Crash Course in Decision Theory

Review of Set Theory

a set is a collection of distinct objects
- \(A = \{1,2,3,4,5\}\) (numbers)
- \(B = \{ pizza, hamburger, soda \}\) (food items)
- \(C = \{ \text{vote for incumbent}, \text{vote for challenger} \}\) (strategies)
- \(D = [0,1]\) (points on a unit interval)
we say \(x \in X\) if \(x\) is an element of set \(X\), and \(y \notin X\) is \(y\) is not an element of \(X\)
- \(1 \in A\) but \(6 \notin A\)
- \(0.67 \in D\) but \(1.1 \notin D\)

Choice and Preferences

choice set \(X\) is a set of things to choose from
- objects (pizza, hamburger, salad, other things to eat for lunch)
  - actions (go to a bar, watch Netflix, study, other things to do on a Friday night)
  - any time you make a decision, you pick an element of some choice set
agent's preference relation \(\succsim\) over elements of \(X\) ranks elements of \(X\)
- if \(x,y \in X\), then \(x \succsim y\) means that agent weakly prefers \(x\) to \(y\)
  - ``\(\text{pizza} \succsim \text{salad}\)" = you'd rather eat pizza than salad
  - ``\(\text{watch Netflix} \succsim \text{study}\)" = you'd rather watch Netflix than study

Choice and Preferences, Continued

\(\succsim\) is similar to \(\geq\), but for individual's preferences
- everyone agrees that \(5 \geq 4\) but \(\text{pizza} \geq \text{salad}\) is an opinion and not a fact
- hence we use \(\text{pizza} \succsim \text{salad}\) -- in this person's opinion, pizza is better than salad
other relations:
- \(\succ\) denotes strict preference (similar to \(>\))
- \(\sim\) denotes indifference (similar to \(=\))
natural connection: \(x \succsim y\) implies \(x \succ y\) or \(x \sim y\)

Rationality (in Actions)

for any economic model, we need to assume rationality in actions and preferences
- economists get bad rep for assuming rationality \(\longleftarrow\) let's debunk this claim
rationality in actions
- what it means: agents choose the best action out of available options
rationality in actions imposes NO conditions on preferences
- agents can be selfish, narcissistic, altruistic, greedy, hateful, etc
in practice, we usually make simplifying assumptions about agent's preferences
- politicians want to win elections, voters want to elect their favorites, etc

Rationality (in Preferences)

preference relation \(\succsim\) is rational if it is complete, transitive, and reflexive
- completeness: agent can compare any two options in choice set \(X\)
  - for any \(x\) and \(y\) in \(X\), either \(x \succsim y\) or \(y \succsim x\)
  - if you asked them to compare two options, they never say I don't know
- transitivity: given three options \(x\), \(y\), and \(z\) in \(X\), if \(x \succsim y\) and \(y \succsim z\), then \(x \succsim z\)
  - transitive example: if \(A\) is \(B\)'s sibling and \(B\) is \(C\)'s sibling, then
  - non-transitive example: ``an enemy or my enemy is my friend""
- reflexivity says that \(x \succsim x\) and assumed to always be true
if \(\succsim\) is rational, then so are \(\succ\) and \(\sim\)

Transitivity

transitivity may seem innocuous, but it may be violated even by reasonable preferences
suppose \(X\) is a (choice) set of \(1000\) cups of coffee
- cup \(c_1\) has \(1\) drop of coffee replaced with water
- cup \(c_2\) has \(2\) drops of coffee replaced with water
- ...
- cup \(c_{1000}\) has \(1000\) drops of coffee replaced with water
unless one has exceptional taste buds, \(c_1 \sim c_2\), \(c_2 \sim c_3\), ..., \(c_{999}\sim c_{1000}\)
- transitivity implies that \(c_1 \sim c_{1000}\) \(\longleftarrow\) do you think this is true for you?

Utility Function

utility function \(u: X \to \mathbb{R}\) represents preference relation \(\succsim\) if \(x \succsim y \iff u(x) \geq u(y)\)
if preference relation \(\succsim\) is rational, then we can find utility function \(u\) to represent it
- rationality (in actions and preferences) = maximizing utility
we will simply assume that all our actors have a utility function that they maximize

Examples of Utility Functions: Politicians

politicians prefer to win the elections
- set of outcomes is \(X = \{ A \text{ wins}, B \text{ wins} \}\)
- candidate \(A\) receives utility of \(1\) if she wins and utility of \(0\) if she loses (\(B\) wins)
- \(u_A(A \text{ wins}) = 1\) and \(u_A(B \text{ wins}) = 0\)

Examples of Utility Functions: Voters

voters prefer policies close to their bliss points (ideal policies)
- set of policies is \(X = [0,1]\), where \(0\) is left (blue) and \(1\) is right (red)
- voter has ideal position \(v \in X\)
- utility of policy \(x \ne v\) decreases with distance between \(x\) and \(v\)
- \(u_v(x) = - |v-x|\)

Utility Maximization

what does it mean to maximize utility?
- choose element with the highest level of utility
actors are usually constrained in some way
- micro: consumers pick best bundle given budget constraint
- micro: firms minimize cost given technology constraint
- labor economics: people decide between work and leisure given 24 hours in a day
what are the possible constraints that agents face in elections?

Uncertainty

we often deal with situations involving risk (= uncertainty)
- exogenous: what is the probability that it rains on election day
- endogenous AKA strategic: what is the probability that government is overthrown a month after election?

Preferences Under Uncertainty

instead of a choice set, we have a set of lotteries
- let \(X\) be the set of outcomes
- let \(\Delta X\) be the set of all probability distributions over \(X\)
- a lottery \(L\) is a probability distribution over \(X\) (\(L \in \Delta X\))
  - each outcome is assigned a probability (possibly \(0\)), probabilities sum up to \(1\)
preferences under uncertainty = preferences over lotteries

Preferences Under Uncertainty: Binary Outcome Space

example: election with two candidates, set of outcomes \(X = \{ A \text{ wins}, B \text{ wins} \}\)
- \(L_1\): \((0.5, 0.5)\) \(\longleftarrow\) each candidate wins with probability \(1/2\)
- \(L_2\): \((0,1)\) \(\longleftarrow\) candidate \(B\) wins for sure
does candidate \(A\) prefer \(L_1\) or \(L_2\)?

Preferences Under Uncertainty: Continuous Outcome Space

set of outcomes \(X\) could be continuous (as opposed to finite)
- examples: \([0,1]\), \([0, + \infty)\), \((-\infty, +\infty)\)
- if \(X\) is continuous, then a lottery could be a continuous distribution
example: \(X = [0,1]\), policy space from ultra-left to ultra-right
- \(L_1\): uniform distribution \(U[0,1]\)
- \(L_2\): \(0\) with probability \(0.75\) and \(1\) with probability \(0.25\)
does voter \(v\) prefer \(L_1\) or \(L_2\)?

From Preferences to Expected Utility

if preference relation over lotteries satisfies rationality conditions, we can find an expected utility function
- rationality conditions (for your reference): complete, transitive, continuous, independent
  - completeness and transitivity are same as without uncertainty
  - continuity: if you change a lottery by a little, the ranking should stay the same
  - independence: your ranking of two lotteries should not depend on a third lottery

Expected Utility Function

expected utility function \(u: X \to \mathbb{R}\) represents preference relation \(\succsim\) if
- \(L_1 \succsim L_2\) is equivalent to \(\mathbb{E}_1 [u(x)] \geq \mathbb{E}_2 [u(x)]\), where
- \(\mathbb{E}_i [u(x)]\) is expected value of \(u(x)\) under distribution \(L_i\)
maximizing expected utility = picking lottery with highest expected utility
- not the same as picking lottery with highest expected value

Examples of Expected Utility: Politicians

set of outcomes (results of election): \(X = \{ A \text{ wins}, B \text{ wins} \}\)
\(A\)'s utility function: \(u_A (A \text{ wins}) = 1\) and \(u_A (B \text{ wins}) = 0\)
what is \(A\)'s expected utility if \(A\) wins with probability 60%?
- \(L_1 = (0.6,0.4)\) so that \(\mathbb{E}_1[u_A(x)]=\)

Examples of Expected Utility: Moderate Voter

set of outcomes (policies): \(X = [0,1]\)
\(v\)'s utility function: \(u_v (x) = -|v-x|\), where \(v = 0.5\)
what is \(v\)'s expected utility if policy is \(0\) w/ prob 60% and \(1\) w/ prob 40%?