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%?