Model thinking
Micro motives ≠ Macro behaviour
That is: observed macrobehavior does not automatically imply biased micromotives!
Schelling model of segregation model (Schelling's Tipping Model -> Agent based model). When deciding whether to stay or move house within a neighborhood each person has a threshold of how much they like to be with alike people based on which they decide to stay or move (e.g. Sonja\'s current threshold is 35%, i.e., she needs at least 35% of her neighbors to be of the same type in order for her to remain at her location.)
Sample Models/Social Science/Segregation
Even
though people might be comfortable with only 30 to 40% of their
neighbors being like them, the mathematics of that preference shows that
sorting of the sort seen in the New York map above is the natural
consequence. The acceptable micropreferences (microbehavior) does not
produce the macrobehavior that is believed to be desireable by
politicians or even what X believes they would like in diveristy.
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Threshold = 40%: => initially 28.8% unhappy and sorting yeilds 79.5% similarity to neighbors and 0% unhappy
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Threshold = 52%: initially 50.6% similar & 57.4% unhappy. Sorting yields 93.8% similar with 0% unhappy. Note 'black' boundary (empty) homes.
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Threshold >= 80%: Initially 50.6% similar and 89.6% unhappy. Sorting is not stable as happiness threshold cannot be met. In reality this suggests people move continuously. Moving continuously, churning
How to make people follow examples (standing ovation model)
Sorting -- People that hang out together tend to act and look alike. People that act and look alike tend to live and hang out together. (as a result, people would move)
PEER EFFECTS- Stop smoking because of peer influences (looking, acting like peers) (as a result people would switch)
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Collective actions
Threshold to buy a purple hat (if x amount has the hat, I buy it too):
| Person | Threshold | Will the buy? | Person | Threshold | Will the buy? | |
|---|---|---|---|---|---|---|
| 1 | 1 | No | 1 | 0 | Yes | |
| 2 | 1 | No | 2 | 1 | Yes | |
| 3 | 1 | No | 3 | 2 | Yes | |
| 4 | 2 | No | 4 | 3 | Yes | |
| 5 | 2 | No | 5 | 4 | Yes | |
| Average | 1.4 | Average | 2 |
Collective Action: More likely (a) if lower thresholds and (b) if more variation in the thresholds\ Note that more variation implies more likely to have some lower thresholds in the population to trigger and epidemic of behavior change. Not just average thresholds but distribution of thresholds determines cascades.
Decision Trees
http://silverdecisions.pl/SilverDecisions.html?lang=en
Imagine the following scenario. You\'re planning a trip to a city and you\'ve got a ticket to go to the museum, let's say from one to two. And suppose the museum is quite a way from the train station. So, you look at train ticket prices and you see you can buy a ticket for the three o\'clock train. For only \$200. But the four o\'clock train is \$400. You know, should I try and save money by buying that ticket for the three o\'clock train before it sells out? Now there\'s a 40 percent chance you\'re not going to make. The train. So now you got to think, oh my gosh, should I but the ticket or not? Give there\'s a 40 percent chance I\'m not going to make it. And if I don\'t make it, then I\'m going to have to buy two tickets. I\'m going to basically throw away the \$200.
.6*200+.4*600 = 360 vs. 400 better go for the buy option.
Inferring probability
Investment opportunity to win 50,000 \$ while investing 2,000\$ what probability of winning do we need to pay off?
50,000p -- 2,000 (1-p) > 0
50,000p -2,000 + 2,000p > 0
52,000p > 2
p > 52,000/2 = 4%
50 * x + -2 * (x-1) > 0
Infer payouts
Assuming that there is a cost to go to the Airport
1/3 * (VisitPayoff -- CostPayOff) + 2/3 (-CostPayOff) \< 0
1/3 VisitPayOff -- 1/3 CostPayOff - 2/3 CostPayOff \< 0
1/3 VisitPayOff - CostPayOff \< 0
1/3 VisitPayOff \< CostPayOff
VisitPayOff \< 3 * CostPayOff (if you didn't go it means the visit PayOff was less than 3 times greater than the cost payoff
Value of information
How much would I pay to know that there is going to be a cashback program?
- Without information (I'm better of buying now with 0\$ pay off)
- With Information (I better rent and with a payoff of \$200)
- \$200 -- 0\$ = \$200
Modeling people
When we construct a model, we want to think about how the agents will be acting: rationally, behaviorally, or according to simple rules. We gather data about how agents will act and use this information to construct the model.
Rational
When to expect rational thinking:
Large stakes involved
Repeated decisions offer learning
Group Decisions
Easy decisions.
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"As if": Intelligent rule-based behavior may be indistinguishable from optimal or near-optimal behavior.
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Benchmark: Optimal behavior provides a benchmark as an upper bound on people's cognitive abilities.
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Logical but people are not always logical.
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Will describe application to decision making and games.
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Actors have an objective and optimize to that objective.
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Firms -- maximize profits,
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Individuals -- maximize utility,
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Political Candidate -- maximize votes
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Apply to: Investments, Purchases, Educational level, Vote for
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Rational does not imply selfish
Optimize revenue
Revenue = price * quantity
Quantity = q
Price = 50 -- q (more supply lowers price)
Optimize q for maximum revenue
Revenue = (50-q) * q
Behavioral
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Observe people are not rational.
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Prospect Theory -- gains and losses viewed differently
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Hyperbolic Discounting -- how much we discount the future
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Status Quo Bias -- tendency to stick with what we have and avoid changes.
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Base Rate Bias -- influenced by what we are currently thinking
Rule based
e.g., Schelling, simple rule that is close to what people do.
Tipping points
Basic Idea: A small change in input makes a very large change in output as seen in the graph. Note: Time charts can be misleading.
http://www.netlogoweb.org/launch#http://www.netlogoweb.org/assets/modelslib/Sample%20Models/Earth%20Science/Fire.nlogo
Tipping point = 59% in terms of tree density in a forest and the spread of fires
Rule of 72
he Rule of 72 is a simple way to determine how long an investment will take to double given a fixed annual rate of interest. By dividing 72 by the annual rate of return, investors obtain a rough estimate of how many years it will take for the initial investment to duplicate itself.
Exponential growth: If you start with 1 penny and double it every day you have more than 10 Million after 31 days
Why do some countries not grow?
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Growth requires a strong central government to protect capital and investment
- BUT that government cannot be controlled by a select few. Extraction (by protected government supporters) limits growth by lowering investment and capital.
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Increases in innovation means less labor is required.
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Growth requires creative destruction
- Growth rates are supported by innovation in the long run.
Innovation
Teams:
Teams have different perspectives and different heuristics, and all that diversity makes them better than individuals on average.
Recombination:
Big Idea: Solutions from one problem and a solution from another problem are combined to make a better solution.
Recombining solutions from a number of subproblems yields better solutions.
Culture
Summary: Basic Cultural properties: there\'s a lot of difference between cultures, and those difference may arise because the fact that people need to coordinate within groups with which they interact. There\'s also consistency within cultures, and that happens because it gets cognitively easier to do the same behavior in lots of different domains. And then third, we see a lot of heterogeneity within cultures. And that happens, as we saw by using a Markov analysis on our model, because if people just make small mistakes or occasionally try an innovation, those differences are going to propagate through the population in two ways; within an individual and across individuals. And that\'s going to give us a lot of within-culture heterogeneity. So cultures differ between themselves. Cultures differ within themselves. But they still have this consistency. They have what you might call a cultural signature. These very simple models combined with our tools of Lyapunov functions and Markov processes have helped us understand why that happens.
Condorcet\'s Jury Theorem
Condorcet\'s jury theorem states that given a group of voters (a
\"jury\") independently choosing by majority vote between a correct
outcome with probability
and
an incorrect one with probability
:
1. If
(so
that each voter is more likely to vote correctly that incorrectly),
adding more voters increases the probability that the majority chooses
correctly and the probability of a correct decision approaches 1 as the
number of voters increases;
2. If
(so
that each voter is less likely to vote incorrectly than correctly),
adding more voters decreases the probability that the majority chooses
correctly and the probability of a correct decision is maximized for a
jury of size one.
Terminal velocity
Terminal velocity is the maximum velocity attainable by an object as it falls through a fluid (air is the most common example).
Falling skydivers reach terminal velocities of 200 mph. Terminal velocities scale with the inverse of mass. Assume that a skydiver has a mass 400 times larger than the stuffed cheetah. For a heavy object, the terminal velocity is generally greater than a light object. This is because air resistance is proportional to the falling body\'s velocity squared. The square root of 400 equals 20. Therefore, the terminal velocity of the stuffed cheetah will equal 200 mph divided by 20, or 10 mph.
| Object | Mass | Terminal velocity |
|---|---|---|
| Sky diver | 100 kg | 200 kph |
| Stuffed animal | 100/400 = .25 kg | 200 kph / Sqrt(100 kg / .25 kg) = 10 kph |
https://hypertextbook.com/facts/1998/JianHuang.shtml
Entropy
Entropy measures the uncertainty associated with a probability distribution over outcomes. It therefore also measures surprise. Entropy differs from variance, which measures the dispersion of a set or distribution of numerical values. Uncertainty correlates with dispersion, but the two differ. Distributions with high uncertainty have nontrivial probabilities over many outcomes. Those outcomes need not have numerical values. Distributions with high dispersion take on extreme numerical values.
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Capture the evenness of a distribution across types. (kinda variation for categorical values)
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Also called Simpson's index
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Does not take the differences between the types into account (e.g. differences between apples and peers vs. elephants and apples)
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E.g. colors of tables
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There is 1/3 red 1/3 white and 1/3 white tables
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Square each result (i.e. 1/3^2^*3) = 1/3
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Take the inverse of the number = 3/1
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The entropy is higher the more types there are
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The entropy is lower if the proportions of the type are unequal (more of one type as compared to the others)
Classes of outcomes
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Equilibrium
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Cyclic
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Random
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Complex
A pencil resting on a desk is in equilibrium. The planets orbiting the sun are in a cycle. A sequence of coin flips is random, so are (approximately) stock prices on the New York Stock Exchange, as we shall learn in the next chapter. Finally, the neuronal firings in a person's brain are complex; they do not fire randomly, nor do they fire in a fixed pattern.
Bernoulli
A player who makes 46% of his three-pointers has about a 1/1000probability of making nine in a row (0.46^9^). If that player keeps taking three-pointers, then in a ten-year NBA career (about 800 games), the odds of not making nine in a row at least once (0.999^800^) are about 47%.
Networks
Degree
Measure of connectedness of a network. For one node it is the number of nodes it is connected to. For a network it is the average degree of all nodes. But can also be measured as:
2 x # Edges (because each edge connects to nodes) / Nodes
A network has 50 nodes and 100 edges. What is the Average Degree of the network? 2 * 100 / 50 = 4
Node = Dots
Edges = Connections
| Node | # friends |
|---|---|
| A | 2 |
| B | 3 |
| C | 2 |
| D | 1 |
| Average | 2 |
| Node | Friends | # Friend’s friends | Average of friend’s friends |
|---|---|---|---|
| A | B C |
3 2 |
2.5 |
| B | A C D |
2 2 1 |
1.66 |
| C | A B |
2 3 |
2.5 |
| D | B | 3 | 3 |
| Average | 2.4 |
Numbers of triangles (see below) over of all possible triangles
Number of possible triangles = number edges choose 3.
E.g. 4 choose 3 = 4 =FACT(4)/(FACT(3)*FACT(4-3))
Small worlds network model http://www.netlogoweb.org/launch#http://www.netlogoweb.org/assets/modelslib/Sample%20Models/Networks/Small%20Worlds.nlogo
Preferential attachment model http://www.netlogoweb.org/launch#http://www.netlogoweb.org/assets/modelslib/Sample%20Models/Networks/Preferential%20Attachment.nlogo
Randomness
Outcome = a (luck) + (1-a) (skill) -- with a 0 \<= a \<= 1
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Use performance to determine a.
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If the outcome is consistent then a is small (luck doesn't play a big role). If the outcome is showing huge variations a is big.
Reasons to use this model
(a) the model helps to assess outcomes
(b) anticipate reversion to the mean (if luck is involved people will not win or lose consistently)
(c) give good discerning feedback (in case the outcome is due to skill)
(d) fair allocation of resources (skill or luck(fairness).
The Paradox of Skill (Maubossin): When you have the very best competing, the differences in their skill levels may be close. So the winner will be determined by luck!
Random walks.
Summary: An outcome may be a series of random events and thus we should expect to see some big winners and some big losers. So we cannot expect past performance to be a good indicator of future performance. Thus key question is -- is this a random walk process -- or not? If so, expect regression to the mean.
Efficient Market Hypothesis helps explain although there may be some deviations / challenges as noted in the critiques.
This is like a sliding window of randomness affects the overall outcome (e.g. employees leaving and new employees getting hired; new and old products, team's of players)
Strategy
Colonel Blotto: A way to analyze competition, along multiple fronts trying to find advantages through strategic mismatch your opponents choices.
Insight: Weaker player improves opportunities by opening more fronts (dimensions). Since this will make the stronger opponent to spread out their troops more so that the weaker opponent has the ability to win on some fronts at least. Example:
differentiation in products, terrorists attack in unexpected places.
~Prisoners'\ dilemma~
Two players who can either cooperate or defect. Collectively they are better off if both cooperate but individually, they are better off if they defect (avoiding 0 pay-off)
Examples
| First move | Second move | Pay-off P1 | Pay-off P2 |
|---|---|---|---|
| P1 = C | P2 = D | 0 | 6 |
| P1 = D | P2 = D | 2 | 2 |
Different application of PD
Firms would be better of with high prices but since they are individually better off with lower (gaining more market share over the competition) prices they will end up with lower prices.
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2 banks decide either not to buy ATM machines or to buy ATM machines
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If they don\'t buy ATM machines Both make profits of, let\'s say, four.
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if one of them buys buys ATM machines, and puts them up all over town, everyone will go to that bank.
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The other bank is forced to do the same. Now, both of them have probably the same customers they had to start out with, but they\'ve spent all this money on ATM machines. So they\'re actually worse off.
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In fact, it could even be worse than that. Because maybe before, part of the reason they made such profits is because they got geographic grants. People who lived around the bank shopped at that bank. Now that there\'s ATM machines anywhere, people can shop whatever bank they want. And that\'s created more price competition, we end up with the consumers being better off.
When player 1 gets the expensive hand made match he gets all the attention.
Ways to achieve cooperation
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Repeated Interactions (direct reciprocity),
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Reputation (indirect reciprocity)
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Networks
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Group Selection Influence
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Kin Selection
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Laws (prohibitions)
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Incentives (fines, fees, and subsidies).
Collective action problems
There is a cost to the individual if they take action but, yet a benefit to all if they do so. (free rider problem)
Assume 10 people with cooperation payoff = 1 and defection payoff = 0 for everyone and ß = 0.6. If x~1~ defects his payoff is 5.4 but if cooperates his payoff is 5.0 so it is his advantage to defect to the expense of his colleagues.
x~1~ takes action: 0 + .6 * 9 = 5.4 - 0 because no cost of taking action; 9 because x~1~ takes no action
x~1~ takes no action: -1 + .6 * 10 = 5 - -1 = cost for x~1~ to take action; 10 = including x~1.~
Common pool resource problems
Limited resource available to be used by many
Mechanism design
Designing incentive structures so that we get the sort of outcomes we want.
Reveal hidden actions/information.
Incentive compatibility = It makes sense for a worker to put in effort if the amount of money received minus the cost of effort must be greater than or equal to the probability that if they slack off, the outcome will be good anyways.
Fisher's theorem
More variation = faster adaptation
Fisher's theorem vs. six sigma
Diversity Prediction Theorem
(a) more accurate individuals imply more accurate predictions
(b) more diversity in crowd implies more accurate predictions.
Mechanism Design
An institution consists of a means through which people communicate information as well as a procedure for making decisions, reallocating resources, or producing outputs based on the information revealed. In markets, people and firms communicate through prices to execute trades and make production decisions. In hierarchies, people communicate through written language to organize work plans. And in democracies, people communicate preferences through votes. Voting rules then decide policies. Well-designed institutions induce communications and actions that produce desirable outcomes. Ineffective institutions do not.
Mechanism Design highlight four aspects of institutions:
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information, what the participants know and should be revealed to them;
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incentives, the benefits and costs of taking particular actions
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how the individual incentives actions translate into collective outcomes;
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computational costs, the cognitive demand placed on participants.
Mount-Reiter diagram
A mechanism consists of six parts:
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an environment (the relevant features of the world)
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set of outcomes,
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a set of actions (called the message space)
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a behavioral rule that people follow to produce actions
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an outcome function that maps the actions into outcomes
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a social choice correspondence that maps the environment into a set of hoped-for outcomes.
Within a set of outcomes, an outcome is Pareto dominated if there exists an alternative that everyone prefers. All other outcomes are Pareto efficient.
For an example of Pareto efficiency, consider the following four payoff profiles for three people:
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{(3, 3, 4), (9, 0, 0), (0, 8, 1), (2, 2, 3)}
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All except (2, 2, 3) are Pareto efficient. The allocation (2, 2, 3) is dominated by (3, 3, 4).
