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
 objects (pizza, hamburger, salad, other things to eat for lunch)
 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 salad than pizza
 ``\(\text{watch Netflix} \succsim \text{study}\)" = you'd rather watch Netflix than study
 if \(x,y \in X\), then \(x \succsim y\) means that agent weakly prefers \(x\) to \(y\)
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 rap 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
 nontransitive example: ``an enemy or my enemy is my friend""
 reflexivity says that \(x \succsim x\) and assumed to always be true
 completeness: agent can compare any two options in choice set \(X\)
 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 \(v\)
 utility of policy \(x \ne v\) decreases with distance between \(x\) and \(v\)
 \(u_v(x) =  vx\)
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 ultraleft to ultraright
 \(L_1\): 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
 rationality conditions (for your reference): complete, transitive, continuous, independent
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 value
Examples of Expected Utility: Politicians
 set of outcomes: \(X = \{ A \text{ wins}, B \text{ wins} \}\)
 \(A\)'s utility function: \(u_A (A \text{ wins}) = 1\) and \(u_A (B \text{ wins}) = 1\)
 what is \(A\)'s expected utility if \(A\) wins with probability 60%? $$ L_1 = (0.6,0.4) \text{ so that } \mathbb{E}_1 [u_A (x)] = $$
Examples of Expected Utility: Moderate Voter
 set of outcomes: \(X = [0,1]\)
 \(v\)'s utility function: \(u_v (x) = vx\), where \(v = 0.5\)
 what is \(v\)'s expected utility if policy is \(0\) w/ prob 60% and \(1\) w/ prob 40%?