# Lecture Notes

PHIL 401C

## Table of contents

## Week 1 Wednesday

- Game thoery - study of strategic interactions between multiple DMs, almost anything counts as a game.
- Descriptive GT - is; Normative GT - ought.
- Equilibrium concepts - used to characterize how agents interact or should interact.
- Ultimatum game \(\in\) bargaining game - dividing resources.
- Subgame perfect equilibrium of the ultimatum game - proposal offers lowest possible amount of money and the second accepts.
- What happens when players carry about things other than money?
- Theories of individual rationality and how it can transfer to group decision-making.
- Check participation instructions.
- In the decision matrix:
- Actions - row headers, what is within the DM’s control.
- States - column headers, states of the world, what is not within the DM’s control.
- Outcomes - cells in the matrix; represent the result of performing an action at a state.

- We assume a DM has preferences over the outcome space.
- Can you attach numbers to preferences to preserve ordering? Philosophically, this should give us pause.
- One act dominates another if it is preferable in all states of the world.
- Weak dominance - an action weakly dominates another if it is preferred to the other in at least one state and not dispreferred in any state.
- Principle of strict/weak dominance - it is irrational to perform a strictly/weakly dominated action.
- Act-State Dependence - the decision maker can influence the chance of various states coming true. If the decision matrix does not have act-state independence, we can end up with bizarre results (e.g. party or study with pass or fail states)

## Week 2 Monday

- Decisions under ignorance - ignorance of objective probabilities.
- What is objective probability?
- What is the probability of a nonrepeatable event?
- What rules can you derive without assigning objective states?
- Maximin - pick the action with the maximum minimal outcome. A pessimistic position.
- Minimax regret - find the decision with the minimum maximum regret - weight
- Principle of insffucueitnr easoning, optimism-pessk9m ri;ze - wait undethe frture/
- Optimism-Pessimism - choose the action which maximizes the weighted average.
- How to understand alpha? Heuristic for weighting cases, don’t think about it past abstraction.

- Principle of Indifference - equally indifferent/unknown certainties are treated as equal.

Discussion

- How do disentangle state from action? Can write using conditional statements.

## Week 2 Wednesday

- What object is probability applicable to?
- Why can we assume that things we ascribe probability to have those mathematical properties?
- Decisions under ignorance do not consider the probability of states of the world.
- Intuitively, we want to weight probabilities.
- There are many different ways to think about probability.

```
Survey Says
1/6, 1/6
1/3
90%
100%
95%
1%
1e-10%
50%
95%
Sure
Maybe
No
Maybe
```

- Discussions about likelihood and probability are everywhere - applied to both observable and unobservable events.
- Some events are repeatable, others are oe-time.
- Probabilities can be applied to propositions and events.
- Probability can be ‘objectively subjective’ or objective.
- Strength of evidence - changes to probabilities are made proportionality to evidence.
- Comparability - numbers can be compared. Can probability be compared? Can probability assessments be precise?

Random idea - infinitely nesting probability distributions. Individual probabilities have probability distributions.

- Probability is bounded. Is there a maximum and minimum probability? Is 0 or 1 probability possible?
- Additivity - probabilities can be added.
- An interpretation of probability answers three questions - how it is measured, its mathematical properties, and its use.
- Interpretations
- Frequency - some percentage of events occur in some population. Nicely explains matheamtical properties of probabilities.
- Hypothetical frequency view - addresses infinite problems

- Propensity - probabilities are properties of objects which cause them to behave or tend towards certain behavior.
- Logical - probability is a measure of an argument’s strength. Probability quantifies the strength of evidence for a conclusion given the premises of an argument.
- Subjective (Personalist) - measure of how strongly we believe in particular propositions.
- Principal Principle - if you know the objective probability of an event is some quantity, you should be confident proportional to that degree.
- Kolmogorov - set of states. A probability assigns numbers to subsets of the state set.

## Week 3 Monday

- To be rational, you must act as if you follow certain calculations.
- Act as if you associated states with probabilities.
- Expected utility - weight outcomes by probability of the state.
- Why is it rational to maximize expected utility? Why do we act as if we assign numerical utilities?
- Where do utilities come from and what do they represent?
- Utilities - strength of preference. Probabilities - strength of belief, or ‘objective’ frequencies.
- Expected Utility Theory is used both as normative and perspective theories.
- “We should act like EU maximizers”

- Is EUT a good descriptive theory? Maybe, maybe not, probably not.
- Normative question - why should a DM behave as if they maximized expected utility?
- Why should a decision-maker’s preferences be represented by numerical utiltiies?
- Utility - strength of preference of a particular decision-maker.
- Utility is not money. St. Petersberg Paradox
- Preferences of money may have certain feature: weigh losses higher than gains, diminishing returns.

- When does a utility assignment adequately represent a DM’s strength of preferences?
- Possible solution: ordinal assignment.
- Maximin, domination, etc. can function only with ordinal preference. This does not apply to minimax regret, optimism-pessimism, PIR, etc.
- Interval Scales - represents ordinally, and are linearly transformable into each other.
- If we can measure a decision maker’s strength of preference on an interval scale & know what decision rule they use, we can predict the decision maker’s outcome.
- The stronger your preference for something, the more you will tolerate a reduction in its probability of uccrrence.
- Lottery - probability outcomes (also called roulettes or Von Neumann Morgenstern lotteries).
- If a decision-maker’s preference over lotteries have certain mathematical properties, we can measure the DM’s preferences on an interval scale.
- Probabilities are objective probabilities; not the probabilities of states. Rather, the probability of outcomes.
- vNM’s theorem is intended to justify replacing outcomes with numbers. It is not a justification for assigning subjective probabilities to states or using probabillistically weighted averages.

## Week 3 Wednesday

- Why should a DM behave as if they maximized expected utility?
- Lottery - a gamble premised on probability, interpreted as the objective type (propensities, frequencies)
- If a decision maker’s preferences among lotteries have certain mathematical properties, then we can measure the decision maker’s preferences on an interval scale.
- Properties of preference - we assume a decision maker’s preference relation has the following properties:
- Transitivity
- Completeness: there must be a relationship between options.
- Independence: \(\forall p \in [0, 1]: (A \ge B) \to (ApC \ge BpC)\)
- Continuity - \((A \ge B \ge C) \to \exists p \in [0, 1] : B \sim ApC\). Preference relations are not infinitely strong.

- Utility as a latent vector?
- vNM’s theorem: the utility function assigns outcomes to an interval scale and all such functions are derivable through positive linear transformation.
- Reduction of compound lotteries - lotteries of lotteries can function as a single lottery.
- There is controversy in how to compare utilities between different agents. The currently described theory does not let you do this - you cannot compare utilities between people.
- How to justify certain axioms? Often derived from the methods of measurement.
- Some aixioms - psychology verified empirical facts about the human condition. Continuityh can be psychologically verified.
- A normative defense of transitivity - some decision theorists assert that intransitive preferences are irrational. Money pump argument - intransitivity can lead to indefinite loss of money.
- Common tactic in normative decision theory - suppose an axiom is violated, suppose that behavior is linked to the axioms; show that this violation makes you exploitable.
- Normative argument for the independence axiom - dutch book.
- Rationality in normative theory does not necessarily contribute to decision theory as a descriptive theory.
- Normativ erationality with vNM: preferences satisfy the completeness accent; empirically preferences follow continuity; our preferences should satisfy transitivity and independence for the skae of rationality. By vNM, we should act like we are choosing outcomes as if we were maximizing expected utility. This is invalid.

## Week 4 Wednesday

- Why shold a decision maker behave as if they maximized expected utility?
- Importance of as-if. Could reconstruct behavior as if you maximized expected utility.
- Why behave as if you assign numerical utilities to options? vNM - if options are roulette lotteries. Why assign subjective probabilities? Anscombe and Aumann’s theorem can help us answer the second or third questions.
- Why should beliefs be represented by probabilities?
- Even if we should act like EU maximizers, does anyone really act like it? - Sometimes not.
- St. Petersberg Paradox
- Money has diminishing marginal utility - people do not behave as expected monetary value optimizers.
- Monetary loss incurs a steep drop in utility.
- Don’t people still maximize utility even given diminishing marginal value of utility?
- Prospect - prizes and associated probabilities.
- Regular prospect - has a possibility of zero, or there is heterogeneity in the rewards in terms of loss / gain.
- Value of regular prospect - weighted sum.
- Irregular prospect - sum to one or homogenous
- Prospect theory as normative?

## Week 5 Monday

- Last class - violations of EUT from the perspective of a normative theory.
- The Ellsberg paradox - people’s preferences are inconsistent with EUT, but really decision making under ambiguity.
- Urns and balls
- Ambiguity - there are objective probabilities but they are not known to the decision maker.
- We end up choosing certain probabilities even when they are inconsistent with EUT.

- Decision making under uncertainty - choosing between horse lotteries, subjective probabilities.
- Objective probabilities don’t exist.

- Next up -
*philosophy*philosophy. *Should*one behave as if they are an expected utility maximizer? To be rational, one must maximize one’s subjective expected utility.- Subjective probabilities - personal likelihoods attached to states
- Utility - strength of preference

- To be rational is to maximize the expected satisfaction of one’s preferences.
- Why not maximize other things?
- Value
- Welfare
- Interest

- What is the relationship between preferences, values, welfare, interests?
- How do we measure these things?
- Value and choice - how direct is the line?
- Economic theory - preferences are revealed by choices.
- Chernoff’s condition / Sen’s alpha - independence of irrelevant alternatives. Even if your option set is contracted, you should still choose the best option you would have chosen before.
- Menu depedence
- Cocaine and tea - the addition of cocaine affects your decision.
- The addition of an option adds new information which changes our understanding about the thing itself.
- We also see the Chernov Condition violated in plurality voting.

## Week 5 Wednesday

- Past two weeks - studied arguments for normative EUT (to be rational, a decision maker must maximize subjective expected utility.
- Utility should represent a single decision maker’s strength of preference.
- This implies that we should be preference maximizers.
- What is a preference?
- Descriptive questions:
- How do preferences influence choice behavior?
- How do we determine what a decision-maker prefers and the strength of preferences?

- Normative questions:
- Does rationality require maximizing something other than preference?
- Are third parties required to help a decision maker from maximizing their preferences?

- Logical positivism - movement in the 1920s in Europe and the United States. Core tenet - the verification theory of meaning. The meaning of a sentence is the set of conditions under which the sentence would be true; that is, its verification conditions (verification principle of meaning).
- What is the meaning of words?
- Logical positivists get a bad rap in the history fo science. Logical positivism renders a lot of discourse as meaningless - people think about a lot of things which are in fact meaningless. We could avoid political conflict if we just got clearer on particular ideas.
- Behaviorism - mental states are states of behavior.
- Feelings are demonstrable by different behaviors.

- Revealed Preference Theory - some versions are equivalent to behaviorism.

## Week 6 Monday

- Primary sources - Hausmann, Thoma, Dietrich & List, among others.
- Goal - to understand whether or not we can interpret expected utility theory as a normative theory.
- Thoma’s paper - primary focus is scientific / social-scientific.
- Does it make sense to interpret preference as behaviorism / revealed to model social behavior?
- Thoma and Hausman - what notion of preference is the one which is good for economists?
- Not interested in understanding what the pretheoretic concept of preference is.
- Normative questions - how should people make decisions as individuals, and how should they behave collectively?
- The discussion of what preferences are informs these discussions - we need to assume and move forward with what subjects are.
- Game matrices populated by preferences as total subjective comparative evaluations vs. actual measured preference will vary.

## Week 6 Wednesday

- A game consists of: players, actions, preference relation or utility function for each player.
- Strategic profile - specifies what actions every player in the game takes.
- Equilibria are strategic profiles - we need to specify an action for every profile
- Two equilibria - Nash equilibria for normal-form games and subgame perfect equilibrium, refinement of Nash equilibria.
- Game theory is full of other equiilibrium concepts.
- Normatively, equilibria describe how agents should behave.
- Descriptively, equilibria describe how players do choose.
- Nash equilibrium - a strategic profile in which
- Each player’s action is a best response to the other players’ actions
- No player is better off by taking a different action.

- Identifying Nash equilibria with two-player games, the quick way: underline best responses for each player. Nash equilibria are action profiles with all responses underlined.
- No pure strategy Nash equilibrium - for games without Nash equilibria.
- Equilibrium selection problem - how to normatively understand which option equilibrium players should choose?
- Mixed strategy Nash equilibrium - randomly choose with probability
- Nash’s Theorem: any game with finitely many players and actions has a a mixed strategy Nash equilibrium.
- There is a way for players to choose randomly such that no agent is better off using a different strategy.

- Is there something wrong with thinking about rationality? Are the sexist problems in game theory inherent to or external to games?
- Elizabeth Anderson - feminism and rational choice theory

## Week 7 Monday

- Prisoner’s dilemma - there’s a situation where both do better (stay silent).
- The strategic profile
- Strict Pareto domination - better in all cases for all players
- Weak Pareto domination - better in at least one case, no one else is worse off.
- Pareto efficient - dominates all other outcomes.
- Gibbard-Satterthwaite Theorem: no voting system that chooses a candidate from ranked ballots has all of the four:
- Non-dictatorial, no voter such that one voter chooses the winner
- The system always chooses a winner
- If a candidate is unanimously ranked as the top choice, that candidate is chosen.
- Non-manipulable, the only Nash equilibria in a game are where voters honestly report their voting preferences.

## Week 7 Wednesday

- Do the game theoretic equilibria concepts pick out the same types of behavior as the prescriptive recommendations of expected utility theory?
- Is there a conflict between the theory of rational interactive theory and rational individual theory?
- Two types of significance of equilibria concepts - descriptive and normative, how players will and should behave.
- Rationalizable - expressable in terms of EUT
- In science, we always understand that they have their place; only physicists make claims about everything, and even that is a publicity stunt.
- A Nash equilibrium is a strategic profile in which each player’s action is a best response to the remaining players’ actions.
- Osborne - each player’s belief is derived from past information playing the game, and that this experience has sufficiently high information density. She is sure of which actions her opponents will choose.
- Osborne’s argument: a player believes they know all others’ strategies; if the beliefs are true, then all the players’ actions are best responses to one another.
- Agents do not need to be expected utility maximizers.

- It does not matter what your subjective probabilities are over the different states are distirbuted
- How do we know things
- What is a real truth?
- Game theorists rarely suggest normative claims clear.
- Harsanyi - scathing reviews of Rawls, exploration of political philosophy
- Ethics, decision theory, game theory - more or less same method.
- Rational behavior described by a set of axioms.
- Utility maximization - a more convenient characterization of rational behavior under certainty than others.

- Iterated elimination of dominated strategies.
- If rational players know each other to be rational, contestants choose strategies which cannot be eliminated through dominance considerations.
- If a unique strategic profile survives the iterated domination of dominated strategies, it is a Nash equilibrium.
- Higher-level knowing - common knowledge of rationality. But all that is required is common belief - that I believe that you are rational. We just have to believe these things.
- How many levels high of common knowledge do we need to go? No more than the number of strategies in the game.
- Iterated domination of weakly dominated strategies - what to do?
- What if rationality is not common knowledge? What can happen when we believe the other person isn’t rational? etc. etc.
- When players have incorrect or misperceived beliefs, we can choose out of equilibria.

## Week 8 Monday

- Not everyone acts, and then things simultaneously happen, like Nash equilibria may suggest.
- Preferences in cases of sequential dependence need to be modeled sequentially
- Goals:
- Understand notation used in the Osborne textbook
- Define what a subgame perfect equilibrium is

- Set of players - unique decision maker.
- Two types of histories - subhistories and terminal histories.
- Terminal histories are histories in which the game concludes.
- Proper subhistory: the game hasn’t ended yet. Contains the empty history/sequence - no one has done anything yet.
- Player function - who plays after each subhistory?
- Strategy - tells a player what they might do even in places which they might never end up with if they followed the strategy.
- A strategy must specify all of the components of decision responses.
- Strategic profiles - all combinations of possible actions, even if they never materialize.
- We can create game matrices after particular subhistories.
- What does the game look like after a player has chosen? Include all strategies which are available even if it has been ruled out.
- Subgame perfect equilibrium - a strategic profile is a Nash equilibrium of every subgame.
- Why does this matter?
- No joining, tree structures
- Backwards induction

## Week 8 Wednesday

- Backwards induction - a technique for finding subgame perfect equilibria.
- Goals - apply backwards induction to find subgame perfect equilibria, identify assumptions about players’ rationality which are necessasry to ‘rationalize’ subgame perfect equilibria.
- Look backwards and see how to maximize each player’s utility given the later players’ decisions.
- If the utilities end up being the same, you just need to be more careful.
- Every choice is a bestr esponse in a subgame which leads this player to have a particular choice to make.
- Backwards domination assumes that players have full knowledge of other players’ utilities.

## Week 9 Monday

- What if a player has multiple best responses?
- When do game theorists believe Nash equilibria can plausibly describe real human interactions?
- Each player’s belief is derived from past experience, and that this is sufficiently extensive.
- A player confidently believes all players will behave in a certain way. If the player is rational, they will choose the best response. All players’ actions are best responses to each other. These work together to form a Nash equilibrium.
- Osborne’s assumptions are weak - it does not require that agents are expeected utility maximizers, and does not require that players know other players’ strategies. They do not need to know what their strategies are available or their preferences.
- Why is always defecting the subgame perfect equilibria? Any strategy which defects on the last round will beat a strategy prescribing always collaborate. Then, the second defecting to last defecting will dominate the former. etc. etc.
- Dominance does not require probability assignment (expected utility maximization) or numerical utilities
- For some item to be strictlyd ominated, you need to know what they will choose
- Main questions - when do game-theoretic equilibria questions describe how people do and ought to behave?
- To answer the normative question, ask what assumptions we need about decision methods.

## Week 10 Monday

- Decision theory has both descriptive and normative interpretations.
- Why might Nash equilibria descriptively be accurate?
- Why do game theorists believe Nash equilibria describe real human interactions?
- We also want a normative claim – when should we make decisions?
- Normative ‘ought’:
- Legal and moral obligations might conflict.
- We are often tempted to infer normative morality from what we are restricted to do.

- Should we distinguish rationality and morality?
- Suppose being rational is maximizing one’s preferences: then obligation by rationality may differ from morality.
- Misalignment between rationality and morality: how to engineer rationality in line with what is morally right.
- Decision theorists aim to describe what rationality requires.
- EUT - characterize the obligation of individuals to rational fulfillment of preferences
- Can we appropriate Osborne to derive a normative conclusion.
- Kant - categorical imperitive. A correctc chracaterization.
- Ac ariant of OSborn’es argument
- Premise 1: a player shodl maximize expected subjective probability.
- A player’s best respond is tha action whihch maximi

- What does it man to act because of resasons? Is there a conssitent rent?
- Economist often read game theory by invidiaulsa; wnr45eon w5yu4e 6hw6 5w6ionwli6y fwnno un6r45e6wnr tully ti5e6-y4w5.

## Week 10 Wednesday

- It can be rational to advance the goals of a group, even if one prefers to act otherwise.
- Don’t think about it in terms of individual preferences – think about preferences and commitments. Commitments are other features which override individual preferences – not something which figures into the preference itself.
- Objection – is there irrationality in maximizing the preferences of someone who is not oneself? Why is maximizing collective interests rational?
- Arguments for normative EUT can be money pumped, dutch booked, subjected to sure loss.
- It might not be irrational for Ann to model her decisions off of Bob’s, if Bob’s preferences satisfy appropriate axioms.
- Is there a positive reason to believe Ann is rational?
- We need to expand the determining bases of rationality.
- Orthodoxy: preferences, beliefs about states of the world, behavior.

- Should mental states factor into rationality?
- Orthodox decision theorists seem to leave out mental states and certain types of non-empirical ad=nd the environment.
- Wht elee does Anderson propose? Can be rational to believe ways if we understand our soical relations as well?
- Wht I mssing from invidiaul-centered inishgts? Why is it not rational? Can a cause be irrational?
- Counterpreferential behavior might be recommende by reasoning – the conclusions of valid arguments that have premises about group identification.
- What can the premises and derivations of arguments in favor of team reasoning tell us?

## Week 11 Monday

- Principle of indifference – a bad principle, you can get different results depending on you assign different probabilities. To apply the principle, you need an inprincipled way of dividing the space to distinguihs between the cases.
- Bertrand’s paradox, Van Fransen’s Cube Factory
- Anderson: it can be rational to advance the interests of a group with which one identifies, even if they prefer to act otherwise.
- Does inconsistency between preferences and behaviors cause irrational behavior?
- IrraiontliY; disobeys certain axioms of rational behavior.
- “Not irrational” shouldn’t necessarily imply “rational”.
- Is there a positive argument that group identification can be rational? Turning “not irrational” (Anderson) into “rational”.
- What is the reasoning involved in justifying such behavior?
- Rationality has a robust association with reason. Raiontliy which can be justified by reasoning.
- Empirical evidence for rationality?
- There are many plausible arguments which might represent “team reasoning”
- Why should arguments presented in Gold and Sugden justify (robust normatively) forgoing dominant strategies for an individual?
- Objection 1: In what sense is there a collective utility function for a group? What are group “preferences”?
- Possible response: experimenter believes she is speaking to and offering a series of choices to a single person behind a computer. Experimenter thinks they are assigning single outcomes to an individual but in fact it is a team. We could ascribe a utility function to a team if you can’t see the inner mechanisms of choice-making.
- 1955 Harsanyi theorem: if you and I choose in such a way that if \(s\) Pareto dominates \(s'\), then we will never choose \(s'\), then it follows that the utility function that the experimenter will attribute to us is a weighted sum of our individual utility functions.

- Objection 2: Even if arguments can justify counterpreferential behavior, do people actually reason in this way? Are people doing waht team reasoning folks advocate?

## Week 11 Wednesday

- Team reasoning vs transformation payoff theories
- Team reasoning and payoff transformation theories are intended to solve equilibrium selection and non-equilibria play.
- We can understand these both descriptively and normatively.
- What theory of rationality allows us to make consistent intuitive normative claims?
- Team reasoning: people behave as if they’re thinking collectively – we need a need a new theory of rationality.
- Payoff reasoning – suggests that we just need to define utilities in the appropriate way.
- What if you never met someone? Would you still play even in the ultimatum game? Do we really form a team? Do we care about each other?
- Payoff transformation theories are interesting, and right in some sense, but many people are unhappy with them.
- Anderson: payoff transformations don’t show that we can cooperate in Prisoner’s dilemma, but rather show that the people were not playing a Prisoner’s dilemma in the first place. But there are real Prisoner’s dilemmas which are not addressed by payoff transformation theories.
- We might restrict ourselves to plausible types of transformations – focus on utility functions by SVOs. However, this provably does not allow us to explain equilibrium selection.
- Low-Low remains a Nash equilibrium even if using any one of the SVOs.
- Anything which retains Pareto dominance still recommends Low-Low.

- Are there plausible transofrmations to explain patterns of choice?
- Players prefer the outcome in which both of them get nothing than when both of them get a sum smaller than the maximum. This is a logical necessity. You can indeed use a payoff transformation, but you have to bite a heavy bullet.

Course Wrap-Up

- “Game theory says it’s only rational to…”, “Rational choice theory says we must…” – always more controversial than people let on.
- Rational choice theory does not require you to be selfish; your utility function is variable. They can represent many different preferences.
- The deployment of equilibria concepts in the social sciences depends on strong assumptions about what players know and believe.
- The normative and descriptive uses of game theory should be distinguished, even if they are often closely related.
- How does normative theory guide empirical work? EUT sets the explanatory goals for descriptive theory.
- Descriptive work of course also guides normative work. “Ought implies can.”