# 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 pizza than salad
- \(\text{watch Netflix} \succsim \text{study}\) = 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 \(=\))

- \(\succ\) denotes
- 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*

- economists get bad rap for assuming rationality \(\longleftarrow\)
**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?*

- transitivity implies that \(c_1 \sim c_{1000}\) \(\longleftarrow\)

## 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 \(v\)
- \(u_v(x) = - |v-x|\)

- set of policies is \(X = [0,1]\), where \(0\) is

## 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

- 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 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%?