't come you would go alone but would not enjou that as much. Going alone would give you a 3 utility. But if your friend comes, you would have an 8 utility. Calculate: 1. 2. **Solution**: Sinceyou know your best friend well, you suspect an 85% chance they are free and would come. The tickets cost \$75, but your friend would only reimburse you fro their ticket if they're free and able to come. [*EV* = (0.85×]{.math.inline} -[\$75) + (0.15×]{.math.inline} -[\$150)]{.math.inline} Further, if they don't come you could go alone but would not enjoy that as much. Going alone would give you a 3 utility. But if your friend comes you would have an 8 utility. [*EU* = (0.85 × 8) + (0.15 × 3)]{.math.inline} Compound Probability -------------------- - - 1. 2. #### Example: Conditional Events Modus formerly had a technology licensing agreement with another company, but the relationship has gone sour and the former partner owes Modus \$50,000. They are, however, refusing to pay, so Modus is considering litigation. It's possible that they could lose in court, and it's possible that they could win but be awarded only part of the total amount owed. Suppose when Modus threatens to sue, the former partner offers to settle the matter for \$10,000. The decision: Should Modus accept the offer, or go to trial? **Solution**: EV of accepting the settlement = \$10,000 1. 2. 3. 4. 1. 2. 3. 4. [*EV* *of* *Going* *to* *Trial* = (0.2 × 0) + (0.08 × 50, 000) + (0.6 × 25, 000) + (0.12 × 10, 000)]{.math.inline} 20% chance of losing in court 80% chance of winning in court: [*EV* *of* *Going* *to* *Trial* = (0.2 × 0) + (0.08 × 50, 000) + (0.6 × 25, 000) + (0.12 × 10, 000)]{.math.inline} **Conclusion**: We want to maximize expected value. That means choosing the option with the highest expected value. - - Does this mean you are guaranteed to get more than \$10k if you go to court? No. ### Summary of Compound Probabilities 1. - - 2. - Expected Value & Insurance -------------------------- - - - #### Example: Warranty costs \$50 for your lab's new \$500 centrifuge. Replaces the device if it dies within 3 yrs. Estimate 5% chance of centrifuge dying wihtin that period (and you NEED to have a centrifuge, so if it dies you need another) Assume you've already bought the centrifuge. Will you ALSO buy the warranty? What is the expected value? **Option 1: Don't buy the warranty** [*EV* = (95% × \$0) + (5%×]{.math.inline} -[\$500)]{.math.inline} **Option 2: Buy the warranty** [*EV* = (95% × \$0) + (5% × \$0) − \$50]{.math.inline} ### Insurance Coverage Types To purchase insurance, you pay a premium, for your chosen policy. Then, insurnace companies often offer 4 types of insurance: #### Example: Deductibles Imagine insurance costs \$6000/year for your \$500,000 warehouse. It replaces the warehouse if it burns down. Assume 1% chance of such a fire per year. The Policy has a deductible of \$10,000. **Option 1: Don't buy insurance** [*EV* = 1%×]{.math.inline} -[\$500, 000) + (99% × \$0)]{.math.inline} [=]{.math.inline} -\$5000 **Option 2: Buy insurance** [*EV* = (99% × \$0) + (1%×]{.math.inline} -[\$10, 000) − \$6000]{.math.inline} **Alternative Calculation**: [*EV* = (99% − \$6000) + (1%×]{.math.inline} -[\$16, 000)]{.math.inline} [=]{.math.inline} -\$6100 #### Example: You have a \$200,000 cottage and a \$10,000 shed. Chance of both buildings burning down: 1% each year; chance of only sed burning down: 3% each year; chance of only cottage burning down: 1% each year **Option 1: Don't buy insurance** [*EV* = (95% × \$0) + (1%× ) + (3%× ) + (1%× )]{.math.inline} **Option 2: Buy insurance policy A (\$4000 premium; \$5000 deductibles)** [*EV* = (95% × \$0) + (1%× ) + (3%× ) + (1%× )]{.math.inline} **Option 3: Buy insurance policy B (\$2000 premium; \$100,000 max coverage)** [*EV* = (95% × \$0) + (1%×]{.math.inline} -[\$110, 000) + (3% × \$0) + (1%×]{.math.inline} -[\$100, 000) − \$2000]{.math.inline} **Conclusion**: - - - Option 3 maximizes the expected value  #### Example: Calculate: the expected value and the expected utility of purchasing the tickets without being able to contact your friend first. Modus has just purchases a new company van for \$55,000 complete with a detachable trailer for \$4000. Linda Hicks, the Senior VP and CFO has asked you to pick the best auto insurance for them. Since the van is driven quite a bit, making deliveries, picking up orders and tkaing staff members to various events - the likelihood of geting into an accident within the year is hgiher than normal. You estimate there is abotu a 4% chance the van is in an accident. If that happens, you think there is a 75% chance that the van will be totally unusable but the trailer would be fine. You think the chance of hte van benign fine, while trialeer is destroyed (in case of an accident) is 5%. And there is a 20% chance that an accident would do full famage to both van and trialer. **State Farm**: Full insurnace at premium of \$1500 **CCA**: \$50,000 max coverage at premium of \$1100 **Geico**: premium of \$1250 with a \$5000 deductible [96%]{.math.inline} chance no accident [4%]{.math.inline} chance there will be an accident **Conditional**: - - - [0.03 + 0.002 + 0.008 = 0.4 = *chance* *of* *accident*]{.math.inline} **No Insurance**: [*EV* = (0.96 × 0) + (0.03 × \[ − 55 000\]) + (0.002 × \[ − 4000\]) + (0.008 × \[ − 59 000\])]{.math.inline} **State Farm**: [*EV* = (0.96 × 0) + (0.03 × 0) + (0.002 × 0) + (0.008 × 0) − 1500]{.math.inline} **CAA**: [*EV* = (0.96 × 0) + (0.003 × \[ − 5000\]) + (0.002 × 0) + (0.008 × \[ − 9000\]) − 1100]{.math.inline} [=]{.math.inline} -{.math.inline} **Geico**: [*EV* = (0.96 × 0) + (0.03 × \[−]{.math.inline} 0-150-8-40-1250 [=]{.math.inline} -1448 ### You Can't Afford to Lose? Why do you buy house insurance even when it doesn't maximize value? - - - ### When Not to Maximize Expected Value? - - - Summary ------- - - - - - - - - - - - 11.25.2024 \| Creative Thinking & Virtuous Arguers ================================================== Creative Thinking ----------------- **Argument:** a set of claims in which some of those claims are offered as reasons in support of another Every argument must have: - - **Cretivity in Argumentation:** - - - - - - - **Crtiical Creative Capacity** We are seeking: 1. 2. 3. **Creativity & Decisions**: What's a decision? - - - ### Why Creativity Matters You're wondering whether your company should invest in BMTech, a new start-up with a patent on biodegradable needles. After conducting some research, you think it is wise to invest, and now you need to express your reasons to convince your boss. The **better the reasons, the stronger the support for your conclusion**, the more likely the boss will be to agree with you. **If you cannot find good reasons, that may be a sign that your conclusion should be changed**. ### Generating Reasons: The ECO Approach **Evidence** - What evidence do you have to support the belief? - **Community** of Experts of Opinion Leaders - What would the smartest people in the field say? - **Objections** - How might someone disagree? What can you say in advance to stifle their objection? - - - **Creating Questions**: This strategy is based on answering good questions. Having better questions can contribute to producing better answers/reasons. **Inquiry Questions**: - - - - - - - - - - - - - ### Creativity, Argument Types and Fallacies - - - - ### The Role of Creativity - - - - - Exemplary Critical Thinkers --------------------------- Resolving Disagreements ----------------------- - ### Disagreement - - - - - - ### Pragma-Dialectical Theory Ideally, a disagreement can be resolved through argumentation - 1. 2. 3. 4. **Confrontation**: - - - - - From Confrontation to Opening: - - - - - - **Opening**: Participants settle on some ground rules: - - - - - - - **Argumentation**: - - - - - - - **Closing**: What's the result? - - - - - ### 10 Rules for Conducting a Critical Discussion The 4 Stages tell you **what** should happen, but they don't say as much about how it should happen - - - - **Different Arguers**: - - - - **What kind of arguer would we like to discuss with?** - - - - - - ### What is a Virtue? - - ### Intellectual Humility - - - - ### Intellectual Empathy - - ### Intellectual Courage  **Where can we virtuously argue?** - - - - - ### Augmentation in Negotiation **Negotiations**: - - - **The Role of Argmentation**: - - - - **Mediation**: - - **The Role of Argumentation:** - - - - **Legal Action**: - **The Role of Argumentation**: - - - - Summary ------- - - - - - - - - - Diagramming Practice --------------------