# 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 salad than pizza
• $$\text{watch Netflix} \succsim \text{study}$$" = 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 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
• 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 $$v$$
• 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 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: $$u_v (x) = -|v-x|$$, where $$v = 0.5$$
• what is $$v$$'s expected utility if policy is $$0$$ w/ prob 60% and $$1$$ w/ prob 40%?